and yea if t=0, dy/dx just equals 0
#help-0
tender anchor
no worries thanks for ur help
sharp sail
you can ask for help for part b though
tender anchor
i figured it out
sharp sail
great
tender anchor
had to make it so denominator was = to 0 so its undefined so i made 3t^2=243 and solved for ts -9 and 9
sharp sail
nice
worldly ermine
help me with the second one, I tried conjugate multiplication but It kinda hard since I cant get 1 - cos^2
tacit arch
help me with the second one, I tried conjugate multiplication but It kinda hard ...
!show
lone heartBOT
help me with the second one, I tried conjugate multiplication but It kinda hard ...
Show your work, and if possible, explain where you are stuck.
@worldly ermine Has your question been resolved?
meager terrace
lone heartBOT
meager terrace
Hello, this is a calculus 1 course and we are learning displacement functions, velocity and accleration. I have provided one definition for the -35km/hr but I'm sure of what a second would be. Anyone have any ideas?
tight locust
this seems unclear
there's only one definition of velocity, not too sure what they want you to do here
meager terrace
i know it makes no sense
maybe it's the accleration ? like it's velocity is slowing with respect to time?
but accleration would be time squared. i think
teachers give no help because it's an online class as well
austere lagoon
The negative sign indicates that the vehicle's direction is opposite to the reference direction. the vehicle is moving in the opposite direction of the positive direction
both seem similar
meager terrace
Yeah, I provided that as one of the two ways it can be interpreted.
or maybe my explanation is actually two
back towards the origin and a movement in the negative direction (opposite of ref)
austere lagoon
its certain that the only thing it means that its displacement is decreasing
u just gotta say that in two diff ways
idk what that
meager terrace
hmm, okay. thanks for the help.
lilac condor
maybe they want m/s?
lone heartBOT
Available help channel!
scenic wing
are the two triangles produced attached to each other do they exist at the same time or do they exist separately
lone heartBOT
are the two triangles produced attached to each other do they exist at the same ...
@scenic wing Has your question been resolved?
fair cobalt
Basically there's two ways to build a triangle with those measurements. There's not necessarily any connection, it's just two distinct possibilities.
Either shape works.
thick lynx
Show that for any functions $u, v$ and $w$, $u \circ (v \circ w) = (u \circ v) \circ w$.
thick lynx
u(vow) = u(v(w))
u(v)ow = u(v(w))
Yes
wind flax
just use your text book's definitions if you want to make it more rigorous
worn fox
writing something like u((v(w))) doesn't really make sense, you want to just evaluate each at a general point in the domain, you will essentially get the same thing but just makes sense when applied to an element
thick lynx
writing something like u((v(w))) doesn't really make sense, you want to just eva...
What do you mean with "just makes sense when applied to en element"?
worn fox
like you want to write what $(u \circ (v \circ w))(x)$ means
ocean sealBOT
ΣΑC
worn fox
$u \circ (v \circ w)$ is the name of a function and so if you want to see if its equal to another function you have to see what it does to things in its domain
ocean sealBOT
ΣΑC
thick lynx
$u \circ (v \circ w)$ is the name of a function and so if you want to see if its...
How would you show that it does the same things as $(u \circ v) \circ w$?
worn fox
**ΣΑC**
well what is this
you've basically already done it you just left out the application to an element
worn fox
you need to be applying it to an element from the start
worn fox
right and then carry on by undoing what you did but in a different order so that you get the other composition
thick lynx
right and then carry on by undoing what you did but in a different order so that...
$u(v(w(x))) = (u(v) \circ w)(x) = ((u \circ v) \circ w)(x)$
worn fox
your first step should be $(u \circ v)(w(x))$
ocean sealBOT
ΣΑC
thick lynx
Oh
worn fox
because again writing something like u(v) in isolation doesnt make sense
thick lynx
your first step should be $(u \circ v)(w(x))$
$u(v(w(x))) = (u \circ v)(w(x)) = ((u \circ v) \circ w)(x)$
worn fox
yes exactly
hybrid ivy
hey guys,
i have this hw question that i cant seem to figure out, just wondering if i can get some help
hybrid ivy
i am unsure how to get the correct answer
raven rover
(number in degrees) * pi / 180 = (number in radians) @hybrid ivy
In other words, 180 degrees = pi radians
worn fox
it already has the pi there for you
hybrid ivy
the website says this is the asnswer
and idk how it got this
since im getting a different asnswer
worn fox
use the formula, and enter the part without the pi
lone heartBOT
@hybrid ivy Has your question been resolved?
quick moth
lone heartBOT
quick moth
Hi can someone work me through the first steps to working this out
quick moth
Thanks
finite flax
"converse of corresponding angles postulate"
quick moth
I still don’t understand
I need someone to run me through it please
@mellow saffron
Mb
<@&286206848099549185>
finite flax
where are you stuck
quick moth
With that first step
I need someone to run me through it
Because IXL is different to what he does
finite flax
With that first step
Remember that you need the distance between the two points?
quick moth
yeah
finite flax
they told you it is equal to the radius of the circle
so you should have a tool
that draws arcs or circles
finite flax
ok, draw a line from the circle center to one of the points
that will be the transversal
quick moth
Meaning it’s wrong
finite flax
undo all that
quick moth
Ok
finite flax
show me.
quick moth
Thanks bro
finite flax
you didn't start over
quick moth
I did
finite flax
why is there so much junk here
finite flax
ok
finite flax
ok, same thing
finite flax
STOP IT
quick moth
Mb
finite flax
if you aren't going to listen I will leave
quick moth
finite flax
I have better things to do
quick moth
Ok
finite flax
alright, that is your "transversal" line.
quick moth
Ok
finite flax
now, construct an arc (or circle, if you must) centered at point C and passing through point B. This arc (or circle) will have the same radius as Circle(A)
finite flax
yes
quick moth
Ok
finite flax
...
in fact, this step might be extraneous since you begin with a circle whose radius is equal to the distance between B and C
anway
quick moth
…
finite flax
draw a point where line AC intersects circle A
quick moth
How?
finite flax
no
that is where AC intersects circle C
anyone could make that mistake
also, we don't need the first circle, sorry
start over
quick moth
Ok
finite flax
this only needs two steps I think
finite flax
when you start over, draw the transversal again
quick moth
Ok
finite flax
the line.
then, where the line intersects the only circle you have, copy the circle at that location. Draw a point there first if you are required to.
show me when you are done.
quick moth
finite flax
good
quick moth
Ok
finite flax
now draw the point D where the circles interect ABOVE line AC
then draw a line through AD.
finite flax
I kinda see where the problem happened
quick moth
Ok
finite flax
start over, draw the transveral again
finite flax
ok, construct a copy of this circle where AC intersects circle A, in the LOWER LEFT area
quick moth
Done
finite flax
why is it so much bigger
quick moth
Idk
quick moth
Just show picture or something
I can’t vc
Otherwise I’ll get the beating of my life
quick moth
Yeah….
finite flax
I'll try to post the steps then
quick moth
Thanks
finite flax
I only have a quarter and an envelope so I found the centers of these circles
Which are just traces of the quarter
Ok, let's draw transversal AC
quick moth
Ok
quick moth
Yeah
quick moth
K
finite flax
Also point E:
Now I need to sweep an arc centered at E with radius BD through circle A...using a compas...which I don't have.
BUT
I can copy that radius length BD to my envelope:
And construct point F:
...and draw a line through AF
quick moth
This wouldn’t work in IXL and I can’t tell u why because I don’t know
So I’m just gonna leave this one
hollow jackal
what tools do you have?
finite flax
finite flax
So I’m just gonna leave this one
Cool. Totally worth being late to family dinner. You do have the tools. You just need points, lines, circles
quick moth
Cool. Totally worth being late to family dinner. You do have the tools. You just...
?
Thanks I guess?
U could of left at any time ?
Your not forced against your will to be here
hollow jackal
he's saying he went through the effort to show you how to do it and you still give up
finite flax
he's saying he went through the effort to show you how to do it and you still gi...
tbh my picture of a "good teacher" never tells their students when they are sacrificing but being real...I would have left late anyway
@quick moth Constructions are either the most fun or least fun you will have in math but if you can understand even a little bit of it you will be in good shape for two very challenging units.
so please don't give up.
lone heartBOT
@quick moth Has your question been resolved?
lone heartBOT
lone heartBOT
Available help channel!
brazen berry
Lin Alg 2:
lone heartBOT
Lin Alg 2:
brazen berry
I'm wondering what i did wrong for matrix B?
Is it because one of the entries is 0?
I worked it out and this solution should work but maybe im forgetting some theory
i'll work on a different solution while i wait
hmmmm
another solution i tried didnt work upon entry but works when i work the math out
lone heartBOT
@brazen berry Has your question been resolved?
brazen berry
i think there may be something wrong with the question.
prof posted earlier saying an error with it was fixed but who knows
scuffed ahh class
our syllabus link redirects to assignment 2 💀
lone heartBOT
@brazen berry Has your question been resolved?
lone heartBOT
@brazen berry Has your question been resolved?
ocean sealBOT
echo socket
,w characteristic polynomial of {{-6, 0}, {43/8, 17/8}}
ocean sealBOT
echo socket
Yeah, doesn't happen to match up though
Oh, wait, phi doesn't get you the characteristic polynomial here, right
My bad
Wait let me check
echo socket
Lin Alg 2:
This should have worked, yes
brazen berry
This should have worked, yes
.close
grave eagle
yo
tight locust
how embarassing!
ocean sealBOT
dios
grave eagle
and did not flip the sign!!!!!!
dont you flip the sign when you divide/mutliply?!??!?!
(both sides)(
grave eagle
yes sir
grave eagle
ok i see what you mean here
tight locust
not quite
now let's take the original statement 2 < 3
this time, let's multiply both sides by -2
then we get -4 < -6
grave eagle
ohhhh
tight locust
is that true?
tight locust
yes. you only flip the direction of the inequality when you mulitply or divide by a negative number
viscid coral
How do I find inverse cos of -0.2 on casio calculator? I typed in cos -1 (-0.2) by pressing shift and cos then -0.2 but it gave me 101.53... instead of the value i want 1.7721...
viscid coral
I got it!
vapid steppe
does $\lim_{x\to \infty}{x-x}$ equal zero or indeterminate
lone heartBOT
does $\lim_{x\to \infty}{x-x}$ equal zero or indeterminate
ocean sealBOT
jashxdlol
vapid steppe
cuz if u do direct sub u get ∞-∞
tight locust
well why don't you try simplifying
naive valley
it equals zero, because x-x equals zero for every x
so you are seeking $\lim_{x \to \infty} 0$
ocean sealBOT
Bungo
vapid steppe
but like if you split up the limit you get lim as x→∞ of x minus limit as x→∞ of x
tight locust
if you want a cool limit that involves a subtraction then try taking the limiting difference between the integral of the logarithm and the trapezoidal approximation
naive valley
cuz if u do direct sub u get ∞-∞
you can't do this substitution, because the limit of a sum only equals the sum of the limits if those limits exist
vapid steppe
you can't do this substitution, because the limit of a sum only equals the sum o...
oh ok
.close
signal violet
how do i do this?
lone heartBOT
how do i do this?
nimble latch
fathom harness
y = a(x-k)^(1/n)
0 = a(1-k)^(1/n)
=> k = 1
y = a(x-1)^(1/n)
-3 = a(-1)^(1/n)
=> a = 3
y = 3(x-1)^(1/n)
6 = 3(8)^(1/n)
2 = 8^(1/n)
@signal violet hey done
fathom harness
I'm starting with assuming the general equation, y = a(x-k)^(1/n)
Then using the different points and plugging in their values, I find their possible values.
clever coral
Hii can anyone explain me what's rational
fathom harness
Hii can anyone explain me what's rational
In maths, anything which can be expressed in the form of p/q (a fraction).
clever coral
In maths, anything which can be expressed in the form of p/q (a fraction).
So like rational can be any numbers?
fathom harness
p and q have to be integers
So like
1 is rational because it can be written as 1/1
69 is too
Because 69/1 or 138/2
But
A number like √2 is not rational
You can try any combination of p/q
But it's value will never exactly be √2
fathom harness
So like rational can be any numbers?
π is also irrational
clever coral
You can try any combination of p/q
Can u show example
fathom harness
Can u show example
For which number?
clever coral
For which number?
The combination one
fathom harness
I mean combination for which number? π?
clever coral
"You can try any combination of p/q" like try any combination what does it mean😭
fathom harness
Oh lol
Like let's say someone gave you the number 6.999999...
... means the 9 keeps on repeating
And asks you to find whether this number is rational or not
clever coral
Ohhh
fathom harness
Wait 1 moment
clever coral
Sure
fathom harness
That was a bad example 💀
Let's take a new one
Let's say someone gives you the number 3.25
And asks you to figure out whether this is rational or not
clever coral
Yeah
clever coral
Idk??? Aren't you supposed to tell me
clever coral
I think it's rational ig
fathom harness
Lol
Actually its a good reasoning
Any number with finite number of digits is rational
Cuz
3.25 is the same as 325/100
We can see that it is in the form of p/q
So it must be rational
What about numbers with inifnite digits tho
Like
3.333333...
... means the decimal part keeps on repeating
here we can do a little bit of guess work
10/2 is 5
Nope too big
@clever coral I actually have to go now
Got an urgent call
I'll just finish this one and then go
ocean sealBOT
Result:
2.5
Result:
3.3333333333333
Eeee it worked
So we know that 3.33333... is rational as well
Generally
Any number with finite number of digits
Or
A repeating pattern
Is rational
3.333... has a repeating pattern
Even 3.5656565656... would be rational
Make a guess yourself how
But something like 1.262861718816868198162.... would be irrational
There's no repeating pattern
Obviously the math behind how to know whether a number is rational or not
Is a not just guesswork
But guesswork is a good step if you are a beginner
clever coral
Hii my wifi die for a second
clever coral
Alr byee tyy
fathom harness
Ttyl!
@signal violet did you understand my answer?
If not, then :p
Ping the helpers role
fathom harness
I gave you all the steps tho
rotund crater
why is zero
lone heartBOT
why is zero
vale wigeon
Since ln(y) is constant with respect to x,
clever coral
Any number with finite number of digits
So like if u want to know if the number is rational or not you divide them with what numbers? Or you just divide with the numbers that's in question?
thorny patio
So like if u want to know if the number is rational or not you divide them with ...
Rational Number is defined as
Integer divided by non-zero Integer
It so happens this includes all numbers whose decimal forms either contain a finite number of digits or form a repeating pattern with their digits
A better question is how to find or figure out something is irrational
lone heartBOT
@rotund crater Has your question been resolved?
empty spade
lone heartBOT
empty spade
,rotate
ocean sealBOT
empty spade
so I got these 6 equations by labeling the unknowns in the pic but idk what to do from there
quick moth
he's saying he went through the effort to show you how to do it and you still gi...
It’s literally impossible for me to do it like I stated I tried for 30mins so don’t give me all your back chat
lone heartBOT
@empty spade Has your question been resolved?
empty spade
<@&286206848099549185>
after adding the first five equations i get
2000 + 5c + 2d + 2e + a + b = 3(a + b + d + e + 1000)
and using the sixth equation
2000 + 5c + 2d + 2e + a + b = 18c
which can be further simplified into
2000 + 2d + 2e + a + b = 13c
correct me if im wrong
fading comet
<@&286206848099549185>
lone heartBOT
@empty spade Has your question been resolved?
west girder
lone heartBOT
west girder
Can I have a hint
austere lagoon
lone heartBOT
@west girder Has your question been resolved?
cyan valve
how do i turn this into a graph on google sheets
lone heartBOT
how do i turn this into a graph on google sheets
cyan valve
pls help
shut pewter
Just select the data you want to create a graph with
It's called chart in google sheets
lone heartBOT
@cyan valve Has your question been resolved?
cyan valve
uhh
hold on
so i drew my results on a line graph
and plotted the avgs
and this is what it looks lole
do i draw the trend line
@shut pewter
i changed the incrmements
i went up by 2
shut pewter
I mean if your research is all about trends
You can use any charts, but if this is a trend for the number of plies and number of water absorbed, then you can use line graph
lone heartBOT
@cyan valve Has your question been resolved?
alpine sable
lone heartBOT
alpine sable
Im stuck
lone heartBOT
@alpine sable Has your question been resolved?
alpine sable
<@&286206848099549185>
lone heartBOT
@alpine sable Has your question been resolved?
ocean sealBOT
kebesque
alpine sable
could someone explain the process to answer
lone heartBOT
could someone explain the process to answer
mellow grail
could someone explain the process to answer
take 3 common and do comparision of coefficients
lone heartBOT
@alpine sable Has your question been resolved?
mental hearth
lone heartBOT
echo socket
Well, could you explain what |x| denotes?
mental hearth
Well, could you explain what |x| denotes?
denotes mod
so it will return the positive value of input
on removing the sign it may give a +ve or -ve
echo socket
Right, so |x| = x if x is positive
mental hearth
inside
echo socket
And |x| = -x if x is negative
echo socket
These two cases insure |x| is always nonnegative
mental hearth
mod of -2 is 2 right
mental hearth
And |x| = -x if x is negative
then how does this make sense
echo socket
then how does this make sense
If x is negative, then -x is positive, right?
warped topaz
lone heartBOT
https://i.imgur.com/Q8kBQsd.png
warped topaz
I know how to solve these kinds of questions once I have an idea of how to rearrange it but im not really sure how to rearrange this one
I dont know any formulas for cos or sin to the 3rd power, and if I make it sin(x)sin^2(x) or something like that I will have 1 sin or cos left over, so I wont be ale to do u-substitution
(sin(x)) (sin^2(x)) (cos^3(x))
(sin(x)) (1 - cos^2(x)) (cos^3(x))
But then I still have 1 sin right?
So I cant substitute u
Nvm I got it
Thank you!
❤️
.close
tender anchor
I gotta use L’hopital rule for
lim x->1 (x^3-1)/(x-1)
lone heartBOT
I gotta use L’hopital rule for
lim x->1 (x^3-1)/(x-1)
tawny crown
Not necessarily
tender anchor
does it start out like 3x^2 / 1
tawny crown
You can factorise x³-1
tender anchor
o my prof wants me to do it using the rule it’s the unit we’re learning
tender anchor
u/v?
tender anchor
yeah
upbeat hornet
does it start out like 3x^2 / 1
Can you finish it from here?
tender anchor
yeah it’s 3
upbeat hornet
Has your question been resolved?
tender anchor
yes thanks guys
upbeat hornet
.close
lone heartBOT
lone heartBOT
Available help channel!
fallen verge
Testing something, i do not need help
fallen verge
Interesting
echo socket
!status
lone heartBOT
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
D:<
long solstice
lol
worn fox
daring totem
lone heartBOT
Available help channel!
bronze ridge
I'm doing some repetition questions in linear algebra and I have some minor questions regarding injective transformation matrices. Namely, what is the preferred way to check if a given transformation matrix A is injective or not? The method I've used involves solving for Ax = 0 and verifying that the only allowed solution is x = 0 (0 vector that is). However, by doing this, am I really verifying that the transformation is injective? Because checking for Ax = 0 only checks the trivial solution, what about the other ones?
I guess the bulk of the question is if the method I described works reliably and in that case why. Also, if there exists alternate ways to check if a transformation is injective.
vale wigeon
However, by doing this, am I really verifying that the transformation is injective? Because checking for Ax = 0 only checks the trivial solution, what about the other ones?
vale wigeon
Av = Aw is equivalent to A(v-w) = 0
waxen osprey
ker=0 means a linear map is injective
vale wigeon
it is an easy exercise to verify that a map is indeed injective if and only if its kernel is trivial
bronze ridge
So in other words, if the null space consists of only the null vector, then the transformation is injective? That's the way I understood it at least
waxen osprey
yes if the transformation is linear
pine fractal
how can i prove: degree of p(x)=n, p(x)*(p(x)+2)=0 at most (n+2) root , where p(x) belong Z[x] integral coefficient
bronze ridge
.close
alpine sable
lone heartBOT
alpine sable
Could anyone please confirm if I did this correctly? I'd appreciate it.
Q is to find the line integral
lone heartBOT
@alpine sable Has your question been resolved?
alpine sable
<@&286206848099549185>
zealous lichen
I think you parametrized the curve in the opposite direction
alpine sable
Oh I was confused about that, should I use pi/2 to 0? (I thought we go from min to max value)
zealous lichen
or just multiply by -1 not to do all the work again
alpine sable
Oh okay will do that, thank you so much!!
zealous lichen
wait it shouldn't be negative on line 2
zealous lichen
-cost(sint)^3 on the circular curve
alpine sable
.close
alpine sable
Hi, I have to find and sketch the domain of the given function
Function is : 1/(x+1) + 1/(y-1) + 1/(x+y-z)
alpine sable
I have found the domain which is x+1=/0
y-1=/0
x+y-z=/0
But I'm not sure how to sketch this
Can anyone please guide me?
Should I make like a 3D plot with these planes
And shade the outer area?
But that's very complicated
tardy stag
sketching something in 3d seems very complicated
especially something like that which is basically the whole universe minus some planes
tardy stag
are you sure that this is a function of 3 inputs? and not two?
your domain seems to be right
alpine sable
Yes it is f(x,y,z)= that stuff
tardy stag
huh
i guess i would sketch the planes that cut out the domain
and just say "everything except these"
alpine sable
f(x,y,z) = 1/(x+1) + 1/(y-1) + 1/(x+y-z)
Oh
Okay, I'll do the same then, thank you so much!
lone heartBOT
Available help channel!
left nimbus
How can the derivative of an integral be equal to the function? For example, if v(t) describes the velocity of an object with respect to time, then shouldn’t the rate that the area increases by be equal to v(t)*dt?
left nimbus
Wouldn’t it be better to say that the derivative of an integral is proportional to the function instead of equal?
echo socket
By definition they are equal
left nimbus
So they just… defined them that way?
echo socket
Unless you are defining with the Riemann sum, then it's not immediate
left nimbus
What do you mean?
I’ve seen the mean value theorem proof of the ftc, it’s just weird how the rigorous definition doesn’t really match with the concept to me
echo socket
Proof using ftc you mean?
There's no proof of FTC 
, it's a set of axioms
Which definition are you talking about btw
peak bough
Unless you are defining with the Riemann sum, then it's not immediate
Why would you not be defining integrals that way 
I can get behind saying the derivative of an antiderivative of a function is equal to the function per definition. But I can't see why you would define integrals like that. Seems too artificial in my opinion 
echo socket
Then I need to look into how the connection between antiderivatives and areas under the curves was discovered
Although yeah Riemann sum is more natural as a definition
left nimbus
I was talking about the first one, where F(x)=integral from a to x of f(t)dt (sorry I don’t know TeX)
But really I’m trying to figure out how d/dx(integral of f(t)dt) = f(x)
echo socket
Uh there's a video of 3b1b explaining that in an intuitive and visual way
left nimbus
Yeah I saw that
peak bough
But really I’m trying to figure out how d/dx(integral of f(t)dt) = f(x)
The only proof I'm familiar with requires a bit of Analysis knowledge I don't know if there exists simpler proofs that still remain fruitful. There probably does but I'm not familiar with them
left nimbus
Ok, it was just weird conceptually to me
Integral of a derivative makes more intuitive sense somehow
ig it’s easier since I can think about adding up velocity * time over an interval
peak bough
I also saw a "proof" a few years back using the squeeze theorem if an area function for a given function exists (the existence is not part of the "proof", sadly), then its derivative is the given function. The proof didn't really connect it with any formal integral-definition, so the proof moreso gave a connection between direct area and the derivative of an area function. It also heavily relied on the given function to be strictly monotone, so it's very restrictive.
arctic salmon
How do i justify that (-1)ⁿ doesn't have a limit?
lone heartBOT
How do i justify that (-1)ⁿ doesn't have a limit?
worn fox
find two convergent subsequences that converge to different values
alpine sable
How do i justify that (-1)ⁿ doesn't have a limit?
Boundedness and monotonicity are necessary here
Verify the boundedness and monotonicity of (-1)^n
lone heartBOT
arctic salmon
Thanks for the help ! I appreciate it !
lone heartBOT
Available help channel!
limpid shore
Let $m$ be any good old integer. Let $p<0$, then if $m2^p$ and $(m+1)2^p$ are not integers, how come $N=\lfloor m2^p \rfloor=\lfloor (m+1)2^p\rfloor $?
ocean sealBOT
sunside_
vale wigeon
what is p
limpid shore
sorry, $p$ is also an integer
ocean sealBOT
sunside_
vale wigeon
ah.
was about to present a counterexample
so letting n = -p for convenience
you're asking why m/2^n and (m+1)/2^n don't have an integer between them
limpid shore
I believe that's correct
vale wigeon
i believe it might not even matter that the denominators are powers of two here lol
but i have taken up the letter n. oops
m/r < k < (m+1)/r would mean m < kr < m+1 which cannot happen for m, r, k ∈ Z
lone heartBOT
lone heartBOT
Available help channel!
deep ember
lone heartBOT
surreal meadow
let 3t = x
deep ember
i know this equals one using the (sin^2t + cos^2t)=1 theorem but in this case im confused as to where 3 goes
surreal meadow
sin^2x + cos^2x = 1
vale crag
you're focusing too much on the symbols
sin^2(something) + cos^2(something) = 1 for any something you put inside
even if it's 3t it doesn't matter
it's still true
deep ember
ahhh okay, so its a general rule that if it falls within that identity, it doesnt matter what value is x right
surreal meadow
the t in "sin^2t" is arbitrary on purpose
it could be t
it could be 1
it could be 3x
it could be anything
lone heartBOT
stuck mountain
hey
lone heartBOT
Available help channel!
rotund crater
how i can solve this ?
lone heartBOT
how i can solve this ?
worn fox
you would want to start by finding y'
alpine sable
You can fill in the x-coordinate to get the slope
After that you can find the b in y=ax+b because it has to go through (1,0)
lone heartBOT
@rotund crater Has your question been resolved?
blissful temple
using log tables
rotund crater
After that you can find the b in y=ax+b because it has to go through (1,0)
what do you mean whith that formula?
alpine sable
But how i can calculate ln or log without calculator it is posible?
Ln(1) is doable because 1=e^0
rotund crater
can illustrate me?
alpine sable
Wait is x-1 your answer for the question a?
rotund crater
i aplly y-y_1 = m(x - x_1)
alpine sable
Did you already find the slope of the tangent line?
alpine sable
Yes
alpine sable
I dont really know how to explain but its just the rule.
The steps for solving the equation for a tangent line to the function f(x) at point (A,B) are:
Find the derivative f’(x).
Plug x=A into the derivative.
The answer is the slope of the tangent line.
Set equation of tangent line to y=ax+b.
You already solved for a (the slope).
Now plug x=A and y=B into the equation of the slope.
Solve for b.
Now you have the equation of the tangenr line
rotund crater
always is y=ax+b?
rotund crater
thx for the help
lone heartBOT
@rotund crater Has your question been resolved?
tranquil osprey
if we have 4 casino chips, and 3 of those are prime numbers (2, 3, 5, 9)
how do we calculate the probability of getting two chips, picked randomly, and both of them being prime numbers?
lone heartBOT
@tranquil osprey Has your question been resolved?
tranquil osprey
9 is not a prime
sorry i meant to write 2 not 1
tranquil osprey
.reopen
lone heartBOT
✅
tranquil osprey
<@&286206848099549185>
foggy copper
that's selection without replacement, right?
tranquil osprey
yeah i think
foggy copper
in that case you'd want the odds of not picking a prime on the first draw times the odds of not picking a prime on the second draw
foggy copper
mathematically it makes no different
tranquil osprey
we want a pair, and we want to calculate the percentage of that pair being composed of 2 prime numbers
foggy copper
*difference
tranquil osprey
mathematically it makes no different
really?
foggy copper
you're just doing draws one and two simultaneously
foggy copper
really?
could you explain the difference between drawing two things at once and drawing one thing after another if no replacement occurs, mathematically speaking?
ig you could model all the different pairings but it really makes no difference for the question of "is either of my chips prime"
tranquil osprey
the odds of not picking a prime on the first time is 1/4, correct? that's 0.25
the odds of not picking a prime on the second draw would be 3/3 because there is nothing but primes left, so 0.25 x 1 x 100 = 25%
im trying to follow ur logic but i may have followed it wrong
foggy copper
no
if you didn't get a prime on the first then your odds of a prime on the second are 1/3
but what you want is the odds of not failing to choose a prime every time
which is 3/4, and then 2/3
do you see
tranquil osprey
which is 3/4, and then 2/3
i dont get, if we dont want to get aprime wouldnt it be 1/4 then 1/3
3/4 would be if we wanted primes
which we do
but u said "not wanting"
im confused?
foggy copper
but u said "not wanting"
i've corrected that now
we want all primes
so the odds of that are (prime on draw 1)*(prime on draw 2)
tranquil osprey
in that case you'd want the odds of not picking a prime on the first draw times ...
so should i ignore the "not" here aswell
foggy copper
yes
tranquil osprey
so the odds of that are (prime on draw 1)*(prime on draw 2)
ye ur right, chatgpt got the same answer using nCr formula
tranquil osprey
no problem
foggy copper
i should have gone to bed ages ago
tranquil osprey
so the odds of that are (prime on draw 1)*(prime on draw 2)
for cases where the set size is far too large to do something like this, i should use nCr right?
also, could you explain why say a set of size 4 has 4! ways of being arranged? i understand that it does, and i understand that we can prove it by counting ourselves manually, but i dont get it mathematically speaking
foggy copper
ye ur right, chatgpt got the same answer using nCr formula
just a word of warning about doing things that way https://fortune.com/2023/07/19/chatgpt-accuracy-stanford-study/
foggy copper
for cases where the set size is far too large to do something like this, i shoul...
yea that sounds right off the top of my head. but i haven't done serious work with probabilities for a while
tranquil osprey
just a word of warning about doing things that way https://fortune.com/2023/07/1...
yeah im aware of chatgpt's unreliability
but it got the same answer my tutor got
but i dont really trust my tutor
plus our lessons r over
foggy copper
also, could you explain why say a set of size 4 has 4! ways of being arranged? i...
in short, because you have 4 choices for your first slot, then 3 choices for your 2nd slot (bc one thing's been taken), and so on until there's only one thing left for the final slot.
which comes to 4*3*2*1 options
which is, by definition, 4!
tranquil osprey
ah okay thank you
lone heartBOT
@tranquil osprey Has your question been resolved?
bronze yew
how might I go about figuring out what percent of numbers between 0 and 96000000 have a sum greater than 66? I really dont know where to start
lone heartBOT
how might I go about figuring out what percent of numbers between 0 and 96000000...
@bronze yew Has your question been resolved?
worn fox
it would be easier to figure out the percentage that have a sum that is less than or equal to 66
bronze yew
it would be easier to figure out the percentage that have a sum that is less tha...
Hmm, wouldnt that just be 1- the percentage of what im asking? I mean id be glad to get that stat as well
worn fox
well yeah the point is knowing one tells you the other
but one is easier to calculate than the other
bronze yew
but one is easier to calculate than the other
Ok. any ideas how I might start?
worn fox
just start counting systematically
also not even sure how to parse your inital question, you're asking for a percentage of numbers "that have a sum greater than 66"
thats not a property a number can have so you need to be more clear
lone heartBOT
@bronze yew Has your question been resolved?
bronze yew
thats not a property a number can have so you need to be more clear
The sum of each digit
worn fox
okay well that wasn't clear at all
lone heartBOT
@bronze yew Has your question been resolved?
bronze yew
okay well that wasn't clear at all
12345678 would be 1+2+3+4+5+6+7+8=36
old moat
how might I go about figuring out what percent of numbers between 0 and 96000000...
It’s in combinatorics
It’s called generative functions if I’m not wrong about the translation
In discrete math
I’d help you but I’m on my phone
lone heartBOT
@bronze yew Has your question been resolved?
rose sigil
12345678 would be 1+2+3+4+5+6+7+8=36
any such number needs to have 8 digits because 9*7 < 66
if the first digit is 9, the possible numbers can be counted by staring at 95999999 and thinking about how many ways you can lower the last 7 digits while keeping the sum greater than 66. since 9+5+9+9+9+9+9+9 = 68, this can only be done by lowering one digit by 1 or not lowering any
for the rest, you can count in a similar, but slightly more complicated way working down from 89999999 and thinking about how many ways you can lower the digits by at most 4
the second case is same as the number of nonnegative integer solutions to x_1 + x_2 + ... + x_8 <= 4
perhaps you have a formula for that already
well it would be a formula for the number of nonnegative integer solutions to x_1 + x_2 + ... + x_8 = n
then you could use it a few times
lone heartBOT
@bronze yew Has your question been resolved?
spark quartz
how do i find the dimension of this subspace?
lone heartBOT
how do i find the dimension of this subspace?
spark quartz
i set up this equation: $\begin{pmatrix} w \ x \ y \ z \end{pmatrix} =
w \begin{pmatrix} 1 \ 0 \ 0 \ 0 \end{pmatrix} + x \begin{pmatrix} 0 \ 1 \ 1/3 \ 1 \end{pmatrix} + y \begin{pmatrix} 0 \ 3 \ 1 \ -3 \end{pmatrix} + z \begin{pmatrix} 0 \ 2 \ -1/3 \ 1 \end{pmatrix}$ which means the dimension should be 4, but its actually 3
ocean sealBOT
severed_toast
spark quartz
what am i doing wrong
winter light
how do i find the dimension of this subspace?
You're in R⁴, so the dimensions of the vector space is 4. Since you have 1 independent equation (try to show somehow that those two equations they give you) are actually the very same, the dim. of the subspace is 4 - 1, hence 3
mortal trellis
from where are you getting those equations
spark quartz
-x + 3y + z = 0 -> x = 3y + z
etc.
oh
theres a mistake in the second row then
theyre stil linearly independent tho
mortal trellis
still not sure where you are seeing these equations
you are playing around wildly with the signs for some reason
you only have two equations
-x+3y+z=0
2x-6y-2z=0
mortal trellis
but that doesnt end up with 4 linearly independent equations somehow
spark quartz
here a similar question
and then i set up something like this
and i tried to do the same thing here
mortal trellis
you cant keep isolating new variables for old ones
at some point you run into loops
if you already wrote x in terms of y and z then you cant next write y in terms of x and z
cause x is already written in terms of y
spark quartz
ah yeah that makes sense
hmmm
so say i rewrite x as x = 3y + z
then i have (w, 3y + z, y, z) right
mortal trellis
yes
mortal trellis
yes
spark quartz
so if i rewrite x as x = 3y + z could i rewrite w as w = z + 2y
(just a general question, not specific to this)
mortal trellis
w doesnt even appear in the equations
mortal trellis
if you had an equation w-2y-z=0 then you could rewrite w=z+2y, yes
but if you have more than 1 equations what you really should be doing is row-reducing
to spot which of the variables are free and which can be written in terms of the other oness
spark quartz
i think that's what the markscheme does here right?
mortal trellis
yes
spark quartz
it just doesnt really make sense to me, writing the equations that define the set (the vectors in the set) into a matrix and row reducing
mortal trellis
if (w,x,y,z) is in the set, then it has to satisfy the linear system of equations
so you have to solve the linear system of equations
aka write in matrix and row reduce
spark quartz
and its just an efficiency choice to write them in as rows instead of columns right?
mortal trellis
not really
that just turns out rather naturally when you write it as a system of linear equations
if you call the matrix you posted M
you want that M*(w,x,y,z)=0
writing those as rows fits perfectly with matrix-vector multiplication
spark quartz
you want that M*(w,x,y,z)=0
sorry could you explain why exactly we want this
mortal trellis
if you read what that matrix vector multiplication says, it is literally that -x+3y+z=0 (first row) and 2x-6y-2z=0 (second row)
spark quartz
yes, but why do we want it equal to 0?
mortal trellis
$\begin{pmatrix} -1 & 3 & 1 & 0 \ 2 & -6 & -2 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \end{pmatrix} \cdot \begin{pmatrix} w \ x \ y \ z \end{pmatrix} = \begin{pmatrix} -x+3y+z \ 2x-6y-2z \ 0 \ 0 \end{pmatrix} = \begin{pmatrix} 0 \ 0 \ 0 \ 0 \end{pmatrix}$
ocean sealBOT
denascite
mortal trellis
the equations in the set definition tell you that it should equal 0
spark quartz
**denascite**
ah this is clear thank you
okay yeah i get it now
thank you for your patience and time
.close
fervent ferry
hey hello!
i'm trying to understand what the prroof is trying to prove really..
fervent ferry
I'm not sure I undertstand what the "q leads to y". I mean i think i understand the general idea.
if i have many inputs, each act as a different "axis" and their sum is a linear combination. This makes so that also the solution is in the form of a linear combination
but where it explains the proof, i don't understand.
are we substituting the formula on the top with a linear combination?
or is there a clear passage that goes from
because the only way i can justify the right hand side is by substituting q.
Also is this a "proper" notation? or it's like pseudo code to illustrate a concept?
lone heartBOT
@fervent ferry Has your question been resolved?
lime sluice
lone heartBOT
lime sluice
near apex
!status
lone heartBOT
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
good start
lime sluice
then found 2 equations
stucked here
lime sluice
near apex
Alright. Can you solve them to find value of a?
near apex
should i put value of a^2 in the first
Yeah. That would work.
lime sluice
I want to know how much time a polynomial intersects x axis
lone heartBOT
I want to know how much time a polynomial intersects x axis
lime sluice
Is there any kind of theory
lime sluice
.close