frigid canopy
Thanks!\
#help-19
odd edgeBOT
odd edgeBOT
Available help channel!
broken crest
translated: Considering the region D and the function f(x,y) write the integral in both cartesian and polars coordinates
broken crest
I dont understand how are the limits set and how is that blue region chosen if, according to the inequation, y>0
Please, tag me if someone answers with the explanation, thank you
wicked olive
I dont understand how are the limits set and how is that blue region chosen if, ...
Blue region is the domain of the integration
broken crest
Blue region is the domain of the integration
Yes but how is it set?
broken crest
lol
wicked olive
y<= -x
broken crest
i understood
wicked olive
So the region is below the line
odd edgeBOT
wicked olive
And also the region is inside the radius
wicked olive
i was seeing the right side of the plot as y>=0
Alright
broken crest
thank you very much
wicked olive
Have a nice day
broken crest
u 2
wicked olive
thank you very much
No need, it was nothingš¤£
broken crest
No need, it was nothingš¤£
XDD
wicked olive
Also close the help forum ticket too if you can, pls
odd edgeBOT
ā Original question: #help-19 message
broken crest
.close
odd edgeBOT
odd edgeBOT
Available help channel!
torn tulip
odd edgeBOT
torn tulip
tall ivy
a*x/100 = b
torn tulip
i tried to solve it but i got a big quadratic
torn tulip
yeah
tall ivy
a + ax/100 = b (Dude this seems correct, but it looks like it doesn't work)
Now we can substitute the value of a here
You gotta solve that big quadratic, I've got the answer, and it's one of the options (after some approximation)
@torn tulip can you show me the quadratic you got
odd edgeBOT
@torn tulip Has your question been resolved?
tall ivy
Alright, now apply the quadratic formula (or simply put it on desmos if you're permitted to do so)
torn tulip
Alright, now apply the quadratic formula (or simply put it on desmos if you're p...
bruh
we are not allowed to use desmos
i meant i know the answer but
tall ivy
Scientific Calculator?
tall ivy
Alright, won't be a problem
odd edgeBOT
torn tulip
.reopen
odd edgeBOT
ā Original question: #help-19 message
tall ivy
Whatever man, should have said you no longer need help before abruptly closing the channel as the helper was typing
torn tulip
i meant we came to the end no?
tall ivy
Well you gotta say that you no longer require help, also I was typing when you just closed it, that's not very nice
Whatever dude, just be a bit more careful next time, like we're here to help you man
torn tulip
i closed after you wrote the calculator
tall ivy
I thought you still needed help to reach the answer (IK you have the answer, you mentioned it, I thought you needed help with the process)
You should clarify things out before just abruptly closing the channel
torn tulip
bruh?
you want me to solve quadratic
for no reason
we do not get 10 minutes for wasting
we need quick methods in the exam hall
which can hit the answer within seconds
maybe i will use options
hope you understand the situation
tall ivy
You should have stated that first man, we treat mathematics absolutely, as if there is no scope of hit and trial
Well anyways, we aren't bots, we're humans, just be a bit more considerate the next time, clarify things out and then close the channel
Now please close the channel whenever you wish to
torn tulip
.close
quasi shard
quasi shard
Can someone give me a hint for this
I have found a = 0 for the no solution case and I need to know look at cases where the are infinitely many solutions but I don't know how to manipulate the matrix
bronze canyon
det(A)=0 implies nonuniqueness
quasi shard
det(A)=0 implies nonuniqueness
What is det(A) as notation represent?
quasi shard
Ok so your saying I need to take the deriminent
bronze canyon
yeah
specifically if det(A)=0, then there are either infinitely many or no solutions
quasi shard
But how can I use this fact to find three specific values of a then?
bronze canyon
do you know how to find determinants?
if so, just find the det and set =0
this will give you all possible values of a which arent unique
bronze canyon
det(A)=0 implies nonuniqueness
i can also explain why this works if you need
quasi shard
Ye
bronze canyon
to start, do you know what det does?
quasi shard
do you know how to find determinants?
Well let me see if I can do this. I haven't done to many of them
bronze canyon
to start, do you know what det does?
not just how to solve it, but what it mechanically does?
quasi shard
It reducea the dimensions of the matrix right?
bronze canyon
im not sure what you mean by that
rref reduces to dimensions
det finds the volume of the vectors that make the matrix
is the best way to put it
so, think about a linear function which takes a whole dimension to 0
its volume is also 0
quasi shard
What do you mean it takes a whole dimension to 0?
bronze canyon
but this transformation forces infinitely many soltuions
bronze canyon
What do you mean it takes a whole dimension to 0?
yes
in fact, if some non zero vector goes to 0 (ie Ax=0 for x neq 0), then a whole dimension must also go to 0
this whole dimension is called the nullspace
<@&268886789983436800>
scammers!!
thank you
anyways
quasi shard
Ok let me make sure I'm understanding we have an operation that transforms a non zero vector to 0. This it's what the det does and what it turns it into is called a nullspace
quasi shard
Isn't the det the same as the magnitude of the cross product
quasi shard
Ok
quasi shard
So 1 and 3
bronze canyon
so these guys
we can make a parallelogram out of them
det literally finds the area of this parallelogram
with signs
so negative area means it follows left hand rule, positive area means it follows right hand rule
this is precisely what magnitude of cross product does in 2d and 3d
but det works in 4d
make sense so far?
quasi shard
Yes
bronze canyon
in fact, this used to be the classical definition of det
so, now when any one dimension goes to 0, we have this 'parallelogram'
quasi shard
I want to make sure I understand det is an operation that you apply
bronze canyon
so, now when any one dimension goes to 0, we have this 'parallelogram'
which has area 0
quasi shard
which has area 0
So one our vectors has become the 0 vector then
bronze canyon
I want to make sure I understand det is an operation that you apply
yes, to the matrix (or any linear function)
quasi shard
Or is it just equal to it
bronze canyon
yes one of the vectors is a 0 vector
quasi shard
Couldn't you also have a parallel vector too?
bronze canyon
also this
yes
but if our linear transformation takes 2d to 1d, it must still map a dimension to 0
i can prove this simply actually
quasi shard
Ok
bronze canyon
say x not =0 and Ax=0
then the whole dimension is given for any t, tx (just scales of this)
then A(tx) = tA(x) = t0 = 0
so it has to give the 0 vector for a whole dimension, if a nonzero vector goes to 0
make sense?
quasi shard
Sorry I'm not following why we need the tx
quasi shard
Just the same vector but times a scalar
bronze canyon
correct
so {tx| for any real t}=span(x)
tx is every single vector in that span
give me any real t (including 0)
so, now consider this
tx is every vector in that direction
let x be a nonzero vector, then tx is also nonzero
let A(x)=0 also
so we have this algebra
A(tx) = tA(x) (by linear A)
= t 0 (by A(x)=0)
= 0 (since t times 0 vector is 0 vector)
so all of the span went to the 0 vector
does that make more sense?
its okay if not, idk where youre at, so you just need to tell me what you dont understand
quasi shard
which has area 0
Ok so when we are referring to tx. I want to make sure we are referring to any parallel vector in this image
bronze canyon
Sure, but thats not what i proved
quasi shard
tx is every vector in that direction
let x be a nonzero vector, then tx is also ...
I am not following how A(tx) = tA(x)
bronze canyon
All i proved was that if Ax=0 for some nonzero x, then all of its scales must also be 0
bronze canyon
I am not following how A(tx) = tA(x)
Thats the definition of linearity
quasi shard
Ah I see just scalar multiplication
bronze canyon
Bot broke?
$A$ is linear iff:
1: $A(t\vec x)=tA(\vec x)$ for any vector $x$ and scalar $t$
2: $A(\vec x+\vec y)=A(\vec x)+A(\vec y)$ for any vectors $\vec x,\vec y$
Oh there we go
quasi shard
Ok so your saying a is a vector since it is closed under addtion and multiplication
bronze canyon
Ig my internet died
Sorry abt that
$A$ is linear iff:
1: $A(t\vec x)=tA(\vec x)$ for any vector $x$ and scalar $t$
2: $A(\vec x+\vec y)=A(\vec x)+A(\vec y)$ for any vectors $\vec x,\vec y$
clever fjordBOT
Cycadellic
bronze canyon
There we go
quasi shard
Ok
bronze canyon
Ok so your saying a is a vector since it is closed under addtion and multiplicat...
Those are vector space axioms, not anything about A
All we know is that A is linear
quasi shard
Ok
bronze canyon
So the whole span of vectors that A takes to 0 are all taken to 0
That gives us infinitely many solutions
But also det(A)=0
Because det is the area of that āparallelogramā where one vector went straight to 0
quasi shard
Ok so if we look back in our matrix your saying that one row should all be 0 for this to hold
bronze canyon
yes, after rref
But also
If we get one row =0 and something on the right hand side not =0, thats a contradiction so there can be no solution
So say that whole bottom row went to 0
It depends on the b_3
quasi shard
yes, after rref
What is rref?
bronze canyon
If b_3=0, then infinitely many solutions
If b_3 not =0, then no solutions
bronze canyon
What is rref?
Two matricies are rref equivalent if
One row is scaled by non zero
Two rows are swapped
One row is replace by its sum with another row
quasi shard
So the case of a being 0 handles this since we are not able to choose what b1,2 or 3 are
bronze canyon
Two matricies are rref equivalent if
One row is scaled by non zero
Two rows are ...
The gaussian operations
bronze canyon
So the case of a being 0 handles this since we are not able to choose what b1,2 ...
We actually dont care what b are
All we about care is if its not unique
b can be any possible value, but if the whole row is 0, then there are infinitely many or no solutions
bronze canyon
Possibly
There may be more
I mean there arent, but its best practice to show it
det(A)=0 must give you all possible nonunique solutions
Did everything make sense here? I can approach it purely algebraically too, but i figured the explanation is more intuitive
quasi shard
Ok so what I understood is this: We can see if there are no unique solutions if we take the det(A) = 0 (also I just realized I am not sure how to compute determinant for row echelon matricies) .
bronze canyon
Ignore row echelon for now
quasi shard
How can we do this?
quasi shard
The thing is I don't know how to solve determinants if there is the line. The only way I know to solve determinants at least at the moment is for caculating cross products. But the way the matrix is set up right now is different.
bronze canyon
Were chopping off that side and ignoring it
You cant find det of a non square matrix, and if you somehow force it, youll get 0 anyways
quasi shard
Ok then let me compute this right now
bronze canyon
The right side of the bar is the solution matrix
The left is the coefficient matrix
Only the coefficient matrix will tell you about uniqueness, so we ignore tje solution matrix entirely
quasi shard
Ok
bronze canyon
Im walking around so youll need to ping me
stiff agate
I am confident in my ability to tackle mathematics problems of any kind with precision and efficiency. With extensive experience across various mathematical domainsāincluding algebra, calculus, geometry, statistics, and advanced topics like differential equations and linear algebraāI can deliver clear, accurate solutions tailored to your needs. Whether you require step-by-step explanations, complex proofs, or practical applications, I adapt to any level of difficulty, from basic arithmetic to university-level challenges. My approach ensures clarity, logical reasoning, and timely delivery, making even the most intricate problems manageable. I stay updated with the latest mathematical techniques and tools to provide high-quality results. Let me handle your mathematics queries with expertise and care, ensuring your complete satisfaction. Please share your specific requirements, and Iāll get started promptly.
quasi shard
@bronze canyon This is what I got
bronze canyon
I am confident in my ability to tackle mathematics problems of any kind with pre...
Is this ai?
quasi shard
Seems like it
bronze canyon
You just unironically use em dashes and parallel sentences like that?
stiff agate
You just unironically use em dashes and parallel sentences like that?
I think it is the grammamrly that is behaving like this
bronze canyon
Mmhm
stiff agate
Sure
bronze canyon
<@330098469597347840> This is what I got
Checking one sec
stiff agate
Do you have a task that I can help you in handling
bronze canyon
Out irl
bronze canyon
Yeah its not right
bronze canyon
<@330098469597347840> This is what I got
Start with the matrix, show your steps
quasi shard
Yeah its not right
stiff agate
Start with the matrix, show your steps
bring a task
stiff agate
Bring the instructions for this
stiff agate
I second you
quasi shard
But the x intercepts should be it then
bronze canyon
Oh i see you did make a mistake
I see the problem you forgot a - sign
You put -8a, it should be +8a
quasi shard
Oh yes sorry
stiff agate
Oh i see you did make a mistake
Sure
quasi shard
Ok 2 and 4 then
quasi shard
0
quasi shard
Ok thx
gaunt estuary
(3+2i) Ā²/2+i = a+ib
odd edgeBOT
(3+2i) Ā²/2+i = a+ib
glossy jay
times the complex conjugate of the denominator on both numerator and denominator
gaunt estuary
times the complex conjugate of the denominator on both numerator and denominator
It's a complex number questions
glossy jay
$\frac{\left(3+2i\right)^2}{2+i}=\frac{\left(3+2i\right)^2(2-i)}{(2+i)(2-i)}=\frac{\left(3+2i\right)^2(2-i)}{5}$
clever fjordBOT
BrandenXia
glossy jay
glossy jay
$(2+i)(2-i) = 2^2 - i^2$
clever fjordBOT
BrandenXia
gaunt estuary
aĀ²+bĀ²= aĀ²+bĀ²+ 2ab
thick roost
Didn't understand
He rationalize the denomintor
glossy jay
aĀ²+bĀ²= aĀ²+bĀ²+ 2ab
$(a-b)(a+b)=a^2-b^2$
clever fjordBOT
BrandenXia
gaunt estuary
I mean (a+b)Ā²
thick roost
aĀ²+bĀ²= aĀ²+bĀ²+ 2ab
Dont forget the i
glossy jay
$(a-bi)(a+bi)=a^2+b^2$
clever fjordBOT
BrandenXia
gaunt estuary
Dont forget the i
Yes
glossy jay
since -i^2 = 1
glossy jay
BrandenXia
just simplify with the normal algebra rules
gaunt estuary
;-;
glossy jay
all other parts should be same as how you simplify expressions in real numbers
thick roost
aĀ²+bĀ²= aĀ²+bĀ²+ 2ab
Just use this with the i try to calculate this
gaunt estuary
just simplify with the normal algebra rules
It's (3+2i) Ā² so can we use this formula? (a+b)Ā²= aĀ²+bĀ²+ 2ab
glossy jay
yeah
glossy jay
It's (3+2i) Ā² so can we use this formula? (a+b)Ā²= aĀ²+bĀ²+ 2ab
$=3^2+12i-4$
clever fjordBOT
BrandenXia
glossy jay
$(a+b)(c+d) = ac + ad + bc + bd$
clever fjordBOT
BrandenXia
gaunt estuary
What
thick roost
And u forget i^2=-1
glossy jay
2(3)(2) term need to contain i, and (2)^2 term need to be negative
gaunt estuary
a= 3 and b= 2i?
thick roost
Yea
gaunt estuary
since i^2 = -1
Is iĀ² always equal to -1?
Alright
Butt
It's 1 right?
i= ā-1
thick roost
Is iĀ² always equal to -1?
Yes
gaunt estuary
In iĀ²= (ā-1)Ā²
In roots so minus minus cancel and 1 is only one lefted
?
Now?
@thick roost
thick roost
BrandenXia
Now use this
thick roost
the denominator should be (2+i)(2āi)=2^2-i^2=4ā(ā1)=5.
odd edgeBOT
@gaunt estuary Has your question been resolved?
odd edgeBOT
@gaunt estuary Has your question been resolved?
whole flame
i am struggling to understand how 5 and 11 are related at all. Alternate exterior makes the most sense but the answer cant just be what fits the best there surely has to be one definitive answer. I asked both another person in the same class and also my older sibiling (who is taking calculus) and neither of them could figure it out. Is the question just wrong? or am I missing something?
whole flame
does... that mean something? We definitely havent gone over that in class nor is it anywhere in the textbook, paper, or notes.
tall ivy
Wait what, the lines aren't even parallel, you can't even have alternate angles
tall ivy
does... that mean something? We definitely havent gone over that in class nor is...
Vertically Opposite Angles are equal, like 5&7 and 6&8
Yeah IG 5 and 11 cannot be related, they could only have been related if c and d were parallel to each other
Then 5 would have been equal to 11
whole flame
Wait what, the lines aren't even parallel, you can't even have alternate angles
the lines dont need to be parallel for there to be alt. angles i thought?
on all the examples ive been given there have been no parallel lines
tall ivy
Ahhhh yeah wait a second
whole flame
Yeah IG 5 and 11 cannot be related, they could only have been related if c and d...
but is says classify it as one of the following and none of the options are "none" and I went ahead and did the rest and they all have answers
tall ivy
Well, I was fairly sure that the lines had to be parallel for alternate angles to be equal
But if it states otherwise
Then 7 would be equal to 9 as they are alternate angles (Dude it feels wrong for me to say this, like I'm kinda sure the lines have to be parallel)
And 9 would be equal to 11 as they are vertically opposite angles, so 5 = 11
Yeah dude I'm fairly sure that the lines have to be parallel for the alternate angles to be equal, I checked it by drawing a figure, it just can't work always, the angle between our intersecting line and stationary line remains same, but if you move the other line away from parallel (basically rotate it), the angle for that differs, hence the angle can only be equal in one case, it cannot be always valid
Do confirm with your brother or someone experienced first, I'm just going with logic as my main take
whole flame
Well, I was fairly sure that the lines had to be parallel for alternate angles t...
well angles dont need to be equal to be alternate, I think your thinking of vertical angles.
alternate just has to do with position, the size literally doesnt matter
tall ivy
Ahhhhhhh wait so it's not about equality, mb dude, we just have to show they are related, apologies for that
whole flame
ill give the definitions rq: "Alternate interior angles are a pair of angles that are both interior, non adjacent, and are on opposite sides of a traversal", "Alternate exterior angles are a pair of angles which are on the exterior, are not adjacent, and are on opposite sides of a traversal"
whole flame
Ahhhhhhh wait so it's not about equality, mb dude, we just have to show they are...
yeah all g
tall ivy
I thought we had to show a relation between them in the sense that how are they related in an equation, which is just not possible for non-parallel lines
whole flame
ah I see
whole flame
ill give the definitions rq: "Alternate interior angles are a pair of angles tha...
also, the angles fit neither of these definitions. 5 is on the interior and 11 is on the exterior so it cant be either of these
tall ivy
Well IG they are alternate exterior angles then
tall ivy
Well it is on the opposite side of the line
whole flame
yeah but they arent both exterior
an alt. ex. angle would be like 2 and 12 because they are exterior and are on opposite sides of the line
tall ivy
Well when I looked it up it just shows two lines
I think you're overcomplicating this, they're on the opposite side of the line each
Also 7 and 9 are alternate interior angles
whole flame
Also 7 and 9 are alternate interior angles
yeah ik
whole flame
I think you're overcomplicating this, they're on the opposite side of the line e...
well then why arent they alt. int. angles
why do they have to be alt. ex. angles?
tall ivy
Well, you probably know the difference between alt. int, and alt. ext. better than I do, considering this is honestly too basic for me
Can you enlighten me please?
whole flame
exterior angles are just like, angles on the outside. Like around angles 4, 3, 5, and 6 you could kind of draw a square around them while angles 1 and 2 you cant, so 1 and 2 would be exterior while 4, 3, 5, and 6 would be interior
tall ivy
Oh wait, so basically what you're trying to say is the angles have to be literally on the exterior, so if I draw 4 lines and a traversal, then the outer angles of the outer 2 lines are the only exterior angles, is this what you're trying to say?
whole flame
yes
tall ivy
Ahhhh well no, that's just not how it works
whole flame
im so lost bro š
tall ivy
Like this is related to 5 and 11 right, so lines c and d, we can basically ignore b
You've overcomplicated this, it doesn't literally have to be the exterior angles for it to clasify as an exterior angle
whole flame
how does that make sense???? it doesnt have to be an exterior angle for it to be an exterior angle????
tall ivy
They just have to be on two separate lines that have a traversal, and the angles are on the outer side of just those lines, and are, well, alternate
whole flame
well then how would they be alternate interior lines
like
any angles
not 5 and 11
tall ivy
how does that make sense???? it doesnt have to be an exterior angle for it to be...
Well it doesn't have to literally be an exterior angle, in the sense, it doesn't have to literally be the outer angle on the outermost line, you could technically ignore the outer line if it's not required
tall ivy
well then how would they be alternate interior lines
Well the angles Vertically Opposite to alt. ext. angles are alt. int. angles, and vice versa
Like dude honestly, you've overcomplicated it so much that even I'm surprised
whole flame
i am still so unbelievably lost. Are you saying that because 5 and 11 arent on the same side of line a that they are alt exterior? that doesnt make sense
tall ivy
But that's fine, you're learning, and it shows that you question things you learn
tall ivy
i am still so unbelievably lost. Are you saying that because 5 and 11 arent on t...
Okay see, is line b in any way related to the relation between 5 and 11?
tall ivy
Okay, so it's not relevant, we can basically ignore it
whole flame
but that doesnt mean it isnt still there
like
are you allowed to just... do that?
just pretend something isnt there if it isnt part of the question
that feels very very wrong
whole flame
i understand what you mean that if b isnt there then yeah 5 and 11 are alt ext.
but why are we allowed to say b isnt there just because its not relevant
tall ivy
Well dude the thing is, the only thing that matters is : There have to be two lines, there has to be a traversal, the angles have to be on the outer side of the line, the angles have to be Vertically Opposite to the alternate Interior angles
Literally nothing else matters, if all these conditions are fulfilled, then the angles are alternate exterior
whole flame
I totally believe you that its the right answer but I don't understand why we are allowed to just say B cant be there, aren't we modifying the equation? Like we are finding the answer and then changing the graph to fit that answer. It feels like finding the criminal and then looking for the crime
tall ivy
Neither a third line matters, nor anything else, calling it "exterior" is done only to state that they are on the outer side of those two lines, it doesn't literally have to be on the outer side if there are multiple lines, the multiple lines do not concern us here
whole flame
ohhhhhhhh, so its exterior because its on the exterior of c?
but its on the interior of b
tall ivy
ohhhhhhhh, so its exterior because its on the exterior of c?
Yes precisely, it just has to be the exterior of c, and the exterior of d, it doesn't literally have to be the exterior of the figure
tall ivy
so its both
Yes, in relation to b and c, angle 5 is interior, but in relation to c and d, the angle 5 is exterior
tall ivy
so then 5 and 6 are both interior *and* exterior?
Yes, depends on which two lines you're looking at
umbral epoch
why is this wrong
odd edgeBOT
why is this wrong
odd edgeBOT
umbral epoch
.reopen
odd edgeBOT
ā Original question: #help-19 message
umbral epoch
nvm idk what im doing wrong
i got
f_x = sqrt(y)
f_y = x/2sqrt(y)
and f(2,9) = 6
and then plugged it into z-z0 = f(x,y) + f_x(x-x0) + f_y(y-y0)
oh nbm
gawsh diddly i forgot to plug P into fx and fy
.clos
e
.close
odd edgeBOT
odd edgeBOT
Available help channel!
mystic saffron
i need help
odd edgeBOT
i need help
odd edgeBOT
@mystic saffron Has your question been resolved?
wide oasis
$\int\frac{dx}{5+4sin x} = ?$
odd edgeBOT
$\int\frac{dx}{5+4sin x} = ?$
clever fjordBOT
Prathmesh
wide oasis
can anybody please tell me how to start?
wide oasis
ohh thankss
somber talon
Make sin(x) in terms of tan(x/2)
if you see 1/(a+bsinx) or 1/(a+bcosx) do this
wide oasis
$\int\frac{dx}{(cos^3x)(\sqrt{2sin2x})}$
clever fjordBOT
Prathmesh
wide oasis
what about this one?
wide oasis
if you see 1/(a+bsinx) or 1/(a+bcosx) do this
alrighhtt
vivid berry
Hi guys, before you ask, no this is not an exam, itās just a practice thing so we can see if weāre okay for the real exam. My question is concerning the first equation. They ask to know if the function is really continuous in zero. Hereās what I did for now : Since we know that x<0 on the first equation, we know that we will have to do the lim when x approaches 0-. Also since thereās a tanx, you can replace it by sinx/cosx and sin2x can be rewritten as 2x (sin2x/2x) to give us two. Now Im pretty sure we have to send the cosx down by doiing 1/cosx times our equation. After that, I donāt know how to proceed. What should I do next?
vivid berry
For context, I have not seen lāhopital rule or more advanced stuff, im really on the beginning side of calculus. I have saw the basic identity of sin^2x + cos ^2x = 1, tanx = sinx/cosx and sinx/x equals one when x approaches 0
somber talon
if it is 0/0 or inf/inf form you can just multiply and divide by the angles in the trigonometric function and evaluate it to 1 by the standard limits
vivid berry
Sorry, im not familiar with those terms, what angles are you taking about?
somber talon
yeah if you get 0/0 or infinity/ infinity form after letting x = to the value that its approaching there are standard limits which evaluate to 1 or some other number
like sinx/x = 1, tanx/x = 1
vivid berry
Okay thanks, it helps a bit, but Im not sure I see any standart limits here except for sinx/x
somber talon
well for x<0 even tanx is valid
somber talon
multiply and divide by x
vivid berry
I see
Alr
But then what do I do with this x ^3 at the bottom?
Because of this,
i cant divise
Divide
somber talon
see tanx/x and sin2x/2x become 1then just simplify the numerator and cancel one of the x
vivid berry
Okay, I have to go to bed rn though, but I will try that, thanks a lot!
somber talon
no problem
vivid berry
.close
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cinder haven
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cinder haven
can anyone help me
wicked olive
Arent you the same guy we just helped couple hours back?
cinder haven
wait let me do that right now
wicked olive
Do it
wicked olive
No, whereād the k term go?
vivid shoal
Hi
cinder haven
So let's go step by step
yea
eternal sorrel
You want to solve the equation xĀ²+2kx-3=0 in terms of k by completing the square
so start by completing the square
How do you do that ?
cinder haven
yea I done that
wicked olive
x + k = root 3 + k
Itāll be root(3+k^2)
And they have asked you to answer in the form of k
If im not wrong
eternal sorrel
yea I done that
Yes but you seem to have a slight mistake at the end, maybe you just copied something wrong at the end
As Limitless pointed out it should be sqrt(3+kĀ²)
Not sqrt(3+k)
cinder haven
isnt root k^2 k?
wicked olive
isnt root k^2 k?
Itāll be +-
eternal sorrel
$\sqrt{k^2+3}\neq \sqrt{k^2}+\sqrt{3}$
clever fjordBOT
yassine
eternal sorrel
Although I don't think that's your mistake
But you have sqrt(kĀ²+3) so you cant just take kĀ² out as k
cinder haven
ohhhh
cinder haven
wait is this part so far right (x+k)^2 = 3+k^2
cinder haven
That way you can spot your mistake
yea
eternal sorrel
Yes that's correct
wicked olive
Or even back track it
eternal sorrel
Ok now what do you do?
cinder haven
what i got after that
eternal sorrel
What did you get
cinder haven
x+k = root 3+k
eternal sorrel
No that's not correct
cinder haven
ok so thats my mistake
eternal sorrel
So you want to take sqrts on both sides of the equation right ?
cinder haven
yes
eternal sorrel
ok so just do that, so $(x+k)^2=k^+3\implies\sqrt{(x+k)^2}=\sqrt{k^2+3}$
clever fjordBOT
yassine
eternal sorrel
So that's what it means to take sqrt on both sides right?
cinder haven
wait am abit confused
eternal sorrel
What is confusing you? Maybe I can help clear the confusion
cinder haven
ok am confused on how there is still the squared and theres a root at the same time
for both of them
like whered the root come from is kinda where am confused basically
eternal sorrel
wait is this part so far right (x+k)^2 = 3+k^2
you have this and want to isolate x on one side of the equation right ?
So you want to reach something like x=.... At the end
cinder haven
yea yea
eternal sorrel
You have ^2 on the left side of the equation so how do you get rid of it?
cinder haven
by square root
cinder haven
ohhh
eternal sorrel
Because there is something important that you should keep in mind before cancelling them
And also because I wanted to show you each step clearly
eternal sorrel
yassine
So now that you agree on this
Let's cancel the sqrt and Ā² on the left side
But be careful
eternal sorrel
Because remember that sqrt(anything) is >=0 right ?
cinder haven
and it makes sense now
eternal sorrel
and it makes sense now
So what did you get
eternal sorrel
Exactly that's it
cinder haven
idk why it took me so longš
eternal sorrel
From Lebanon, what about you
cinder haven
Egypt
eternal sorrel
Ohhh nice to meet you
eternal sorrel
np, have a great day/night
cinder haven
you 2
eternal sorrel
If you are done please type .close to close the channel
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broken crest
why is that region painted if the title says y>=0
odd edgeBOT
why is that region painted if the title says y>=0
broken crest
<@&286206848099549185>
narrow crypt
Can we get more context
I believe the author is trying to show without the restrictions first?
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random thunder
for part B) i understand the solutions, but why doesnt (2/3 * 2/3) + ( 1/3 * 2/3) also give you 8/9?
random thunder
shouldnt what i did still be a valid method
oh is it because i shouldve done (2/3 * 2/3) + ( 1/3 * 2/3) + ) + (2/3 * 1/3)
drawing the tree diagram rlly helps
pls ping btw!
austere jolt
oh is it because i shouldve done (2/3 * 2/3) + ( 1/3 * 2/3) + ) + (2/3 * 1/3)
(2/3 * 2/3) + ( 1/3 * 2/3) = 6/9. this is more accurate.
remember that you have HH, TH, HT, and TT to consider
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random thunder
(2/3 * 2/3) + ( 1/3 * 2/3) = 6/9. this is more accurate.
wait so shouldnt the method be this (2/3 * 2/3) + ( 1/3 * 2/3) + ) + (2/3 * 1/3)
which considers 3/4 scenarios that include H
austere jolt
that's why I also replied to the more accurate one
your initial method is wrong. should have made that clear though, sorry
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rancid canyon
Guys can someone help me angain?
odd edgeBOT
Guys can someone help me angain?
silk stream
Ask
errant dew
wooden python
Guys can someone help me angain?
send your question here
rancid canyon
Ignore the letters a and b
Wait
the total length of satin ribbon needed to finish the supports by bordering them
how do i do this
rancid canyon
Ignore the letters a and b
the total area for 1 coaster is 50,24cm
silk stream
How
errant dew
What is the radius of the coaster?
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@rancid canyon Has your question been resolved?
rancid canyon
What is the radius of the coaster?
50,24cm
errant dew
Iām asking for the radius
So we can work out the area
Oh nvm
Sorry I thought youād gotten it wrong
Yeah that area is right but maybe use . since the question says pi is 3.14 not 3,14?
errant dew
Ignore the letters a and b
By area of fabric does it mean the coaster is fabric?
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zinc warren
Hey i have a question, can this shortcut of dividing the percentage by 10 and the number by 10, then multiplying them, be used for any percentage problem?
zinc warren
also yes i know its easy but im really stupid
steep matrix
ye
brazen hare
Hey i have a question, can this shortcut of dividing the percentage by 10 and th...
the % sign just represents division by 100 i.e. x% = x/100. So you can prove or disprove this algebraically
proven harbor
divide 25 by 10
divide 200 by 10
and multiply 2,5 by 20
Yes
zinc warren
alright thanks guys
proven harbor
That is correct
zinc warren
You can even 200/100 = 2 and now 2 * 25
wow my math teacher makes stuff way too complicated
proven harbor
Lol
zinc warren
welp % are pretty eazy
proven harbor
!done
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zinc warren
.close
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proven harbor
Good luck!
zinc warren
Good luck!
i appraciate u i have exams in 4 weeks itll help
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tired crescent
Hey, i'm trying to get hands on PohligāHellman algorithm, I got till the "n - 1 = product of prime number" but I just can't get my head around how it derived g^pā1/p^n(i)
tired crescent
if someone can help me get intuition for this, it would be awesome :)
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@tired crescent Has your question been resolved?
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timber river
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timber river
dont understand this
like why are two sides 10N and 8N
why 30 and 105 degrees
looks nothing like the original force diagram
ivory raven
I think that yellow box might be pointing to the wrong diagram
it matches the diagram above, not to the left
but it's the same process either way, so maybe it's just illustrating the idea without the specific numbers
sturdy cape
it matches the diagram above, not to the left
It does match; the thing is structured shittily, because the rest of the example working out is on the next page
I recognise the textbook
timber river
It does match; the thing is structured shittily, because the rest of the example...
yeah for the blue diagram
sturdy cape
well, yes - the blue diagram is the working out
That should have some extra WO on the next page
The yellow box is an explanatory note
You should have done some vector addition from the Pure textbook
timber river
yeah i know how to do cosine rule, i just dont know how the blue triangle is formed in the first place
sturdy cape
Can you see how to get from the original diagram to the yellow box?
sturdy cape
timber river
ah
sturdy cape
You've got two vectors, OP and OQ; their resultant force (because they represent forces) is OP + OQ
You add them however you add vectors
Connect O with that new green P, and you end up with the blue diagram
Now, we don't exactly know what theta is (heck it could be negative, so the resultant force might actually be downright rather than up-and-right)
But we have enough information in the triangle to figure out how to calculate R
timber river
Now, we don't exactly know what theta is (heck it could be negative, so the resu...
wouldn't this fuck up the triangle
sturdy cape
Not as much as you might think
The 8 and 10 will still be there (they have to be there or else the forces aren't the same)
And that 105 degrees, by construction, must also be there
timber river
yeah im tripping it would still be a triangle
timber river
Connect O with that new green P, and you end up with the blue diagram
couldn't you just
find the resultant vector and find the angle it makes with the x axis
timber river
yeah i know
i just worded it badly
i was asking how much more benefitial is using this method (triangle law), compared to raw dogging it
sturdy cape
How would you do it then? (also please don't call it rawdogging 
)
timber river
How would you do it then? (also please don't call it rawdogging <:girlbleak:1195...
like (10cos45i + 10sin45j) + (8cos30i - 8sin30j), finding the magnitude of that, and what angle it makes with the x-axis -> arccos([10cos45 + 8cos30] / magnitude of resultant)
sturdy cape
You can do it like that, then, yes
There's not necessarily one explicit way to do things in maths
The core point however is to make your working out clear, is all
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noble olive
This is a simultaneous equation with three variables.
I simply have a question
Why does it work out for only some numbers while others donāt?
How do you know which letter to get rid of as the second last step? Is it simply trial and error?
(Left hand side is what I tried first, and I couldnāt get an answer so I tried cancelling out the other one. But I got it right so Iām confused why the first one didnāt work)
orchid willow
If there exists a solution, it shouldn't matter which order you solve for it
the behavior of linear systems like this are well described by math called linear algebra
noble olive
Also what does it mean by hence, solve
Is the bottom also a simultaneous equation
orchid willow
when you have a system that "doesn't work," I think you might mean one that has rows that are not linearly independent. there's a lot of interesting reading I can send you on this
noble olive
when you have a system that "doesn't work," I think you might mean one that has ...
Sure Iād love it
orchid willow
Also what does it mean by hence, solve
Is the bottom also a simultaneous equati...
the bottom is similar in the sense that
ok so
basically
being "linear" is a property with respect to a variable
like, 3x^2 is not linear with respect to x, since you're writing the expression in terms of cx^2, not cx (a constant times x)
but it's linear with respect to x^2
noble olive
Is this correct so gar
orchid willow
so the idea is, the equations really are in a similar form. just take x=a^2, y=b, z=c+1
orchid willow
https://textbooks.math.gatech.edu/ila/matrix-equations.html
this is a little tricky and has some deep ideas, so I would only recommend reading it out of interest
orchid willow
Is this correct so gar
it's hard to tell since a lot of the work is out of frame and blocked by your pen
orchid willow
I donāt know how these are connected
connected to what? the solutions for a,b,c?
noble olive
connected to what? the solutions for a,b,c?
Since the question says hemce
orchid willow
the bottom is similar in the sense that
please read what I said starting from here
and let me know if anything I said doesn't make sense
noble olive
So thereās an easier way
orchid willow
and plus, you can't know if you have learnt it or not necessarily, since it could be something you've learn but perhaps in a "form" you don't recognize
and also, much of what you're taught cannot necessarily be "explicated" in the sense that you just overall get better at seeing forms of the same problem and solving problems. It's this sense of similar form that I'm trying to show you exists between these two systems of equations right now
orchid willow
So thereās an easier way
I would say try to read and understand what I am saying first, and let me know if any part is confusing
noble olive
I would say try to read and understand what I am saying first, and let me know i...
Dude I donāt get it at all!!
orchid willow
Dude I donāt get it at all!!
hmmm ok. let me act like chatgpt for a second and try to break it down. I want to explain 3 things
- what does it mean for an equation to be linear (in terms of some variables)
- why is the second system of equations "similar" to the first
- how you can use solutions solutions from the first system of equations to solve the second
actually, in fact, we can ignore 1. It's not actually important
going to point 2, consider the substitution $x=a^2, y=b, z=c+1$. I would suggest you to rewrite the first system of equations with this substitution, to see how the second and first systems are similar if you don't see so already
clever fjordBOT
大é»é¢å ę£åæ32
orchid willow
now for point 3
you have that z=-1, y=2, and x=4, right?
you also have that x=a^2, y=b, z=c+1
substitute in your values for x,y,z into these substitutions that relate x to a, y to b, z to c
for instance, you know that x=4, x=a^2, right?
by substitution, you can then conclude that 4=a^2
now, you can proceed to solve for $a$ and see that $a=\pm 2$
clever fjordBOT
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swift heron
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swift heron
D((a_11)) = a_11D((1)) by multilinearity right, but how do we know that this is the only such function
idek if my question makes sense
brain fucked
forest sky
well that forces D( (1) ) = 1 and every other output can be calculated by linearity
e.g. D( (5) ) must equal 5 and so on
swift heron
wait are we assuming D(I) = 1 here?
forest sky
well that's part of the theorem
forest sky
so the theorem says that if we want a function D satisfying the properties:
- domain M_n(F), codomain F
- alternating
- multilinear
- satisfies D(I) = 1
then there is only one such function. it would not be unique if we dropped the last requirement
swift heron
hm lets pretend we dropped the last argument
then any function D(a11) = ca_11 would work right
@forest sky oh i se enow lol
im just stupid
D((a11)) = a11D((1)) by linearity
but D((1)) always = 1
so D((a11)) = a11
.close
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swift heron
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swift heron
@woeful briar
woeful briar
sec
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swift heron
Let $r \in \text{Row}(AB)$. Then $r \in \text{Col}((AB)^T) = \text{Col}(B^TA^T)$. We must show $r \in \text{Col}(B^T)$. Observe that if $r \in \text{Col}(B^TA^T)$, then $r \in \text{Span}{B^T(a_1^T), B^T(a_2^T), ..., B^T(a_j^T)}$. Then we can represent $r$ as a linear combination such that $r = \displaystyle\sum_{i=1}^jB^T\alpha_i(a_i^T)$. We can represent $(a_i^T)$ in terms of the standard basis such that $a_i^T= \displaystyle\sum_{s=1}^n\gamma_{i,s}e_s$. Then $B^T(a_i^T) = B^T\bigg(\displaystyle\sum_{s=1}^n\gamma_{i,s}e_s\bigg) = \sum_{s=1}^n\gamma_{i,s}B^T(e_s).$ Then $r=\displaystyle\sum_{i=1}^j\alpha_iB^T(a_i^T) = \sum_{i=1}^j\alpha_i\sum_{s=1}^n\gamma_{i,s}B^T(e_s) = \sum_{s=1}^n\bigg(\sum_{i=1}^j\alpha_i\gamma_{i,s}\bigg)B^T(e_s)$. We have represented $r$ in terms of the columns of $B^T$, so $r \in \text{Col}(B^T)$, then $r \in \text{Row}(B)$. Therefore, $\text{Row}(AB) \subseteq \text{Row}(B)$
clever fjordBOT
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robust mirage
what did i do wrong here ?
odd edgeBOT
what did i do wrong here ?
robust mirage
zinc glacier
sign
zinc glacier
you put two negatives when seaparating
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idle bridge
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idle bridge
howd it go from this to
- 1/8 sin 4x
cold sage
integrating introduced a 1/4 factor, so 1/16
then theres the 2, so 1/8
the two - cancel out
and cos goes to sin
idle bridge
wait
idle bridge
integrating introduced a 1/4 factor, so 1/16
then theres the 2, so 1/8
so heres what i have
so far
its (- ) (-1/4 ) integral of cos 4x (2dx)
is that good so far
cold sage
which is int ( 1/2 cos(4x) ) yeah
cold sage
sure
idle bridge
what do i do from there
cold sage
do the integration
cold sage
$\frac{1}{2} \int cos(4x) dx$
clever fjordBOT
Aā¤Ć
cold sage
there was no 2x, just a 2
idle bridge
oh okay right
cold sage
but if you dont know the result of that integral straight away, you can do it by doing a substitution of u=4x
cold sage
yeah?
idle bridge
so take away -1/4 then take away 2
cold sage
thats a 2
idle bridge
yeah dont u put it to the side
cold sage
you can factor out 2 from the integral
that doesnt make it become 1/2
thats like saying 2x = 1/2 (x)
cold sage
1/4 times 2
?
yeah
cold sage
yeah
you just multiplied by 1, not 2
2 * 1/4 = 2/4 = 1/2
thats where the 1/2 i wrote comes from
idle bridge
okayyy i see now
idle bridge
2 * 1/4 = 2/4 = 1/2
the final answer is 1/8
idle bridge
oh okay
cold sage
it was always a 1/2
cold sage
we can move constant factors in and out of integrals
so we originally had a 1/4 and a 2, but that was always just a single factor of 1/2
cold sage
indeed so
cold sage
integrate cos(4x)
cold sage
good decision
cold sage
thats right
cold sage
just got to be careful
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remote cloud
How would I go about this proof? No answers just kinda need a way to approach it
odd edgeBOT
How would I go about this proof? No answers just kinda need a way to approach it
@remote cloud Has your question been resolved?
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autumn flare
I had a doubt regarding concept in linear algebra,
I understood the need of two vectors(linearly independent or invertible) to form a plan in 2D space.
But why do we need three vectors to cover each and every point in 3d space, why can't it be just 2 indepdent vectors with 3 dimensions?
autumn flare
*independent
forest sky
but then those two vectors will form a plane in 3D space
autumn flare
why is there a seperate need of a extra vector
Could you be more elaborative?
lets say i have v transpose = [ 1 2 3 ] , and w tranpose = [ 4 5 6]
forest sky
let's say you have two vectors which point along a table like this (the table is mathematically a plane). we can add and scale them together in such a way as to reach any other point on the table, but that couldn't somehow point away from the table in any way
autumn flare
the combination of 3c + 6d = b
autumn flare
let's say you have two vectors which point along a table like this (the table is...
understood but, the the combination fo 3c + 6d = b should ideally give me all the points in the the 3rd dimesnion right
as both of them lie in the 3rd row
wooden python
mate we have no idea what you mean by any of b, c or d ...
forest sky
lets say i have v transpose = [ 1 2 3 ] , and w tranpose = [ 4 5 6]
can you describe the linear combination of v and w which gives [5 7 0] ?
autumn flare
let me write in the matrix form ,
[1 4] [a]. [b1
[2 5 ] [b] =[b2
[3 6 ][c]. [b3
the 3rd row is the 3rd dimesnion right?
autumn flare
can you describe the linear combination of v and w which gives [5 7 0] ?
hmm
1a + 4 b = 5
2a + 5b = 7
3a + 6b = 0
i think it will result in imposisble equation
as we will get different values of a and b?
so is that why 3rd vector is needed?
forest sky
autumn flare
1 a + 4b + 7c = 5
2 a + 5b + 8 c = 7
3 a + 6b + 9 c = 0
forest sky
so is that why 3rd vector is needed?
yes, you can't possibly reach every vector in 3d space just with two vectors
autumn flare
and we will also get impossible equation?
autumn flare
1a + 4 b = 5
2a + 5b = 7
3a + 6b = 0
i think it will result in imposisble equa...
when we solve this?
autumn flare
https://www.desmos.com/3d/gzztkgurdi
let me ahve a look at it
forest sky
it will have no solution, yes