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stoic galleon
So in Rogawski 4th edition for multivariable calc it says (paraphrasing right now) that curl is a vector quantity that which has a magnitude equal to the maximum rate of rotation of putting some kind of paddle in a vector field and then the direction is the orientation of the paddle
stoic galleon
Doesn't really make sense for this analogy like isn't curl supposed to be related microscopic circulation
I don't really have an intuition of this. Can someone explain?
I'm asking since Green's Theorem is the idea of summing the microscopic circulation in some domain being equal to macroscopic circulation
In other words we use curl to make a line integral in a plane easier to compute as a double integral
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stoic galleon
<@&286206848099549185>
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stoic galleon
Bruh
gritty rose
screenshot/picture some paragraphs/formulas to give more context
stoic galleon
screenshot/picture some paragraphs/formulas to give more context
This is it.
This website though here talks about curl as microscopic circulations
Circulation though AFAIK is defined as line integral of some vector field along a curve (most of the time closed)
How is curl related to it when curl is a measure of velocity basically?
It sounds like it makes sense but things just don't fit together for me
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lean rock
I am learning this now, also. Check this video, it might help in gaining a conceptual understanding of curl: https://www.youtube.com/watch?v=rB83DpBJQsE
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lean rock
@stoic galleon there is a link in the article that you mentioned to "get intuition".
There are animations and explanations on GeoGebra as well.
I asked ChatGPT for many analogies to these abstract ideas, which seems to be an effective way to to learn.
For example, an atmospheric system, like a tornado can have many smaller circulations in it. All these microscopic circulations make up the overall circulation of the tornado, which is the macroscopic circulation.
There seems to be a difference between "magnitude vector curl(F)(P)" and "magnitude curl(F)(P)". Magnitude vector curl is the velocity of rotation at point P, the speed at which the paddle rotates. Magnitude curl is a scalar quantity, it shows the strength of the rotation of the paddle in point P.
At least this is how I understand it at this point, it might be wrong.
Curl is used in 3D, while Green's Theorem is used in 2D.
lean rock
So for the 2D plane, only one component of the curl is used, it is isolated with • k. In 2D, circulation can only happen around an axis parallel to z.
<@&286206848099549185>
Curl shows the tendency of rotational motion in a velocity field. It captures the presence and strength of vortices or rotational motions.
lean rock
What is the difference of circulation and curl?
stoic galleon
@lean rock I understand this stuff but if we capture this in terms of physical quantities the "circulation" macroscopically is the vector line integral, which is work (measured in Newtons*Meters). But curl magnutude is not work. Curl's magnitude would represent velocity, so dimensionally we it doesn't connect
So how can we say "sum of microscopic curls" (measured in meters per second) = "sum of microscopic circulations" (measured in Newtons*Meters) = macroscopic circulation (measured in work)
We need a conversion factor between the two and I don't see it there
violet oyster
hi
Someone who speaks Spanish to explain to me the congruence of triangles please?
stoic galleon
Bro there's already a question in this one post in help and get your own channel
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wicked zodiac
how do you do this without graph
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how do you do this without graph
full marsh
Find f(g(x)) and g(f(x)) and make sure they're both x
torn jolt
how do you do this without graph
you can find inverse of f(x) and see if its equal to g(x)
wicked zodiac
you can find inverse of f(x) and see if its equal to g(x)
How do you do that
Can u do a example
full marsh
Find f(g(x)) and g(f(x)) and make sure they're both x
I'm pretty sure that easier.
torn jolt
alright, i doubt you've learned this yet or not
full marsh
But do whatever you're comfortable with
torn jolt
i'd do
let y=f(x)
x=f^-1(y)
let x=h(x)
h(y)=f^-1(y)
replacing y by x
h(x)=f^-1(x)
here's an example
f(x)=x-1
let y=x-1
x=y+1
let x=f^-1(y)
f^-1(y)=y+1
replace y by x and there it is
ig i made a mistake
full marsh
That's a lot of work for just verifying that the functions are inverses of one another
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lone flint
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grave elm
you would need lambert function to solve e^x * x = -1
full marsh
No
e^x here is the slope
You need to sub in x = 0
your slope at (0, 1) will be e^0 which is 1
torn jolt
You need to sub in x = 0
why?
grave elm
ahh ic
torn jolt
your slope at (0, 1) will be e^0 which is 1
oh
lone flint
your slope at (0, 1) will be e^0 which is 1
Damn
grave elm
why?
e^x is derivative, it gives you slope of tangent at point x, so you would have to plug it in
lone flint
Gimme a min
torn jolt
e^x is derivative, it gives you slope of tangent at point x, so you would have t...
i see thanks
lone flint
Intersection points (-1,0) (0,1)?
Aye so distance is 2
Ahem sqrt(2)* my bad
Ok thanks
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torn bane
is there no variable x?
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is there no variable x?
torn bane
just unique solution of y=-5, z=10?
or infinite solutions
cuase rank is 2
and theres free variable
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pure sandal
is there an original question?
fast peak
x can be whatever it wants
and it still satisfies the system
so yes, infinitely many solutions
grand jacinth
U can also interpret it as the intersection of two planes. It will always result in a line where every point is a solution. As a line is made by an infinite number of points, it has infinite solutions
silk veldt
(3a-b)^4 pascal triangle
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(3a-b)^4 pascal triangle
silk veldt
im confused on how the (-)(+) change thru the equation
onyx glen
treat 3a - b as (3a) + (-b)
grave elm
when you treat the second term as (-b), (-b)^1 will be same as -b
so 108a^3 b will have negative sign
rest of signs is fine
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grave elm
,w expand (3a-b)^4
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lone flint
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lone flint
I can deduce the equation of the tangent but I don't know what to do after
vast fossil
The slope of the normal line is -1/3
Meaning f and g both have the tangent with slope of 3 at x = 1
lone flint
So I can just do f'(1)*g'(1)?
lone flint
Meaning f and g both have the tangent with slope of 3 at x = 1
I don't know how to get from this to the slope of tangent of f*g
vast fossil
So I can just do f'(1)*g'(1)?
No, you need to apply the product rule
(f * g)' = f' * g + f * g'
clear vigil
i don't quite understand wdym by common tangent at x=1, does that mean both the curve intersect at x=1 and that tangent has slope 3, or that a random common tangent two the two functions has a slope 3 at x=1?
vast fossil
f'(1) = g'(1) = 3, like I said, so the expression (f * g)'(1) turns into 3(f(1) + g(1))
vast fossil
i don't quite understand wdym by common tangent at x=1, does that mean both the ...
They intersect at x = 1 and have the same derivative at x = 1
lone flint
i don't quite understand wdym by common tangent at x=1, does that mean both the ...
Can't imagine anything beside intersect
clear vigil
Oh now I understand, I thought you had to find f×g(1)
vast fossil
f'(1) = g'(1) = 3, like I said, so the expression (f * g)'(1) turns into 3(f(1) ...
But f(1) = g(1), so this is just 6f(1)
clear vigil
So you have to find (f×g)'(1)
vast fossil
Let us now find f(1)
The tangent line to f at x = 1 is y = -x/3 + 3/2, it must have the same value as f at x = 1
So f(1) = -1/3 + 3/2
lone flint
7/6 I think
clear vigil
Can't imagine anything beside intersect
if they intersect at x=1, then x=1 in equation of line will give f(1)
vast fossil
So 6f(1) is just 7
lone flint
Ye
lone flint
The tangent line to f at x = 1 is y = -x/3 + 3/2, it must have the same value as...
We are using the same line and the same point so same value for g(1)?
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maiden pike
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maiden pike
does 1 represent 360?
real relic
What do you mean?
rotund summit
real relic
180 + the angle = 1
the value of tan =1
viral zenith
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viral zenith
Hey, thanks for replying! What does M and N stand for?
So I just need to solve this?
It is not
Hence my confusion
Hmm okay, it is a weird question
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ashen pewter
how to do b?
dapper oak
its midnight lol
dapper oak
pst
torn jolt
Suppose that $A$ and $B$ are sets where $\abs A \le \abs B$ and $\abs B \le A$. This means that there are injections $f: A \longrightarrow B$ and $g: B \longrightarrow A$. To prove this, we must show that there is a bijection $h: A\longrightarrow B$ implying that $\abs A = \abs B$.
\vs{3 mm}
To build $h$, we construct the chain of an element $a \in A$. This chain contains the elements [
a, \map f a, \map g{\map f a}, \map f{\map g{\map f a}}, \map g{\map f{\map g{\map f a}}}, \hdots
]
It also may contain more elements that precede $a$, extending the chain backwards.
\vs{2 mm}
So, if there is a $b\in B$ with $\map g b = a$ then $b$ will be the term of the chain just before $a$. Because $g$ may not be a surjection, there may not be any such $b$, so that $a$ is the first element of the chain. if such a $b$ exists, because $g$ is an injection, it is the unique element of $B$ mapped by $g$ to $a$; we can denote it by $\map {g^{-1}} a$ (This defines $g^{-1}$ as a partial fraction from $B$ to $A$).
\vs{2 mm}
We extend the chain backwards as long as possible in the same way, [
\textxs{adding} \map {f^{-1}}{\map {g^{-1}}a}, \map {g^{-1}}{\map {f^{-1}}{\map {g^{-1}}a}}, \hdots
]
Knowing all of this, how do we verify that $h$ is a bijection?
glossy valveBOT
bitter lagoon
Is this the exercise?
torn jolt
Yes?
bitter lagoon
Ok why would you need to show that there is a bijection in the first place
|A| <= |B| and |B| <= |A| can only be true if |A| = |B|
torn jolt
sure but this isnt proving it
also it should depend on the type of chain that u may have, no?
there are 4 types of possible chains
chains that form a loop, that is, carrying them forward from every element in the chain so that it will eventually return to that element (type 1)
chains that go backwards without stopping (type 2)
chains that go backwards and end in the set A (type 3)
chains that go backwards and end in the set B (type 4)
bitter lagoon
Well if its a bijection any chain can only be of length 2 before looping, as for $a \in A$ $f(a)$ must be in $B$. If h is a bijection $g(f(a))$ must be $a$ again no?
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Cyrol
bitter lagoon
Because if $g(f(a))$ isn't $a$ then there is an Element $x \in B$ for which $\vert g(x) \vert > 1$ meaning $g$ isn't injective and $h$ isn't bijective
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Cyrol
bitter lagoon
At least as far as my understanding of this problem goes
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torn jolt
I'm completely lost on how to solve this problem
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I'm completely lost on how to solve this problem
torn jolt
I've been thinking about using triangular cross sections to compute the volume of the shape
,rotate
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torn jolt
if the left is 0 and the right is $$ \sqrt{\frac{2}{3}} L$$
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Brandon H
torn jolt
the area of the triangular cross sections would be $$\frac{1}{2} * \frac{l}{n} * idk$$
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Brandon H
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stiff summit
can you make an integral using these?
torn jolt
can you make an integral using these?
yes? but idk how
$$\int_0^{\sqrt{\frac{2}{3}} L} \frac{1}{2} * ? * ? dx $$
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Brandon H
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pure sandal
the proof is geometric right
torn jolt
idk
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runic spruce
for part c i got the range as : 0 <= h(X) <= root5/5
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for part c i got the range as : 0 <= h(X) <= root5/5
gilded quarry
Yeah, your answer is correct
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pearl ermine
A random sample of 20 college graduates revealed that they worked an average of 4 years on the job before being promoted. The sample standard deviation was 16 month. Calculate the margin of error for confidence interval (Notes: Use 3 digits after decimal point) is 99% confidence interval.
pearl ermine
just a question with regards to the formula for margin of error where MOE = critical value * stand dev / sqrt(n)
in this case, would we just do critical value * 16 or * 16/sqrt(20)
and 2nd question, do we change the 16 months to years
pearl ermine
<@&286206848099549185>
ember shadow
you can change to years if you want, doesn't seem like they ask for it
and yes, include sqrt(n)
ember shadow
because it's the standard error
pearl ermine
and another question to clarify, we use t-distribution since n < 30 right
pearl ermine
because it's the standard error
so if we're given a population standard dev instead of a sample standard dev, does that change things
ember shadow
and another question to clarify, we use t-distribution since n < 30 right
I think you use the z-score for MOE, but tbh I don't remember for sure
ember shadow
so if we're given a population standard dev instead of a sample standard dev, do...
you won't know the population standard deviation
runic spruce
stuck on 8
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onyx glen
!status
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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'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above
1 or 2?
runic spruce
1 really, dont see how u can get that area
onyx glen
the pink region is the symmetric difference of two quarter-disks
so if you can get the area of each quarter-disk and their intersection, getting the pink area is not hard from there
pure sandal
hi bonx
runic spruce
so if you can get the area of each quarter-disk and their intersection, getting ...
hmm
echo pendant
I tried answering 46) using u-substitution. Why does the solution equate to that? My current answer isn't even close to the solution. Am I missing something? If so, what should I study?
Ps. I substituted a with 5
fathom saddle
Can't just let an x out of the integral like that, since the integral depends on x, by depending on u which is x but in disguise
echo pendant
Can't just let an x out of the integral like that, since the integral depends on...
How do i remove the x then?
fathom saddle
In order to have a successful u-sub:
- Every x has to be turned to a u
- dx has to be turned to du
It's possible that your choice of sub cannot work
echo pendant
So u substitution doesn't work in this equation?
What are the other options available? Especially in the forms from 45-52?
fathom saddle
I wonder if IBP is the right first step, yeah.
echo pendant
I wonder if IBP is the right first step, yeah.
Ibp?
humble steppe
What are the other options available? Especially in the forms from 45-52?
Hint: Completing square following by a trig sub
candid girder
Help
rough tundra
Could you possibly specify what you want help with?
spice orchid
read
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candid girder
no
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glad prairie
if i have a bilinear function then why f(a+b, c) != f(a,c) + f(b,c)
why i can't use the linearity on the first variable?
spice orchid
who is saying you cant?
glad prairie
chat gpt lol
spice orchid
well there you go, dont listen to chatgpt
glad prairie
- a teacher
spice orchid
you need to provide more context
obviously if you have a bilinear function it is linear in both slots
viral jasper
👏 chatgpt 👏 does 👏 not 👏 do 👏 math 👏
glad prairie
the context is the determinant
ie det(A+B) != det(A) + det(B)
because det is n linear well i don't really get the argument
glad prairie
ok then det(A) = det(a1,...,an)
so det(A+B) = det(a1+b1, ... , an+bn)
can i use the n-linearity here?
spice orchid
sure, but you are going to have a bunch of terms
glad prairie
ah ye it's not det(A) + det(B)
spice orchid
you could also have just proved this with a counter example
its nearly always not true
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glad prairie
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glad prairie
ah yes just to be sure
det(a1, ... an) != 0 <=> the column vector a1 .. an are independent
is it really an equivalence?
brittle steeple
yeah
fast peak
always a good exercise to try to find a sketch of the proof again
brittle steeple
glad prairie
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dusky tendon
hello
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glad prairie
To calculate a determinant is it right to do that?
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To calculate a determinant is it right to do that?
fossil stump
That's the first part of it
Gotta do the same for each of the other entries in the first row
glad prairie
when there are it's 0* .... and it simplify it ok
fossil stump
1 det(3x3) - 2 det(3x3) + 3 det (3x3) - 4 det(3x3)
stiff musk
side note: if you get really lucky you might notice that the rows are linearly dependent..
(Row2 - Row1)/4 = (Row2 - Row4)
glad prairie
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little merlin
help please
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little merlin
So far, I have only found these
hollow sable
the ratio is the same as the height of the small triangle to the height of the big one
hollow sable
So far, I have only found these
youre almost done, use the fact that OD=BO
little merlin
Oooh i got it
little merlin
the ratio is the same as the height of the small triangle to the height of the b...
though, i couldn't really understand this part tbh
how are the heights in the same ratio as sides?
hollow sable
try finding FE and AC in terms of the heights
well not in terms of the heights, in terms of x
but youll realize that hte side lenghts are the heights multiplied by sqrt3/2
so the sqrt3/2 cancels when you divide
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proven magnet
I have no idea how to sketch for this question. Sally (S) is standing on Pride Rock (R), 50 m above ground level. She sees her friend Tim (T) standing on the ground, with an angle of depression of 25°. Behind Tim, Sally sees Prue (P), with an angle of depression of 17°.
fathom saddle
I think the only thing that can trip you up is "angle of depression"
You'll want to know what that means
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proven magnet
I think the only thing that can trip you up is "angle of depression"
depression angle is like the angle of something looking down on something else
right
fathom saddle
Yeah, from the horizontal
If Sally looks horizontally, then lowers her gaze by 25°, she'll see Tim
tough thicket
Hey, guys. I apologize if this question is somewhat elementary, but does anyone know if I can divide both sides of 2e^x·cos(x) = 0 by 2e^x? (Is there a mathematical "rule" preventing me from, or advising against, doing that?)
plush egret
the only thing that would stop you is division by 0
like for example, you could divide both sides by cos(x) here, too
but youd be making assumptions about the solution in doing so
so you just have to ask if 2e^x is ever 0
and if so, account for that assumption
sharp flame
Or if it ever is 0, you can just keep it
There would be no real reason to get rid of it
plush egret
yea, it gives you easy solutions here
sharp flame
If it's on both sides of the equation you can factor it out
tough thicket
I am trying to find the zeroes of this equation, given that x must be between [–π,π]. So, I am guessing dividing by 2e^x is the best choice.
plush egret
I am trying to find the zeroes of this equation, given that *x* must be between ...
maybe an even better way to think about it is like
one of the two factors must be 0
if the thing is equal to 0
so consider separately where each is 0
i get approaching it in an equation way too though 
tough thicket
one of the two factors must be 0
I like that approach. However, there is nowhere within [–π,π] where 2e^x = 0 (at least, not that I know of); there are, however, values where cos(x) = 0
plush egret
I like that approach. However, there is nowhere within [–π,π] where 2e^x = 0 (at...
those are your solutions, then
tough thicket
e^x is never 0
Ahh. Do you mind explaining why? I know this is an aside from my original question, but I am interested in knowing why.
plush egret
Ahh. Do you mind explaining why? I know this is an aside from my original questi...
well on one side, it looks like e times itself a bunch of times
tough thicket
those are your solutions, then
Hmm, so, x = No solution or x = ±π/2 would be the correct way of denoting the zeroes of the function, then, right?
plush egret
and on the other side, it looks like 1 divided by e a bunch of times
buncha math with positive numbers
plush egret
Hmm, so, x = No solution or x = ±π/2 would be the correct way of denoting the ze...
hrm? theres a solution, right
tough thicket
hrm? theres a solution, right
Yes. x = ± π/2 is the solution to cos(x) = 0 over [–π,π], but 2e^x = 0 has no solution, right?
plush egret
sure
tough thicket
Or maybe writing "DNE" is better?
tough thicket
Lolol ok. I am curious, though. How would you write the solution to 2e^x = 0?
plush egret
no sol
tough thicket
Lolol ok ok
left granite
DNE is for fancier stuff
wait, I thought that the way you might write so in a formal proof is to use $\nexists$ on whatever represents the solution
glossy valveBOT
Stole
tough thicket
and on the other side, it looks like 1 divided by e a bunch of times
Do you mind elaborating on this? I am looking at the graph of e^x, and I'm not seeing the latter part of your explanation; I see how the graph of e^x looks like e multiplied by itself a bunch of times, but I don't see how the other side of the graph looks like 1÷e a bunch of times
plush egret
just think like a negative exponent
its like division
so as you divide 1 by more and more e's
it gets smaller and smaller
sharp flame
Ahh. Do you mind explaining why? I know this is an aside from my original questi...
Exponentials are never 0
What Jan Niku said makes sense
But only for integers
plush egret
you can also just remember it as a rule
plush egret
But only for integers
yea im just talking about loosely
how youd interpret why it might be true
sharp flame
You can also prove it with the logarithm
$$e^x = 0$$
$$\ln e^x = \ln 0$$
$$x = \text{BOOM}$$
glossy valveBOT
NEONPerseus
plush egret
yea but then you have to know domains
sharp flame
Ah
tough thicket
Lmbo. x = Boom. Ok, so since e is about 2.72, e^x can never be 0, since e is a positive number being raised to some power.
plush egret
just think e times any number of other e's is still just some positive number
sharp flame
Lmbo. x = Boom. Ok, so since e is about 2.72, e^x can never be 0, since e is a p...
Well technically the closer x gets to 0, ln x gets to negative infinity
It blew up because it's undefined in our case
left granite
hm but also proving the series converges everywhere
plush egret
im too tired 
sharp flame
Why is that necessary, the Taylor series is already well established, we need only show it has no real roots
plush egret
looks like someone did it using existence uniqueness of ode 
sharp flame
Which isn't any easier
tough thicket
Assuming OP is familiar with Taylor series
Nope. I'm still a math noob 😐
left granite
Why is that necessary, the Taylor series is already well established, we need on...
haha idk
sharp flame
Nope. I'm still a math noob 😐
I think you should just stick to what Jan said
left granite
or you can just say e^x -> inf when x -> inf, so e^-x = 1/e^x goes to 0 when x-> inf
sharp flame
e^(something) is just e multiplied by itself some number of times
sharp flame
You can't multiply numbers and end up with 0 unless one of those numbers is 0
sharp flame
If e^x is 0, then is it legal to use e^(-x)
sharp flame
plush egret
assume 1/0
idk im trying to find a clean one
i think question is answered either way
tough thicket
It is not 😔
kind jay
Find me a point where it is 0
,w e^x=0
glossy valveBOT
kind jay
Damn thought there would be imaginary solutions
glossy valveBOT
NEONPerseus
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@tough thicket Has your question been resolved?
deep forge
Help pls
pine osprey
Just apply one of the trigonometric functions
last scaffold
is 3/25 the answer for 36000000/3 x 10^8?
full forumBOT
is 3/25 the answer for 36000000/3 x 10^8?
torn jolt
,w 36000000/3 x 10^8?
glossy valveBOT
torn jolt
Well I guess so
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@last scaffold Has your question been resolved?
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@last scaffold Has your question been resolved?
torn jolt
Does anyone know where this equation for the simpson rule came from?
torn jolt
It comes from this page of my book https://math.libretexts.org/Bookshelves/Calculus/CLP-2_Integral_Calculus_(Feldman_Rechnitzer_and_Yeager)/01%3A_Integration/1.11%3A_Numerical_Integration
Is it related to lagrange interpolation?
https://en.wikipedia.org/wiki/Polynomial_interpolation 
i can't find it
rare dock
Is it related to lagrange interpolation?
yes
torn jolt
oh i see
Still super confused
So it's not based on lagrange interpolation?
It comes from https://en.m.wikipedia.org/wiki/Newton–Cotes_formulas ?
Can i replace the simspon rule thingy with lagrange interpolation?
oh
rare dock
One derivation replaces the integrand
f(x) by the quadratic polynomial… One can use Lagrange polynomial interpolation to find an expression for this polynomial,
torn jolt
I might have misread
So P(x) is the lagrange interpolation function.
and the derivative of it is this part?
torn jolt
antiderivative* ?
glossy valveBOT
💜𝓁𝒶𝓎𝓁𝒶💜
rare dock
which is just a number
torn jolt
true
rare dock
P and f are supposed to be “similar” in shape (on [a,b])
rare dock
**💜𝓁𝒶𝓎𝓁𝒶💜**
and this is the area under P from a to b
so that approximates the area under f from a to b
rare dock
yay 🙂
This took me an entire day to understand
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rare dock
<a:bigbrain:694298751354798121> This took me an entire day to understand
haha that’s ok, math is hard
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elfin stream
Simplify: ((a ^ (1 / m) - a ^ (1 / n)) ^ 2 + 4a ^ ((m + n) / m))/((a ^ (2 / m) - a ^ (2 / n))(root(a ^ (m + 1), m) + root(a ^ (n + 1), 3)))
junior geode
,rotate
glossy valveBOT
elfin stream
Tyvm
stiff summit
!nosols
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As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.
junior geode
Sorry
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@elfin stream Has your question been resolved?
lone sleet
in determining limits, if we get 0/0 at f(x) as x->3, we don't immediately conclude that limit DNE because maybe y doesnt exist at that point but both sides are approaching at the same y value. On the other hand if we evaluate f(x) as x->3 and get a number, for exmple 5, we immediately conlcude that the limit exists. Why is that? How can we assume that the left and right are approaching the same value, havne't we only evaluated that a y value exists at f(3). Why aren't we using the same 'do not assume it's exists or doesn't just by doing f(3), as we did for when we get 0/0
lofty island
in determining limits, if we get 0/0 at f(x) as x->3, we don't immediately concl...
Wait, what's y here?
y=f(x)?
lone sleet
yes
limber flicker
in determining limits, if we get 0/0 at f(x) as x->3, we don't immediately concl...
for a continuous function f(x) the limit at x=a is f(a)
limber flicker
in determining limits, if we get 0/0 at f(x) as x->3, we don't immediately concl...
is the function is discontinuous and after evaluating f(x) at x=a you find some indeterminate form then you have to find other reason to conclude the limit
lone sleet
except for a piecewise function, if a function exists at f(a) it immediately implies that it is continuous at that point and the limit exists regardless of not knowing how the function behaves to left or to the right of it ?
limber flicker
not entirely sure but I would disagree, thinking about like the 'edge' of a piecewise function that is for example, 0 for 0->1 , 1 for 1->2 etc
,w plot floor(x) for 0<x<10
glossy valveBOT
bitter beacon
we have 2 red and 2 white balls in a box . how many different ways you can choose 3 balls out of that 4 ?
wild sleet
i don't know why you think it's relevant
bitter beacon
ok then say we had 14 red and 24 white . Then what would you do ?
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@bitter beacon Has your question been resolved?
bitter beacon
<@&286206848099549185>
lofty island
4C3/(2!*2!)?
bitter beacon
this equals to 1 ?
this does not makes sense
rrw and wwr are at least two different ways
bitter beacon
but your formula is not right ?
lofty island
I think that should have been multiplied by 2. I am not very sure as I haven't done combinatorics in a while
bitter beacon
y but i need exact formula to do this type of tasks in general
lofty island
Let me revise and come back then
bitter beacon
i can count all possible ways when balls are 4
but not id there is larg amout of balls
stiff summit
we have 2 red and 2 white balls in a box . how many different ways you can choos...
is this for probability?
bitter beacon
well in book yes but i am interesting only how many different ways is that possible , But why it does matters wheter it is for probability or mot ?
torn jolt
making a table is only possible way, i guess
well there can be other ways, which i am unaware of
bitter beacon
well there can be other ways, which i am unaware of
there must be
bitter beacon
maby this whould help guys : in a book there is this question : we have 11 black and 5 white balls. what is probability that we take 2 balls and both will be white ?
and the answer is 1/12
stiff summit
well in book yes but i am interesting only how many different ways is that possi...
because if you’re doing it, then the order of choosing matters
bitter beacon
which is 5C2/16C2
made mistake there
So i wonder why all possible ways of choosing 2 balls out of 16 is 16C2 ?
if these balls were all dofferent than i agree
but we have same coloures balls there and that is confusing
wild sleet
the word ways is very general, it doesn't define what to count
another reasonable answer is one
there's is only one way to take 3 balls: take 3 balls
for probability purposes, balls behave like they are different
bitter beacon
for probability purposes, balls behave like they are different
why
wild sleet
idk
stiff summit
for probability purposes, balls behave like they are different
It would be more like “order matters” than “balls are different”, wouldn’t it?
lofty island
for probability purposes, balls behave like they are different
They do?
lofty island
It would be more like “order matters” than “balls are different”, wouldn’t it?
I think this is what you meant to say
bitter beacon
ok lets leave probability alone for now
wild sleet
it's like an empirical fact, that the nature cares about each ball separately, and doesn't care about the color
bitter beacon
and just count how many ways i can take 2 ballss out of that 16 if i wont to get different results
wild sleet
when you pull them randomly, that's how it works: the chance can tell the balls apart when you can't
wild sleet
yeah, if the color affected the chance, in real life, we would use that for probability instead
bitter beacon
and we should think as if ballss were different or numerated
so should we imagine numbers on each ball?
bitter beacon
ok problem solved 🙂
wild sleet
the color doesn't matter, you can tell the balls apart
doesn;t change the result if there's color in addition to 16 numbers
wild ruin
What is meant by M^(T)?
full forumBOT
What is meant by M^(T)?
twilit leaf
,w transpose matrix
glossy valveBOT
twilit leaf
Ok sure thats an example
^T means to transpose the matrix
Reflect it across the negative diagonal
wild ruin
Ah okay
I thought it was a power, thx for the help
I think I can solve the rest of it now
torpid monolith
Hello
torpid monolith
So the final answer of the thing im answering is this
oh wait wrong pic
this
but isnt this
can be simplified further?
into this?
onyx glen
you cannot cancel out the e^2's as you attempted.
formal timber
torpid monolith
yeah so it really doesnt simplify more than this
thank you @onyx glen @formal timber
.close
static field
what is this xD
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what is this xD
vale beacon
!status
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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'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above
Which step are you on @static field?
static field
1
vale beacon
Ok, what information do you get from this problem?
For example, what is the height the ball is dropped from?
static field
h? next bounce = 2/3 h something like h! but with 2/3 till the end i guess
vale beacon
ok, so the next bounce is 2/3 h. What is the next bounce after that?
static field
h/3*2 to the infinite
vale beacon
not quite.
If every bounce is 2/3 the one before it wouldn't it be a seqence that looks like this?
$h, \frac{2}{3}h, \frac{2}{3} \cdot \frac{2}{3}h, \frac{2}{3} \cdot \frac{2}{3} \cdot \frac{2}{3}h...$
glossy valveBOT
jpsz
vale beacon
Does this make sense?
static field
mm i get it like this h/3*2 = h/3*2 = h/3*2 = h/3*2 = .....
i don't get the "," or the multiplication of fractions there
vale beacon
The "," is there to seperate each bounce distance
pseudo cape
\begin{enumerate} \item $\frac 2 3h$ \item $\frac 2 3 \cdot ( \frac 2 3 h)$ \item $\frac 2 3 \cdot (\frac 2 3 \cdot ( \frac 2 3 h))$ \end{enumerate} where each element represents the height after the corresponding bounce, for example $\frac 2 3h$ after the first bounce
vale beacon
^ @static field
pseudo cape
That's an alternative way of writing it, if you don't want it as a sequence as jpsz wrote it
balmy tusk
How do you multiply congruence classes? ex [5]8 * [3]8
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How do you multiply congruence classes? ex [5]8 * [3]8
spice orchid
[5*3]_8
balmy tusk
so its just [15]_8? and then reduce?
spice orchid
yes
balmy tusk
ok perfect. and what's an efficient way to reduce congruence classes?
spice orchid
just keep taking away multiples of whatever you're moding by
if the numbers are massive then long division will always work
hearty cipher
Can someone help me find the x and y intercepts?
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Can someone help me find the x and y intercepts?
torn jolt
Can someone help me find the x and y intercepts?
do you know what an x intercept is
and a y intercept
torn jolt
i understand what you mean but that's not the right way to say it
hearty cipher
Oh
torn jolt
you have to express it in coordinate form
ember shadow
(6, 0) and (0, 4) specifically
torn jolt
^
torn jolt
yeah
hearty cipher
Thank you guys so much
torn jolt
you're welcome
hearty cipher
.close
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hearty cipher
How the mcrib am I going to solve this (I did) Is my answer correct?: x = 1.96218928, y = -sin (1.96218928) + 4
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How the mcrib am I going to solve this (I did) Is my answer correct?: x = 1.9621...
Show your work, and if possible, explain where you are stuck.
@hearty cipher Has your question been resolved?
hearty cipher
.close
tidal sierra
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tidal sierra
i need help on this question because I struggle in School on this question. My expected grade is a Grade 7
I really want to achieve this grade
hasty meadow
2y-30=70
tidal sierra
y = 50
hasty meadow
yes
now
just gimme 1 sec
we can say that the angle i have marked in HUGE red for some reason
is equal to 2y-30
tidal sierra
yeah
tidal sierra
what is an isosceles i kinda forgot
hasty meadow
2 sides and 2 angles are equal
tidal sierra
ok
hasty meadow
So that massive red angle is equal to Y
tidal sierra
ok
tidal sierra
80
hasty meadow
x=80
tidal sierra
the answer was 60 degress
tidal sierra
ok
hasty meadow
i am so stupid
sorry
i typed the wrong thing
no did i
get someone elses help rq let me collect my calm
hasty meadow
ok
sorry mate i didnt read it properly
that huge red angle is equal to 2y-30
meaning its 70 aswell
so its 50+70+X = 180
i am really sorry
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@tidal sierra Has your question been resolved?
median cedar
in how many ways can we arrange a group made of 1 owner, 1 helper and 1 co-owner out of 5 people
median cedar
i see you use permutations, but we don't take order into account in this case, shouldn't we've used combination?
hidden spire
median cedar
what is that
hmmmm
why?
we use this formula for other stuff
as i was taught
my real question is: how many ways if there are 2 people who can't be in the same group
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@median cedar Has your question been resolved?
torn jolt
need help to solve this equation:
x² / (x² + 1) = 2
torn jolt
i guess make it [
\f{x^2}{x^2+1} -2 = 0
]
and add up the fractions, then you can completely ignore the denominator since the expression is equal to 0
glossy valveBOT
kind jay
$\frac{x²-2x²-2}{x²+1}$ do u have this?
glossy valveBOT
Fucktalogist
torn jolt
yeah... and that's
-x² - 2 = 0
ignore denominator
kind jay
is that equal to x²=-2?
torn jolt
no solution
yeah this equation has no solutions over R
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torn jolt
uh
torn jolt
sick
kind jay
,w plot x²/(x²+1) -2=0
glossy valveBOT
torn jolt
this is the original equation I need to solve
i used substitution
yeah that works
and got
i used math app and it says that the solution to this whole equation is ALSO +-sqrt(2), workbook also says this is the solution (besides +-1/sqrt(3))
oh... i see
it's x/(x²-1)
and I wrote t = x/(x²+1)
Thanks for helping!
.close
exotic shoal
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lofty island
<@&286206848099549185>
You can't abuse the helpers ping like this
You can only use it after 15mins of no help
Also, you should use relative velocity here
exotic shoal
no i dont want help from u
its something to do with SUVAT
and sim equations
not 'relative velocity'
noble horizon
you're just being rude at this point
exotic shoal
okay..
noble horizon
BC will be 240-s
noble horizon
now literally plug everything you know about the particles and solve for s
exotic shoal
240-s
noble horizon
AB is 240 apart
exotic shoal
who said ac is s
exotic shoal
What would the inital and final velocity be, just 5?
exotic shoal
u have no time
@noble horizon wait so am i wanting to find t
or what
cos im confused on what i am actually trying to find
noble horizon
you have to solve for t and s at the same time
you'll end up with 2 equations that you'll have to solve simultaneously
or if ur smart there is a way to cancel out t
noble horizon
or if ur smart there is a way to cancel out t
actually no there really isnt
exotic shoal
wait
noble horizon
so just solve simultaneously
exotic shoal
so for BC
is s 240-s
u 5
v 5
a 0.75
and for AC
Is
s s
u 4
a 0.75
?
Because thats what i used and shoved into s=ut+1/2 at^2
and formed 2 sim equations but i got wrong answer
noble horizon
what answer did u get
exotic shoal
12.9 for t and then 56.3 for s
exotic shoal
man
exotic shoal
So we cant use suvat?
because suvat works for constant acceleration only
for BC
?
noble horizon
no you can just set a = 0
karmic grotto
noble horizon
or again use v = s/t
@karmic grotto start a new help channel
this is someone else's
exotic shoal
no you can just set a = 0
why does that work
noble horizon
why does that work
because constant speed means 0 acceleration
exotic shoal
does it?
noble horizon
yes
exotic shoal
ohhhh yeah
because
acceleration is change in velocity if velocity doesnt change
a is 0
I LOVE U
ITS ALL FALLING INTO PLACE
DAMN I LOVE U SO MUCH
noble horizon
😅😅
exotic shoal
Ok im giving it another go
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