<@&286206848099549185>
#help-10
static patio
obtuse pebbleBOT
@static patio Has your question been resolved?
obtuse pebbleBOT
Available help channel!
safe spade
Anyone know what equation this is?
obtuse pebbleBOT
Anyone know what equation this is?
short spruce
looks like some awfully written physics equation
something about black holes according to wikipedia
why are you working with an equation you've never seen before
er - expression, rather?
wtf
safe spade
wtf
its from a manga
middle right is hawking radiation just need ot figure out how to use it because it deals with large extra dimensions (n)
schwarzschild radius is pretty easy
Mbh is black hole mass
obtuse pebbleBOT
@safe spade Has your question been resolved?
nocturne creek
This question again? I'm no physicist, but my best guess would be that this is Planck's law for black body radiation. Basically, a black body emits radiation, but it does not emit all frequencies at equal amounts. You use the equation to work out how much of a given frequency will be emitted if the body is at a certain temperature.
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
timid silo
obtuse pebbleBOT
misty terrace
hint: try counting the number of cubes from 1 to n instead of 2^9+1 and 2^27+1 first
obtuse pebbleBOT
@timid silo Has your question been resolved?
languid jay
I need help on this question I can’t remember how to find the sin cos or tan for something not on the unit circle
obtuse pebbleBOT
I need help on this question I can’t remember how to find the sin cos or tan for...
@languid jay Has your question been resolved?
tardy epoch
Show the original question
languid jay
Show the original question
Ok
The question itself I have no issue with
I just can’t remember how to find sin cos or tan of something not on the unit circle
<@&286206848099549185>
Please I beg of you
spiral maple
I just can’t remember how to find sin cos or tan of something not on the unit ci...
calculator
languid jay
calculator
We can’t use calculators
That’s the problem
Also it has to be in radians
spiral maple
you maintain the correct units
languid jay
Yeah Ik how to turn it into radians and Ik that cos is the x value
spiral maple
But anyway, you know $$\sin(337.5)=\sin(360-22.5)=-\sin(22.5)$$
warm shaleBOT
Mosh
spiral maple
through simple identity work
languid jay
through simple identity work
Ok can you explain the identity work
spiral maple
1st equality is addition, not explaining that
period of sine is 360deg, so $\sin(x)=\sin(360+x)$
warm shaleBOT
Mosh
spiral maple
and sine is odd so $\sin(-x)=-\sin(x)$
warm shaleBOT
Mosh
obtuse pebbleBOT
spiral maple
and you can easily find sin(22.5).
obtuse pebbleBOT
Available help channel!
distant raft
Let $\mu$ be an absolutely continuous probability distribution with distribution function F and density f. For any Borel set $\Lambda$ and $a<b$, let v($\Lambda \cap [a,b]) = \int_{a}^{b} \mathbb{I}{\Lambda}(x)f(x)dx.$ Show that this defines a probability measure v on the Borel sets of $\mathbb{R}$, and that v=$\mu$. Conclude that for positive Borel functions g, $\int{a}^{b} g(x)dF(x)=\int_{a}^{b} g(x)f(x)dx.$
obtuse pebbleBOT
Let $\mu$ be an absolutely continuous probability distribution with distribution...
warm shaleBOT
Neko
obtuse pebbleBOT
@distant raft Has your question been resolved?
mint patio
(this isnt a test btw this is just a part of the review that my teacher gave. It is an optional assignment and does not count toward my grade.)
obtuse pebbleBOT
(this isnt a test btw this is just a part of the review that my teacher gave. It...
mint patio
im not even sure where to start on this question
civic zealot
do you know the definition of a geometric sequence?
mint patio
isnt it like multiplicative
civic zealot
yes, but more specific than that
civic zealot
civic zealot
basically, each terms is the previous term multiplied by a specific value
civic zealot
Since each term is just the previous multiplied by a specific number, you can find the ratio by doing division of subsequent terms
i.e. $f(n) = ar^{n-1}$ and $f(n+1) = ar^n$, so $f(n+1)/f(n) = ar^n/ar^{n-1} = r$
warm shaleBOT
Zybikron
civic zealot
So since you know $f(2)$ and $f(6)$ you can find $f(6)/f(2) = r^4$
warm shaleBOT
Zybikron
civic zealot
$f(6)/f(2) = ar^5/ar = r^4$
warm shaleBOT
Zybikron
civic zealot
but you have values for f(6) and f(2). So you can find r by setting the two equal and solving
civic zealot
sorry for the questions 😅
🤷♂️ it's part of learning
civic zealot
so what is the value of f(6)/f(2)?
timid silo
civic zealot
if you have a question take an unoccupied room. See #❓how-to-get-help for more information
obtuse pebbleBOT
@mint patio Has your question been resolved?
haughty aspen
How to solve this question?
obtuse pebbleBOT
How to solve this question?
haughty aspen
Should i find the cost first?
obtuse pebbleBOT
@haughty aspen Has your question been resolved?
timid silo
@civic zealot
obtuse pebbleBOT
<@!334482942506041345>
timid silo
are you good with sequences zybikron?
hybrid gull
Just ask your question and don’t ask for someone to be ready
timid silo
I have half of it figured out
well like I have a platform set out for the problem
I saw that zybikron helped someone with a sequence problem above so I asked
hybrid gull
Assign variables
hybrid gull
What does your work look like so far?
hybrid gull
r is for geometric sequences
timid silo
hybrid gull
Use d for arithmetic sequences
timid silo
is x ok?
hybrid gull
That shall be sufficient for now
hybrid gull
Arithmetic: 9, 9 + x, 9 + 2x
Geometric: 9, 11 + x, 29 + x
Now you can find x given that geometric sequences have a common ratio
hybrid gull
The quotient of one term divided by the next term is always constant
hybrid gull
is x the same tho because its being used for both the AP and GP
I believe it’s the same given that the arithmetic series is modified (adding 2 and 20) to create the geometric series
The geometric sequence comes from the arithmetic sequence
timid silo
hmmm alright alright
hybrid gull
29 + 2x - sorry about that
timid silo
hybrid gull
I’m making all sorts of mistakes here 🤦
hybrid gull
Arithmetic: 9, 9 + x, 9 + 2x
Geometric: 9, 11 + x, 29 + 2x
timid silo
soooo x is equal so what can be do with that?
hybrid gull
Arithmetic: 9, 9 + x, 9 + 2x
Geometric: 9, 11 + x, 29 + 2x
How can we relate the terms in the geometric sequence we have here to each other?
hybrid gull
We can use the fact that geometric sequences have a common ratio to solve for x
What equation can we make relating these terms together using this fact?
timid silo
$$\frac{9}{11+x} = \frac{11+x}{29+2x}$$
warm shaleBOT
痛苦
timid silo
this..?
hybrid gull
Yup 🙂
hybrid gull
Yes - can you explain how you got this?
timid silo
uh yes
hybrid gull
You’re correct
timid silo
I have $$121 + 22x + x^2 = 261 = 18x$$
warm shaleBOT
痛苦
timid silo
timid silo
Yes - can you explain how you got this?
well because in a geometric sequence consectuive terms have a common ratio and if you divide it, you will get the common ratio
hybrid gull
well because in a geometric sequence consectuive terms have a common ratio and i...
Nice
warm shaleBOT
痛苦
timid silo
using GF its 2sqrt143
oh its 140 oops
so 14 and -10
are my solutions?
or is it -14 and 10
hybrid gull
Which pair?
hybrid gull
Solve the equation - what are its solutions
hybrid gull
They should follow the original rules that the problem gave you
If they don’t, then maybe we made a mistake
timid silo
so is it -14 or 10
its -14 because we want the smallest right?
so its 1
@hybrid gull
hybrid gull
Whichever gets you the smallest
Check x = -14 here
Arithmetic: 9, 9 + x, 9 + 2x
Geometric: 9, 11 + x, 29 + 2x
Do you get arithmetic and geometric sequences?
timid silo
yes
its asking for smallest possible value for the third term in the geometric sequence which is 29 + 2x
so when its -14 its 29 + 2(-14) = 1
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
pearl rover
its been kind of confusing for me, so I just wanted to confirm that:
the rank of a matrix = the dimension of a column space = pivot of the column space
but they do not equal the dimension of the space that the matrix is a subspace of (i.e. a 3x4 matrix whose column space is a subspace of R^3 but may have a rank of 2)
obtuse pebbleBOT
its been kind of confusing for me, so I just wanted to confirm that:
the rank o...
@pearl rover Has your question been resolved?
wicked bolt
yeah i think that is
say the matrix is 3x4 then it sends vectors from R^3 to R^4 right
but if the rank is 2 then the column space would just be a plane in R^4
pearl rover
got it, thx! was confused since my book says the dimension n of the subspace = R^n
wish they didnt give the definition this way
obtuse pebbleBOT
@pearl rover Has your question been resolved?
rough bough
obtuse pebbleBOT
strong vale
Which one bro
rough bough
How do we do (b)
strong vale
I believe they are perpendicular if the dot product of the normal vectors to the plane are zero
quaint glen
idk what a "direction vector" would be for a plane
strong vale
yup sorry edited
rough bough
quaint glen
way overthinking it
as azeem said, they're perpendicular iff their normal vectors are perpendicular(that is, their normal vectors have dot product 0)
rough bough
My bad but the question about non Skegness is question 2
strong vale
,w define skew line
warm shaleBOT
rough bough
Wait they cannot be parallel?
raven spire
._.
restive swift
._.
rough bough
But these are vector equations though? If they are not parallel, wouldn’t they will interscect?
raven spire
x + y + z = 1 and x + y + z = 2 are two parallel planes
strong vale
a circle about orgin is $x^2+y^2=r^2$
warm shaleBOT
azeem321
strong vale
So what do you think a sphere equation would be
rough bough
Yeah x^2 +y^2 + z^2 = 1 is a sphere
strong vale
yh
rough bough
So how would we do that question?
Since the x axis are parallel they are not skewed?
Since if they were skewed then they must not intersect and are not parallel
rough bough
So if they are parallel then they are automatically cannot be skewed?
In the first one, it is parallel in the z axis?
raven spire
(3k + 2, k - 1, 2k + 3) and (3t - 1, -t - 4, 3t + 2)
(3, 1, 2) . (3, -1, 3) = 9 + 6 - 1 = 14 not zero
so at least they're not parallel
Now, you need to show that they intersect
rough bough
👌
raven spire
So yeah, they intersect and hence are not skew
rough bough
Wait how did you get k and t values
rough bough
Okay got it
I thought that -t was a positive t
What happens if it is a positive t instead?
raven spire
wdym?
rough bough
Like if the equation for y makes it so that it becomes t - 4
And then you are trying to find t and k, how would you find it then?
rough bough
Wait you removed the two different equations?
raven spire
which ones?
timid silo
which ones?
He is probably talking about the messages u deleted just now
raven spire
Given two line equations, do you know how to check whether they intersect or not...
And I asked something as well.
rough bough
raven spire
Thank you very much for taking the pain to write it down
rough bough
How do we know when two line equations intersect?
Like in 2d plane, i can see that if they are parallel they wouldn’t intersect
But in the 3d, Is it so that even when they are parallel they still intersect?
raven spire
Do you know where these two equations came from?
raven spire
Yes
The x-, y- and z- coordinates to be same for both lines at the point of intersection.
If the two x-, y- equations don't give a result, or have no solution, then it's sufficient info to conclude that the lines don't intersect
rough bough
Yeah
raven spire
If the two x-, y- equations, however, have multiple solutions, then the two are parallel, and you'd have to check with the z- equation
for a particular solution
rough bough
Multiple solutions?
raven spire
Depending on that, you can conclude whether both line coincides with each other, or intersect exactly once
rough bough
If they have multiple solutions wouldn’t that only exist for degrees of 2z/
raven spire
3t + 6 = 3k + 2
and
t + 2 = k + 1
rough bough
How are these multiple solutions?
raven spire
(3t + 6, t + 2, t + 2), (3k+2, k + 1, k + 1) coincide
and wtf do you mean by multiple solutions?
when I say multiple solutions, I clarified, very clearly, about me talking about "EQUATIONS"
If two x-, y- equations have multiple solutions, then you can't really say anything until you check with the z- equation
royal basin
what problem are y'all doing rn
rough bough
So from an earlier example, wouldn’t that be multiple solutions]\
raven spire
what problem are y'all doing rn
Given equation of two lines, to determine if it's skew. Ex. 2.4, 2 b)
royal basin
ok so parameterize them both, write down a system of 3 equations in 2 unknowns, and establish that it is inconsistent
what's the big deal exactly
raven spire
OP received the answer is now curious what happens if the equations were: $$\frac{x-2}{3} = y + 1=\frac{z-3}{2} , , \text{and} , , \frac{x+1}{3} = y - 4 = \frac{z - 2}{3}$$
warm shaleBOT
royal basin
do the same thing
rough bough
Yeah
royal basin
the first line is parameterized as (2 + 3t, -1 + t, 3 + 2t) and the second as (-1 + 3s, -4 + s, 2 + 3s)
raven spire
use the definition of a pair of skew lines
rough bough
If they are skewed then the third one should have the LHS and RHS invalidated
Meaning they cannot equal
raven spire
Yes
rough bough
But didn’t you say something about if the first two x and y are not equal to each other then the z is not equal
And you should only go to the z only if you have multiple solutions
raven spire
Are we done discussing the original question?
rough bough
Yes
raven spire
okay
(2 + 3t, -1 + t, 3 + 2t) and the second as (-1 + 3s, -4 + s, 2 + 3s)
Here, for example: second equation gives -> t = s - 3
Plug into the first, you get: -> 3s - 7 = 3s - 1, which is true for no t, s
hence, the two lines never intersect
rough bough
Okay so this is using one of the equation to find the intersection using the other one
obtuse pebbleBOT
@rough bough Has your question been resolved?
royal solar
(2i+4j-6k ) . (i+4j+3k) = 2+16-18 = 0
since the normals are perpendicular, so are the planes
@rough bough
royal solar
Given equation of two lines, to determine if it's skew. Ex. 2.4, 2 b)
l1 = (2i-j+3k) + a (3i+j+2k)
do the same for l2
equate i,j and k to get 3 linear simultaneous eqns
solve to get a and b using 2 of the 3 eqns
use those values in the third eqn. if they’re equal, they don’t skew. if they’re not equal, they don’t intersect
general idea of approach to this problem
obtuse pebbleBOT
@rough bough Has your question been resolved?
timid silo
Ok. So I was trying to find out how the formula of lx works for this graph example here. It’s for one of my assignments.
obtuse pebbleBOT
Ok. So I was trying to find out how the formula of lx works for this graph examp...
timid silo
I’m trying to find out how I got for example .915 from 1.000
Here is what it represents but looking more on on how the formula of Ix works.
obtuse pebbleBOT
@timid silo Has your question been resolved?
timid silo
.close
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
half marsh
Hi! On this image the angle alpha appears in "two places". I know there's a mathematical rule that explains why (something about parallels I think?) But I don't remember which one. Could someone explain this to me or at least tell me the name so I can look it up?
obtuse pebbleBOT
Hi! On this image the angle alpha appears in "two places". I know there's a math...
tulip kestrel
in the triangle with alpha and P vector
half marsh
in the triangle with alpha and P vector
yes
tulip kestrel
you have angles alpha and pi/2 and pi/2 - alpha
if we are talking in radians
the y- component is perpendicular with surface
half marsh
yes
if we are only given alpha between the x axis and the stripped surface, we can deduce the angle between the P vector and Y axis being alpha, no ?
tulip kestrel
yes
half marsh
you have angles alpha and pi/2 and pi/2 - alpha
in that scenario, we would know it's pi/2 - a certain angle but not necessarily alpha right ?
tulip kestrel
i think i got you confused im going to make a drawing
half marsh
ah thank you very much!
half marsh
yes!
tulip kestrel
since BC hits the ground perpendicular
the angles of a triangle should equal 180
that means the sum of the three angles should equal 180
we have two angles given alpha and 90
half marsh
right
tulip kestrel
so the last angle would be 180-alpha-90
that would be 90-alpha
in triangle ACD C would be 90 degrees because we choose the y axis perpendicular with the slope
we already know 90 - alpha
then we find the angle on the other side to get 90 degrees
do you understand?
half marsh
so we know angle ACB is 90-alpha and angle ACD is 90 degrees is that right
tulip kestrel
well its dangerous to say that lets say divide c in c1 and c2
acd is 90 degrees
and ac1b is 90-alpha
we need to find c2 for wich c1+ c2 is c
so 90-alpha +c2 = 90
tulip kestrel
no problem :))
half marsh
.close
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
obtuse pebbleBOT
spiral maple
Yeah, you can draw it
use a graphing software
desmos exists
geogebra exists
wolfram exists
surreal estuary
desmos exists
.
surreal estuary
geogebra exists
.
surreal estuary
wolfram exists
.
sweet spoke
obtuse pebbleBOT
sweet spoke
ellooo
I need help with this
I'm stuck at 2n²-2/(n+1)(n-2)
and the ans is 2(n-1)/n-2
alpine raven
n^2 - 1 = (n-1)(n+1)
sweet spoke
oh
riiiiight
oh ma god
thank you lol
♡♡♡
so simple yet hard lol
ty again~
.close
fickle ivy
Hello I got a problem here and I don't really know how to solvei t
obtuse pebbleBOT
Hello I got a problem here and I don't really know how to solvei t
fickle ivy
The question goes: You score 9 out of 10 goals on a penalty. During one season you get to take 10 penalties. What are the odds on scoring minimum 9 goals?
spiral maple
Let X be the number of goals. X follows a binomial distribution with what parameters?
fickle ivy
The way i tried to solve it was i calculated the odds on scoring 9 goals then one miss and after that multiply by 10 since the order of the miss can occur any penalty out of the 10
spiral maple
$P(X\geq 9)=P(X=9)+P(X=10)$
warm shaleBOT
Mosh
fickle ivy
isnt there any other methods to calculate probability?
spiral maple
not really
fickle ivy
for example a coin. If you flip it 4 times what are the odds on it being heads 4 times in a row, i usually do (1/2)^4
fickle ivy
ok but back to my first problem, how am i supposed to solve it?
fickle ivy
sorry english is not my first language, what do you by that?
spiral maple
$X\sim \operatorname{Bin}(n,p)$ what are n and p?
warm shaleBOT
Mosh
spiral maple
n being the number of trials
p being the probability of success
fickle ivy
10 and 9
fickle ivy
0,9
spiral maple
They always score a goal....?
fickle ivy
0.9
spiral maple
yes
so then just plug everything into the formula
$P(X=x)=\binom{n}{x}p^x(1-p)^{n-x}$
warm shaleBOT
Mosh
obtuse pebbleBOT
@fickle ivy Has your question been resolved?
thorn kite
Answer is B
obtuse pebbleBOT
Answer is B
spiral maple
F_gsin(theta) is component down the incline
thorn kite
Yeah
spiral maple
yeah
thorn kite
Wdym
thorn kite
Fg = mg and canceled the m
spiral maple
You wrote cos instead of sin
thorn kite
Oohhh
spiral maple
Did I really need to repeat that?
thorn kite
F parallel is cos tho
spiral maple
acceleration is down the incline
cause it's assumed to not leave the incline
so ma only considers forces along the slope
$\sum F_x=ma \ \sum F_y=0$
warm shaleBOT
Mosh
thorn kite
So I use sin for Friction ?
spiral maple
friction is a whole other story.
Please: read what I am saying
Cause you're clearly not
The component of gravity is wrong
Nowhere did I say friction was wrong
thorn kite
So I don’t cancel m and keep it as fg
spiral maple
OMFG
Read
You used cosine instead of sine
And don't change the subject.
For the component of gravity down the incline, you used cosine instead of sine.
thorn kite
I use sine for both ?
spiral maple
Are you purposefully not reading?
thorn kite
I am I just don’t understand what u mean by down the incline
thorn kite
Yes
spiral maple
So the vector components along the incline.
The gravity component down the incline uses sine not cosine.
thorn kite
Okay I understand that part but what gravity component is down the incline
spiral maple
The one that points along the incline
thorn kite
Which point is thattt
thorn kite
Where do I put sine in
Sorry man but we haven’t directly learned about what you are talking about
I just need to find the coefficient of friction
obtuse pebbleBOT
@thorn kite Has your question been resolved?
tardy epoch
do you have an example? the right hand side is already as simplified as it gets
oh you mean $f(k) = f(1) + ... + f(k-1)$ for all integers $k$
warm shaleBOT
riemann
warm shaleBOT
riemann
compact shadow
$a_{n}=S_{n-1}$ so $a_{n}=S_{n}-S_{n-1}=a_{n+1}-a_{n}$, $a_{n+1}=2a_{n}$
compact shadow
Oh yes S_n is the sum of a_k k from 1 to n
warm shaleBOT
Mathgeek007
compact shadow
Definition of S_n
Sum of first n terms minus sum of first n-1 term, that leaves you only the n th term
Oh that’s you. Goodbye
timid silo
Compute the sum
$$\frac{2}{1 \cdot 2 \cdot 3} + \frac{2}{2 \cdot 3 \cdot 4} + \frac{2}{3 \cdot 4 \cdot 5} + \cdots$$
obtuse pebbleBOT
Compute the sum
$$\frac{2}{1 \cdot 2 \cdot 3} + \frac{2}{2 \cdot 3 \cdot 4} + \f...
warm shaleBOT
kaniii
timid silo
so,using partial fraction decomposition, I was able to write every term as $\frac{1}{n-1}+\frac{-2}{n}+\frac{1}{n+1}$
warm shaleBOT
kaniii
timid silo
now, im not sure how to use telescoping to combine and eliminate terms
timid silo
does your sum start at n=2?
yes
timid silo
write out the sum for n=2, 3, 4, 5
4/3 -> 59/24 -> 141/40 -> 137/30 ->
tardy epoch
i mean don't simplify. you should see cancellations if you write out the summands
$(a_1 + b_1 + c_1) + (a_2 + b_2 + c_2) + (a_3 + b_3 + c_3) + (a_4 + b_4 + c_4) + ...$
warm shaleBOT
riemann
tardy epoch
**kaniii**
where this is $a_n + b_n + c_n$
warm shaleBOT
riemann
timid silo
i mean don't simplify. you should see cancellations if you write out the summand...
i did that but i couldnt really find any
let me show
$(\frac{1}{1}- \frac{2}{2} + \frac{1}{3}) + (\frac{1}{2} - \frac{2}{3} + \frac{1}{4}) + (\frac{1}{3} -\frac{2}{4} + \frac{1}{5}) + (\frac{1}{4} - \frac{2}{5} + \frac{1}{6}) + ...$
there
warm shaleBOT
kaniii
timid silo
write -2/n = -1/n - 1/n
but we dont have 3 fractions of any same denominator
i still dont see it
tardy epoch
**kaniii**
look for 1/4
tardy epoch
**kaniii**
what's $c_2 + b_3 + a_4$?
warm shaleBOT
riemann
obtuse pebbleBOT
@timid silo Has your question been resolved?
timid silo
what's $c_2 + b_3 + a_4$?
0
tardy epoch
i dont really see how to cancel it out
Generalize it to any n now
It also holds for 1/3
And if you write more you'll cancel more
tardy epoch
in the end, what does it not cancel?
Write it out
tardy epoch
3/2 and 1/(n+1) will not be cancelled out <@!261933205387477002>
Are you supposed to evaluate a series
timid silo
Take the limit of your partial sum
i dont know the limit
tardy epoch
i dont know the limit
like you dont know what limits are?
tardy epoch
3/2 and 1/(n+1) will not be cancelled out <@!261933205387477002>
or you don't know how to take the limit of this
timid silo
like you dont know what limits are?
this
so what do i do
u there?
@tardy epoch (sorry to ping)
raven spire
**kaniii**
?
raven spire
so,using partial fraction decomposition, I was able to write every term as $\fra...
?
Does that decomposition help you in the slightest?
timid silo
yes?
raven spire
then
can you see the terms cancelling?
for some n, there exists a n - 1, and a n + 1, so 1/(n+1) + 1/(n - 1) would be present as long as n > 2, and those terms will cancel -2/n 
here's a better decomposition that let's you see the telescoping series
$\frac{2}{n(n+1)(n+2)} = \frac{1}{n(n+1)} - \frac{1}{(n+1)(n+2)}$
warm shaleBOT
raven spire
now, I believe you can follow up and hope you'll remember this decomposition
timid silo
can you see the terms cancelling?
yes
timid silo
3/2 and 1/(n+1) will not be cancelled out <@!261933205387477002>
and trhis is what i got
timid silo
now, I believe you can follow up and hope you'll **remember** this decomposition
ok
but that still doesnt help me find the sum
royal basin
when n goes to infinity, what happens with 1/(n+1)?
timid silo
when n goes to infinity, what happens with 1/(n+1)?
get smaller?
approaches 0 i think
so we can say the sum is 3/2
royal basin
yes, exactly. it approaches 0
tardy epoch
3/2 and 1/(n+1) will not be cancelled out <@!261933205387477002>
did you do this wrong?
craggy pagoda
obtuse pebbleBOT
craggy pagoda
how did they get 25/4
nocturne minnow
mint tide
Currently learning about MMS (Manufactured Solutions) and having a bit of trouble figuring out where to start for this problem. If someone could help me that'd be great!
obtuse pebbleBOT
Currently learning about MMS (Manufactured Solutions) and having a bit of troubl...
obtuse pebbleBOT
@mint tide Has your question been resolved?
mint tide
.close
woven basin
obtuse pebbleBOT
leaden ibex
Do you know how to calculate the cross-product?
woven basin
yes
leaden ibex
Well then calculate (1,3,-1) x (2,-1,3)
And you'll get (8,-5,-7)
numpy agrees with the text
silk condor
Is it not (-1 * 2) - (1 * 3)?
silk condor
Happens sometimes
obtuse pebbleBOT
warm shaleBOT
Yeetus
compact shadow
Doesn’t make any sense
Looks like E is a subset of X right?
But S is a subset of P(X)
Oh nvm
I didn’t see “collection of”…
Oh so you need to prove two things
Proving first, that S’=collection of subsets having that form, is a sigma algebra containing Y and {Z}
Next proving any sigma algebra T containing Y and {Z}, S’ is contained in T
Proving those two things I think
The second is trivial right
That leaves us the first one
Proving S’ is a sigma algebra
No
You only need to prove S’ is a sigma algebra now
Originally You want to prove that S’ is the minimal sigma algebra generated by Y union {Z}, which is equivalent to it being the minimal sigma algebra containing Y union {Z} , which is equivalent to it being the intersection of all sigma algebra containing Y union {Z}
compact shadow
Proving first, that S’=collection of subsets having that form, is a sigma algebr...
That’s why you need to prove the following two things, and the second one is trivial
Doing with you
No something missed
Actually three things I put the first two together
1, S’ contains Y union {Z}
2, S’ is a sigma algebra
3, for any sigma algebra T containing Y union {Z}, T contains S’
And 3 is trivial
Maybe 1 is also trivial
Yeah 1 is also trivial
Any A from Y , A=AnZ union An Z^c
And Z=XnZ union empty n Z^c
Any non empty sigma algebra of X must contain X and empty
Yeah only 2 left
Yeah that’s the only thing left to be done
We do it by definition I think
Just proving it satisfies what a sigma algebra should satisfy
Complementation , countable union or intersection
?
You need to prove that the complementation of AnZ union BnZ^c must also be of this form
By definition
Yeah
By definition
Idk, but
It’s the only thing left cause I found
Countable union is easy
Countable union of (A_k nZ)U(B_k n Z^c) is (AnZ)U(BnZ^c) where A / B is the countable union of A_k / B_k respectively
So only complementation left
Using algebra now
AZ+B(1+Z)+AZB(1+Z)+1=AZ+B+BZ+AZB+AZB+1=AZ+BZ+B+1 and I need to write it as some EZ+F(Z+1)+EFZ(Z+1)=EZ+FZ+F
No a union b is a+b+ab
Complement of a = 1+a
Nvm I still need to translate back into set theory anyway
I got it
It is (A^c n Z)U(B^c n Z^c)
Left hand side: AZ+BZ+B+1 , this right hand side: (1+A)Z+(1+B)(1+Z)+(1+A)(1+B)Z(1+Z)=(1+A)Z+(1+B)(1+Z)=Z+AZ+1+B+Z+BZ=AZ+BZ+B+1
They are equal
Left hand side I mean
compact shadow
AZ+B(1+Z)+AZB(1+Z)+1=AZ+B+BZ+AZB+AZB+1=AZ+BZ+B+1 and I need to write it as some ...
This
So it’s done
Jacobson basic algebra volume 1 chapter 8 has it though, one to one correspondence between Boolean rings and Boolean lattices
Anyway you can check it within set theory
obtuse pebbleBOT
@warped fulcrum Has your question been resolved?
compact shadow
Maybe try venn diagram
alpine heath
dont need work for this answer, but would be appreciated
or an explanantion so i can create work for it
alpine heath
nah
obtuse pebbleBOT
leaden ibex
ohok
balmy mortar
I don't think that's the issue.
leaden ibex
First you need to consider whether 8 points on a quiz/exam/test are worth breaching academic ethics to you
alpine heath
u right my b
balmy mortar
looks diophantine -
reverse euclidean algorithm?
If you havent heard of it, look up how to do it
shrewd quartz
Can anyone help guide me regarding the following question: https://math.stackexchange.com/questions/4389756/identify-a-region-given-its-restraint
obtuse pebbleBOT
Can anyone help guide me regarding the following question: https://math.stackexc...
@shrewd quartz Has your question been resolved?
long sinew
I don't think that's the issue.
Lmao he took out the 8 points as if now it's not a quiz anymore
shrewd quartz
.close
long sinew
obtuse pebbleBOT
long sinew
Ball lands means we set this equal to zero and then solve for d?
junior inlet
yea, just find root of this thing
long sinew
yea, just find root of this thing
Oh I was overthinking this a lot...
I got 0??
Oh my dumbass should have known, since theres no constant...
.close
long sinew
obtuse pebbleBOT
forest sinew
what u tried
long sinew
what u tried
The problem is, I don't understand any of it, "Assume that the ratio between the populations was constant"
forest sinew
yea
long sinew
How does that help me
forest sinew
you want to envision in your mind
damn i was hoping i could find a gif
u ever see a cell split,
so there are now 2, where there were one
then both of those split, so now theres 4
gilded needle
It's poorly worded, but it means, assume that the ratio (population in year N+1 / population in year N) is the same for any N
gilded needle
i.e. that the population grows by the same percentage each year
long sinew
It's not?
forest sinew
no
gilded needle
No, it'll be exponential
forest sinew
its exponential
forest sinew
you need two things in your head
first you need the image of cells growing in number by splitting
and second u need to know that two points define an exponential curve
long sinew
and second u need to know that two points define an exponential curve
That I did not know sadly
gilded needle
They actually tell you what to do, write an equation of the form $b = ar^n$. Here a = the population in 1997, b = the population in 2017, n = the number of years between 1997 and 2017, r = the unknown rate you want to compute
warm shaleBOT
OurBelovedBungo
forest sinew
they are given in your formula
you need an initial population and a ratio, its equivalent to having two points
long sinew
7.52B = 5.88B(r^n)?
gilded needle
where n = ?
gilded needle
Those two populations are how many years apart?
gilded needle
(btw, I had to google because I didn't believe the population was below 6 billion in 1997, it's insane how fast it grows..)
Yep, just need to solve that equation for r
long sinew
(btw, I had to google because I didn't believe the population was below 6 billio...
Yes
7.52/5.88 = r^20
twentieth root???
gilded needle
Haha, yes
long sinew
Gross
Well i guess the question only asks for a formula...
Do you have more of these types of problems so I can practice?
long sinew
I see, it parenthesized incorrectly
whoops
gilded needle
had to add my own brackets
Pretty scary what 1.2% growth can do when the numbers are that huge
gilded needle
Me? I could probably make one up but do you have a textbook? There should be more in there..
long sinew
Me? I could probably make one up but do you have a textbook? There should be mor...
No I don't, sadly
This seems a bit too easy, but maybe I missed something
gilded needle
Try look for exponential growth (or exponential decay) practice problems
long sinew
Try look for exponential growth (or exponential decay) practice problems
Alright
.close
obtuse pebbleBOT
long sinew
Thanks btw
obtuse pebbleBOT
Available help channel!
storm ocean
obtuse pebbleBOT
@nova marsh Has your question been resolved?
compact shadow
yes from the fact that j th column of AB is Σb_ij (i th column of A) also i th row of AB=Σa_ij (j th row of B)
(Sum in terms of i / j respectively )
compact shadow
Okay. i and i’ row of A are equal, then i and i’ column of A^T are equal then i, i’ column of B^TA^T are equal, which are i ,i’ column of (AB)^T which are i, i’ row of AB
Don’t understand your symbols
You mean i th row of A?
[A]_i
What do you mean line , row?
Then it’s wrong
It should be Σa_ij [b_j]
j from 1 to n
thorn kite
Was in here earlier but couldn’t find the chat
obtuse pebbleBOT
Was in here earlier but couldn’t find the chat
thorn kite
The answer is B
cedar lichen
Am I mistaken in saying the force of gravity along the slope is F_g *sin(34), not cos(34)?
thorn kite
That’s what the other guy said but I didn’t understand
Your saying F parallel should be sine?
cedar lichen
Yes. Look at the right triangle you drew. We want the side parallel to the slope. The top angle can be found to be 34, same as the incline. Then you just use trig
cedar lichen
Yes
zenith nymph
How come
x = 4y
6y + x/2 = 212 isnt right?
obtuse pebbleBOT
How come
x = 4y
6y + x/2 = 212 isnt right?
cedar lichen
Looks right to me
zenith nymph
Oh ok cuz the answers on google said it was 6x + y/2 = 212 so i was confused
cedar lichen
They just used different letters. y = 4x and 6x + y/2 = 212. They're identical in meaning
zenith nymph
Ohh
but the way they solved it they did some let statement saying x was the bigger number and y was the smaller number
but when it said "six times the smaller number" they put 6x
obtuse pebbleBOT
Available help channel!
wicked epoch
obtuse pebbleBOT
@wicked epoch Has your question been resolved?
obtuse pebbleBOT
@wicked epoch Has your question been resolved?
wicked epoch
.close
digital fjord
obtuse pebbleBOT
digital fjord
hi for 9f)
Could someone tell me what im doing wrong
I wrote the derivative of f)
and got
(x^1/2) / (x^1/3)
and subsituted 1
and i got the wrong answer so
<@&286206848099549185>
i tooked derivative to help me find m
high lily
show work for your derivative
obtuse pebbleBOT
@digital fjord Has your question been resolved?
wicked bolt
you could split it up as $y=\frac{\sqrt{x}}{\sqrt[3]{x}}-\frac{2}{\sqrt[3]{x}}$
warm shaleBOT
worm
wicked bolt
try doing the derivative again
also you can't take the derivative of the numerator and the denominator separately you have to either combine it or use the quotient rule
digital fjord
ohh okok
west stratus
you need to know X ?
digital fjord
i wanted to know x to find m
dense dagger
But then I can't get into here
obtuse pebbleBOT
But then I can't get into here
dense dagger
I don't know which inequality i have to use in order to prove part a of the question
obtuse pebbleBOT
@dense dagger Has your question been resolved?
dense dagger
<@&286206848099549185>
twilit loom
I don't know which inequality i have to use in order to prove part a of the ques...
the mean value theorem is useful here
dense dagger
So I take derivatives of arctanu/u w.r.t. u?
twilit loom
hm, I was thinking just using the MVT for arctan(u)
royal solar
a = arctan(u)
da/du = lhs
da/du < a/tan(a) < 1
that’s just first thoughts and stuff?
do you reckon we could prove that tho?
idk
royal solar
the mean value theorem is useful here
ahhh
twilit loom
for some $a\in(0,u)$ $$\frac{\arctan{u}-\arctan{0}}{u-0}=\frac{1}{1+a^2}$$
warm shaleBOT
dino
obtuse pebbleBOT
@dense dagger Has your question been resolved?
twilit loom
@dense dagger how's it going?
low fossil
Just take it =0 is ok
low fossil
dense dagger
Ok let me try try
obtuse pebbleBOT
@dense dagger Has your question been resolved?
obtuse pebbleBOT
@dense dagger Has your question been resolved?
dense dagger
I think I got stucked
So my argument is as follows
Let f(x)=arctan(x) be a cont. function on (0,u)
f'(x)=1/(x^2+1)
And hence f'(a)=1/(a^2+1)
For a>0
Then by LMVT,
There exists some real number a belongs to (0,u) such that (f(u)-f(0))/(u-0)=f'(a)
And we have: (arctan(u))/u=1/(a^2+1)
Since 1/(x^2+1) is positive for all x belongs to real, then f(x) is strictly increasing
dense dagger
But then I can't get into here
Can anyone tell me how to obtain this?