you can summarize it up with " you can compose limits of continuous functions"
#help-0
slate jolt
tulip oak
Because it's really just a whole bunch of transformations applied to continuous functions
Ah fair enough
slate jolt
its the only limit law you need almost
tulip oak
They're all the same pretty much, just different operations
slate jolt
yeah
slate jolt
continuous operations
tulip oak
Thank you so much
slate jolt
but for example you cant use this to compose when theres a discontinuity
you need to be careful
sometimes you might
but its case by case
tulip oak
got it
slate jolt
but for example you cant use this to compose when theres a discontinuity
discontinuity at the point
i mean
tulip oak
👍
slate jolt
.close
scenic wing
im stuck on 332
lone heartBOT
im stuck on 332
stuck cape
332?
scenic wing
yes
wanton tusk
let u=cosθ
scenic wing
ZAMNNNNNNNNNn
lone heartBOT
@scenic wing Has your question been resolved?
crimson crag
where did I go wrong
lone heartBOT
where did I go wrong
crimson crag
tall shuttle
Seems correct
subtle birch
where did I go wrong
you didn't
storm ridge
4 dot 8 is 4*8 right?
It's all correct then
scenic wing
$-\int \frac{1}{u} du$
lone heartBOT
$-\int \frac{1}{u} du$
ocean sealBOT
putridplanet
tardy stag
yes integrate
scenic wing
idk how genius
tacit arch
Damn sassy helpee
tardy stag
the antiderivative of 1/x is ln |x|
ocean sealBOT
putridplanet
alpine sable
- C
tardy stag
absolute value
scenic wing
and if i substituted u as cosx then its $-ln(cosx)+c$
ocean sealBOT
putridplanet
tardy stag
still missing an absolute value
alpine sable
absolut value
ocean sealBOT
putridplanet
scenic wing
idk
tardy stag
try a substitution
alpine sable
yes
scenic wing
i already substituted
alpine sable
what did you get
ocean sealBOT
putridplanet
scenic wing
u=sinx
tardy stag
i already substituted
scenic wing
is that allowed
alpine sable
yes
tardy stag
of course
scenic wing
but was my work right
scenic wing
**putridplanet**
to get this .
tall topaz
Yes
scenic wing
ok v =lnu ?
tardy stag
v = ln u
tacit arch
ok v/u now what
That's not complete substitution
scenic wing
right
tacit arch
u = stuff in terms of v
tardy stag
and you need to consider the du
scenic wing
$\int\frac{v}{u} xdx$
tacit arch
Absolutely not
tardy stag
x?
ocean sealBOT
putridplanet
tacit arch
u = stuff in terms of v
du = (stuff with v) * dv
scenic wing
du = 1/u dv
tacit arch
x?
.
Meaning you shouldn't have any x
You only want to go forwards in substitution, not backwards
scenic wing
$\int\frac{v}{u} udv$
ocean sealBOT
putridplanet
ocean sealBOT
putridplanet
scenic wing
Hello ???
alpine sable
**putridplanet**
no
tall topaz
**putridplanet**
That is correct but remember what v is you have to sub back in v =ln u
scenic wing
That is correct but remember what v is you have to sub back in v =ln u
so $\frac{lnsinx^2}{2}+c$
tall topaz
and what is u
tall topaz
beim?
ocean sealBOT
putridplanet
ocean sealBOT
Pure
ocean sealBOT
Pure
peak mist
lone heartBOT
lone heartBOT
Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
peak mist
Where is matrix B?
limpid turret
where did the 2(3)(-1) come frome
2 comes the $2C_2$ operation. 3 comes from the $3C_2$ operation. -1 comes from the $C_2\leftrightarrow C_3$ operation.
ocean sealBOT
SWR
peak mist
what about this
limpid turret
Iirc, adding one column to another does not change determinant.
But I would need to look it up to be certain
Yeah. Adding or subtracting rows preserves the determinant
lone heartBOT
@peak mist Has your question been resolved?
peak mist
Yeah. Adding or subtracting rows preserves the determinant
thank you
vivid tusk
I got the answer -5/6 when finding the definite integral, which is the net area, but how do i find the area? (total area?)
tacit arch
I got the answer -5/6 when finding the definite integral, which is the net area,...
Find the area above the x axis and below the x axis separately
vivid tusk
so i think my first step would be to evaluate y at 0
tacit arch
Had a typo. Should have been x axis for both
onyx cove
With regard to partial derivatives, should you treat variables that you're not differentiating as 0, or just leave them as the variable? E.g does f'_x(x,y) = x + y equal 1+y or 1+0?
onyx cove
I've seen conflicting information on this, some sources go with 0, and some keep it as the variable
ivory pivot
they are just constants
onyx cove
This causes problems when you take second partial derivatives
limpid turret
This causes problems when you take second partial derivatives
How?
onyx cove
If f'_x(x,y) = 1+ythen f''_y_x(x,y) = 1, but if f'_x(x,y) = 1, then f''_y_x(x,y) = 0
limpid turret
If `f'_x(x,y) = 1+y`then `f''_y_x(x,y) = 1`, but if `f'_x(x,y) = 1`, then `f''_y...
The first one here is just wrong.
onyx cove
Because the y is left in?
If so then I don't know why half of online resources do it like that
limpid turret
Because the `y` is left in?
Yes
limpid turret
If so then I don't know why half of online resources do it like that
Like where?
onyx cove
Here's a problem from https://www.youtube.com/watch?v=JAf_aSIJryg
Khan Academy also leaves the non-differentiated variable as is
limpid turret
Because, in both cases, y is a constant attached to x.
onyx cove
My course textbook also has this for x+y+xy
onyx cove
Because, in both cases, y is a constant attached to x.
How does that change things? If the non-differentiated variable is to be treated as a constant, then surely it should be set to zero and multiply away the whole term?
hushed locust
if we have a full derivative, then we treat the 5 in 5x differently than the 5 in x + 5
limpid turret
My course textbook also has this for `x+y+xy`
Because there's an xy term
fathom grove
With regard to partial derivatives, should you treat variables that you're not d...
you treat the variable as a constant
so if it’s a coefficient to the variable you’re differentiating by, it stays, but if it’s by itself with other numbers or non-differentiated variables, it becomes 0
onyx cove
Oooh, ok. I was taking treat it as a constant a bit too literally. Thank you all
.close
green gulch
Question, how do we know that z is in the second quadrant?
lone heartBOT
Question, how do we know that z is in the second quadrant?
green gulch
I assume because it is in the second quadrant we subtract the angle from pi
but how do we know it is?
if x is > 0 and y < 0 shouldnt it be the 4th quad?
here is the example from my textbook
he solves this in the second quadrant
do you flip the signs of x and y to find the quadrant?
ocean sealBOT
ionic jewel
unless i mistyped (or wolfram is wrong!) thats Q1
green gulch
really, how?
Im supposed to determine the quadrant first to know what to subtract my angle by
lone heartBOT
@green gulch Has your question been resolved?
daring cipher
so I got the negation of the first statement
lone heartBOT
so I got the negation of the first statement
lone heartBOT
@daring cipher Has your question been resolved?
daring cipher
<@&286206848099549185>
jagged forge
<@&286206848099549185>
If you go back one step you can see easier
daring cipher
I don't understand
jagged forge
Using the identity on the bottom
daring cipher
still don't understand
prime badge
i'm at least 70% sure it's d)
daring cipher
same
thats why I wanted to make sure
but I didn't know where to start or what to do after the answer I got
jagged forge
i'm at least 70% sure it's d)
yea idk ive been trying to get to A but once I started writing it down I dont see it
Cause your simplification is correct, but its not equal to A
serene moon
lone heartBOT
serene moon
can any1 help me
absolutely no idea where to start
wait
nvm
it's pi/3
so ig i just gotta get cos and sin from that
.close
harsh spear
How do i sum a scalar number for example the value 3 with a matrix for example of type 2x2 or 3x3?
winter light
You can't do that
Because dimensions do not match
Unless you disguise that scalar as a matrix full of ones multiplied by that element
lone heartBOT
@harsh spear Has your question been resolved?
rotund crater
lone heartBOT
ionic jewel
you could just look at K_5 and count tbh, but the better idea is to look at it and figure out the pattern why
rotund crater
you could just look at K_5 and count tbh, but the better idea is to look at it a...
Isn't there a formula to determine how many there are?
ionic jewel
surely you know how to do #19
then #20 you can think - wow, I can start at any vertex, then go to any other vertex, then for the second step I can go to any vertex except the first one
this pattern extends into 3 and 4
watch out for double counting paths (its distinct, so AB and BA are the same)
rotund crater
surely you know how to do #19
i dont understand how i can get 19
lean forge
Do you know matrices ?
You can do a path matrix and M^n gives you the number of paths of length n from point a to point b
rotund crater
Do you know matrices ?
yes but how i get path simple
lean forge
yes but how i get path simple
rotund crater
https://medium.com/@serii.hmeli123123/path-matrix-warshalls-algorithm-afe331147d...
adjacency matrix?
lean forge
Just sum the coefficient
All path in the adjacency matrix are simple though
Correct me if I’m wrong otherwise there’d be infinite path
rotund crater
Correct me if I’m wrong otherwise there’d be infinite path
I have an exam, they give me undirected graphs
rotund crater
Just sum the coefficient
can tell me a example?
lean forge
If you want the number of simple path of length two
Take M^2
Sum the coefficients
And you have your number
Draw a simple graph
And try it 👍🏻
rotund crater
If you want the number of simple path of length two
I get 20
lean forge
Try it with a simple exemple fort
First
Like a straight line
With three points
And two lines
You should get one
rotund crater
19 , i get 20
rotund crater
Try it with a simple exemple fort
Okay wait
ionic jewel
i dont understand how i can get 19
all paths of length 1 on K_5?
do you know what a path is and what K_5 is?
rotund crater
do you know what a path is and what K_5 is?
Yes but i dont know how i can calculate that
rotund crater
Try it with a simple exemple fort
I get M^1 if i sum all elements = 4
lean forge
I get M^1 if i sum all elements = 4
That would be correct
Four path of length one
From 1 to 2, from 2 to 3, from 2 to 1, from 3 to 2
So you see that it works
rotund crater
sooo my k^5 is 20?
lean forge
Idk didn’t check it
You could count
Or trust the formula
Or try to prove it to understand why it works
lean forge
Yes
lean forge
What does it say
rotund crater
What is that formula for?
lean forge
For what 😭
rotund crater
The paths of at least length
n are found by adding all the lengths up to n.
At least length 3:
Include paths of length 1, 2, and 3.
I do not understand what it means
rotund crater
ok dont worry but thx for help me
lone heartBOT
@rotund crater Has your question been resolved?
heady sky
can anyone help out with this
lone heartBOT
can anyone help out with this
heady sky
or this
lone heartBOT
@heady sky Has your question been resolved?
lone heartBOT
@heady sky Has your question been resolved?
marsh knot
or this
average rate of change is just f(b)-f(a) over b-a
which is basically the slope formula
so you could plug in 4 and 1 into f(x) and subtract those
then subtract 1 from 4 then divide the 2 results
marsh knot
can anyone help out with this
you can use the same formula for average rate of change for this question as well. basically just calculate each thing the question is asking for
(which is basically just reading a table)
vernal kite
Why is A, B, E correct?
Why is C, D, F incorrect?
lone heartBOT
Why is A, B, E correct?
Why is C, D, F incorrect?
vernal kite
Fyi I’m a complete beginner to this stuff so idk 😔😔😔
snow cobalt
f(n) = 3*(1/3)^n
f(n+1) = 3*(1/3)^(n+1) = 3*(1/3)^n*(1/3)
Divide both
What did you understand so far by the rest of it?
lone heartBOT
@vernal kite Has your question been resolved?
patent turtle
I'm investigating the sum of complex exponentials. The resulting angle in particular. For a sum e^(ia1) + e^(ia2), the angle is given by:
x = arctan((sin(a1) + sin(a2))/(cos(a1) + cos(a2)))
I've observed numerically that as long as the sum of angles a1 + a2 falls within [-pi, pi]:
x = (a1 + a2)/2
Now I'm stuck trying to prove this. I'm not sure where to start, I've searched for trig identities to help me out here and have failed to find any. I would prefer a nudge in the right direction here if possible but I understand if it's quicker or easier to just give an explanation.
lone heartBOT
I'm investigating the sum of complex exponentials. The resulting angle in partic...
@patent turtle Has your question been resolved?
patent turtle
<@&286206848099549185>
tacit arch
What is "the angle" you're referring to
patent turtle
The angle between the resulting complex number and the x-axis/real number line
tacit arch
Just use Euler's identity
Then group real and imaginary parts together
Then find that angle using arctan
patent turtle
How is Euler's identity applicable here?
tacit arch
How is Euler's identity applicable here?
Do you know what it is
Oh I don't
I meant Euler's formula
,tex .demoivre
ocean sealBOT
riemann
tacit arch
With n=1
patent turtle
This is how I derived the arctan formula outlined in my question. I'm not sure what you're suggesting here.
mellow grail
I meant Euler's formula
Its called the De Movieres formula 😐
mellow grail
I'm investigating the sum of complex exponentials. The resulting angle in partic...
If a1 is between pi and -pi, then the range of its sine is?
Range might help here, so could sum to prod identities
plain ember
lone heartBOT
plain ember
how do u go about solving this
celest terrace
use the FTC
celest terrace
fundamental theorem of calculus
plain ember
what is that
(i just started integrals)
i learned how to calculate anti derivative
and the difference between definite and indef integrals
idk what FTC is
celest terrace
let's start with the question first
plain ember
yes
celest terrace
$\int^3_1[9f(x) - 5g(x)]dx$
ocean sealBOT
mud ツ
celest terrace
for integrals you can split it when it's being added or subtracted
$\int^3_1[9f(x) - 5g(x)]dx$ = $\int^3_1 9f(x) dx - \int^3_1 5g(x)dx$
ocean sealBOT
mud ツ
plain ember
for integrals you can split it when it's being added or subtracted
oh
u dont just find anti deriv and plug in the limits?
celest terrace
that works too, but it's not 9(2)
plain ember
oh cause limits are different?
celest terrace
yes
plain ember
ooooooooooo
celest terrace
that's where this comes in
celest terrace
yes
celest terrace
$\int_b^a f(x) dx = -\int_a^b f(x) dx$
ocean sealBOT
mud ツ
plain ember
hm
idk what to do for the first one cause its not like a function
it just says fx=2
celest terrace
you know $\int_3^1 f(x) dx = 2$
ocean sealBOT
mud ツ
plain ember
correct
celest terrace
flip it and you get $-\int_1^3 f(x) dx$
ocean sealBOT
mud ツ
plain ember
correct
ok so for first one
we get f(3)-f(1)=2
so i guess when we flip
we get -1+3=2 again?
what i do wrong
celest terrace
from $-\int_1^3 f(x) dx$ you should get -(2)
ocean sealBOT
mud ツ
celest terrace
you just substitute the integral with the number
plain ember
i thouight it was f(b) = f(a) for def integrals
which would be f(-3)-(-1) = -3+1=-2
ooh
mb
celest terrace
you don't have to, and you also don't know what f(x) is
celest terrace
you evaluated the integral from 1 to 3 of f(x)
constants can be taken out of integrals
celest terrace
yes
plain ember
constants can be taken out of integrals
elaborate on this one
celest terrace
$\int^3_1[9f(x) - 5g(x)]dx$ = $\int^3_1 9f(x) dx - \int^3_1 5g(x)dx$ = $9\int^3_1 f(x) dx - 5\int_1^3 g(x)dx$
ocean sealBOT
mud ツ
plain ember
woah
celest terrace
and you can substitute $\int_1^3 f(x) dx$ and $\int_1^3 g(x) dx $ with the given numbers
ocean sealBOT
mud ツ
celest terrace
so then, $9\int^3_1 f(x) dx - 5\int_1^3 g(x)dx$ = 5(-2) - 5(-6)
ocean sealBOT
mud ツ
dusk umbra
is the complex conjugate of the product of two functions just the product of the complex conjugate of those two functions
dusk umbra
so the complex conjugate is "distributive"?
bleak crest
.reopen
dusk umbra
???
bleak crest
find x, y and z
dusk umbra
????
bleak crest
i dont understand how to solve this can you help me
dusk umbra
this channel is in use bro
bleak crest
yes..
dusk umbra
i think i might be able to help you
give me a second to think
do x y and z need to be integers?
bleak crest
yes
dusk umbra
negative 1
positve 4
and 3
what i did
was find the first 5 integers negative and positive
and cubed them
then i mixed and matched until i got the solution
i cant figure out a more elegant solution than just brute forcing it
alpine sable
Hi, could someone help me delimit this orange thing in Geogebra? I’m supposed to do a bridge but since my homework is due 11:59 I’ll stick to this simple design. If someone could help me, I would be very grateful
dusk umbra
bro what the fuck is going on in here
dusk umbra
yes
-1 cubed is negative 1
wait
i think i fucked up
yeah its -1 2 and 3
-1 cubed is -1
yeah i fucked up
sorry
idk man
dusk umbra
im unsure but im gonna go ahead and close this beause this channel has been hijacked lol
.close
dapper raven
help
dapper raven
how do i find the range and domain of the function f(x)=x^8/3 without a calculator
foggy pecan
$f\left( x \right)=\frac{x^{8}}{3}\text{ }\text{ }\text{ then:}\\dom\text{ }f=\mathbb{R}\text{ }\text{ and}\text{ }\text{ }rng \text{ }f=[0,\infty )$
ocean sealBOT
Joanna Angel
foggy pecan
this function is trivial
you can set any real number x
hence domain is an antire set of all real numbers
dapper raven
im sorry i meant x^(8/3)
foggy pecan
🙂
ok moment
$f\left( x \right)=x^{\frac{8}{3}}\text{ }\text{ }\text{ then:}\\dom\text{ }f=[0,\infty )\text{ }\text{ and}\text{ }\text{ }rng \text{ }f=[0,\infty )$
ocean sealBOT
Joanna Angel
foggy pecan
the base of the power must be non-engative
greater ro equal to zero
then results of such oeprations are the same
dapper raven
but the function is defined when i graph it? i get y values when i put in negative values
for x
foggy pecan
you cant set
y negative
neither x
power function is defined onyl for arguments non-negative
unless
you write it in such way: but that is other function:
$y=\sqrt[3]{x^{8}}$
ocean sealBOT
Joanna Angel
foggy pecan
that is same only for non-engative
but for negative, you cant use first fucntion you wrote
dapper raven
ok
what about the range im having trouble justifying why its [0,oo)
would the value of the numerator have to do with it? or the numerator/denominator being even/odd
foggy pecan
$\forall _{x\ge 0}\text{ }\text{ }x^{\frac{8}{3}}\ge 0$
ocean sealBOT
Joanna Angel
foggy pecan
the difference wud be like this :
$\forall _{x> 0}\text{ }\text{ }x^{-\frac{8}{3}}> 0$
ocean sealBOT
Joanna Angel
foggy pecan
negative exponent for such power has infleunce
dapper raven
i know but for example if it was x^(5/3) and x^(8/3) why would the ranges be different from each other
inner haven
what about the range im having trouble justifying why its [0,oo)
i think to justify the range, you can think abt it how the range is the y-values that you get from your function. If you were to plug in -infinity into your equaiton, you would get a infinite value back, if you plug in infinity into your equation, you also get a infinite value back
inner haven
i know but for example if it was x^(5/3) and x^(8/3) why would the ranges be dif...
degree of the graph
when u have x^8/3 its basically a parabola (U shape) and when u have x^5/3 its a odd power
dapper raven
how do you know its a parabola without graphing
dapper raven
but wouldnt the degree for x^5/3 be positive as well?
inner haven
not exactly since the degree is x^5
dapper raven
is it because its an odd number?
dapper raven
so an odd degree is defined by an odd numerator
inner haven
so an odd degree is defined by an odd numerator
basically yea
dapper raven
does that mean the denominator being odd/even is irrelevant if the numerator is odd
so if it was odd/even its still odd degreed?
inner haven
so if it was odd/even its still odd degreed?
only if its odd
dapper raven
so odd/odd?
inner haven
Odd/odd would give u a odd degree, even/odd gives u a even degree
dapper raven
what about odd/even
this is my justification for the range pls tell me if its ok or if im missing something: the range is [0,oo) because any value of x raised to a rational exponent where the numerator is even and the denominator is odd will result in only positive y-values
inner haven
That sounds fine to me @foggy pecan What do u think abt it
foggy pecan
That sounds fine to me <@1035542975503138836> What do u think abt it
If the exponent of the power function is a rational number, then there are not any problems with positive numbers
On the other hand, there are problems with the extension of these definitions to bases that are not positive real numbers.
to avoid it, I define power functions with rational exponents only for non-positive numbers, unless exponent is negative rational one, then i define it fo positive numbers
so $\left( -2 \right)^{\frac{8}{3}}
$\left( -2 \right)^{\frac{8}{3}}$
ocean sealBOT
Joanna Angel
foggy pecan
is disallowed for me, unless we use radical forms
then yes
$\left( \left( -1 \right)^{2} \right)^{\frac{1}{2}}=1^{\frac{1}{2}}=1\neq \left( -1 \right)^{2\cdot \frac{1}{2}}=\left( -1 \right)^{1}=-1$
ocean sealBOT
Joanna Angel
lone heartBOT
Available help channel!
daring pilot
I am getting 5 but that is wrong
lone heartBOT
I am getting 5 but that is wrong
daring pilot
I am getting 5 but that is wrong
such that x+y+z>0 and xyz = 2***
i did
[det(a)]^2 = 1
lone heartBOT
@daring pilot Has your question been resolved?
daring pilot
<@&286206848099549185>
daring pilot
Well, your method is better
yeah but the answer isnt matching
cyan seal
Can you show your work?
daring pilot
1 sec
cyan seal
Why ±1?
daring pilot
square of det(a) is 1
so to remove square i need to take both positive and negative one right?
i initially didnt do that and only got 5 as the answer
cyan seal
square of det(a) is 1
Wrong
cyan seal
Wrong
I mean, it's right, but that's gives you only 1 and not -1
√x² = x
So √|A|² = |A| = 1
daring pilot
I mean, it's right, but that's gives you only 1 and not -1
i need to take -1 to even get 7 as an answer tho, if we just take +1 then our only answer is 5
daring pilot
So √|A|² = |A| = 1
how i saw it is
when x^2 = 1 then x = +1,-1
then why is it any different in
[det(a)]^2 = 1
you can also write that as [det(a)]^2 - 1 = 0
which then becomes (det(a)-1)(det(a)+1) = 0
that gives us det(a) = +1,-1
daring pilot
when you calculate x^3+y^3+z^3 it should come out as 1+6
can you please tell in more detail
visual trench
sorry, no clue
daring pilot
sorry, no clue
np thank you for the help, I need to figure out the flaw in my method because it came to my mind naturally so there must be something conceptually incorrect with my understanding
lone heartBOT
@daring pilot Has your question been resolved?
tight nest
lone heartBOT
tight nest
I was working on this question
And i got
Horizontal stretch by factor of 1 / e^2
Is that correct?
indigo coral
Horizontal stretch by factor of 1 / e^2
It’s a vertical stretch by e^2
tight nest
wouldn't both make sense??
indigo coral
wouldn't both make sense??
Yes you are right
barren portal
If the differential equation is of the form $Mdx+ Ndy=0$, when do we have an integrating factor $\frac{1}{Mx+ Ny} ?$
ocean sealBOT
.doc
barren portal
An important thing to add here is, both M and N are homogenous functions of same order
tight nest
Im solving 2 b) and value of k is -0.0726
This is my working and im stuck and i think i did something wrong..
barren portal
Channel occupied
lone heartBOT
@barren portal Has your question been resolved?
tribal wolf
Anyone here familiar with M/M/1 Queueing models?
lone heartBOT
Anyone here familiar with M/M/1 Queueing models?
tribal wolf
Suppose a customer arrives at a random time and finds the queue empty, but you are unable to determine if the server is busy or not. How would you calculate the probability that the server is busy?
lone heartBOT
@tribal wolf Has your question been resolved?
vocal tapir
excuse the bad writing
lone heartBOT
excuse the bad writing
lone heartBOT
@vocal tapir Has your question been resolved?
exotic canopy
excuse the bad writing
no clue what's going on here
$\lim_{x\to 0 } \ln\left[\cos(ax)\right]^{\frac{1}{\sin^2(8x)}}$?
ocean sealBOT
artemetra
scenic wing
im stuck on 393 idk where to start
lone heartBOT
im stuck on 393 idk where to start
storm ridge
You could directly integrate it first
And then substitute
That's the integral for inverse tangent, are you allowed to directly write it? Or do you need to show it first and then substitute?
quartz coyote
^ it's the same as writing
scenic wing
ok
full lantern
lone heartBOT
full lantern
hey Im looking at a solution for a q and was wondering why the limit is 0 for all points in the domain
alpine sable
hey Im looking at a solution for a q and was wondering why the limit is 0 for al...
I don't understand your doubt
According to the limit if the number is irrational f(x) = 0
If not f(x) = 1/q
So that makes the function discontinuous
cold ivy
can you write de function
full lantern
dont know how on latex
is the picture not working
unless you mean draw the graph of f
alpine sable
The only explanation I can think of is c tends towards an irrational number
That would make sense
full lantern
but not all values in the domain of f is irrational
why are the limits of the rational numbers 0?
cold ivy
i mean the definition of function f(x)
alpine sable
i mean the definition of function f(x)
It is given in the question
Unless ur asking for the function itself
Which usually isn't given in limits
lone heartBOT
@full lantern Has your question been resolved?
novel estuary
is there a proof for why the limit is always 0
I think that you should use a proof by contraddiction. Suppose there exists an $c_0$ in the domain such that the limit isn't 0, call that limit L. Suppose L>0, and apply the limit definition with epsilon=L, you'll find $f(x) >0$ for all x in a neighborhood of $c_0$, but it is impossible because in the irrationals $f(x) =0$
If L<0 choose epsilon=-L
Not 100% sure but it should work
ocean sealBOT
milo_schwartz
full lantern
OHHHH
that makes so much sense
wow thank you mate, dk how you came up with that
.close
rocky juniper
lone heartBOT
rocky juniper
!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 was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
1
magic drum
can I ask for homework....
stable bison
Do you agree that the set {0, 1}^N is isomorphic (bijective) to P(N) ?
rocky juniper
ik that N is not bijective to P(N)
stable bison
It basically asks for an infinite subset of N such that there exists a total ordering on it.
stable bison
ik that N is not bijective to P(N)
Yes, but {0, 1}^N is bijective to P(N).
Think of it this way - a subset of N is the same as a function from N to {0, 1}, which tells, for each element of N, whether it belongs in the said subset of N.
Thus, {0, 1}^N, the set of all such functions, is the set of all subsets of N (also explains why |P(S)| = 2^|S|).
The question thus asks for an infinite set of subsets of N which have a total ordering.
rocky juniper
could you give me a small example or a way to show that a set C will be a total order where C is a subset of {0,1}^N?
im strugglign with this
novel estuary
I think you can choose the following set of functions from N to {0,1} :
The first function will be the null function a(n) =0 for all n
The second one will be b(1)=1, b(n) =0 for all n greater than 1
The third one c(1)=1,c(2)=1,c(n)=0 for all n greater than 2
And so on
rocky juniper
or a way to filiter out the subsets of N that do not have a total order
stable bison
or a way to filiter out the subsets of N that do not have a total order
Do you know what a well ordering is ?
novel estuary
In this way the set of these function, call it C, should satisfy the requests
rocky juniper
no
stable bison
Okay. Here is another idea - can you check that a(n) = n implies a total ordering on N ?
Then, can you check that a(n) = n implies a total ordering on N \ {0} ?
Then the same for N \ {0, 1}.
So on so forth.
And since there exactly N such subsets, you have a countably infinite amount of subsets for which there is a total ordering.
Does it make sense ?
rocky juniper
Then, can you check that a(n) = n implies a total ordering on N \ {0} ?
i dont get why u ask this
stable bison
i dont get why u ask this
The question asks for total orderings - a(n) = n implies such a total ordering.
stable bison
Yes - the question asks for an infinite amount of subsets for which there is a total ordering. I am trying to show that if you have a total ordering implied by a(n) = n, then removing the "last" element (if it exists, which in our case, it does), then you will also have a total ordering.
rocky juniper
is a(n) = n a well known thing for total orders or is it from the context of the question
stable bison
is a(n) = n a well known thing for total orders or is it from the context of the...
No - you have to prove that a(n) = n implies a total ordering.
rocky juniper
from the definiton of total ordering $x \preceq y \lor y \preceq x$ how do u prove that a(n) = n implies total ordering
ocean sealBOT
luke1
rocky juniper
is it just the = ? from <=
stable bison
from the definiton of total ordering $x \preceq y \lor y \preceq x$ how do u pro...
The question says that x is smaller than y iff a(x) <= a(y).
Since a(n) = n, we have that x is smaller than y iff x <= y.
So proving that a(n) = n implies total ordering is the same as proving that the "conventional" ordering <= of the natural numbers is a total ordering - that is, proving that for any two natural numbers x and y, either x <= y, either y <= x (or both, but never neither).
Ugh, I guess this is quite difficult to prove unless you take for granted the fact that the ordering of the naturals is total. We could try to do it if you know about Von Neumann ordinals, though.
rocky juniper
no idea what that is
stable bison
Oof, okay. I hope (but don't gurantee) that it is not a problem if you take for granted the fact that the ordering of the naturals is total.
All you then have to show is that the ordering implied by a(n) = n is the same as the conventional ordering of the naturals, thus it is a total ordering.
Then all you have to show is that no matter how many times you take the first (smallest) element, you're still left with a total ordering. And since N is a countably infinite set, you can take the first element a countably infinite amount of times, thus form a countably infinite amount of subsets for which there is a total ordering.
stable bison
Depends, I don't know what it asks for.
Wait, wait, wait... I think that I've misinterpreted the whole question. 😐
rocky juniper
Wht!!!
stable bison
Yup, I've misinterpreted the whole question, I'm really sorry. I'll try to help you with this too.
rocky juniper
dw its fine i need the help i can get
ur misinterpreted answer did help me a bit to understand some stuff
stable bison
So basically, we have the set of functions from the natural numbers to {0, 1}. We call a function smaller than another function if for any n, the value of said function at n is smaller than or equal to the value of the other function at n. Then the question asks us for a countably infinite amount of such functions.
Let's think a bit about this. Do you agree that if you take a function, and all you do is to increase the value of one of itss assignment from 0 to 1 (but never the other way around), the new function is smaller than the old function ?
Then, you could start with the function which assigns all natural numbers to zero, then always assign the smallest natural number assigned to 0, to 1.
So the first function is a(n) = 0, the next one is a(n) = 1 if n = 0, and a(n) = 0 otherwise, the next function is a(n) = 1 if n = 0 OR 1, and a(n) = 0 otherwise.
And since we have the natural numbers, you can do this a countably infinite amount of times.
Of course, this is quite an informal proof, though hopefully it is enough to point you in the right direction. If you need help formalizing the proof feel free to ask.
(It is the same as @novel estuary's answer, now I noticed).
Does it make sense, @rocky juniper ?
rocky juniper
im reading it several times to understand it
stable bison
Oh okay. If something is unclear don't hesitate to ask.
rocky juniper
Then, you could start with the function which assigns all natural numbers to zer...
i dont rly understand this
"then always assign the snalest natural number assigned to 0 to 1"
stable bison
Okay, you have a function which assigns all natural numbers to zero.
We want to generate a new function, starting from this function, which will be bigger than it.
What do we do ? We search what is the smallest number of the first function that is assigned to zero, and in the new function, we assign it to 1, while keeping all the other values the same.
This new function will obviously be bigger than (or equal) to the first function, while being different from it.
Then, we generate a third function from this new function, which will also be bigger than or equal to it.
Then we generate a fourth one etc. and by induction, we generate, in the end, |N| such functions, thus a countably infinite amount of such functions.
stable bison
and then C will just be those sets of functions
C will be the set of such functions, yes. All that is left to do is to prove that those functions are totally ordered.
Do you need help with this ?
rocky juniper
is it just to do with the f(n) <= g(n) and the definition of total ordering is x <= y or y <= x which f(n) <= g(n) satisfies?
or have i just gotten that completely wrong
stable bison
Yes, kind-of. If f < g, then for all n, f(n) < g(n). Let's try to prove that if it is not a total ordering, we get a contradiction (thus, by the law of the excluded middle, implies that it is a total ordering.).
Suppose that C is not totally ordered. Then, there are two functions f and g such that neither f <= g, neither g <= f. So, not(f <= g OR g <= f). Thus, not(forall n f(n) <= g(n) OR forall n g(n) <= f(n)). Thus, by De Morgan's law, not(forall n f(n) <= g(n)) AND not(forall n g(n) <= f(n)). Thus, exists n (not f(n) <= g(n)) AND exists n (not g(n) <= f(n)). Thus, exists n (g(n) > f(n)) AND exists n (f(n) > g(n) -- relation 1.
BUT, f <= g if and only forall n (f(n) < g(n)) -- relation 2.
Thus, by proof of contradiction (since relations 1 and 2 are contradicting themselves), C is totally ordered.
I think it is quite like what you stated, but formalized.
lone heartBOT
@rocky juniper Has your question been resolved?
rocky juniper
thanks for this
stable bison
Actually you could try to prove by induction that C is totally ordered.
We start with that function assigning everything to zero. Obviously that singleton set is totally ordered.
Then, when we add an extra element, it also is totally ordered.
So on so forth.
rocky juniper
btw this was the part b i was talking about
stable bison
btw this was the part b i was talking about
Oooh, this one is interesting.
I'll think about it since it is definitely harder.
stable bison
btw this was the part b i was talking about
Okay, I think there is indeed a such uncountable set.
Here is how to construct it - do you agree that P(N) is uncountably infinite ? We will have to construct a set {0, 1}^N that is bijective to P(N).
We could do it this way:
- Take an arbitrary member of P(N) - that is, a subset of N.
- Construct an element of {0, 1}^N based on that subset - that is, any element in that subset is assigned to 1, and anything else is assigned to 0.
- Prove that this set of functions (Z) is bijective to P(N) (all there is to do is to prove that there is a function which undoes step 2 - that is, construct a subset based on that function). Thus, {0, 1}^N has the same cardinality as P(N), thus it is uncountably infinite.
- Prove that Z is totally ordered.
Does it make sense, @rocky juniper ?
rocky juniper
yea
for step 3, do we just do the opposite on what step 2 does i.e element in that subset is 0 anything else is 1
im rly bad at abstract alg
stable bison
for step 3, do we just do the opposite on what step 2 does i.e element in that s...
Recall that a function is bijective if and only if it has an inverse function.
You can thus prove that the function from P(N) to {0, 1}^N is a bijection by constructing its inverse function, a function from {0, 1}^N to P(N) which "undoes" it.
rocky juniper
and to prove that Z is totally ordered do we follow similar steps to part a ) ?
stable bison
and to prove that Z is totally ordered do we follow similar steps to part a ) ?
Yes, you could follow the same steps.
Though actually I think it is a bit different.
Let me think a bit about it.
WAIT
This set Z actually is not totally ordered.
Because, for example, take the functions f and g generated by the subsets {0} respectively {1}. Then, neither f <= g, neither g <= f (you can check this yourself).
And since Z (which is equal to {0, 1}^N) is the only uncountable subset of {0, 1}^N we can construct, and since it is not totally ordered, we have that any totally ordered subset of {0, 1}^N is countable.
My bad!
lone heartBOT
@rocky juniper Has your question been resolved?
winter tangle
show that if G is a topological group and A,B⊆G are compact subsets, then AB is compact.
am i right in using tynchonoff theorem and the continuity of the map?
winter tangle
<@&286206848099549185>
lone heartBOT
@winter tangle Has your question been resolved?
winter tangle
<@&286206848099549185>
novel estuary
I don't know what a topological group is, but I'm currently studying compactness. I know that if A,B are compact then AxB is compact for Tychonoff theorem as you said, and we know that the continuos image of a compact set is compact. Therefore if in the field you're studying there exists some continuos map $f\colon AxB \to AB$ then it should work. I studied group theory and I think that AB is something of the form $z \in AB \iff z=a+b$ with $a\in A b\in B$ and $+\colon AxB \to G$ should be an operation
ocean sealBOT
milo_schwartz
novel estuary
if + is continuos then you're done, but as I said I didn't study topological groups so I don't know if it's the case
winter tangle
well a topological group is a Hausdorff topological space G which is also a group, whose
group operation $\dot : G \times G \to G$ and the inversion ${}^{-1} : G \times G \to G$ are both continuous functions
ocean sealBOT
張嘉棋
winter tangle
$G \times G$ has the product topology
ocean sealBOT
張嘉棋
novel estuary
well a topological group is a Hausdorff topological space G which is also a grou...
Oh okay, so if they're continuous you're done for the reason above. You can also say that AxB is compact for the kuratowski theorem if you don't want to use the tychonoff theorem
serene moon
can any1 help me
lone heartBOT
can any1 help me
frail grove
sin/tan = cos
frail grove
why?
serene moon
if I do it it's 0/0
frail grove
$\frac{\sin \theta}{\tan \theta} = \frac{\sin \theta}{\frac{\sin \theta}{\cos \theta}} = \sin \theta \times \frac{\cos \theta }{\sin \theta}=\cos \theta$
cinder egret
start with tanx = sinx/cosx
ocean sealBOT
Adam Chebil
serene moon
.close
slender tangle
Help
slender tangle
This is an easy question
But I feel like its a tricky question because all the other questions were hard
,rotate
ocean sealBOT
slender tangle
Did I do it right??
storm ridge
Why subtract it with 18 tho?
rose sigil
no?
slender tangle
Ig they're asking the total population
So + 18
storm ridge
Also wait I think...
slender tangle
Also wait I think...
The weird thing is that Im doing integrals
storm ridge
You need to count the rate for the first 3 days too
slender tangle
I dont know why my teacher gave me this exercise
Im probably doing something wrong
storm ridge
You need to count the rate for the first 3 days too
At the end of day 1, it would've grown by f(1)
By day 2, f(2) and so on
slender tangle
At the end of day 1, it would've grown by f(1)
By day 2, f(2) and so on
Oh yes
So I have to multiply everything
By 1 2 and 3?
Like I repeat the same formula 3 times with 1,2 and 3
slender tangle
It should be that ig
Hold on
slender tangle
@storm ridge what you think
Hold on
Its 188
Instead of 224
And its 56 instead of 92
Apparently this is the right procedure
But I don’t understand it
Can someone explain it to me please?
shut pewter
Apparently this is the right procedure
this is integration
basically it assumed that the rate of growth is a derivatve of a function
which it obviously is which is the number of insects
henc dN
slender tangle
this is integration
Im pretty sure my math teacher hasn’t even explained it
shut pewter
yeah then this wouldn't work
slender tangle
I swear she always give us stuff she never explains
shut pewter
No, dN/dx is the rate itself
dN is the change in the number of ants
and dx is the number of days
so change in the number of ants/ change in the number of days
Just like a car
Change in distance / change in time = speed
@slender tangle wait what grade are you?
slender tangle
<@801562285889028127> wait what grade are you?
I think our school system is different
Im 18 (2005)
I’ve never failed
shut pewter
ok ok
slender tangle
a class
shut pewter
do you know calculus already?
slender tangle
I feel like my teacher give us exercises she never taught us
shut pewter
because either this is an introduction to calculus
slender tangle
do you know calculus already?
I know integrals and derivates
shut pewter
or this is calculus itself
shut pewter
Apparently this is the right procedure
Ok then just do this
slender tangle
Wait galm
Give me 2 mins please
N’(x) is 20+6x+3x^2 (rate of growth)
So Im supposed to do the integral of N’(x)?
shut pewter
yah
shut pewter
yeah
remember that dN/dx is just the same as N
N'(x)
just two completely different notations
slender tangle
∫N’(x) dx= ∫20+6x+3x^2 dx?
shut pewter
yeas
shut pewter
yeah because when you sub 0, it cancels everything
since 18 = bla bla bal + C
since x = 0
then that will be your new C
use that and sub 4 to get the answer
slender tangle
use that and sub 4 to get the answer
shut pewter
yeah
so just use the equation again but this time replace C with 18
and then have N(4)
limber bison
Help
How much variation dr in the radius of a coin can be tolerated if the volume of the coin is to be within 1/1000 of their ideal volume?
lone heartBOT
Help
How much variation dr in the radius of a coin can be tolerated if the vol...
@limber bison Has your question been resolved?
wheat sparrow
lone heartBOT
lone heartBOT
@wheat sparrow Has your question been resolved?
wheat sparrow
<@&286206848099549185>
lone heartBOT
@wheat sparrow Has your question been resolved?
alpine sable
@wheat sparrow
wheat sparrow
yes?
wheat sparrow
hmm
I read "the distance from the axis of rotation to x (i.e., the radius of a sample shell)"
I assume it means y-axis to the x value
which is just x
i think
alpine sable
@wheat sparrow , thinking more, I think you're right
Although I don't know the shell method, it seems logical to me
Sorry for answering even though I don't know the shell method very well
wheat sparrow
its all good
alpine sable
I avoided to answer because of this
But as I saw no one was answering, I decided to try to help
wheat sparrow
.close
lone heartBOT
lone heartBOT
Available help channel!
wide onyx
lone heartBOT
wide onyx
how do i graph this
lone heartBOT
@wide onyx Has your question been resolved?
hybrid epoch
I have a exercise but I translate for y'all.
The Pic du Midi de Bigorre cable car (Pyrenees) has two sections.
The first section leaves from La Mongie (M) located at an altitude of 1753 m and arrives at Taoulet station (T). On this section, the cable car travels 1800m (MT).
The second section leaves Taoulet (T) and arrives at Pic du Midi station (P). On this section, the cable car travels 2630m (TP).
At what altitude is the Pic du Midi station located? Round to the nearest meter.
hybrid epoch
The exercise 7-7
lone heartBOT
@hybrid epoch Has your question been resolved?
tulip oak
Question number 10,
These problems are for squeeze theorem and Trigonometric limits, but I can't see where the strategy lies for this, since you cannot make a sinx or cos x less between -1 and 1 inequality to create the "trap"
Also, I've deduced that the right side expression is a step function
lone heartBOT
tulip oak
Can anyone help with that question?
lone heartBOT
tulip oak
Question number 10,
These problems are for squeeze theorem and Trigonometric li...
My question is here