help-28

[slate night]

I tried using prrof by induction, which is the method the marking scheme says to use

[glacial pasture]

is the o just notation for this operator

[slate night]

but it didnt work out

[slate night]

yes

[glacial pasture]

show your induction

[slate night]

phone died

just as i was sending it

basically

i did

S(2) = x o x = (x1 + i)(x2 + i) - i

which is true

so then, assuming S(k) is true

I did S(k+1)

which is where i messed up pretty sure

@glacial pasture

[slate night]

i ended up getting a o xk+1 = a(xk+1 +i)

which is completely wrong ig

all the numbers after the x, before a space, are meant to be subscript fyi

[glacial pasture]

so you assume x1 o x2 o ... o xk=(x1+i)(x2+i)...(xk+i)-i
doing o x(k+1) on the previous should get us
xk o x(k+1)=
(x1+i)(x2+i)...(xk+i)-i+i-i

which simplifies to (x1+i)(x2+i)...(xk+i)(x(k+1)+i)-i

which is what we wanted

[slate night]

why do you find out xk o x(k+1)?

[glacial pasture]

because we want to show that
x1 o x2 o ... o xk o x(k+1)=...

we know x1 through to xk

so we just do o x(k+1) on that

writing xk may not have been the right choice, but the general idea is there

[slate night]

yes thats what i did

and since

x1 o x2 o ... o xk = (x1+i)(x2+i)...(xk+i)-i

i rewrote them as a

[glacial pasture]

yeah thats sensible

[slate night]

so then what would that result

because i think i messed up the algebra then

[glacial pasture]

then you use this definition

a o x(k+1)

[glacial pasture]

which is this bottom line here

[slate night]

ok yah makes sense

that gave me the same result

alrighty

thanks

/close

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[severe crater]

why is D1 an interval but S1 not two ponts

[fast peak]

what are D1 and S1?

[severe crater]

Disk and sphere

D^n is the n dimensional disk

and S^n is an n dimentional sphere

I don't understand why S1 is a two dimensional circle but then D1 is a one dimentional interval

[fast peak]

well S1 is still a line, hence the 1

it just "lives" in 2-d space usually

[severe crater]

why isn;t the same true for D1

[fast peak]

well you could also draw D1 in 2d space

but how you draw stuff isnt the important point

[severe crater]

is D1 an interval or a disk

[fast peak]

intuitively, you need 1 number to describe either where you are on S1 or D1

hence they are both 1-dimensional things

D1 is the 1-dimensional version of a disk

[severe crater]

okay so it's an interval

[fast peak]

which is an interval

[severe crater]

okay

by that logic S1 would be a 1 dimensional sphere

witch is two points

but it's not tho

its a circle

that's what I'm not getting

[fast peak]

to describe where you are on a circle you just need the angle. aka 1 number

of course intuition always tends to fail at some point, which is the case here with S0

[severe crater]

I don't get it

I mean the way I'm seeing it is that you need 1 number the radius to define any sphere

so in 1 dimensional space

we would get two points

I don't see how the fact that you need an angle to specify a point fits into this

[fast peak]

it doesnt matter in which space we are living

we care about the set itself

not where it lives

S0 is points. points are "0-d things"

S1 is a line. lines are "1-d things"

S2 is an area. "2-d thing"

thats why they are named that way

[severe crater]

okay

[fast peak]

dont forget that these are names we made up

[severe crater]

I think I'm getting it

D1 is a line because it is one D

wait

so to define a sphere I need r

I get it

kind of

So if we looked at D1 in two dimentions

[fast peak]

we dont care about the radius

[severe crater]

if we embedded D1 in two dimensions wed get a disk

[fast peak]

the circle "looks" the same either way. the important things we care about arent impacted by the radius

[severe crater]

lines are a one D thing but a circle is a 2d thing

I apologise if I'm missing something obvious

[fast peak]

a circle is a line

[severe crater]

in what sense

like I'm pretty sure they arent homeomorphic

I mean they aren't homeomorphic

[fast peak]

well I am using line in the intuitive english sense

[severe crater]

okay

[fast peak]

locally it looks like a line

[severe crater]

are you saying it like if we were a 2d viewer then a circle would be a line

in the same sense that as 3d viewers all spheres are circles

[fast peak]

I suppose? not really what I meant tho. if you zoom in really closely to the circle, you cant differentiate from a straight line

like we on earth thing about the surface we live on as an area

[severe crater]

how does that entail a circle is a line tho

I agree locally a circle is a line

[fast peak]

well thats what we mean when we say a thing is 1-dimensional

[severe crater]

okay well in euclidean space locally everything is a line

[fast peak]

well no. not the unit square in R^2

thats an area

[severe crater]

okay but a circle is an area as well

[fast peak]

are you confusing the words disk and circle

circle is just the boundary

its just the line

[severe crater]

I thought square was just the boundary as well

[fast peak]

oh sry did I use the wrong word?

I meant [0,1]^2

[severe crater]

I don't know what [0,1]^2 is

are you talking about [0,1] cross [0,1]

[fast peak]

yes

the usual notation for the cartesian product of a set with itself

anyway, I have to go. maybe someone else can help better

[severe crater]

I messed up notation my bad

thank you for your help

I agree that an area isn't a line

okay

I think I got it

thank you

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[supple jay]

guys i calculate like this

is it wrong?

expoenent 10 is for how much each economy periods would grow in 10 years, given their rate of change 1% in old economy and 2.6% per year in new economy

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[grave patio]

Hello, pls help to solve it, i solved, but my answer is wrong. This is condition:Find all values ​​of the parameter a for which the equation
has exactly 1 root per interval [0; 5]. Also as you can see, it's translation my exercise.

[gritty rose]

!show

[full forum]

Show your work, and if possible, explain where you are stuck.

[amber bone]

I can not read that languages, is this russian?

[grave patio]

I translated russian into english

[grave patio]

But answer is

Idk, where i have mistake

I looked at all 4 conditions and there is no solution to all of them, very strange

I was completely confused, because it seemed like I decided correctly, found everything, believed with the condition, but nevertheless this can't be

[full forum]

@grave patio Has your question been resolved?

[grave patio]

<@&286206848099549185>

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@grave patio Has your question been resolved?

[night timber]

attempted this many times and have no idea

[full forum]

@grave patio Has your question been resolved?

[grave patio]

So

I am still waiting

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@grave patio Has your question been resolved?

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[graceful peak]

Two diagonals of a regular polygon intersect at an angle of measure 147 degrees. What is the minimum number of sides of the polygon?

[graceful peak]

How do i approach this problem?

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@graceful peak Has your question been resolved?

[graceful peak]

<@&286206848099549185>

The progress ive had is that the number of sides must be greater than 120/11

Because the greatest angle of intersection is gonna be equal to one interior angle of the polygon

Im not sure though

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@graceful peak Has your question been resolved?

[graceful peak]

Hello

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@graceful peak Has your question been resolved?

[graceful peak]

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[sullen salmon]

\ln\left(x^{2}+1\right)-e^{-x}+1

[sullen salmon]

ln(x^2+1)-e^-x+1

how is it an always an increasing function from R TO R

[sly sky]

Notation not workin?

[sullen salmon]

when we take the derivate of the fxn

[sullen salmon]

here mate

[sly sky]

Oh mbad

[sullen salmon]

we got

2x/x^2+1 +e^-x

how it is always +ve?

[sly sky]

Ngl its kinda hard to read it like that

Hol up lemme write it out

[sullen salmon]

wait

the derivative of the fxn

[sly sky]

Whaddya mean when you say “R to R”?

[sullen salmon]

its is defined for

Whole Number Line

or say

real number

[sly sky]

Ohhhh you mean the bounds

[sullen salmon]

yes

in short we can put any value of x we want

[sly sky]

Are you asking for the algebraic proof or more so a geometric intuition of it?

[sullen salmon]

algrebric

b'coz

i check on desmos

the graph of the

[sly sky]

Well first take the derivative of the function

And see what you get

[sullen salmon]

the graph is always increasing

[sly sky]

Thats the derivative?

[sullen salmon]

nah

thats the real fxn

[sullen salmon]

this is derivative

[sly sky]

Ohhhh

[sullen salmon]

like this isn't always +ve

if i put x=-10

it will throw some negative value

[sly sky]

Well i mean if you graph it out, doesn’t it show you why it works?

[sullen salmon]

like yeah i saw the graph on desmos

that is inc

but

theoritcaly

i feel that it should be inc

all the time

[sly sky]

Im confused about what ir asking again

[sullen salmon]

wait

[sly sky]

So it works algebraically, but it doesnt show up on the graph?

[sullen salmon]

let me start from start

i have a fxn

[sly sky]

Mkay

[sullen salmon]

i want to know about that how the fxn is always increasing algebrically

not graphically

[sly sky]

And when you say that, are you referring to the function itself, or its derivative?

[sullen salmon]

this fxn

[sly sky]

Mkay

Well think about it like this: are there any values you can think of, that would cause any of those terms in the function to not exist?

[sullen salmon]

for above fxn?

[sly sky]

Yep

[sullen salmon]

no the domine is clear

we can put anything we want

[sly sky]

Yep

[sullen salmon]

so now

[grave patio]

And also my question didn't solve

[sullen salmon]

if we take the derivate of the fxn

[sly sky]

Request in another help ill help you there

[sly sky]

Mhm

[sullen salmon]

we got this

[sly sky]

Yep

[sullen salmon]

so if the fxn the real fxn

not the derivative one

[sly sky]

Mkay

[sullen salmon]

if that fxn has to be

always increasing

this derivative

have to be always +ve

[sly sky]

Im not familiar with the term +ve help me on that 💀

[sullen salmon]

ok

so this derivative

when we put any value of x

[sly sky]

You mean always positive?

[sullen salmon]

this has to be >0

[sullen salmon]

eggxatly

[sly sky]

Okay sweet

And it should

[sullen salmon]

why

if i put

x=1-0

-10

i wil get -20/101+e^10

oh fuck

now i realise

[sly sky]

Yep

[sullen salmon]

god damm it

[sly sky]

LMAO

[sullen salmon]

bruh

wtf

i was thinking that

it will make that negative

but didn't actually put the value

🤡

[sly sky]

LUL

Yep the second term increases much faster than the first term can decrease

[sullen salmon]

yeah

b'coz e^x

[sly sky]

Best way to think about it is to compare the rate of changes for each terms

[sullen salmon]

if i put

[sly sky]

Yep

[sullen salmon]

x to -x

[sly sky]

I make the mistake all the time as well dw 💀

[sullen salmon]

great

thanks man

for da help

🫡

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[sly sky]

Ofc ofc

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[fallen stream]

i figured out the diagram but can't figure out how to solve it

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[urban plume]

Hi! :) I would like some help with my homework, or at least this part:

[torn jolt]

what number/letter?

[urban plume]

wha?

[torn jolt]

which part

[urban plume]

OH

I don't understand how b-d were solved

a is fine but Idk the rest T~T

[torn jolt]

okay

given f(x) = 2x^2 + 4x

f(x-5) = ? how would you start

or let me give my own example

f(2x-3) how would you go about this

[urban plume]

mmm

subtract 5 from just the x^2??

I'msorryIdkxD

[torn jolt]

nope

[urban plume]

not surprised :')

[torn jolt]

you treat (2x - 3) as x

and substitute where there was x

[urban plume]

O H

[torn jolt]

so can you do that for me for

f(2x-3) = ....

[urban plume]

it is 1 am so it might take a tiny teensy bit but yes

1 moment dkhasld

[torn jolt]

sure no rush

lmk if something is not clear

[urban plume]

ok ok

f(2x-3) = 2(2x-3)^2 + 4(2x-3)?

[torn jolt]

[urban plume]

:D!!!!!

[torn jolt]

$$f(x) = 2x^2 + 4x$$
$$f(2x-3) = 2(2x-3)^2 + 4(2x-3)$$

[urban plume]

this makes a lot more sense than I thought 😭

ok I think

[torn jolt]

you're okay with the expansion and simplifying right

[urban plume]

expansion as inn

[torn jolt]

and so on

[urban plume]

I believe so

[torn jolt]

thats all they did here

[urban plume]

that makes so much sense, sob

c does too but d seems a bit odd

actually

no

this makes

[torn jolt]

how would you tackle d

[urban plume]

so much sense suddenly-

ok ok

2(-5x)^2 + 4 (-5x) ?

:D!!!!!

omg ok this is clicking tysm

could I also ask abt this?

the switch between x and a is confusing me a tad

[torn jolt]

you treat whatever you were given in the parentheses as x

do C(a)

[urban plume]

ok that makes sense

but then where does h come in

[torn jolt]

what do you get for C(a+h)?

[urban plume]

I have no clue without looking at the solution T~T

[torn jolt]

dont look at it

any ideas?

[torn jolt]

first lets start with

[urban plume]

uhhh

something like (8x + 5 + h) ??

[torn jolt]

for C(a)?

[urban plume]

oh just C(a)

[torn jolt]

.

[urban plume]

8x + 5 right?

[torn jolt]

yeah C(x) = 8x + 5

[urban plume]

:D

[torn jolt]

C(a) = ?

[urban plume]

ooh right C(a) = 8a + 5 ?

[torn jolt]

yes

[urban plume]

woops xD

[torn jolt]

okay now do C(a+h)

[urban plume]

ehm

I have no idea what to do with the h

just 8a + 5 + h ?

[torn jolt]

what did you do here

[urban plume]

plugged in the formula

[torn jolt]

no

yh do it to here

[urban plume]

sob

[torn jolt]

but dont forget brackets

[urban plume]

but what do I plug in for h?

[torn jolt]

you're dealing with (a+h) not just h

[torn jolt]

what will you input here

replacing x

[urban plume]

C(a + h) = 8(a + h) + 5?

not x

frick

[torn jolt]

[urban plume]

.......................

omg that is so obvious, I'm a fool fdnsakldsa

I blame it on the sleep deprivation T~T

[torn jolt]

one more and you sleep

[urban plume]

D: but I'm so close to finishing this assignment

[torn jolt]

$$\frac{C(a+h) - C(a)}{h}=$$

[urban plume]

Idk what that means but it looks important so I screenshotted it

[torn jolt]

what?

[urban plume]

oh wait there's another question lmao

I'm sorry, I'm

very brain empty rn

[torn jolt]

finish this

write on a paper

and send pic

[urban plume]

terrifying, hm

[torn jolt]

its just what you did for 1 and 2

[urban plume]

oh duh

..... u h

I did not know there could be two F(x) thingys

[torn jolt]

you know the values for each right

[torn jolt]

you found them

[torn jolt]

.

[urban plume]

ok wait this makes

sense

lemme do this on paper rq

[torn jolt]

[urban plume]

8 (a+h) + 5 - 8(x+5) / h ?

or is the second 8 in the parenthesis

[torn jolt]

$$\frac{8 (a+h) + 5 - 8(x+5)}{h}$$

no

[urban plume]

no to the 8 thing or no to the equation

[torn jolt]

$$\frac{C(a+h) - C(a)}{h}= \frac{8 (a+h) + 5 - 8(x+5)}{h}$$

[torn jolt]

write on paper and take a pic

[urban plume]

I can't really transfer pics to Discord, but I can draw it out digitally

[torn jolt]

everything

oh nvm

i misread ur question

[urban plume]

me xD

[torn jolt]

what is C(a+h)

[urban plume]

8 ( a+h) + 5 ?

[torn jolt]

okay and C(a)

[urban plume]

8x + 5 ?

[torn jolt]

okay and C(a+h) + C(a)

[torn jolt]

what

[urban plume]

I'mverytired

8a + 5 ?

[torn jolt]

yes and

[urban plume]

ok I see what I did wrong dnjsakld

8 (a+h) + 5 + 8a +5

?

[torn jolt]

i meant C(a+h) - C(a)

[torn jolt]

but correct for C(a+h) + C(a)

[urban plume]

dnsjad oop

8 (a+h) + 5 - 8a +5

[torn jolt]

okay go on

[urban plume]

oo wait I'm kinda good at this

kinda

keyword kinda

8a +8h + 5 -8a + 5 then uh

[torn jolt]

also dont forget all of this is divided by h we will substitute later

[urban plume]

8h + 10 ?

ofc ofc

I do not like h, but alas, I finally understand h xD

[torn jolt]

8(a+h) + 5 - (8a+5)

the brackets

[urban plume]

ohyeah

sec xD

[torn jolt]

so 8a + 8h +5 -8a -5

[urban plume]

ok this is making sense

so that's uh

just 8h ?

[torn jolt]

[urban plume]

;D

:D*

[torn jolt]

8h/h?

[urban plume]

8 :D

tysm <333

I'll try the rest on my own

[torn jolt]

anything that wasnt clear?

[torn jolt]

okay

[urban plume]

I think I'll have to work on actually remembering all the steps, but I am now well on my way

tysm, you're amazing

how do I end the chat thingy again? ;-;

[torn jolt]

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[torn jolt]

practice practice practice

[urban plume]

ooo pog

🫡

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[iron sand]

how do i proove this?

[iron sand]

$\sum_{i=1}^{n} i \binom{n}{i} = n \cdot 2^{n-1}$

[onyx glen]

$\sum_{i=1}^{n} i \binom{n}{i} = n \cdot 2^{n-1}$

[glossy valve]

Ann

[onyx glen]

are you allowed to use generating function shit?

[glossy valve]

Slowaq

[iron sand]

no i am supposed to proove it by induction

[onyx glen]

$\sum_{k=0}^n \binom{n}{k} x^k = (1+x)^n$, take the derivative of both sides then put x = 1

[glossy valve]

Ann

[onyx glen]

ah damn.

gonna be painful

[iron sand]

xd

yea and i dunno how to du it

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[untold vessel]

how is this calculated?

how do I in general calculate when its not dydx, dydt, or d/dx or d/dt

y * ?

[primal stag]

can u give an example so i can explain it?

[kindred grove]

the derivative dx/dt is a function

y is also a function

multiply those two yes

[untold vessel]

function is: x * y
y = 2
x = 0.5

calculate x * dydt + y * dxdt

x * dydt = 0.5 * 0

[kindred grove]

and where's t ?

[untold vessel]

dunno about the other one

[untold vessel]

not there

thats why its 0

[primal stag]

its like x(t)?

[kindred grove]

you sure

[untold vessel]

yes

[kindred grove]

can we see the whole question then ?

[untold vessel]

its spread out over many many pages

[kindred grove]

yeah right it's the multivariable chain rule

[untold vessel]

some more stuff but yeah its very very long

im just tasked with calculating the last part

but I cannot for the life of me figure out how to compute y * dx/dt

[kindred grove]

where do those values come from

the 2 and 0.5

[untold vessel]

we are just tasked with calculating based on those

[kindred grove]

ok so dz/dy

[untold vessel]

yes in the end

[kindred grove]

there's no chain rule here ?

like z depends directly on y

[untold vessel]

okay but

how do I calculate dx/dt of (x*y)

thats my main question really

[kindred grove]

it's d/dy

stay focused

dz/dy = d/dy (xy + y cos(pi x))

[untold vessel]

well im asking specifically for this one

with y * dx/dt

[kindred grove]

it's just an example

they never gave you an actual formula for x(t) or y(t)

so of course you can't compute that, as in get the numbers

[untold vessel]

they wrote that the answer is: da/dt = 0.5 * 0 + 2 * 1

where y = 2 and x = 0.5

So thats why I am confused

[kindred grove]

ah seems like they're saying dx/dt = 1 and dy/dt = 0 for this example

it's so spread out wtf

[full forum]

@untold vessel Has your question been resolved?

[untold vessel]

Yeah hahaha

That’s why it was hard for me to figure out

But thanks

Question here is what is db/dt then lmao

[kindred grove]

any thoughts or no ?

[untold vessel]

No thoughts except “1”

[full forum]

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[kindred grove]

.reopen

[full forum]

✅

[kindred grove]

so $\pdv{b}{t} = \pdv{}{t} \left(\pi x\right)$

wtf

[glossy valve]

aPlatypus

[kindred grove]

pi is just a multiplicative constant (3.1415....) so you can get it out

[untold vessel]

Hmmm okay

[kindred grove]

$\pdv{b}{t} = \pi \cdot \pdv{x}{t}$

[glossy valve]

aPlatypus

[untold vessel]

Aha

[kindred grove]

and dx/dt is 1

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[torn nebula]

How does this satisfy the hypothesis that g is continous on the closed interval [a,b] and differentiable on the interval`s interior (a,b)

[rapid rain]

"f is continuous on the closed interval [a,b] and differentiable on the interval`s interior (a,b)" is correct, right?

[torn nebula]

yh thats the rolle`s hypothesis

i just dont know how the function g satisfies that

[rapid rain]

if $f$ and $h$ are two continuous functions and differentiable functions, then $f+h$ is ...?

[glossy valve]

rafilou2003

[torn nebula]

idk

[torn nebula]

is continuous?

[rapid rain]

yeah

on [a,b] here

and differentiable on (a,b)

well, apply it to $h(x) = \frac{f(b)-f(a)}{b-a}(x-a)$

[glossy valve]

rafilou2003

[torn nebula]

oh i didnt know that equals to h(x)

[torn nebula]

don`t know what to apply, all i know is that this guy wrote f(x) minus f(x)

which is 0

so g(x) = 0

wha

[rapid rain]

?

If you didn't fully understand my hint, take $h(x) = -\frac{f(b)-f(a)}{b-a}(x-a)$

[glossy valve]

rafilou2003

[rapid rain]

this way g(x) = f(x) + h(x)

(whereas g(x) = f(x) - h(x) had you taken the first "h")

in any case, g is the sum (or difference) of two continuous functions on [a,b] and differentiable functions on (a,b)

[torn nebula]

damn

why didnt they say that in the hypothesis

[torn nebula]

oh

[torn nebula]

will g(x) be discontinuous on the boundary intervals

like g(a) and g(b)

[rapid rain]

no, because both f and h are continuous

[torn nebula]

oh alr

[rapid rain]

and a sum of continuous functions is...

[torn nebula]

continuous

yh its just imma tryna understand this line

i understand it as g is discontinuous on closed interval [a,b] and differentiable on (a,b) because f is.

and im like wha

but yh its cool now cuz he shows its continuous i think

[torn nebula]

anyways your help, helped me a lot, just one more thing. Why did you times (x-a) to make it h(x).

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[delicate quartz]

hello, everyone!

I am trying to draw conclusions about the convergence of a series.

I have a sequence defined as in the first rectangle; I'm wondering what conclusions I could draw from the sum

[delicate quartz]

In particular, I'm having trouble getting a start on this 😩 the recursive definition trips me up a bit.

does anyone have any idea?

I already know that the sequence vn is bounded and monotone. could I conclude it is convergent?

[wide sundial]

Is the bound on the right side of the monotonicity

If so then yes

[delicate quartz]

hmm. what do you mean on the right side?

I know that for all n, 0 ≤ vn < 1

[wide sundial]

Well if you have a_n = 1/n

[delicate quartz]

that, and that the sequence is decreasing (vn > vn+1, for all n)

[wide sundial]

Then you say oh this is bounded from above at 2

That’s not very useful

That’s what I mean by “on the right side”

[delicate quartz]

hmm

[wide sundial]

If you have monotonicity and a supremum bound (or inf)

Then you can use monotone convergence theorem

[delicate quartz]

I think that would apply in this case, yes?

the inf would be 0, it is decreasing thus monotone, thus by theorem converges?

[wide sundial]

How do you have equality on the left

[delicate quartz]

oops. my bad 🥲

[wide sundial]

Lol right

You can say

[delicate quartz]

hehe. I'm rusty

you got a sharp eye as well

[wide sundial]

$\text{inf}\left({v_n}_{n=1}^\infty \right)= 0$

[glossy valve]

Frosst

[wide sundial]

And also $v_{n+1} < v_n, \forall n\in \mathbb N$ hence $v_n$ is monotonically decreasing

[glossy valve]

Frosst

[wide sundial]

Then by the monotone convergence theorem, this sequence converges to 0 as n tends to infinity

[delicate quartz]

right. oof

is there any way to establish that the limit of the series is 0?

I think my textbook doesn't have the monotone convergence theorem

[wide sundial]

You can do it via epsilon delta as well

[delicate quartz]

oh god no

[wide sundial]

You need to prove that

[delicate quartz]

then that is not a possibility 😁

[wide sundial]

$\forall \varepsilon > 0, \exists N\in\mathbb N: \forall n > N, |v_n - 0| < \varepsilon$

[glossy valve]

Frosst

[wide sundial]

This implies that $v_n \overset{n\to\infty}{\to} 0$

[glossy valve]

Frosst

[delicate quartz]

I see. it's weird that the monotone convergence theorem is not stated anywhere

[wide sundial]

Well, I think it’s a real analysis thing

[delicate quartz]

hmm

because without it, my argument is much less solid

and this seems like an easy exercise

[wide sundial]

You can easily prove the monotone convergence theorem

Just use the definitions of sup/inf

[wide sundial]

Then this statement

But replace 0 with the sup/inf

[delicate quartz]

🥵

[wide sundial]

What the definition of supremum

[delicate quartz]

the least upper bound?

so, 1?

[wide sundial]

I meant in general

The least upper bound means it is an upper bound, and it’s the smallest one

That’s all you need to prove convergence of a monotonically increasing sequence

[delicate quartz]

okay, so in this case, it would be the other way around

[wide sundial]

You know it has to keep increasing (monotonicity)

You know it can’t go bigger than the supremum

[delicate quartz]

it is decreasing, so I could state the infimum

[wide sundial]

Yes that’s why I’ve got inf here

[delicate quartz]

I see. this was really helpful

I'll give another go at flipping through the textbook to see if I can find the monotone theorem

but this is a good backup plan 🦸

if I may, do you have any idea about this one too?

[wide sundial]

You need the bounds on the right side but also that it’s an inf/sup not just any bound

[wide sundial]

What’s v_n

Oh the same as before?

[delicate quartz]

yep!

i mean, I was expecting to solve some of these by calculation criterias

like

[wide sundial]

Probably you can do ratio test on this

[delicate quartz]

i mean, this is probably applicable here

hmm

in fact, isn't it applicable on the first one as well? let me check real quick

[wide sundial]

I don’t know

We didn’t do the first one

[delicate quartz]

errrr

i'm not sure I'm going anywhere with this

this is very confusing

I think I can compare the first one to a dirichlet series, maybe?

anyway, thank you @wide sundial for the help 🙏

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[runic spruce]

For part a do you assign a vector zero and by doing that you open the possibility that the scalar multiplied by that vector is not zero

I’m just confused because surely that scalar could be zero as well, so I don’t see how the above proof is sufficient

[full forum]

@runic spruce Has your question been resolved?

[devout valley]

It could be zero, but if there’s any which isn’t, that’s enough to show linear dependence

Note that the definitions say “there exist” rather than “for all”, so you just need to find one and you’ve shown dependency

[devout valley]

So basically this is how you’d show it: in fact any nonzero number you’d like would work, by the fact that the zero vector multiplied by any vector is zero

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[inland jackal]

Hello 🙂 I have a question with this exercise

[inland jackal]

Do I have to prove that it's uncountable by creating an injective function or is this far too complicated?

Because I don't see any or can I use cantors diagonalarument?

[fast peak]

cantor works

injective function also works if you choose the sets appropriately

[inland jackal]

If I choose the set of semi-infinite bitstrings?

Then I don't have do define a specific function right?

[fast peak]

you still have to define some function

[inland jackal]

shit ok, thanks 🙂

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[inland jackal]

Do you think this proof is sound?

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[simple lichen]

was wondering if anyone can help with one accounting question? its math-adjacent..

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[outer dust]

how can i answer 3?

[outer dust]

there is a solution provided but it doesnt make much sense to me

.close

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[full forum]

Please don't occupy multiple help channels.

[twin wolf]

smooth brain botter

anyways

How do I find the other solution of -5pi/4

when sin(x-pi/4) = 1

From -pi to pi

I could only find 3pi/4

But not the negative solution

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[torn jolt]

Hello?

[torn jolt]

.close

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[desert musk]

S = {p in P_3 | p(2) = 0 }
show that (x-2,x(x-2),x^2(x-2) is a basis for s

[desert musk]

c1(x - 2) + c2(x(x - 2)) + c3(x^2(x - 2)) = 0

c1(2 - 2) + c2(2(2 - 2)) + c3(2^2(2 - 2)) = 0
c1(0) + c2(0) + c3(0) = 0
0 + 0 + 0 = 0

c1(0) + c2(0) + c3(0) = 0

does this now mean that it is linearly indepdent?

[full forum]

@desert musk Has your question been resolved?

[fathom saddle]

Well, in order to show that they are linearly independent, you need to show that the only solution to that top equation is (c1, c2, c3) = (0,0,0). I don't see that being done here.

You're probably allowed to use that 1,x,x² are linearly independent

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[round plank]

how do i solve this?

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@round plank Has your question been resolved?

[fathom saddle]

What's Rθ?

Rotate the vector by θ?

• What maps to 0? That is, what vectors can rotate to become the 0 vector?
• What vectors can be mapped to? That is, what vectors can you get/not get via rotations?

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[round plank]

.reopen

[round plank]

not sure

[round plank]

this is what the answer was

[fathom saddle]

Well nobody can answer the question if we don't know what Rθ is

But I think I'm right, it's a rotation by θ

[fathom saddle]

Have you thought about these questions?

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[worldly hollow]

This is a pretty easy Geometry question, I'm sure you guys are going to get it in a second, it's just that I don't know exactly how to do this.

Any help is appreciated.

[olive yoke]

Start with angle 5. Remember that all the angles on a straight line like that must add up to 180

You also know angle 1 since the outer lines are parallel

[worldly hollow]

so how would I find angle 1 and angle 5

[grim skiff]

They told you how to find angle 5

Start with angle 5. Remember that all the angles on a straight line like that must add up to 180

[worldly hollow]

does that mean angle 3 and 5 are both 90 degrees?

[grim skiff]

No

[worldly hollow]

they don't "look" like right angles

I don't know how to do this lol

[grim skiff]

Angle 5 is located here

On a straight line

And the sum of angles on a line equals 180

[worldly hollow]

oh

ok

then angle 5 equals 81

How would I try to find angle 1?

I know the lines are parallel but I don't know how I can connect that

[olive yoke]

use the sum of angle 5 and the 47 degrees angle

[worldly hollow]

why does that work though?

[olive yoke]

the lines are parallel

transversal something something

[grim skiff]

It's more of the fact because of the same side interior theorem shown here, I believe

[olive yoke]

yeah

[worldly hollow]

yeah

which one is it though

man I suck at geometry

I can only figure these things out when its exactly like that

[olive yoke]

same-side interior

[grim skiff]

I told you in the statement

because of the same side interior theorem

[worldly hollow]

Ok I got all of them

Thanks

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[grim skiff]

What exactly are you asking?

[full forum]

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.

[fathom saddle]

"Type 4" is not a universal term. What is "Type 4"?

[runic spruce]

Regular upper triangular, additional column and one/more rows of zeros

[runic spruce]

It’s long

[quaint prawn]

even more reason to include it

so we have the context

[devout valley]

How do you define regular(ity), for context?

[runic spruce]

I can send my working, there could be a rule for GE I’ve broken without realising

[runic spruce]

No zeros on the diagonal

I think I’m looking for an absurdity on the bottom row to prove there’s no solutions.. rather than all zeros, I think I should end up w something else

[devout valley]

So does that break it?

[devout valley]

Please do

[runic spruce]

I’ve changed one row at a time

[runic spruce]

I’ve written it there but I’m using the original thing at the top to get the new row at that point

[devout valley]

Looks like to get rid of that (2,1) entry, you'd have subtracted (3/2) * the first row, or?

@viral jasper

[runic spruce]

That’s one way but I didn’t use that

[viral jasper]

@devout valley

[devout valley]

Alright how did you change the second row? not sure i get what you meant

[runic spruce]

First row- 2/3 * second row

Or would that not be changing the second row

Bc I’m doing first-second?

[devout valley]

Oh so you replaced row 2 with that, you mean?

[runic spruce]

Yeah

[devout valley]

Don't think that's legal can only swap rows, multiply a row by a scalar or subtract a multiple of a row from an already existing one

If i recall right

[runic spruce]

Wait so

Do I need a multiple x whatever I’m subtracting from

Instead of a multiple x what I’m subtracting

[devout valley]

Basically multiply the thing you were planning to subtract, then subtract that multiplied version

[runic spruce]

That’s what I did tho, no?

[devout valley]

E.g. subtracting any multiple of row 1 from row 2 is fine

[runic spruce]

So that means my last two steps are both wrong?

First step is fine?

[devout valley]

Erm, basically, replacing R2 with R2 - (3/2)R1 is a fine step to do, but replacing R2 with R1 - (2/3)R2 isn't [the first subtracts some scaled row from what was there, the second basically doesn't, if you get what i mean?]

[devout valley]

No, those are fine (but combinations), and...

[devout valley]

...your first step isn't

[devout valley]

[You multiplied a row by a scalar, and then from the result, subtracted another row from that]

[runic spruce]

Ok that does make sense ngl

Thank god ur here I’m gonna redo it now

Thankuuu

[devout valley]

Yep let me know how it goes

[runic spruce]

For this if I want to replace R2 for example can I do a multiple of R2- multiple of another row

[devout valley]

Yep that would be fine to do

[runic spruce]

New version

It doesn’t change my answer, I don’t know why wolfram alpha is saying there’s no solutions when I get infinite solutions

[devout valley]

,w rref [[2,4,1,1],[3,5,0,1],[5,13,7,4]]

[glossy valve]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by chartbit

[devout valley]

What did you put into wolfram?

[runic spruce]

I didn’t change the first row

Don’t see why it’s be necessary

[devout valley]

That shouldn't matter if you kept it the same?

[runic spruce]

Yh I kept it the same

[devout valley]

[wolfie does rref where it goes a couple of steps further]

[runic spruce]

I got same as row 2 but multiplied by -1

[devout valley]

,w solve [[2,4,1],[3,5,0],[5,13,7]][[x],[y],[z]] = [[1],[1],[4]]

[runic spruce]

Ugh it says no solutions fml

[glossy valve]

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[devout valley]

Hmmm

[runic spruce]

Wtf

[devout valley]

,w solve 2x + 4y + z = 1, 3x + 5y = 1, 5x + 13y + 7z = 4

[glossy valve]

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[devout valley]

[runic spruce]

Idek what that means

[devout valley]

This is basically the equivalent set of equations (if you let x be be vector (x, y, z) basically) and this is basically infinitely many solutions, as is...

[devout valley]

...what this basically says too

[devout valley]

x can be anything you like, y and z are dependent on the value of x

[runic spruce]

Oh ok so it is infinite solutions?

That’s fine then, I was just confused by what calc was saying

Getting harder to check answers

[devout valley]

Yep infinite solutions for this one, you're fine

[devout valley]

Does be like that sometimes

[runic spruce]

Tysm I’m actually done now I have a morning lecture

Ty again saving grace fr

[devout valley]

Wow, that's dedication right there to still be working despite that hope it goes well (and you don't end up feeling too tired!)

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[limpid glen]

hello everyone, does anyone know a statistics worksheet website here? im tackling hypothesis and parametric testing right now

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[gritty rose]

Just Google this

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[untold vessel]

I don't know how to calculate the gradiant of this

😦

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✅

[kindred grove]

@untold vessel you know the multivariable chain rule ?

[untold vessel]

not well

[kindred grove]

it's fine we can go a bit through it

[untold vessel]

Thanks

[kindred grove]

I'm going to compact a bit the notations for simplicity

I'll write $L$ instead of leaky relu, and I'll call $z(w_0, w_1, w_2, w_3)$ the whole expression inside of it

[glossy valve]

aPlatypus

[untold vessel]

okay:)

[kindred grove]

with that our function becomes $L( z (w_0, w_1, w_2, w_3) )$

[glossy valve]

aPlatypus

[kindred grove]

see how we have a composition of 2 functions

L depends indirectly on the weights

i.e. it's a good candidate to apply the chain rule

[untold vessel]

wait wait wait

where does z come from?

[kindred grove]

I just called w_0+w_1x_1+..... , z

[untold vessel]

hmm okay

[untold vessel]

Yes

[kindred grove]

so in that context, if you had to find $\pdv{L}{w_0}$, what would you say it is ?

[glossy valve]

aPlatypus

[untold vessel]

Can you explain what it means for something to be on bottom and top of a derivative?

[kindred grove]

have you solved "normal" derivatives before ?

[untold vessel]

Yes

[kindred grove]

what notation are you used to ?

[untold vessel]

But only dydx

Or d/dx

[kindred grove]

I mean it's the same as dy/dx

the top is a function

the bottom is a variable of that function

[untold vessel]

Mmm okay

But you wrote z(..) as if it’s a function

[kindred grove]

yeah

well it is

[untold vessel]

Alright

[kindred grove]

it depends on w_0, w_1, ...

[untold vessel]

Right

Hm

So we need to say dL/dz * dz/dw0

Right?

[kindred grove]

yeah

[untold vessel]

Question is… how is dL/dz calculated

[kindred grove]

L is a one variable function

[untold vessel]

And why are we leaving out f(x1, x2, x3)? That’s also a function

[kindred grove]

f = L(z) if you think about it

[untold vessel]

Right

[kindred grove]

I didn't invent these L and z from nowhere

[untold vessel]

Okay

[kindred grove]

so dL/dz is just your typical derivative

now one problem indeed is that the function is piecewise

and you can't really do much about it, the answer will also be piecewise

[untold vessel]

Right okay

Piecewise meaning?

[kindred grove]

defined in multiple parts

depending on the input

[untold vessel]

How about dz/dw0

Assuming w0 = 1

How will this be calculated

[kindred grove]

there's no need to assume a value for w0

$\pdv{z}{w_0} = \pdv{}{w_0} \left(w_0 + w_1x_1+w_2x_2+w_3x_3\right)$