do the thing i asked
#help-28
hot lark
Yes im subsituting on the formula
restive kelp
ok, show me
restive kelp
how
restive kelp
yes
restive kelp
Pi/3/3= 3xpi / 3
where are these coming from?
hot lark
The angle is pi/3
restive kelp
The angle is pi/3
no
hot lark
also, this would be pi/9, but we dont need it
Why 9?
Nvm i get it why
So cos(pi/3)
1/2
+isin(pi/3)
Sqrt3/2
Ok it checks out
Ty bro
restive kelp
yw
hot lark
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feral tundra
Im not sure where to start
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Im not sure where to start
restive kelp
so to calc the value of g, you need to substitute in 3, right?
feral tundra
so would i put 3 in for the x?
feral tundra
sooo 2(3)+4?
restive kelp
g(3)=10
restive kelp
then use what u get further for f
.
feral tundra
so i do the 2(3)+4 now
feral tundra
okayy
how do I find w tho?
cause its not given like the previous
or would it be 3
restive kelp
you cant find it
feral tundra
so itd stay 3w
restive kelp
why the first?
feral tundra
cause the w cant be changed no?
restive kelp
yeah
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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
not sure how to begin kinda
i get its the same concept
but like the not =4 messing me up
restive kelp
dont worry about it, its just there so we dont divide with 0
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cyan citrus
Yo
cyan citrus
Ok so the question is
"Solve the following system of linear equations by using gauss jordan method"
This is my work
I almost did everything correctly
The answer is x= 1 , y=3 , z=5
But I got x= 1/3 , y= 3, z= 5
I got the value of x wrong
But I did something wrong in the second last step
Can someone help me out on how I can operate on row 1 in this case?
Can someone help me in pointing out my mistake?
idle kelp
I can't follow what your strategy is, why are you changing your pivot columns that are already 1's into 3's and other numbers
cyan citrus
I can't follow what your strategy is, why are you changing your pivot columns th...
thats what i learnt in my school
afaik im doing that cuz the aim is to get a diagonal matrix in the end
i think i found my mistake
idle kelp
I mean, yes, if you have a diagonal matrix, you can get the final answer even if they aren't 1's but you are needlessly complicated it. I have a hard time believing they taught you to do that because you're just adding extra steps and then undoing them later
cyan citrus
I mean, yes, if you have a diagonal matrix, you can get the final answer even if...
ik its a long ass method
gauss jordan method
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stiff oak
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stiff oak
There are a million ways to solve this, and I can solve it easily, it's just that I don't know when I try FACTORING it doesn't work.
thorny horizon
how do u factor it
hot herald
you can't factorise it over the reals
sharp vine
we're talking about complex space?
hot herald
but you could factor over complex
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thorn crescent
blue is my attempt
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blue is my attempt
thorn crescent
um i realize i got it wrong bc it says u' = 2xe^x + x^2e^x
but why does u' = 2xe^x + x^2e^x?
shouldnt it be 2x*e^x
eh ill figure it out
.close
main forge
how to reverse engineer the distance formula? like I got a distance and I want to get 2 x,y points
main forge
how to reverse engineer the distance formula? like I got a distance and I want t...
formula:
sharp vine
notice that
there are infinitely many pairs of such the points
knowing only the distance between the two points isn't sufficient to find them exactly
atomic blade
main forge
and how do I get a triangle's 3 edge points? I already calculated almost every other data of the triangle
sharp vine
maybe show us the original problem
sharp flame
What's the original question
main forge
I used the sqrt(3) thingy and it wasn't accurate
main forge
and how do I get a triangle's 3 edge points? I already calculated almost every o...
this is the original question
sharp vine
so then what's "other data"
main forge
area
perimeter
a_height
b_height
c_height
a_median
b_median
c_median
alpha
beta
delta
a
b
c
inside circle
outside circle
sharp flame
well then do you have any data that would lay down the location of this triangle on a coordinate plane
main forge
like what?
sharp flame
equations/coordinates
main forge
no
sharp vine
main forge
like how does this website calculate the Vertex coordinates?
https://www.calculator.net/triangle-calculator.html?vc=90&vx=4&vy=3&va=&vz=&vb=&angleunits=d&x=83&y=4
sharp vine
I think it assumes A lies at the origin
main forge
I think it assumes A lies at the origin
okay then I do too now how to calculate with that?
sharp vine
what's more B seems to be positioned at x-axis
so then A, B are very easy to find
C can be find in many ways, one of them is
y coordinate of C is just hc
main forge
okay so how to calc B?
sharp vine
B = (side c, 0) in this case
main forge
and how to get C[x]?
sharp vine
e.g. distance formula, AC is already known
Let's say A = (x1, y1), C = (x2, y2)
x1, y1, y2 are known (and d also obviously)
find x2
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sharp vine
yes it's an algorithm, easy to automatize
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tacit iris
Hey,
I'm having a hard time understanding eigenspaces
Correct me if im wrong, but from what i understand its a subspace of the vectorspace on which the linear transformation was defined, and it is spanned by the eigenvectors of a certain eigenvalue
rapid rain
Yes, it's the subspace that contains all eigenvectors (including 0) of a certain eigenvalue
tacit iris
It contains 0 because every subspace must contain 0, right?
What I dont understand is how to find that subspace, how to use it to find the eigenvectors of a eigenvalues
rapid rain
well quite literally, finding the eigenvectors for an eigenvalue is the same thing as finding the corresponding eigenspace
tacit iris
Sorry I meant the number of linearly independent eigenvectors.
How do I construct it? (the matrix of which its columns are the independent eigenvectors)
rapid rain
Usually you just do it solving for the equation $AX = \lambda X$
glossy valveBOT
rafilou2003
rapid rain
you get a system of equations in the coordinates of X
reduce the system to echelon form
tacit iris
isnt that finding the eigenvalues though?
rapid rain
no, once you found the eigenvalues, you don't solve for $\lambda$ anymore, you solve for $X$
glossy valveBOT
rafilou2003
rapid rain
you plug in $\lambda$ = the eigenvalues that you found
glossy valveBOT
rafilou2003
tacit iris
Right, that makes sense.
And so if I have more than 1 eigenvector, when I reduce the matrix to its echlon form I get that the rows/columns of the matrix are linearly dependent?
rapid rain
As soon as you have 1 eigenvector, resulting rows will be linearly dependent
More precisely, if A has n rows and n columns :
With 1 eigenvector, A will have n-1 independent rows,
With 2 linearly independent eigenvectors, A will have n-2 independent rows,
etc
tacit iris
I dont really understand
rapid rain
Ok I should rephrase a bit
tacit iris
If I have 1 eigenvector, should the system have only 1 solution? Meaning all the rows/columns of the matrix are linearly independent
rapid rain
No, if you have 1 eigenvector, name it x, then all vectors "cx" are also eigenvectors
So the system has an infinity of solutions
tacit iris
ohhh right...
rapid rain
so A has n-1 independent rows (assuming you cannot find another eigenvector than "cx")
The more linearly independent eigenvectors A has (for a same eigenvalue), the less independent rows it will result it
raw nebula
tacit iris
I think I get it, i just need to play with it more
where should I ask if i have any more questions? here?
wait one last question:
if I have eigenvector (1,1,1), when I multiply it with the matrix assosiated to the linear transformation I should get the span of (1,1,1) right?
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glad whale
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glad whale
i have to find inverse or somethign like this one
?
they explain this in 18 sec ๐
maybe i just over think
.close nvm
wintry ether
How do I graph the reciprocal function of x^2 - 4?
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How do I graph the reciprocal function of x^2 - 4?
wintry ether
I am having trouble sketching the asymptotes.
empty sapphire
hi there. What's the problem with the asymptotes?
wintry ether
I already graphed y= x^2 - 4 between the vertical asymptotes of x= 2 and x=-2 but the direction of my hyperbola is wrong.
empty sapphire
can you show your steps on how you're graphing it? What comes next after finding the asympotes?
wintry ether
I graphed my parabola which is a concave downwards. Then I found points near the asymptote and plotted them into the equation and it formed a hyperbola. I don't understand what is the purpose of finding points near the asymptotes in the first place.
empty sapphire
to see whether or not it's increasing or decreasing. If the points are increasing, then you know that your function is increasing towards the asymptote, then decreasing if it's going down,etc.
As an example, let's take the asymptote x = -2.
For the recipricol of f(x) = 1/(x^2 - 4),
If we do:
f(-5) โ 0.0476
f(-4) โ 0.08333
f(-3) โ 0.2
f(-2.5) โ 0.444
f(-2.1) โ 2.439
(fixed a value)
And so on and so forth, we see that it's increasing on the left hand side as x approches -2
like so
What class is this for? Can you use derivatives? That would make predicting the behavior slightly easier with a sign chart ๐
wintry ether
this is pre-calculus year 10
empty sapphire
ah ok
wintry ether
but yes I understand
empty sapphire
so no deriviatives
wintry ether
no, idk what that is
empty sapphire
you'll learn in calculus
wintry ether
is it hard
empty sapphire
I will say the calculus is the easy part, what trips people is the algebra, so make sure you know your algebra properties and you'll be golden
Basically what calculus does is study curves. So like we know that for a linear equation the slope is rise/run and it's consistent. But waht about something like sin(x) or x^2 or ln(sin(cos(2x + 3))? Derivatives are tools to help understand behavior in curves and slope ๐
wintry ether
looks scary
empty sapphire
it's not as bad as it seems
there are some amazing tools that take some time to get used to but make it easy work
you got it ๐
oh
back on track
actually scratch that
that's inverse equations, my bad lol
wintry ether
empty sapphire
yeah sorry scratch that it doesn't work here lol
empty sapphire
that's if the function has an inverse ๐
But you'll learn that down the line
wintry ether
that's *if* the function has an inverse ๐
But you'll learn that down the line
thank you for the help and the extra info
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alpine yoke
Hmm i'm not getting 2 after simplifying. any help?
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Hmm i'm not getting 2 after simplifying. any help?
primal thicket
im lookin at this beau gimme a min ๐
alpine yoke
thanks, take your time
primal thicket
Do you by any chance have the original question? It could be possible something was mixed up, because I'm not getting the same result
primal thicket
ah okay I see, you just condensed it
alpine yoke
Yeah
primal thicket
hopefully you're expected to be able to use a calculator, otherwise simplifying this wont be so fun:
primal thicket
ok, start by tearing each number into individual factors, and see what you can cancel
alpine yoke
Yeah i've tried that about 3 times now, didn't get 2 though
lemme try again
no luck.
primal thicket
damn, i tried factoring it myself but seems i got a bigger result than expected
alpine yoke
Yeah i'm getting an absurd value;
(20^7/3^5)
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velvet shadow
@primal thicket do you still need the problem solved?
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<@370032376702697494> do you still need the problem solved?
velvet shadow
idk how any of the chat commands work but i broke the porblem down into something that looks like
( 8^7 * 2^7 * 5^5 * 5^5 * 3^3 * 3^3 * 3^3 ) / ( 8^7 * 3^7 * 8^5 * 3^5 * 8^3 * 8^3 * 2^3 )
before just eliminating like terms (?) sorry if thats the wrong terminology. anyways it wokred out to 16/8 in the end doing it that way. hope that first part helps
@primal thicket
normal thicket
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rain sundial
normal thicket
Ahh no doubtnut topper solutions please
umbral obsidian
How do you generally solve an (infinite)-(infinite) indefinite limit
tacit siren
Error this question might be off
I don't see the right answer as a choice
I checked on wolfram alpha just to be sure 
But anyways to find the right answer try completing the square
normal thicket
Nevermind i solved it by rationalization
normal thicket
How do you generally solve an (infinite)-(infinite) indefinite limit
Turtle after so many days
I will make it 0/0
normal thicket
A is right answer
tacit siren
No look
umbral obsidian
If you make it infinite/infinite
tacit siren
The expression you want is $x - 1/2$
glossy valveBOT
992qqoloy
tacit siren
That would require a to be -1 and b to be 1/2
full forumBOT
That would require a to be -1 and b to be 1/2
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.
umbral obsidian
Okay then why are you asking this question
tacit siren
tacit siren
The way the problem is written since it says - (ax + b) then 1, - 1/2 should be wrong cus that would give you - x + 1/2
normal thicket
,w limit of {โ(x^2-x+1)-x+1/2}
x tends to infinity
formal scarab
X x
tacit siren
https://www.wolframalpha.com/input?i=Lim+x+to+-infinity+sqrt((x+-+1%2F2)^2+%2B+3%2F4)+-+x+%2B+1%2F2 idk why it's being stubborn lol
But yeah goes to infinity
tacit siren
yeah
glossy valveBOT
normal thicket
When we change -infinity to infinity what does it affect our function?
I swamped the limit and checked it is giving me option A
@tacit siren
,w limit of {โ(x^2+x+1)-x-1/2}
x tends to infinity
glossy valveBOT
normal thicket
So by the question answer should be a=1 and b =1/2 right?
,w limit of {โ(x^2-x+1)-x-1/2}
x tends to -infinity
glossy valveBOT
tacit siren
Hmm well the way I did it is I noticed the square root term tended to $\sqrt{(x - 1/2)^2}$ as x went to negative infinity, which would be 1/2 - x when x is a big negative number
glossy valveBOT
992qqoloy
tacit siren
I'm still tryna see what goes wrong with your way tho 
tacit siren
When we change -infinity to infinity what does it affect our function?
Oh wait it shouldn't change anything, there's no reason to change the -x to a +x
Wait hmm
OK nvm still tryna figure out what goes wrong with rationalizing 
$(\sqrt{x^2 - x +1})^2 = x^2 - x + 1$ as it's always positive, I think . I think the danger is in the denominator then? Cus that doesn't tend to infinity when x goes to negative infinity
glossy valveBOT
992qqoloy
tacit siren
Cus the $ax + b$ term becomes increasingly negative
glossy valveBOT
992qqoloy
tacit siren
But the square root positive
Well if $a$ is positive it becomes increasingly negative I mean
glossy valveBOT
992qqoloy
tacit siren
So from $a^2 =1$ you take the negative solution
glossy valveBOT
992qqoloy
tacit siren
And then that also switches the sign of $b$ from $-2ab = - 1$ when x goes to -infinity compared to when it goes to infinity
glossy valveBOT
992qqoloy
tacit siren
So yeh that's it, it's that you want the denominator to tend to $\infty$, and the choice of $a$ and therefore $b$ that does that depends on whether $x$ goes to negative or positive infinity
glossy valveBOT
992qqoloy
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normal thicket
,w limit of {โ(x^2-x+1)+x-1/2}
x tends to -infinity
I checked all the options into wolfram now
You are the right mistake in the denominator
glossy valveBOT
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agile vigil
can someone check whether this is correct?
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can someone check whether this is correct?
@agile vigil Has your question been resolved?
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@agile vigil Has your question been resolved?
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@agile vigil Has your question been resolved?
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@agile vigil Has your question been resolved?
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balmy bone
sorry
balmy bone
just one more question
austere pond
balmy bone
@strange ether
strange ether
ye?
balmy bone
I understand from before that you factor 999 from itself resulting in +1
but here
shouldn't +1 be 999/998?
strange ether
yes
balmy bone
it's from a tomrocksmaths video
i was just trying to understand the steps
so it was a mistake?
strange ether
probably
balmy bone
okay thank you
or most likely just an approximate to express it more easily i guess
anyways thanks
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torn jolt
Given a set with a cardinality of n
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Given a set with a cardinality of n
torn jolt
How many subsets in the powerset of the previous set have a cardinality of I where I is less than or equal to n?
$$\abs{A} = n$$
$$ B = { x | x \subseteq P(A), \abs{x} = I }$$
$$\abs{B} = ?$$
glossy valveBOT
help me
oak basin
so basically you are asking for the cardinality of P(A)?
torn jolt
so basically you are asking for the cardinality of P(A)?
no
The cardinality of B which is a subset of P(A)
The amount of sets that contain I elements in P(A)
oak basin
oh alright, so basically how many sets in P(A) that have cardinality of I
hmm couldn't you just think of this as ways of choosing I's element out of A (without order)?
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torn jolt
oak basin
well I was kind of asking a leading question but yes, it's basically asking you how many ways you can choose I elements out of a set of n elements
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glossy valveBOT
help me
torn jolt
Make sense
TY
I just finished the part of my book that goes over permutations, did not have this information before
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upper nimbus
could someone check if this equation works?
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could someone check if this equation works?
rocky vale
yeah, that is an improper integral with an infinite discontinuity at x=0
hollow tendon
someone
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woeful girder
What is echelon and non echelon matrices and what's the difference between the two?
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What is echelon and non echelon matrices and what's the difference between the t...
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atomic vortex
Would this be valid?
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Would this be valid?
onyx glen
probably more verbose than necessary but fine
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cursive light
Can someone correct this exponential equation?
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Can someone correct this exponential equation?
white smelt
Can someone correct this exponential equation?
yyou forgot the x in second line exponent
cursive light
Oh wrd thanks g
cursive light
yyou forgot the x in second line exponent
But the answer is still 1 right?
silent quest
So the whole thing will be:
3x = 4x-4
silent quest
But the answer is still 1 right?
yes... techincally
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gaunt lance
If f is an even function and g is an odd functioon, determine wether the composites
f o f f o g g o f g o g
are even, odd or neither
I know that f o f is still even and g o g is still odd, but how would i work out f o g or g o f?
onyx glen
check them against the defns of even/odd functions
sterile crag
f(x)
is even
g(x) is odd
then
f(x) = f(-x)
g(x) = -g(-x)
h(x) = (f o g)(x) = f(g(x))
gaunt lance
I subsituted in -x but for f o g and g o f it came back as both even so wasnt sure if it was the right aproach
sterile crag
if h(x) is even
function
well, let's check
h(-x) = f(g(-x))
=f(-g(x))
=f(g(x))
gaunt lance
sterile crag
=h(x)
sterile crag
what?
sterile crag
well
i supposed
f(x) is even function
and g(x) is odd function
and proved (f o g)(x) is even function
sterile crag
west mango
mighty ridge
long time no see renato
west mango
godess melo, good morning
mighty ridge
night in my country lmao
so you're counting permutations of (1, 2, 3, ..., n) such that the inequality is satisfied ?
west mango
yes
mighty ridge
when you have something like this start by checking small cases
like n = 3 or n = 4
west mango
I already tried, for 1 and 2, first permutations are satisfied
(1) and (1,2), (2,1), for 2 it works bothways
west mango
like n = 3 or n = 4
yeah I was trying for >= 3
for n >= 3 I think has to be of the form
a_n = n or (a_n-1 = n and a_n = n - 1)
unreal coyote
this ends up not being the case
i got the answer with some code that just bruteforces it and the pattern is obvious after some higher n
still thinking abt a proof tho
ah i see
think about having a solution for n=k and consider n=k+1
you need to 'insert' the k+1 into the old list, but it turns out theres not many places you can put it...
west mango
think about having a solution for n=k and consider n=k+1
well, for n = k, a_k = k , which is of the form a_n = n
i think
im still thinking about the n = k+1
unreal coyote
well you can clearly always add the k+1 to the end, correct?
west mango
well, if
n = k + 1
then , (k+1)a_k+1
and given that a_k+1 = k+1
this becomes?
(k+1)^2
however, does this satisfy the inequality?
unreal coyote
the maximum that the old k_ak can be is k*k
this is certainly less than (k+1)^2
so adding k+1 to the end works for all valid sequences of length k to produce one which is valid as length k+1
now consider what happens if you insert it 1 before the end
west mango
you mean
$$a_k = k + 1 \quad \text{while} \quad a_{k+1} \in {1, 2, \ldots, k}$$
In that case, this must hold true:
$$ka_k \leq (k+1)a_{k+1}$$
But given $$a_k = k + 1,$$
$$k(k+1) \leq (k+1)a_{k+1}$$
I see, so we can divide by $$k+1$$ since it's a positive integer, yielding:
$$k \leq a_{k+1}$$
So, for any valid permutation of length $$k$$, there are two ways to extend it to a valid permutation of length $$k + 1$$:
Either by adding $$k + 1$$ to the end?
Or inserting $$k + 1$$ at position $$k$$ while having $$a_{k+1} = k$$?
glossy valveBOT
ใฌใใ (renato , ping if reply)
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west mango
<@&286206848099549185>
west mango
<@&286206848099549185>
upbeat comet
a_k+1โ k unless a_k+1<k
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mighty ridge
hi again renato, I was thinking about your problem, and I think I've more or less figured out a proof, I still have some things to do right now but I'll be available after work, ping me
another hint to make you wait: check for n = 1, 2, 3, 4
you should see a link with fibonacci numbers, and then have a rough sketch of proof with strong induction
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west mango
mighty ridge
more hints: P(k): we can find F_k (kth fibo number) such permutations for 1, 2, ... k
strong induction, assuming P(u) = F_u for all integers u before k+1, so in particular P(k-1) = F_k-1 and P(k) = F_k, and want to show P(k+1) = F_k-1+F_k = F_k+1
for that, show that a_1 ... a_k+1 being valid implies either k+1 = a_k+1, which the leaves the F_k possibilities for this scenario
or k+1 = a_k and then k = a_k+1, which leaves the permutations of P(k-1) as choices, so F_k-1
and then sum these 2 cases
the only tricky part being proving the "either"
but it's of reasonable difficulty if you try by contradiction
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west mango
goddess mรฉlo, im sorry im dropping the question, this is too hard, ngl, and its taking much longer than I expected, I want to get better at math, but this is simply too abstract for me, at least for this point in time, even with a solution I found in the internet , I still cannot comprehend for now, I will come back at it later, hope you are not mad goddess mรฉlo, this question was combinatorics c1, taken from IMO 2020 shortlist problems
hope you have a great day, and sorry for dropping the question, once again goddess mรฉlo
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lavish tapir
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lavish tapir
i need help with question part (iii)
and ive tried part (i) & (ii) but im not sure if its correct
<@&286206848099549185>
proper hawk
calculated the total amount donated by Dalila in terms of A
then set the result equal two times Farisha's donation
fossil umbra
@lavish tapir
fossil umbra
From the information provided:
Year 1: $150
Year 2: $150
Year 3: $170
This means that from year 3 onwards, there's an increase of $20 each year.
(i) To find the amount of money she donated in year 10:
We can find out how much she donated each year based on the pattern.
The difference between year 3 and year 10 is 7 years.
So, she increased her donation 7 times.
Increase = 7 x $20 = $140
Therefore, donation in year 10 = Year 3 donation + Increase
Donation in year 10 = $170 + $140 = $310
(ii) To calculate the total amount of money she gave over the 20-year period:
Years 1 and 2 = 2 x $150 = $300
From years 3 to 20, it's an arithmetic series:
First term (a) = $170
Common difference (d) = $20
Number of terms (n) = 20 - 2 = 18 (since she donated the same amount in the first two years)
Sum of an arithmetic series is given by:
Total amount for 20 years = Years 1 and 2 amount + Years 3 to 20 amount
= $300 + $6120 = $6420
So, Farisha donated a total of $6420 over the 20-year period.
d = $170 - $150 = $20
For Year 10:
a_{10} = $150 + (10-1)($20)
a_{10} = $150 + $180 = $330
glossy valveBOT
Finicalfire
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thorn fjord
I cant seem to find my mistake
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thorn fjord
(vectors)
tacit siren
First line u factored out the 3 incorrectly
thorn fjord
First line u factored out the 3 incorrectly
How should it be factored out then?
tacit siren
if u have a(b +c) then that equals ab + ac
So you'd expect a 3 in both terms to factor it out like you did
But since there isn't
You probably don't need to factor anything to begin with
What are your knowns?
thorn fjord
but can i apply the || a + b || rule then?
tacit siren
Pls tell me ur tired rn :p
thorn fjord
Pls tell me ur tired rn :p
lol
Actually, for once, im not tired
๐
And its correct btw
thxx
tacit siren
np
thorn fjord
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midnight elm
hey i took arethemtic suequcnes, and geometric sequences, what other sequences might i study? I have a summative B investigation, which is an exam where they give us a new rule or whatever it is based on what we studied, and we have to learn its patterns, and deduce a rule veryfiy, justify it ect, what are some things related to arethmetic and gemorteric, that may be in my exam?? any ideas??
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midnight elm
<@&286206848099549185> lol
solemn solar
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torn jolt
I don't understand this at all, As they want resultant, i used R=(a^2+b^2+2abcostheta)^1/2, cos theta for min and max value
But in the solution they seem to just add and subtract those ๐
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fresh ledge
Does anyone know about the Venn diagram?
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fresh ledge
According to your chosen literal, shade each of the operations given in the exercises.
thorny horizon
fresh ledge
Is this right?
thorny horizon
does the triangle stand for the symmetric difference?
fresh ledge
does the triangle stand for the symmetric difference?
Yes
fresh ledge
(๐ต โ ๐ถ) ๐ represents the complement of the symmetric difference between B and C. This means that it includes all elements that are not in B or C.
Yes
Complement.
thorny horizon
Is this right?
what u have shaded in this diagram is just $B \Delta C$
glossy valveBOT
SilverSoldier
thorny horizon
okay that just A
fresh ledge
Yes
thorny horizon
Okay, so now I need A
u need to find the common regions between this one
fresh ledge
Is it right?
thorny horizon
yes 
glossy valveBOT
Finicalfire
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Finicalfire
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onyx glen
you don't need the \begin{document} and \end{document} here
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Ann
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onyx glen
lmfao
anyway like
at this point
why not throw it into overleaf or something and get a pdf out of it
and then screenshot the pdf or send it as is here
brittle steeple
Calculate the eigenvalues, eigenvectors, determinant, and inverse of ( M ).
\Suppose ( E ) is an elliptic curve defined by the equation ( y^2 = x^3 - x ). Determine the torsion points of( E(\mathbb{Q}) ), the rational points on ( E ).
\Explore the properties of the Riemann zeta function
[
\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}.
]
Analyze its convergence and mention any special values or relationships you know.
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thewizardofOU
torn jolt
brittle steeple
sturdy mango
find a formula for the nth partial sum of the series
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find a formula for the nth partial sum of the series
stable plover
because it's not a geometric series
stable plover
try partial fractions
sturdy mango
try partial fractions
how?
onyx glen
$\sum_{k=2}^n \frac{1}{k(k+1)} = \sum_{k=2}^n \paren{\frac{1}{k} - \frac{1}{k+1}}$
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Ann
stable plover
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dim gyro
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dim gyro
I'm confused with the last step
I got lim of fx as n when x = 1
And 0 when 0 <= x < 1
I'm confused about the integral
Ofc they're the same right????
A is going towards 1 on 1^- limit right
stiff musk
I got lim of fx as n when x = 1
how can it be n, you are taking the limit as n->infinity
your answer cannot depend on n
$\lim_{n \to \infty}f_n(1) = \infty$
glossy valveBOT
Bungo
stiff musk
presumably that's why they're fiddling with the limit as a->1-
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@dim gyro Has your question been resolved?
dim gyro
Well yes it would go to Infinity
But it would be whatever value of n is in the sequence
dim gyro
presumably that's why they're fiddling with the limit as a->1-
Is all I meant. If you look at the value at 1 youd would get n. As n goes to Infinity it equals Infiniti
stiff musk
right, so if you let f(x) = lim f_n(x), then
$$f(x) = \begin{cases}\infty & \text{if }x = 1 \ 0 & \text{otherwise} \end{cases}$$
glossy valveBOT
Bungo
stiff musk
and presumably they're using riemann integrals, so f(x) is not properly riemann integrable (because it is not bounded)
so they're using an improper integral
with a lebesgue integral you don't need to do that
dim gyro
Yes, ahhh hell ol its going to Infinity as the integral goes to one
On the rhs
Yeah we're not doing any measure Theory in this course
stiff musk
yea that makes sense
but you're essentially correct that the infinity at x=1 doesn't have any effect
the value of the function at a single point (even if infinite) can't change the value of the integral
dim gyro
Well points have no length and therefore no measure :p
I think is why a point doesn't matter
stiff musk
yes that's it in a nutshell
that becomes more formally/"officially" true with measure theory
dim gyro
Esp in regard to measure Theory and the outter measure
And getting into the like idea that measurable sets are those open sets
stiff musk
basically in measure theory, we allow 0 times infinity and we define it to be 0
dim gyro
Yeah it's The extended real lines right
stiff musk
right
stiff musk
cool!
dim gyro
Okay gotta study Norwegian for the rest of the night
vague tapir
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vague tapir
hiii
so this is one of my linear algebra problems
im not really sure how to proceed
or if my tries make sense
heres what i have so far
<@&286206848099549185>
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@vague tapir Has your question been resolved?
vague tapir
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charred nest
Need help learning a couple of topics
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Need help learning a couple of topics
plain fox
undefined = anything divided by 0
so just use those x values in the denominator and if u get 0 that value is undefined
onyx glen
im so sorry but the shadow from your phone is REALLY unfortunate in how it is shaped
undone vector
im so sorry but the shadow from your phone is REALLY unfortunate in how it is sh...
For real
charred nest
I KNOW
onyx glen
but anyway
charred nest
I WAS LIKE WTF
onyx glen
the only thing that can cause an undefined here is division by 0
charred nest
THAT SHIT LOOKS SO FUNNY LMFAOO
onyx glen
so find where that happens
charred nest
so just use those x values in the denominator and if u get 0 that value is undef...
only denominator
?
rare dock
iโm dead
plain fox
u can use in numerator but 0 in denominator is undefined
twin wolf
plain fox
and u only want to find values where the function is undefined
charred nest
ok so if it equals 0 we are good
onyx glen
only denominator
yes only the denominator
0/(whatever) is fine
(whatever)/0 is not
you want the not-fine case
charred nest
ok so the answer is 4
answer # 4
x=0 x=6
ima need help w another question hopfully the camera dosent show some weird shape
#3
rare dock
does*
undone vector
Na but low key need help w this <@&286206848099549185>
factor numerator and denominator
charred nest
so it becomes -2/(x+5)?
undone vector
Yeah
undone vector
You can take out x from both parts of the fraction
charred nest
how do i do that
undone vector
For the top x^3-16x=x(x^2-16)
undone vector
Then you can factor the quadratic parts further
charred nest
how so?
undone vector
Same way you did for the other problems
charred nest
so like for the denominator (x+4) (x+8)
charred nest
and the top part is x (x-4) (x+4)
undone vector
Yeah
charred nest
ohhh
that was helpful
Im gonna be asking a lot of questions so bare with me lmao
5
undone vector
Try multiplying one of the fractions (either one) by -1/-1
This will make the denominators equivalent then you can add
charred nest
so I got a+b/a-b
undone vector
b/(b-a)=-b/(a-b)
Seems like you multiplied the right side denominator by negative 1 but not the numerator
undone vector
Yeah
charred nest
bet
need help with # 8 <@&286206848099549185>
<@&286206848099549185> if anyone could help
stray ether
multiply by (y+2)(y-4)
charred nest
make that the common denominator?
stray ether
Multiply both left and right sides by (y+2) and (y-4)
You will get a linear equation in two variables with respect to y
and then solve the equation.
charred nest
so like y+2/(y-4)(y+2) + y-4/(y-4)(y+2) = y^2 -4y -y -4 /(y-4)(y+2)
stray ether
a little bug
stray ether
so like y+2/(y-4)(y+2) + y-4/(y-4)(y+2) = y^2 -4y -y -4 /(y-4)(y+2)
the constant of denominator on the right is equal to positive 4.
so it should be y^2-4y-y+4
stray ether
you're welcome
hollow wharf
Itโs just factoring
charred nest
yes
stray ether
it's a little bit complicated
charred nest
ya thats the only one i really am having trouble with
charred nest
Itโs just factoring
@hollow wharf do you know how to do it?
hollow wharf
<@341586602503569418> do you know how to do it?
Pick: numerator or denominator
stray ether
numerator first
charred nest
i figured it out its y^2 + 3y/y+5
cinder gust
Not sure what I'm doing wrong but I can't solve this math problem
cinder gust
I used product rule and got -5sec3xsin5x + 3cos5xsec3xtan3x but I am not sure where to go from there
minor crater
Not sure what I'm doing wrong but I can't solve this math problem
๐ง simply substituting $\pi/2$ gives a result
glossy valveBOT
artemetra
cinder gust
whats the result?
oak basin
by product rule, did you meant you tried to differentiate sec(3x)cos(5x)?
cinder gust
yes
oak basin
well that doesn't do anything, nor help you find the limit either
I'm going to suppose you were trying to use L'Hopital's rule, which is not applicable (yet) in this case
cinder gust
yes I was
oak basin
also, there's another way of doing this without using L'h.
so which method would you want to use?
cinder gust
what methods are there?
oak basin
well one way you can do is to use sum of angles and triple angle identity to simplify the limit
or, if you somehow know the identity cos(nx), or more specifically cos(5x), you could use that as well
But the main idea is that you want to simplify the cos(x) present in sec(3x)
cinder gust
not sure if I've learned about sum of angles or triple angel identity yet in class(im taking calc 1)
the paper said to solve the problems with L'Hopitals rule so that's what I was trying
oak basin
are you sure you have not heard of something like cos(x+y) = cos(x)cos(y) - sin(x)sin(y)?
alright, nvm then, if it specifically asks you to use l'h then just use it
anyway, can you tell some of the indeterminate form in which you can apply l'h?
Or if you can, tell me how you can use l'h in limits? Since it seems like you are not that familiar with l'h
cinder gust
well I know you can only use L'H with fractions, and prior to doing so you should plug in the limit and see if it evaluates to 0/0 or inf/inf
oak basin
right, so basically if we have an indeterminate form of either 0/0 or inf/inf, we can apply l'h
That's correct
now, given that we can apply the theorem, how would you apply it?
Let's say we are trying to do the limit $\lim_{x\to c} \frac{f(x)}{g(x)}$ where this limit has an indeterminate form mentioned
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waler
cinder gust
well you'd get f'(x)/g'(x) and plug in the limit, if it's still indeterminate you apply L'H again until the answer is no longer indeterminate
oak basin
great, so basically the theorem tells us that $\lim_{x\to c} \frac{f(x)}{g(x)} = \lim_{x\to c}\frac{f'(x)}{g'(x)}$ given the condition above
glossy valveBOT
waler
oak basin
alright, you seem to be on track with the theorem. Now let's try to apply that on our problem
So firstly, you can already tell that we can't use the theorem in our problem yet. Can you somehow manipulate sec(3x)cos(5x) into a fraction?
cinder gust
I was trying to do that possibly by applying the product rule but got stuck, so if there is a way I haven't noticed yet
oak basin
again, just forget about differentiating and stuff, since we do not have the necessary conditions for l'h here
like I said, we can't use the theorem in our problem yet
by using just algebra and trigonometry, try to rewrite sec(3x)cos(5x) as a fraction
if you get stuck again, ||try to rewrite sec(3x) in terms of cosine||
cinder gust
if you get stuck again, ||try to rewrite sec(3x) in terms of cosine||
im sure I've learned about whatever you're referring to but I've forgot about it
oak basin
alright then, do you know that sec(u)=1/cos(u)?
cinder gust
I believe I remember noting that down a few weeks ago but it wasn't on my formulas page so I forgot about it
oak basin
sure, the trigonometric secant function is defined as sec(x)=1/cos(x)
this is just a definition basically
cinder gust
I see, what is cosecant defined as?
oak basin
csc(x)=secant(pi/2-x), "co-" stands for complementary, as in complementary angle
hence csc(x)=1/sin(x)
anyway, that's off-topic, so with that in mind, we know that sec(3x)cos(5x) = cos(5x)/cos(3x)
now can you verify that the limit of cos(5x)/cos(3x) as x tends to (pi/2)^- is an indeterminate form?
more specfically, tell me what indeterminate form it is of
cinder gust
0/0
oak basin
great, and technically we have 3 more conditions to check for, but just trust me in it that we can apply l'h here
so tell me what limit do you get after applying l'h?
cinder gust
after appling l'h I get -sin(5x)/-sin(3x)
oak basin
eh not quite, your derivatives are not correct
oak basin
parenthesis, but yes
anyway, so we have that the limit in our problem is equal to the limit of 5/3 * sin(5x)/sin(3x) as x tends to (pi/2)^-
now, what is the result of this limit?
cinder gust
-5/3 because sin(5x)/sin(3x) = -1
cinder gust
makes much more sense now that i know sec(x) = 1/cos(x), thanks for your help
strange scarab
Need help for precal
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left bone
!da2a
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short crest
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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
and which number?
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strange scarab
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all numbers...
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oak berry
so are you familiar with the cartesian plane
and what graphs represent mathematically
if you are not sure where to start, for question 1
i recommend making y the subject then graphing it
if you are not sure what to do from there you might want to look at the vertex form of a parabola
sullen scroll
Im not sure but I think this is the graph for no. 1
that's all I can do im really tired at the moment
austere cove
@strange scarab for 2, this is a parabola in y, rather than x, so the parabola will open on its side instead of top or bottom. If you set x to 0 you can see that y = 0 or 6, and setting y to 3 we can find x = -3, which gives you three points that you can draw a parabola through.
You solve 3 with a similar approach
For 4, you need to use the equation for the focus, do you know it?
For 5, you need the equation for the directrix, same question
strange scarab
For 4, you need to use the equation for the focus, do you know it?
Uhmm 4. F(h,k) -> F(-3,-1)
is it right?
strange scarab
For 5, you need the equation for the directrix, same question
Is the directrix y=k-p?
austere cove
Is the directrix y=k-p?
Yes.