#help-13
cedar kilnBOT
copper crest
mighty drift
A is indep. of B iff P(A|B) = P(A), i.e. iff P(A and B) = P(A) P(B)
the questions asks about A = B
intuitively, when does knowing the value of A not tell you anything new about the value of A ?
copper crest
the questions asks about A = B
=>P(A intersection A) =[p(a) ]^2 ?
upper ruin
If you call x = p(A) you get a nice and simple equation: x = x² which has x = ?
mighty drift
If you call x = p(A) you get a nice and simple equation: x = x² which has x = ?
it's pretty straightforward when you actually set it up as an equation
upper ruin
it's pretty straightforward when you actually set it up as an equation
Exactly
copper crest
n(a) /n(s) = n(a) ^2/n(s)^2?
upper ruin
If you call x = p(A) you get a nice and simple equation: x = x² which has x = ?
@copper crest
upper ruin
n(a) /n(s) = n(a) ^2/n(s)^2?
Don't complicate yourself
copper crest
Yea
mighty drift
n(a) /n(s) = n(a) ^2/n(s)^2?
also don't do that because it only holds for finite universes
upper ruin
If you call x = p(A) you get a nice and simple equation: x = x² which has x = ?
Do you agree you get this? @copper crest
mighty drift
hence the answer being {0, 1}
copper crest
Do you agree you get this? <@1035917542792908830>
K thanks guys
I got confused coz the educator said the options had decimal points
.close
lavish plank
cedar kilnBOT
quartz frost
how can a location be given by a matrix?
i feel like we lack context here
or this is a field i have not touched, i dont know
tiny wing
quaternions 
tiny wing
my understanding is invT2 x T1
this
karmic mirage
GM - GigaMeter = 10⁹
IM - ?
FM - FemtoMeter = 10⁻¹⁵
CM - CentiMeter = 10⁻²
Does something stand for IM?
runic garnet
I’ve never seen an IM before
candid minnow
https://en.wikipedia.org/wiki/Metric_prefix those are the prefixes for the metric system, where did you get IM from ?
runic garnet
Where did u hear about it
high spruce
I don't think IM exists, and google doesn't think so either
wait
hmm this is suspiciously in the ranking of chess titles
troll?
high spruce
?
karmic mirage
so is there i-meter?
runic garnet
No
cedar kilnBOT
high spruce
your goal is beyond my understanding
cedar kilnBOT
Available help channel!
bold mauve
cedar kilnBOT
bold mauve
How can I show this Divergent (D) using Limit Comparison Test
I tried to use $$a_n = \frac{n+1}{n\sqrt{n}}$$ $$b_n = \frac{n}{n^3/2} = \frac{1}{n}$$
wraith daggerBOT
donut
bright bridge
why is there two $$ ignore me
bold mauve
cuz i dont know how to put new lines in the tex bot since it doesn't accept \ in inline tex
but the limit $\lim_{n \to \infty}{\frac{a_n}{b_n}}$ diverges
wraith daggerBOT
donut
bold mauve
so I cant apply L.C.T with my choice for b_n
twin quiver
Try b_n = 1/sqrt(n)
bold mauve
how did you pick that
oh, I just made a simple algebra mistke and simplified my b_n wrong
🤦
thank u
.close
torpid urchin
cedar kilnBOT
torpid urchin
could someone help me fill in these blanks? not too comfortable with complex conjugates
pretty sure i got the first one right though (:
cedar kilnBOT
@torpid urchin Has your question been resolved?
high kernel
Hi can someone help me do this step by step
cedar kilnBOT
Hi can someone help me do this step by step
high kernel
This is very different from the other practice questions
I’m ready to explain what I think btw
dire geode
Show your work
high kernel
Ok
This is how I’d think we’d do it
Would this be correct
<@&286206848099549185>
.close
crimson sedge
im trying to do the following question
cedar kilnBOT
im trying to do the following question
crimson sedge
to express the full powerseries, i did x^n * 4/(8^(n-1)) but it says the answer is incorrect and im confused why
cedar kilnBOT
@crimson sedge Has your question been resolved?
cedar kilnBOT
@crimson sedge Has your question been resolved?
cedar kilnBOT
cedar kilnBOT
Available help channel!
turbid spoke
A local maxima and local minimum is critical point type 1(where the derivate is 0 ) only?
turbid spoke
undifined
turbid spoke
whats type 2 lol
I do not understand why the local maximum or minimum, there are occasions if the 0 is included and others not
modern compass
if there is a maximum or a minimum at a certain x, the derivative is 0 at that x.
the derivative being 0 does not guarantee that there is a max or min there
turbid spoke
type 1 only?
modern compass
for example, f(x) = x^3 has no max or min, but f'(x) = 3x^2 is 0 at x = 0
turbid spoke
for example, f(x) = x^3 has no max or min, but f'(x) = 3x^2 is 0 at x = 0
you broke my head as i guarantee the point
turbid spoke
for example, f(x) = x^3 has no max or min, but f'(x) = 3x^2 is 0 at x = 0
Do you mean that if it is 0 the result is not local?
modern compass
no. I mean the derivative being 0 does not mean there is a maximum or minimum at that point
if there is a max or min at a point, then the derivative is 0 (the reverse of the other statement)
so finding where the derivative is 0 tells you possible locations for a max or min, you still have to check it.
<@&268886789983436800>
crimson sedge
Could someone help me with finding the intervals through the f'(x) of a function?
crimson sedge
cedar kilnBOT
@crimson sedge Has your question been resolved?
rain drift
,rccw
wraith daggerBOT
rain drift
hi @crimson sedge have you found f'(x)? Do you know how in this case?
rustic verge
Does this sum being equal to 'e' anyhow relates to the taylor series of e^x, (x^n)/n!
zealous compass
evaluate the Taylor’s series at x=1??
radiant topaz
Substitute x=1 amd sww
zealous compass
What does sww mean
cedar kilnBOT
@rustic verge Has your question been resolved?
cedar kilnBOT
cedar kilnBOT
Available help channel!
orchid widget
Course: Real analysis
Problem: find the limit x -> 1 ((sqrt(x)-1)/(x-1))
The first thing I did was multiply the conjugates sqrt(x)+1 on the numerator and denominator, to get (x-1)/(xsqrt(x)+x-sqrt(x)-1).
Am I on the right track?
orchid widget
The issue here for the next step is that I simply cannot substitute x with 1 or the denominator is zero
tiny wing
try multiplying by the conjugate of (x-1) instead
orchid widget
try multiplying by the conjugate of (x-1) instead
Thank you, I will do that
I don't know what difference it would make, though
upbeat frigate
cedar kilnBOT
upbeat frigate
,r
rain drift
,rccw
upbeat frigate
idk how to rotate it
rain drift
lol ugh
rain drift
,rccw
wraith daggerBOT
rain drift
there we go
upbeat frigate
it's a simple question im just struggling to set it up
I need to turn this in pretty soon
What do u suggest
rip i think ima submit
i still wanna figure this question out though
even if i dont get to submit it
cedar kilnBOT
@upbeat frigate Has your question been resolved?
delicate garnet
yo, what is UPP!
cedar kilnBOT
yo, what is UPP!
delicate garnet
anyone there?
dire geode
anyone there?
delicate garnet
I did the integration and got an indeterminate
so Lo Hossed it and still got indeterminate
because the e^x
shell method because ya know, in terms of x rotating around y
cedar kilnBOT
@delicate garnet Has your question been resolved?
real glen
rip i think ima submit
The answer is pretty dumb tbh, it's fine if you couldn't answer it
cedar kilnBOT
@delicate garnet Has your question been resolved?
delicate garnet
.close
inland roost
We are drawing dollar bills out of a bag consisting of R red balls and B blue balls. We get 1 dollar for every blue ball, but
whenever we draw a red ball, the game stops and we retrieve the amount of money we have made so far.
(a) Find the expected value of the amount of money made from this game.
inland roost
is this just 1/(P(red ball))?
inland roost
hmmm how would i know if its w or w/o replacement
fair geyser
guess you can't
inland roost
how would i do it if i put the ball abck in the bag
fair geyser
maybe it's the same answer either way, i didn't verify the feel
inland roost
bc of symmetry you don't know what ball you pulled out so dont the probabilties stay the same?
fair geyser
e.g. 4 B, 7 R
the probability of winning exactly $3 is (4/11) (4/11) (4/11) (7/11) if you put it back, that's what your answer does
if you don't, it's (4/11) (3/10) (2/9) (7/8)
not the same answer
well that doesn;t prove it, but the feel became a conviction
weak pecan
Can someone help me when this question.. Given that g(y) = y² - 1, find g(a - 1). Give your answer in its simplest form.
royal loom
if I asked you to find g(2) could you?
royal loom
very nice
weak pecan
Thanks man
acoustic heath
ik there is 146 left but idk what to do
cedar kilnBOT
ik there is 146 left but idk what to do
livid hound
wdym by "there"
steel heart
It's not 146
acoustic heath
wdym by "there"
like 180 total
steel heart
Angle sum property of triangle
livid hound
do you know what that square marking/shape at A indicates?
steel heart
<A+<B+<C=180deg
acoustic heath
angle?
livid hound
there are angles everywhere
do you know what's special specifically about a square shape being used compared to a regular arc
acoustic heath
no
livid hound
well that indicates that BA is perpendicular to AC
and that <A is a right angle
(its common geometry notation that's usually taught at the start of a geo course)
acoustic heath
well that indicates that BA is perpendicular to AC
and that <A is a right angle
what does it mean if its perpendicular to ba and ac
livid hound
you butchered the wording
cedar kilnBOT
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.
crimson sedge
Oh sorry
acoustic heath
livid hound
if this is a knowledge gap, i suggest looking up perpendicular lines and right angle notation
acoustic heath
it probably is
crimson sedge
In my country right angle is noted as a dot inside the angle instead of a square
So i guess it varies
acoustic heath
well thank you for helping
livid hound
In my country right angle is noted as a dot inside the angle instead of a square
germany?
vivid sundial
Hello
vivid sundial
so this questions is already solved in class but I dont get it
tropic oxide
image on its way?
vivid sundial
yea
so i get it when we take the p-series it div
because it is less than 1 but
if you take the limit as it goes to infinity its 1/1 does that mean it conv beause im taking the limit test
tropic oxide
what's the goal? to determine the convergence of the series $\sum_{n=1}^{\infty} \frac{\sqrt{n+1}}{n+2}}$?
wraith daggerBOT
Ann
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
vivid sundial
yes
tropic oxide
if you take the limit as it goes to infinity its 1/1 does that mean it conv beau...
you're overusing the word "it".
vivid sundial
mb
tropic oxide
you're applying the limit comparison test, comparing your series sqrt(1+n)/(2+n) with the series 1/sqrt(n).
the limit of the ratio of these two series' terms is worked out to be 1, about which you care that it is neither zero nor infinity.
this result means that the two series (sum sqrt(1+n)/(2+n) and sum 1/sqrt(n)) have the same convergence status.
tropic oxide
precisely the point.
vivid sundial
but for sqrt(1+n/2+n
tropic oxide
you have that your series has the same convergence status as another series whose convergence status is known.
vivid sundial
but my question if you an/bn
and you do the limit the other series conv
why is that
vivid sundial
mb english isnt my first language
tropic oxide
what is your first language?
vivid sundial
arabic
tropic oxide
... okay, i don't speak arabic.
when you apply the limit comparison test to two positive series a_n and b_n, you calculate the limit of a_n/b_n.
if this limit is neither 0 nor infinity, then you get that the two series have the same convergence status, meaning they either both converge OR both diverge.
tropic oxide
if the limit of a_n/b_n is 0, then a_n decays faster than b_n. if you also know sum b_n converges, then you get that sum a_n converges, but not the other way around
vivid sundial
so your saying because the answer equals to one that means its inconclusive
but because 1 / sqrt(n) div we can assume the other one diverges as well
tropic oxide
so your saying because the answer equals to one that means its inconclusive
NO
HEAVENS NO
tropic oxide
but because 1 / sqrt(n) div we can assume the other one diverges as well
we don't need to assume this we KNOW this, it's what the LIMIT COMPARISON TEST tells us
vivid sundial
I understand everything except for this part
this part equals 1 right ?
and one is a finite number so this part converges
tropic oxide
i don't know how else to explain it.
i have said it as clearly as i could, 2 or 3 times, and it wasn't enough.
vivid sundial
last question does this part conv or div
nvm
i just watched a video now i understand
alright thanks for the help
stiff surge
cedar kilnBOT
stiff surge
How do i do b ii)
wraith basin
Hi
wraith basin
I'm trying to examine continuity at 0 here but no matter how I approach things I can find the limit
<@&286206848099549185>
red pumice
!15min
cedar kilnBOT
Please read #❓how-to-get-help
wraith basin
my bad, 3 mins off
wraith basin
1/e^{inf} so 0
red pumice
$\lim_{x \to 0} e^{-\frac{1}{|x|}}$
wraith daggerBOT
heavy
wraith basin
1/|x| goes to infinity
red pumice
$e^{-\frac{1}{ | x | }} = \frac{1}{e^{\frac{1}{ | x | }}}$
wraith daggerBOT
heavy
red pumice
1/|x| goes to infinity
yes so -1/|x| goes to -infinity
wraith basin
that's exactly what I meant, so we get 1 over a very big number
wraith basin
which goes to 0
red pumice
my bad youre right
put it in the exponent
$\frac{x-1}{x} = e^{\ln (\frac{x-1}{x}) }$
wraith daggerBOT
heavy
wraith basin
allright that may lead somewhere thanks
red pumice
actually you might want to just use lhopital
wraith basin
yeah I've messed around with the ln idea and didn't really get anywhere
hm lhopital also didn't really help, I've gotten
so 0 times infinity after canceling x^2
cedar kilnBOT
@wraith basin Has your question been resolved?
wraith basin
<@&286206848099549185>
cedar kilnBOT
@wraith basin Has your question been resolved?
cedar kilnBOT
@wraith basin Has your question been resolved?
icy kelp
@wraith basin can you tell me the derivative of e^(1/|x|)?
upper ruin
<@1036642946709192785> can you tell me the derivative of e^(1/|x|)?
Split the function in two cases, based on the absolute value. Then you compute the derivative at the two branches (be careful of x = 0, what happens to the derivative there?)
icy kelp
Split the function in two cases, based on the absolute value. Then you compute t...
Please let @wraith basin answer
upper ruin
Please let <@1036642946709192785> answer
Ahn sorry I thought it was you who needed help 🙈
icy kelp
No problem :D
cedar kilnBOT
@wraith basin Has your question been resolved?
ashen flume
cedar kilnBOT
ashen flume
What exactly do they mean by the expected value?
Would i do (0 x 5/50)x(1x7/50) * (2x14/50) * (314/50) * (4x7/50)(47/50)(5*3/50)?
ashen flume
yes
ashen flume
or the number of patients?
it jut says expected value
median goblet
I mean it's a variable
so we need to know what it is
Its a free variable
ok
Listen out
median goblet
Mu (The upside down h) is basically x*P(x)
from a specific value
and u would add those up
so 0*5/50 = 0
ashen flume
Mu (The upside down h) is basically x*P(x)
ok
median goblet
so it would be multiply all of those x*P(x) and add all of those terms
so 0+7/50.... = Mu
So on and so forth
μ=∑(x∙P(x)) You can also look at it likes this
μ=∑(x∙P(x))
The sum of x*P(x)
got it?
or not really
@ashen flume
ashen flume
<@291212806303252481>
yes, i understand
median goblet
alright please close this using .close
ashen flume
.close
storm junco
Hey quick question
Im learning about linear approximation, and was taught the derivation of it looked like this
storm junco
but im a little confused why triangle f=f(x)-f(x_0)
as when looking at basic derivatives, triangle f is completely different
dusk finch
as when looking at basic derivatives, triangle f is completely different
Im pretty sure it should be $f(x+\Delta x)-f(x)$ in the numerator
wraith daggerBOT
MathIsAlwaysRight
storm junco
Im pretty sure it should be $f(x+\Delta x)-f(x)$ in the numerator
but regardless do you know why delta f differs in these formulas
tropic oxide
as when looking at basic derivatives, triangle f is completely different
delta not triangle, also this limit is written incorrectly
num should be f(x+Δx) - f(x) not f(x+x_0) - f(x)
storm junco
yep sorry about that
tropic oxide
also difference in notation as to whether you consider x_0 or x to be the point at which you want the derivative
which is unimportant-ish but can trip up
storm junco
yeah its important to cover all basises
storm junco
num should be f(x+Δx) - f(x) not f(x+x_0) - f(x)
regardless do you know why delta f changes in the derivative formula vs linear approx formula?
dusk finch
delta just means sth like "change in". It's definition depends on context. Also it's not so different. Just let $\Delta x = x-x_0$, Now $f(x)-f(x_0)=f(x_0+\Delta x)-f(x_0)$.
wraith daggerBOT
MathIsAlwaysRight
dusk finch
so the x was just replaced with x_0, thats all
storm junco
so essentially the context is different in both the expressions, so what about the context specifically is causing the difference
dusk finch
Wait how did they even get from third line to last line?
it seems incredibly strange. As well as the first line
storm junco
are you knowledgable about this topic?
the last line is just adding f(x_0) to both sides
dusk finch
I mean it's definitely algebraically incorrect
storm junco
its not
dusk finch
Oh wait sorry, I didnt notice the f(x)
Kind of strange expressing f(x) in terms of f(x) tho...
Shouldnt it be f'(x)?
storm junco
but do you understand this topic, because if your claiming the difference is context specific, and you dont know the context and your trying to help, it seems difficult for you to help
dusk finch
I know the linear approximation, I am just saying that this seems a bit different
this is linear approximation
dusk finch
In first line, start with f'(x) instead
storm junco
yeah
my b i misswrote
twice in one problem
anyways though, do you understand the specific context that changes delta f
dusk finch
Lemme first rewrite your derivation of linear approximation
Alright, so the reason is: in the definition of derivative, you are taking derivative at point x, while in here, the derivative is being take at x_0
lemme clarify this a bit
storm junco
oh nevermind i figured it out
for delta x you have any point x - some point x_0
because we are comparing any point to some point
dusk finch
yep, x_0 and x have different meaning
storm junco
and in delta f we can compare f(x), any point at a certain y value, to some other point thats a y value
while in a derivative, we are comparing an x value extremely close to a variable x
thus the difference occurs
alr cool
.close
sacred jetty
How do u find the functions of the graph?
cedar kilnBOT
How do u find the functions of the graph?
sacred jetty
Idk which values i should pick
tropic oxide
notice you do not need to know the equations of either graph itself
only the asymptotes
which can and should be drawn on the chart
cedar kilnBOT
@sacred jetty Has your question been resolved?
wild sierra
In this image, I want to calculate the coordinates and angle of the arrow, such that the originating point is the center of the yellow circle (coordinates and radius of yellow circle are known).
The yellow and blue circles are on a canvas and their coordinates are known. My challenge is finding the correct coordinates to place the arrow and the correct angle so that it looks like the arrow came from the yellow circle.
Can explain further if needed.
humble marsh
the arrow needs to lie on the line that goes through the centers of the circles
wild sierra
the arrow needs to lie on the line that goes through the centers of the circles
So let's say the line is drawn using these values:
aX = 50
aY = 100
bX = -10
bY = -100
How can I calculate the angle?
wraith daggerBOT
humble marsh
^ that many radians
imagine drawing a right triangle with a horizontal line segment, a vertical line segment , and the line segment between the circles as the hypotenuse
then you can use trig to find the angle
wild sierra
got it. thanks for the help
raw barn
That's what I ended up with, I am unsure if it's correct or not
runic garnet
Show ur work
@raw barn
Aight I’ll give it a look
Ye looks good I think
Just gotta double check smth
Yep looks good
swift anchor
i dont have a specific problem i just want to ask are the roots of a quadratic equation the values for x or is it the square root of b^2-4ac
granite knoll
values for x
solid juniper
the roots of a quadratic are the values of x that make the quadratic equal to 0
cedar kilnBOT
runic garnet
(b^2 - 4ac) is called the discriminant
cedar kilnBOT
Available help channel!
jade stag
i dont understand what they mean by "largest"
cedar kilnBOT
i dont understand what they mean by "largest"
muted bear
largest in value
muted bear
f'(x) when x=-1?
muted bear
what do we know about f'(-1)?
muted bear
no
muted bear
no
jade stag
-1,-2
muted bear
what does f(-1) equal
what is the y coordinate of the function at x=-1
thats all i want to know
cedar kilnBOT
@jade stag Has your question been resolved?
merry hornet
hi
merry hornet
cedar kilnBOT
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
mostly i dont know where to begin
the given taylor polynomial equation is
$P_m(x) = f(x) = f(0)(x_0)(x-x_0)+...+f(m)(x_0)/m!*(x-x_0)^n$
but this doesnt look quite right
wraith daggerBOT
Infectia
faint dirge
... are you sure that's the right equation?
from what I recall taylor polynomials are based around imitating every derivative (in a degree 2 taylor polynomial, the first 1-2 derivatives iirc)
merry hornet
merry hornet
yea i realized that. i solved it out w derivatives this time
merry hornet
theres no way to check atm, but i was wondering, the next step would be plugging in 1.5 to e^2x and then 1.5 into this polynomial correct?
faint dirge
nope, just use your P2(1.5)
the whole point of this is to assume you don't know how to get e^(2*1.5), even if you know its derivatives
marble sluice
https://www.cuemath.com/algebra/square-root-by-long-division-method/ Uhh what is the value of a quotient in this context? Like with this when the quotient is a decimal how do you multiply it ad a decimal or as a whole number…
The instructions don’t say.
cedar kilnBOT
https://www.cuemath.com/algebra/square-root-by-long-division-method/ Uhh what is...
@marble sluice Has your question been resolved?
cedar kilnBOT
marble sluice
.reopen
cedar kilnBOT
✅
marble sluice
<@&286206848099549185>
cedar kilnBOT
@marble sluice Has your question been resolved?
static fern
cedar kilnBOT
static fern
i feel like this has something to do with the quadratic formula and/or discriminant but i'm not sure in which way...
ancient lodge
i feel like this has something to do with the quadratic formula and/or discrimin...
What do you know about the condition for a quadratic to have at least one real root
static fern
the discriminant needs to be $\geq 0$
wraith daggerBOT
FireBlazer
cedar kilnBOT
@static fern Has your question been resolved?
royal loom
cedar kilnBOT
royal loom
Okay so here's what I've been able to think out for this so far
The weight function is the response of a system at rest to a kick with unit-impulse
so in this case, the weight function would have to be the unit step function
and
wraith daggerBOT
AustinU
royal loom
so I now need to know, the Laplace inverse of what, gives the unit step function
which is the delta function
But, I don't know what this tells me about the differential operator
royal loom
since I showed that the weight function must be the delta function
so D=P(D)?
<@&286206848099549185>
If someone does respond to help, please tag me! Thank you
.close
edgy comet
i have a question for completing a square to solve the equation
granite knoll
edgy comet
i get stuck for example in the 4 step of said thing above or in short i give a write example in almost chinese
rain drift
what is the problem?
edgy comet
well i can
this is an exercise for solve the equation by completing the square
i get most of it but i always get stuck in the 5 part of the process
granite knoll
what is "the 5 part"?
edgy comet
step 5 Factor the perfect square trinomial
rain drift
so we won't know these off the top of our heads. Could you post the steps for our context?
edgy comet
yea sure
step 1/0 divide the equation by a not used in this one step 2 move the constant term to the right side of the equation step 3 take half of the coefficient for x and square it step 4 add the square to both sides of the equation step 5 factor the perfect square trinomial s step 6 take the square of both sides
granite knoll
I can't understand what step 1 is saying. but for your posted question, what would you do?
rain drift
I think what it's saying for step one is for $ax^2 + bx + c = 0$, we divide both sides by a to get $x^2 + \frac{b}{a}x + \frac{c}{a} = 0$
wraith daggerBOT
MellowDramaLlama
edgy comet
I can't understand what step 1 is saying. but for your posted question, what wou...
so in step 1 wat its trying to say is divide every thing on the problem by ( a)
granite knoll
I think what it's saying for step one is for $ax^2 + bx + c = 0$, we divide both...
ahhh, I got it
edgy comet
here it would be example x^2 but here how its not 7x^2 its not necessary
that the first step but i messed it up with my opinion
if i am to bad at explaning my stuff and i cant be helped its fine i can understand all let another person who might need more help than me be able to take my spot
.close
steady dawn
cedar kilnBOT
steady dawn
The first one
bright bridge
use 3.14 for pi?
cedar kilnBOT
Show your work, and if possible, explain where you are stuck.
steady dawn
Bruh
Alr here
All I did wa
90 x 3.14
Which equals
282.6
But I just chose D
Did I do anything wrong with the question?
livid hound
where's 90 * 3.14 coming from
why are you multiplying those two values together, what formula/reasoning are you applying
cedar kilnBOT
@steady dawn Has your question been resolved?
oak wagon
cedar kilnBOT
runic garnet
Notice that u can write it as a^2 - b^2
oak wagon
Factoring
runic garnet
U see it?
oak wagon
Yeah difference of squares
wraith daggerBOT
Stephen
runic garnet
Nah don’t expand it like that
wraith daggerBOT
Stephen
runic garnet
The first part is a^2, the second part is b^2
And to factor a^2 - b^2, it’s (a+b)(a-b)
Are u understanding or no?
oak wagon
No
runic garnet
**Stephen**
Look at this
oak wagon
Yeah
runic garnet
Replace the m^2 - 5m with “a” and 6 with “b”
rain drift
no. As an example, consider if we had this:
$(5x^2 - 3x)^2 - 25$. We can do $(5x^2 - 3x)^2 - 5^2$, which is a difference of squares. So we can write this as $((5x^2 - 3x) - 5)((5x^2 - 3x) + 5)$
wraith daggerBOT
MellowDramaLlama
rain drift
is that making more sense?
oak wagon
Yes
rain drift
great! Apply the same logic to your problem
wraith daggerBOT
MellowDramaLlama
rain drift
which are both factorable
rain drift
might be jumping the gun a bit. What is the factorization of $m^2 - 5m - 6$?
wraith daggerBOT
MellowDramaLlama
oak wagon
Oh shi, I thought we were taking out a gcf
wraith daggerBOT
MellowDramaLlama
oak wagon
No
oak wagon
Nah all good g
rain drift
$(m-6)(m + 1)$
wraith daggerBOT
MellowDramaLlama
rain drift
there we go
oak wagon
No
rain drift
I'm pretty sure that expands out to $m^2 - 5m - 6$
wraith daggerBOT
MellowDramaLlama
rain drift
I just did? $(m + 1)(m - 6) = m^2 - 6m + m - 6 = m^2 - 5m -6$
wraith daggerBOT
MellowDramaLlama
rain drift
it's factorable
oak wagon
How bro
rain drift
**MellowDramaLlama**
look right here
wraith daggerBOT
MellowDramaLlama
oak wagon
Oh yeah
rain drift
okay glad we got that cleared lol.
So, we turned $(m^2 - 5m - 6) = (m-6)(m+1)$.
wraith daggerBOT
MellowDramaLlama
oak wagon
rain drift
We can do the same for $m^2 - 5m + 6$
wraith daggerBOT
MellowDramaLlama
rain drift
that's also factorable
rain drift
yep so you see it works out
rain drift
perfect!
So then we have the following:
$\\(m^2 - 5m)^2 - 36 \\= (m^2 - 5m)^2 - 6^2 \\= ((m^2 - 5m) - 6)((m^2 - 5m) + 6) \\= (m^2 - 5m - 6)(m^2 -5m+ 6) \\=(m-6)(m+1)(m-3)(m-2)$
that's about how far we can take it
wraith daggerBOT
MellowDramaLlama
rain drift
ok so that one is just a distribution. It helps to remember the rules "a negative times a positive is a negative" and "a negative times a negative is a positive".
when you see a negative in front like that it's basically saying multiply that by -1
for example, -80 is really -1 * 80
so then you follow distribution laws
$-(1 - y^4) $ is the same thing as $-1 \times (1 - y^4)$
wraith daggerBOT
MellowDramaLlama
rain drift
then you simply distribute and keep your signs accounted for 😉
oak wagon
Aren’t the signs just switched then?
wraith daggerBOT
MellowDramaLlama
rain drift
which, believe it or not is also a difference of squares since 1^2 = 1
so you have $y^4 - 1$ = $(y^2)^2 - 1^2 = (y^2 - 1)(y^2 + 1)$, of which you can further break down since $y^2 - 1$ is also a difference of squares
wraith daggerBOT
MellowDramaLlama
oak wagon
How did my teacher get that as an answer
rain drift
what what I was saying above
$-(1 - y^4) \\= y^4 - 1 \\ = (y^2)^2 - 1^2 \\= (y^2 + 1)(y^2 - 1) \\= (y^2 + 1)(y^2 - 1^2) \\ = (y^2 + 1)(y - 1)(y + 1)$
wraith daggerBOT
MellowDramaLlama
rain drift
because 1^2 = 1, so we have a difference of squares there
rain drift
same reason. $1^2 = 1$, so we can re-write $y^2 - 1$ as $y^2 - 1^2$ and it becomes a difference of squares
wraith daggerBOT
MellowDramaLlama
cedar kilnBOT
@oak wagon Has your question been resolved?
azure trench
cedar kilnBOT
azure trench
How did 1/16 become -ln(16)? Is that a log rule that I'm not aware of?
Is -ln used when the exponent is negative?
rain drift
$ln(\frac{a}{b}) = ln(a) - ln(b)$. In this example, $ln(\frac{1}{16}) = ln(1) - ln(16)$, but $ln(1) = 0$, so we're just left with $-ln(16)$
wraith daggerBOT
MellowDramaLlama
azure trench
OH! That makes so much sense, that is so helpful! Thank you!
rain drift
yep np!
another way to look at it is with this rule $\frac{1}{16} = 16^{-1}$, so $ln(\frac{1}{16}) = ln(16^{-1}) = -ln(16)$. Either way works!
wraith daggerBOT
MellowDramaLlama
azure trench
Okay that make a lot of sense, thank you for your help I spent a while trying to wrap my head around that.
azure trench
Thank you! Enjoy your night!
radiant topaz
Thank you! Enjoy your night!
Wet pillow message incoming
cedar kilnBOT
Available help channel!
kind orchid
teach me step by step how to solve this
cedar kilnBOT
teach me step by step how to solve this
kind orchid
im here and idk what to do next
royal loom
, rotate
wraith daggerBOT
kind orchid
i dont know how to use this for that
cedar kilnBOT
@kind orchid Has your question been resolved?
violet kettle
can someone help or assist me executing this?
cedar kilnBOT
can someone help or assist me executing this?
violet kettle
wait
crimson sedge
i think 4,4,4
plain breach
well 4,4,4 can make an equilateral triangle
crimson sedge
cant all of them
fair geyser
this is easy, you always look at the largest number and check if other two are larger when added
only true with 4,4,4
fair geyser
this is wrong, 3+6 equals 9
fair geyser
yeah others are right
proud canopy
Hello, can someone help me with this? I've been trying to study the conic sections in a while and I still couldn't get it I have to finish this until tonight 😭
cedar kilnBOT
Hello, can someone help me with this? I've been trying to study the conic sectio...
@proud canopy Has your question been resolved?
cedar kilnBOT
@proud canopy Has your question been resolved?
storm mural
Hello, can someone help me with this? I've been trying to study the conic sectio...
conic sections in standard form:
no squared term: line
x^2 OR y^2: parabola
ax^2 + by^2, where a and b are positive: ellipse
special case where a = b: circle
ax^2 - by^2 or by^2 - ax^2, where a and b are positive: hyperbola
to get it in standard form you gotta move all the x^2 and y^2 over to one side
hopefully that helps
it says to use 2 methods is the other a graphing calc or smth?
proud canopy
The methods that they want us to use is circle and parabola
storm mural
The methods that they want us to use is circle and parabola
hm?
proud canopy
Is this correct?
left dirge
The methods that they want us to use is circle and parabola
what is that method?
cedar kilnBOT
cedar kilnBOT
Available help channel!
teal flume
I need help with a. I wrote the dimension as dimP(F) - dimU. dimP(F) =degp + 1 and dimU=(degp + 1)(degq - 1).
teal flume
combining this i get dim(P(F)/U) = -degpdegq-degq-2.. where am i going wrong?
i managed to solve it by showing P(F)/U is isomorphic to P_(degp-1)(F), but i still want to know where i went wrong with the method over?
cedar kilnBOT
@teal flume Has your question been resolved?
cedar kilnBOT
@teal flume Has your question been resolved?
cedar kilnBOT
@teal flume Has your question been resolved?
crimson sedge
how do i solve for delta?
cedar kilnBOT
how do i solve for delta?
austere hull
how do i solve for delta?
there are three thetas which one are you looking for
opaque root
I thought he was asking for the green one
opaque root
!status
cedar kilnBOT
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
hearty sun
Let A be a 2 x 2 matrix with real entries. Assume that A has rank 1. then the operator A sends R^2 to R^2 given by left multiplication with A is an orthogonal projection. Is this statement always, true? sometimes true ( if so what is the condition) or false?
cedar kilnBOT
Let A be a 2 x 2 matrix with real entries. Assume that A has rank 1. then the op...
@hearty sun Has your question been resolved?
hearty sun
<@&286206848099549185>
cedar kilnBOT
@hearty sun Has your question been resolved?
stiff anvil
Hi all, I’m curious if anyone knows how to solve this, or better yet- knows a good video on YouTube that would explain this question well. Thanks!
crimson sedge
Hi all, I’m curious if anyone knows how to solve this, or better yet- knows a go...
While I do watch a lot of math content on YouTube, I don't know a video on this exact topic. However, I can guide you through it
First notice that "Suppose a fish population is currently 6700" means suppose $P_0=6700$
wraith daggerBOT
Labyrinth
crimson sedge
Then "7 years later" means $t=7$
wraith daggerBOT
Labyrinth
crimson sedge
And "later the population is 16000" means $P_t=16000$
wraith daggerBOT
Labyrinth
crimson sedge
On second though I should've probably asked "What's P_0? What's t? What's P_t", I kinda regret just telling you, but too late...
@stiff anvil Anyways, do you know what to do with these bits of information?
stiff anvil
My teacher didn’t exactly explain it, but don’t I go 16,000/6,700?
Or other way around
crimson sedge
My teacher didn’t exactly explain it, but don’t I go 16,000/6,700?
Hmmm that'd work I guess
stiff anvil
I don’t really know the steps to get R as a %
crimson sedge
So first of all, using the exponential model, by how much did the population grow?
crimson sedge
I don’t really know the steps to get R as a %
Don't you know how to express a rate of change in percentages?
stiff anvil
I know how to solve these equations, but not find the r as a %. Or find r if I am not given it within the equation.
crimson sedge
Suppose for the sake of explanation that r=1
That means that for each fish in the population, a single new fish will be present the next year
Or in other words the population doubles every year
You could also state that as "100% of the fish from the current year's population will get assigned corresponding fish the next year"
This probably makes more sense if you think of it as cells dividing
r=1 means that for each cell, 1 copy of that cell will be added the next year, or in other words 100% of the cells will divide in a year
If r=0.5 instead, for each cell, 0.5 copies of that cell will be added the next year on average, or in other words 50% of the cells will divide in a year
r=1 => 100%
r=0.5 => 50%
Notice a pattern yet?
stiff anvil
Yes
stiff anvil
I’m unsure
crimson sedge
How did you know that 1.5 would correspond to 150%?
stiff anvil
Oh I would just move the decimal over 2 to the right
crimson sedge
Oh I would just move the decimal over 2 to the right
Right... Don't you know of an operation that does the same thing?
stiff anvil
X100
crimson sedge
Now make sure you don't forget that because you'll need it
Hopefully making you find it yourself was enough to stop you from forgetting it but it would be really embarassing if you forgot this fact in the middle of an exam
Fun fact: % is secretly a mathematical constant equal to 1/100
Hence why 1.5=150%
Oh and don't make the mistake of writing 1.5*100=150% (* means multiplication), that's not true, just write 1.5=150%
It's a common mistake from students
Anyways back to the question
So I just told you what P_0, t, and P_t are (should've made you tell me instead but oh well...), any idea what to do with this information?
Actually nevermind that, let's do it the way you mentionned, the one where you do 16000/6700
Using the formula, can you write an expression for the factor by which the population grew?
,calc 16000/6700
wraith daggerBOT
Result:
2.3880597014925
So for example in our case the population grew by this factor 
crimson sedge
Using the formula, can you write an expression for the factor by which the popul...
@stiff anvil I'll let you work on this puzzle
stiff anvil
$6700(1+2.38)^7$ ?
crimson sedge
$6700(1+2.38)^7$ ?
16000/6700 isn't the rate of change, it's the factor by which the population grew
There's 2 main differences
First of all one is a rate of change and the other a factor which are different things
Second the factor is over 7 years, whereas the rate of change is yearly
@stiff anvil Scrap that, can you find a formula that involves P_t and P_0 only?
Remember, you're meant to find a formula for the factor by which the population grew over the given time period (in this case 7 years)
What does porker mean
Someone who porks
Obviously
royal loom
stiff anvil
$P_t=6700+2.38t$ ?
wraith daggerBOT
Porker
crimson sedge
Hmmm okay no
And it doesn't answer the question
Let's start simple, where does the 2.38 come from?
stiff anvil
It’s the change from 16k from 6.7k
crimson sedge
It’s the change from 16k from 6.7k
Yes, so what calculation did we do?
stiff anvil
16000/6,700
crimson sedge
Basically we just divided the population size after the 7 years by the population size before the 7 years
If you try to generalize this, what do you get?
What formula do you obtain?
@stiff anvil If you don't know just say so
stiff anvil
Not sure
crimson sedge
Not sure
Sorry for the late reply
Let's go even simpler
What is P_0 in our situation?
crimson sedge
Or is 6700 p_1
P_1 would be the population after a year
Now, what is P_t?
Hmmm alright then
In that case I'll just leave a full, unguided explanation in case you come back
cedar kilnBOT
@stiff anvil Has your question been resolved?
crimson sedge
,calc (16000/6700)^(1/7)-1
wraith daggerBOT
Result:
0.13241719255953
why
eh
Oh wait nevermind my PC was just dying
Or perhaps we got rate limited
The general version of $\frac{16000}{6700}$ would be $\frac{P_t}{P_0}$.$\newline$We have the formula $P_t=P_0(1+r)^t$, and you might realise that a clever trick to pull out a $\frac{P_t}{P_0}$ is to divide both sides by $P_0$, which would give:$\newline\frac{P_t}{P_0}=(1+r)^t$.$\newline$Hence, plugging in $t=7$ and $\frac{P_t}{P_0}=\frac{16000}{6700}$, we get:$\newline(1+r)^7=\frac{16000}{6700}$.$\newline$This becomes an equation where we have to solve for r.$\newline$To solve this, we first start by putting both sides to the $\frac17$th power:$\newline\left((1+r)^7\right)^{\frac17}=\left(\frac{16000}{6700}\right)^{\frac17}\newline\Rightarrow(1+r)^{7\frac17}=\left(\frac{16000}{6700}\right)^{\frac17}\newline\Rightarrow1+r=\left(\frac{16000}{6700}\right)^{\frac17}\newline\Rightarrow r=\left(\frac{16000}{6700}\right)^{\frac17}-1\newline$We can then find an approximate value for r by simplying plugging this into a calculator, and we get:$\newline r\approx0.132=13.2%$
wraith daggerBOT
Labyrinth
crimson sedge
.close
upbeat meadow
Find x accordingly to a :
(a^a)^x=a(a^(a+1)+a)
cedar kilnBOT
Find x accordingly to a :
(a^a)^x=a(a^(a+1)+a)
upbeat meadow
We have a^ax
So i guess we've to rewrite the right side so that it has a base of a
bright bridge
it does?
upbeat meadow
Not rn
cedar kilnBOT
@upbeat meadow Has your question been resolved?
open pike
yo
open pike
.close
fleet solstice
Is there a way to find the probability of getting k heads out if n coin flips? I have no idea where to even begin
slow thicket
Is there a way to find the probability of getting k heads out if n coin flips? I...
yes
this is the binomial distribution
fleet solstice
Is there a way to find the probability of getting k heads out if n coin flips? I...
Oh wait I mean k heads in a row sorry
slow thicket
okay so for that its just basic probability rules
if k = n-1
there are two ways you can get k heads in a row
you can determine how many ways there are as k decreases
and P(k in a row) is straightforward as well
@fleet solstice
fair geyser
i re-solved this thing about 7 times because it doesn't stick
fleet solstice
So for n-1, you can simply shift the heads over. But for n-2, You also have to add on the added possibility for a head at the beginning or end right?
fleet solstice
Oh wait I mean k heads in a row sorry
Actually also at least k heads in a row
But that’s just a sum anyways
fleet solstice
But that’s just a sum anyways
Ohhhh you know what that makes sense
fair geyser
here's my method, although the fact that i keep forgetting it may suggest a flaw
we're going for number of outcomes without k heads in a row
by summing the outcomes that look like
[no k heads in a row]T
[no k heads in a row]TH
[no k heads in a row]THH
[no k heads in a row]THHH
up to whatever HHHHH you need
so one less than k
at the end divide by 2^n and flip