help-49

[hallow belfry]

is this 11ft help

help

[zenith kettle]

i dont understnad from so

[steep prawn]

Hey can anyone help me understand how these implications should go?

[hard ore]

For the first one, you know that sometimes two different numbers have the same absolute value

Like 5 and -5

So that should give you a hint

For the second one, it goes both ways cause you can manipulate each equation to get the other pretty easily

Third one is similar to first

And fourth is similar to second

I can give more hints if you want

[steep prawn]

maybe thats why hold on

actually would the third one be a contradiction?

or maybe am i putting the implication the wrong way for the first and third?

[hard ore]

Yeah

a=b implies |a|=|b|

Cause they’re the same number so ofc they’re gonna have the same absolute value

[steep prawn]

Thank you so much

[hard ore]

No

Np *

[steep prawn]

but before you go lol

well nevermind i can figure that one out thanks

.close

[naive adder]

hello

is this channel available?

[midnight plank]

@steep prawn Has your question been resolved?

[midnight plank]

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[dark gorge]

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[dark gorge]

How can I know which one is correct?

[dark gorge]

I am confused whether its the first one or last one

they both have a hole in their middle
How can I be sure which one it is?

[neon osprey]

What do you mean

You have to match them to the equns

They are already matched

[dark gorge]

The contourplot is given of one of the three graphs
I need to select the right graph

[neon osprey]

Oh

I didnt see the contour

[dark gorge]

I know it's not the middle one, that's obvious

But I can't seem to figure out whether its the first or last one

[neon osprey]

Obviously the first one

The last one is shifted a little to the left

That means the contour wouldnt be symmetric

[dark gorge]

is it because of the deep hole in it?
Because the contour has those lines at each corner and that me doubt it

[dark gorge]

oh yeah you are right

thanks for pointing it out

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[split lichen]

Hey guys, I need help with the following problem: I need to find an example of a set M ⊂ R(all real numbers), so that M but also R \ M are "overcountable". I dont know if I used the correct terms, I'm from germany. this is first semester analysis.
I dont even know where to begin with this one because if the set I choose is overcountable, how can R also be overcountable?

[steel trout]

Overcountable?

Do you mean uncountable?

[split lichen]

yeah probably

i dont know the term in english

in german its "überabzählbar

[steel trout]

Can you just give me the definition maybe?

Ok yes it seems that it’s an uncountable set

[split lichen]

a not finite and not countable set

thats the definition in my textbook

[steel trout]

Well you could do the interval from 0 to 1 and then the complement of that interval is still clearly uncountable

Since it is infinitely many intervals

[split lichen]

oh okay

[steel trout]

There’s a theorem, can’t remember the name, but it says that an interval in the reals is uncountable

[split lichen]

ah, i thought that an interval always is countable

so the set (0,1)

[steel trout]

No definitely not, diagonalization can prove that it isn’t

[split lichen]

wait i dont understand

[steel trout]

Look up cantors diagonal argument

[split lichen]

ohhh yeah

i know that one

so it turns out my question was pretty easy

[steel trout]

Yeah it’s not too bad

[split lichen]

thanks 🙂

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[steep quest]

Can someone explain me how does this end up being √64-28 instead of 64-28

[fast igloo]

since they're both inside the root you can multiply those first then take the root

[stable dawn]

$\sqrt{(8 - 2\sqrt{7})(8 + 2\sqrt{7})}$

[grand pond]

ColdTee

[stable dawn]

$(a + b)(a - b) = a^2 - b^2$

[grand pond]

ColdTee

[slender walrus]

radical bars by themselves don't disintegrate when they clash

$\sqrt{a}\sqrt{b} \redneq ab$

[grand pond]

ℝαμΩℕωⅤ

[steep quest]

thanks i got it

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[tight zinc]

What I did was multiply up and down by radical 7 is this correct?

[jovial nebula]

yo

take the 7 to the left side

[lavish venture]

yes

[jovial nebula]

now root 7 x root 7 = 7

cancel one (root 7 )

[tight zinc]

Thanks

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[last slate]

How do I find imaginary part of z

I can't think of any manipulations

[fallow scarab]

,rotate

[fallow scarab]

what are you even given

[last slate]

Uh

[tardy wyvern]

arg z1-z2 = pi/4

[last slate]

Real part of Z = | z -1|

[last slate]

And this

[fallow scarab]

just that?

[last slate]

yes

[fallow scarab]

i guess write z = a + ib, then expand |z-1| in terms of a and b

[last slate]

no

[fallow scarab]

what are z1 and z2

[last slate]

it didn't work

[last slate]

complex numbers

any

[fallow scarab]

why not

then what's z?

[last slate]

I tried putting rCistheta

[tardy wyvern]

u can use the mod

z-1

as

[last slate]

Z1 and z2 are complex numbers satisfying that relation

[tardy wyvern]

rootx-1 whole square + y^2

[last slate]

Sorry

[fallow scarab]

then it's not any complex number

[last slate]

Yeah

[last slate]

I got equation of a parabola

I tried solving it using coordinates

with parameters t1 and t2

But that went too long

[tardy wyvern]

could I see the working

cuz u dont need to solve real part

[last slate]

Wdym

[tardy wyvern]

I think u can put the real part aside

[last slate]

But I'll have to find z1 + z2 anyway right ?

[tardy wyvern]

and collect iota terms

[last slate]

OK I'll try it

@tardy wyvern I got this now

[tardy wyvern]

mB I was on another guy

[last slate]

Ok

[tardy wyvern]

ur given z1 and z2

2x1=1+y2

and

well idk hwo to write it

this is better

also arg(z1-z2) = pi/4

use it to get the value

[last slate]

Lol

Ok

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[boreal parrot]

Would this be correct as a final answer

My limits are 0 and 2π

[last slate]

you said the limits are 2pi yet you've added 2pi to all your solutions

[boreal parrot]

Oh

Right

[boreal parrot]

If I leave as is, would it be alright?

[slate canyon]

yes your ans is correct

but in 4th line

is that 2t(t+1) -1(t+1)=0?

and (2t-1)(t+1)=0 in 5th line

[boreal parrot]

Yes

Ohh

I get wym

Bracket mix up

[boreal parrot]

I fixed

Tysm

[midnight plank]

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[reef solar]

My calculator got this wrong
This function isn't differentiable at x=5 as it is a sharp corner

[reef solar]

Why did it get it wrong, is it a bad calculator? It is what my school allows in exams

[placid spoke]

the calculator can't understand instantaneous rate of change

so it calculates an approximation to the derivative at 0

i'm not sure which one it uses, but there is a formula for the derivative that is sometimes used:

$f'(x) \approx \frac{f(x+h)-f(x-h)}{2h}$

[grand pond]

doaby

[placid spoke]

if the calculator does this, it makes sense why it would give you 0

[reef solar]

Oh I never saw this formula

But it makes sense

[placid spoke]

it's more commonly used for derivative approximations, not theoretically

it's quick and accurate so

[reef solar]

Since the average rate of change of a small h value is ≈ the rate of change

[reef solar]

Like
Giving a table of values of displacement and time and needing the instantaneous velocity, right?

[placid spoke]

yes very similar to approximating the derivative like that

just a different formula

[reef solar]

I see I see
Thanks

[placid spoke]

no problem, I can't be sure that's what it does but it's my best guess

good question, that's interesting

[reef solar]

Btw it applies only to small values of h i assume

[reef solar]

Maybe there is some algorithm of some series that it uses

[placid spoke]

oh yes absolutely

[placid spoke]

calculators do use Taylor series a lot if you're familiar with that, but they wouldn't here since |x| isn't differentiable

[reef solar]

I am still studying calculus 1 but yeah I have some understanding of it

[reef solar]

Sorry but |x| is just not differentiable at x=0 but differentiable at the rest of it domain, right?

[placid spoke]

that is correct yeah

[placid spoke]

series stuff is super fun, you'll like it a lot I bet

[reef solar]

I hope so lol

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[celest forge]

Can someone help with example 2 like I don’t get it at all I’ve tried to read it but it makes no sense to me (first pic is the question and second pic has the solution)

[last slate]

ok do u want a walkthrough a step at a time thru it

[celest forge]

I don’t understand anything so

[last slate]

ok

,,\4{e^{x+1}}{e^{y+2}} = 3

[last slate]

lets start with this

to solve the system its usually a good idea to solve for one of the variables in one of the equations and then plugging it in in the other

[last slate]

do u agree with this @celest forge

[celest forge]

yes

[last slate]

ok so lets try and isolate y

a good first idea is to get rid of the fraction

you can apply $\ds\4{a^n}{a^m} = a^{n-m}$

[celest forge]

I only understood the part where they just subtracted the powers

[celest forge]

Yea

[last slate]

ok do that what do u get

[celest forge]

e^x-y-1

= 3

[last slate]

ok

so now

you want to get rid of the e

[celest forge]

LN

[last slate]

what is the thing that undos the e

ok

[celest forge]

Multiply ln on both sides

?

[last slate]

take ln of both sides

[last slate]

it's not multiplication

,, a = b \Implies \6\log a =\6\log b

[celest forge]

I forgot log rules

it’s fine continue then I’ll search them up!

[last slate]

i mean i kind of need you to understand along the way

,, e^{x-y-1} = 3

[last slate]

this is where we are at right now

[celest forge]

Yeah

[last slate]

can u try performing this. If u r unsure on how to proceed after just like show me whst fhis looks like rigjt after performing this

[celest forge]

I don’t get it

[midnight plank]

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[fair depot]

is abs(sinh(x)) = sinh(x)

[fair depot]

if x>=0

[fallow scarab]

do you know when sinh(x) < 0?

[fair depot]

?

[misty tundra]

it’s 5

i think

[fair depot]

no

[fallow scarab]

plug in the definition of sinh(x) into sinh(x) < 0

using exponential functions

[fair depot]

and then what

[fallow scarab]

did you do this

[fair depot]

ys

[fallow scarab]

show

solve for x

[fair depot]

you just mean putting a number bigger than 0 into x?

both terms are positive for sinh(x) when x > 0 so it will also be positve

so no need for the abs

[fallow scarab]

no

solving for x in sinh(x) < 0

by plugging in the two exponential functions

[fair depot]

@fallow scarab

i need ot prove this

[fallow scarab]

!original

[midnight plank]

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.

[fair depot]

ok

prove that

for |x| < 1/2

[fallow scarab]

you already showed that

[fair depot]

is the entire question

i know that exp(x)-1 <= 3|x| so i can maybe use that for |x| < 1/2

but i am not sure how

[midnight plank]

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[peak herald]

having trouble learning trig sub integrals

in this process for this example problem, i understand almost everything except two things:

where is the triangle coming from?
how to go from 4(tan(t)-t) to the next term where you undo the substitution

[lapis dew]

sec(θ) = hip / adj = x / 4

[peak herald]

oh wait that makes sense tho. but where does the triangle come from in the first place? like how do i know what the sides are

[lapis dew]

a^2 + b^2 = c^2 using c = x and a = 4

what's b

[peak herald]

wait how do you know c = x in this case? i get how to find the other side from this but how do you get here?

x = 4 * c/a, i dont see how that easily translates to a^2 + b^2 = c^2

[lapis dew]

because hip / adj = x / 4

[peak herald]

oh. yeah. im stupid

[lapis dew]

i mean, technically you could do something like hip = (x/4) and adj = 1, or nest everything too many times

but that one is the easiest to use

[peak herald]

ok yep tysm i just worked it out again myself w/ that knowledge and it all makes sense now

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[crystal rune]

Ik this is very simple but I think im over complicating it

I keep getting decimals and just want to see if it was correct

I tried

4x-14=2x an got 2.3

[orchid vault]

How'd you get 2.3?

4x - 14 = 2x is correct.

But not 2.3

[orchid vault]

Have you written 6x = 14!?

How?

[crystal rune]

I added the 2 and 4

[orchid vault]

Buy you can't add them like that.

If anything, you have to subtract 2x from both the sides.

[crystal rune]

It should be 2x

[orchid vault]

It should.

[crystal rune]

And then x = 7

[orchid vault]

Correct.

[crystal rune]

Thank you

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[tall hull]

can someone guide me

[autumn cobalt]

Why is there (5^3)^x-1 in denominator?

[tall hull]

mb it's suppose to be x only

[autumn cobalt]

The rule you need btw is $(a^x)(a^y) = a^{(x+y)}$

[grand pond]

ZapMighty

[autumn cobalt]

But in the first line you have just 125, with no exponent..

[tall hull]

I was still testing on how to write so I ain't notice

just had the x in my mind

[autumn cobalt]

So it's supposed to be $\frac{5^x \cdot 25^{x-1}}{5 \cdot 125^x}$?

[grand pond]

ZapMighty

[tall hull]

yuh

as the next step is this aight

[autumn cobalt]

But then you did $=\frac{5^x\cdot(5^2)^{x-1}}{5\cdot(5^3)^{x}} = \frac{5^x\cdot5^{2x-1}}{5\cdot5^{3x}}$

[grand pond]

ZapMighty

[autumn cobalt]

But $(5^2)^{x-1} = 5^{2x-2}$ not $5^{2x-1}$

[grand pond]

ZapMighty

[tall hull]

ahh

[autumn cobalt]

And tbh, you should just combine everything into $5^{\text{some expression with x}}$

[grand pond]

ZapMighty

[autumn cobalt]

I think it is $5^{-2}$

[grand pond]

ZapMighty

[autumn cobalt]

(That is without x tho 😅 )

[tall hull]

🫡

got it

thanks

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[autumn cobalt]

🫡

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[exotic stirrup]

Hi I'd like to get some help with these linear algebra quesitons. I have asked them previously in this server but not gotten a sufficient response. Please can you help

[exotic stirrup]

First, I'd like to ensure that I have done part i) correctly.

These are my workings.

Is this correct so far? If not, where have I gone wrong and why?

[midnight plank]

@exotic stirrup Has your question been resolved?

[exotic stirrup]

<@&286206848099549185>

ah i think i have found a n error here

i will show my new workings

[exotic stirrup]

<@&286206848099549185>

is that alright now?

[exotic stirrup]

<@&286206848099549185>

[midnight plank]

@exotic stirrup Has your question been resolved?

[exotic stirrup]

<@&286206848099549185>

[midnight plank]

@exotic stirrup Has your question been resolved?

[exotic stirrup]

<@&286206848099549185>

<@&286206848099549185>

[long viper]

hlow

[exotic stirrup]

HI

okay

so i think ive done this one now, id just like you to help me check through it

let me take a photo

theres my workings

sorry first page abit messy

oh wqait last line of last question should have coefficient 1 not 2 infront ofb_3

[exotic stirrup]

hi

<@&286206848099549185>

[exotic stirrup]

hi

[midnight plank]

@exotic stirrup Has your question been resolved?

[midnight plank]

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[strong ridge]

hey guys what does the y0 mean?

[strong ridge]

doing ODEs

[fallow scarab]

$y_0$ is a number

[grand pond]

riemann

[fallow scarab]

the important part is it's fixed

[strong ridge]

how are we meant of solve for a constant given the only information is that it equals to another constant?

@fallow scarab

[fallow scarab]

yes that's fine

"suppose ...." means you can assume you're given y_0

it's not a variable you find

it's just part of the answer to the question

[strong ridge]

Really appreciate the help @fallow scarab

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[nocturne shore]

Hello. We are tasked with proving that the collection of open sets in the real numbers form a topology. We need to prove with the following subproofs.

1. The empty set and the set of real numbers are open.
2. If we have a collection of open subsets of the real numbers, then the union of all of the subsets in the collection is open.
3. If we have a collection of open sets in the real numbers, then the intersection of all of the open sets is open.

For one, we know we should prove that the set of real numbers is both open and closed. First, to prove the set of real numbers is closed, we can use sequences and converges, but we do not know how to further this proof. Then since the set of real numbers is closed, the empty set, its complement is then open. Then, to prove that the set of real numbers is open we know we can choose some delta for any x and if we create a neighborhood anywhere, it will be contained in the reals. We do not know how to word this formally.

For two and three, we are fairly lost at how to get started.

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[bitter tartan]

help wit h'2 i did g(f) * g' gf(2) * 2 g(1) * 2 2 * 2

[dark tusk]

$H'(x)=f'(x)g'(f(x))$

[grand pond]

javier

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[last slate]

mine

[last slate]

So, every real number can be represented on the number line.

so, all the irrationals, are in included in the number line

like the pi

but

they took the approximation of pi, so like, 3.14 and then labelled it on the number line but how close is the approximation actually to the pi? like 3.14 is not exactly equal to pi so like ..yea basically how close is the approximation actually is? coz the approximation just shortens the long finite pi up

[lusty elm]

you can’t do more than give a bound on the error

[jagged saffron]

Well the distance between them would be π - 3.14, I'm not sure exactly what kind of answer you're looking for

[last slate]

pi - 3.14?

wa

[twilit field]

π>3.14

the error is thus π-3.14

[last slate]

so there is an error..

so you just calc the error you mean

and then label the result on the no line?

[slate canyon]

you mean result's label?

= error's label

[last slate]

okay

Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational.[1] what does countable and uncountable mean here?

.close

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[slate canyon]

what

why did you close the channel

oh i see

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[unreal pelican]

In 11 why would U * W. = -1/2

[unreal pelican]

i get how 1/2 is arrived at but why is it negative

[plush prism]

we can visually see it

if we project w onto u, then it points into the opposite direction of u

[unreal pelican]

so is dong a vector projection the only way to determine that

[plush prism]

we could also use the definition of the scalar product

[unreal pelican]

also when i do U * V = |u| |w| cos(theta) would i make w negative?

[plush prism]

a * b = |a| * |b| * cos(alpha)
here, alpha is between 90 and 180°
cos is negativ on that domain

to get the angle, we get them next to each other

[unreal pelican]

the angle isnt 60?

oooh

[plush prism]

here we see that alpha is 120°

[unreal pelican]

ahh

so is this visually what the dot product is doing?

[plush prism]

you can think of it as a projection or as the definition
only problem with the projection is that the vectors aren't always normed/unit, so we can get problems

[unreal pelican]

isnt the vector projecting different

given you divide by magnitude

[plush prism]

yes

if we project w onto u, we divide by the magnitude of u
but here, u is a unit vector, so it's magnitude is 1

[unreal pelican]

ooh

i guess im confused where the W line comes from

at 120*

[plush prism]

you mean why i was able to move w?

[unreal pelican]

ues

yes*

[plush prism]

if we want the angle between two vector, we want the vectors's base to be at the same place

otherwise we can't really tell what their angle is

[unreal pelican]

right

aaah

[plush prism]

you could also do this

[unreal pelican]

im thinking about hte traingle too much

these are separate vectors

ok

i get it

tyvm

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[last slate]

How do I know that row=1/2 cos phi is a sphere?

[last slate]

i put it into desmos to see its a sphere but How do I get a sphere from the given equation?

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[last slate]

<@&286206848099549185>

[last slate]

<@&286206848099549185>

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@last slate Has your question been resolved?

[last slate]

<@&286206848099549185>

[midnight plank]

@last slate Has your question been resolved?

[last slate]

<@&286206848099549185> please

[pure shadow]

Hey

[ashen venture]

It depends only on phi, so intuitively it is composed of stacked circles, of radius 1/2 cos(phi).

And when only considering the radius of those disks along the vertical axis, cos(phi) gives exactly a circle

so you rotate that around the vertical axis, and you have a sphere

[last slate]

so is row the quivalent of the raidius that's why cos(phi) is radius?

is there any way to prove or like write it out or is it merely off intuition

@ashen venture

[ashen venture]

like thus basically

[ashen venture]

technically, you can prove that it is indeed a sphere, but the proof itself is the process of finding it's a sphere

Rogorously (as far as I'm aware), there aren't any methods which are actually easier than just trying to figure out what it looks like with drawings on a draft

[ashen venture]

add a 1/2 but it's just a scaling factor

[last slate]

hmm ok well thank you for the explanation

[ashen venture]

you're welcome

[last slate]

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[analog mortar]

Ok, so i am doing raycaster

[analog mortar]

and i can get intercept of ray and wall

but i need to draw image from that

if wall is number line i need to know where in that number line intercept is

so i can start reading from "brick.png"

also i do not know much about vector operations, or linear alg 😢

https://www.youtube.com/watch?v=v6adtBlOtSc&t=60s

here is video where guy talks about this concept but doesn't go into implementation

i would prefer sources and names, i dont actually need some one to explain to me in chat. That would take long

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[viral dagger]

there are 2 groups of troops with diffrent ammount of troops,in each group half of the people wear a black uniform and the othwr half wears a green uniform, both groups make a rectangle where the ones with the black uniform are on the edges of the rectangle and the green uniforms are in the inside of the rectangle, how many total troops are there?

[viral dagger]

the choices are
105
106
107
108

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[pearl hull]

Have you solved your question?

[viral dagger]

.reopen

[midnight plank]

✅

[viral dagger]

no

[pearl hull]

okay

I’m trying

[viral dagger]

kay

[pearl hull]

The answer is 108?

I’ve done my calculations

[viral dagger]

how did you get that?

[pearl hull]

I guess the question is requesting for possible quantities, right?

[pearl hull]

You have to assume the numbers of soldiers

[viral dagger]

huh

[pearl hull]

Interesting question, I’m impressed

[pearl hull]

You can get ab-(a-2)(b-2)=(a-2)(b-2) with this graph

ab-(a-2)(b-2) represents the number of black
(a-2)(b-2) represents the number of green

Since peoples are equally separated into two groups

So the equation stays true

After that, just start plugging in numbers orderly

ab = the total number of a certain group

[viral dagger]

so you just try stuff out until it works?

[pearl hull]

Starts from a=1

[pearl hull]

No, I observed it and started from a=4

a=1 and a=2 don’t exist since you won’t get the middle rectangle in that case

3 is just too small as well, only one chunk in the middle, which is still far from the given selections

You can try out a=4 to a=8
Don’t go far because a certain group will be greater than the given answers

Not the results down

And just look at your results and add up two of them to get the answer

What I tried was
a=4 ab=36
a=5 ab=60
a=6 ab=48

And 60+48 is one of the selections

So I stopped

My work

@viral dagger

[midnight plank]

@viral dagger Has your question been resolved?

[viral dagger]

how bout this?

[pearl hull]

Dude, are you going to attend Math competition? 💀

[viral dagger]

kinda

[pearl hull]

That’s good, I’ve never attended one so far

[viral dagger]

oh i actually got it using telescoping nvm

[pearl hull]

Any more tricky questions? Xd

[wanton shore]

Can you tell the solution?

[viral dagger]

it was 1012

[viral dagger]

none anymore sadly

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[cosmic shuttle]

i am trying to find the probablitiy of getting a '3 of a kind' when being delt 3 cards. Do i have to divide by 3! or not?

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[unique cedar]

hi

[unique cedar]

specifically for the first one, my first answer was (1,3) but that was incorrect

[last slate]

the derivative of a position function is a speed function (disregarding vector stuff)

so, asking where the figure is "speeding up" is akin to asking where the function has positive-valued derivatives

and positive-valued derivatives is akin to asking where the function is "rising"

does this help in any way

[unique cedar]

yes

it is rising at (0,1) and (3,4)

[last slate]

yeah

thats correct

[unique cedar]

but when i typed that in it was also wrong

[last slate]

show what u typed

ah wait

the issue is that the curve doesn't end at 4

it goes on to infinity

so the curve is increasing from 3 to infinity as well

well

[unique cedar]

like (0,1),(3,infinity)? i tried that also but it didnt work. do u want a pic?

[last slate]

maybe not. I guess they are not denoting the end of it arrowed so I'm inclined to think it is 4

[last slate]

sure

[unique cedar]

ill try using brackets

by the three

[last slate]

thats not interval notation

thats the issue

Infinity is wrong it ends at 4

but like

,, (0,1) \cup (3,4)

[unique cedar]

yea i did that but it didnt work

want me to show the ones i got right?

[last slate]

ok

[last slate]

wtf

[unique cedar]

i got 9/50 tries so we're good lol

[last slate]

this seems so contradictory. If what they mean by "speeding up" is acceleration then neither of them make sense. If it is meant to mean just speed itself it still doesn't make sense

ok ill leave it to someone else to verify this because i think this is just wrong. But I could be misunderstanding i guess

[unique cedar]

thanks for the help though

[sturdy tendon]

Cna I aks for help on this

[exotic quest]

still having trouble on the original a) ? its been a hot minute but i think i understand the error?

!occupied

[midnight plank]

Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).

[unique cedar]

yes i am

[exotic quest]

my guess is that what its really asking for is for absolute increasing/decreasing functions?

let me try and figure out the interval it might just be the same lol

[unique cedar]

ok thank you

[exotic quest]

im pretty sure they want acceleration

when they say speeding up

so instead of s (position) 'going up' at (0,1) and (3,infinity) you're more looking at the change in change in s lol

its 'increasing in total change in either direction' in intervals (1,2) and (3-infinity)

again not 100% on that but if its not (0,1) then its probably that

do you understand how you deduce which is up and which is down

[unique cedar]

ok ty i got it right

tysm!

[exotic quest]

(if you were curious im pretty sure this is why)

[unique cedar]

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[toxic flower]

I've got an answer for this, but its completely wrong and idk why

[fading niche]

for no solutions you want the lines to be linearly dependent for a unique solution make sure that the determinant exists and for infinite solutions make the determinant 0 i believe

[toxic flower]

Det(A) or det(A|b)

[fading niche]

the determinant of the transformation

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[last slate]

what are the steps to solve for X?

[frail heron]

couldn't the RHS be made simpler in this answer (I + B^-1A^-1)^-1

Steps are similar to algebra, but without commutativity

[last slate]

IDk

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[last slate]

my logic was to find the derivative of both equations and set them equal, effectively giving us the same slopes aka making them parallel

was having some math complications though doing that process

^ would like someone to check if this method of solving is even right

and if it is how to go about the application

$\frac{1}{2(\sqrt{x+1})} = \frac{x+4}{6}$

[grand pond]

wyldinwilliam

[last slate]

$\frac{6}{2(\sqrt{x+1})} =x+4$

[grand pond]

wyldinwilliam

[slender walrus]

you shouldn't be setting the derivative equal to (x+4)/6

[last slate]

how come?

oh wait I think I know what you mean

you're supposed to get the slope of that eq'n and use that to find the original?

or set it equal to it atleast

[slender walrus]

you want the slope of x-6y+4 = 0
and set the derivative of the function to be equal to that

[last slate]

oh okay that makes more sense than what I was doing originally

so the slope would be 1/6 right

[slender walrus]

yes

[last slate]

gotcha

let me try working that out real quick

I'm getting x be 10

wait that's wrong

x=8

so the equation should be y=4x+b

[slender walrus]

no

[last slate]

okay I might've used a rule that isn't a rule

can you inverse both sides?

I don't think you can right

wait nvm you can

okay here's my work

$\tfrac16 = \frac{1}{2\sqrt{x+1}}$

[grand pond]

wyldinwilliam

[last slate]

$6=2\sqrt{x+1}$

[grand pond]

wyldinwilliam

[last slate]

$3=\sqrt{x+1}$

[grand pond]

wyldinwilliam

[last slate]

$9=x+1$

[grand pond]

wyldinwilliam

[last slate]

$x=8$

[grand pond]

wyldinwilliam

[slender walrus]

getting x=8 was fine
but where's

so the equation should be y=4x+b
coming from

[last slate]

oh my apologies

x=8, we'll use that to find our slope

actually

no we already do have our slope

we'll use it to get our y value

which is convinently 3

$3=\tfrac16 (8) +b$

[grand pond]

wyldinwilliam

[last slate]

$b=\tfrac53$

[grand pond]

wyldinwilliam

[last slate]

$y=\tfrac16 x+\tfrac53$

[grand pond]

wyldinwilliam

[last slate]

seems they want it in standard form

I suppose you could multiply both sides by 6

$6y-x-10=0$

[grand pond]

wyldinwilliam

[last slate]

that's it, thank you

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[wispy sonnet]

how do I solve this

[placid spoke]

there is no solving, that's the definition

[junior flower]

people trying to troll with the riemann hypothesis are so bad at it

this happens like every time

[slender walrus]

if you intend to continue using this server as a resource for learning, its a very bad idea to troll

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[grave locust]

can someone tell me how i would graph this inverse quadratic?

like what should i do to get the vertex

[inner wolf]

do you know the general form of quadratic

y^2=4ax

you just have to complete the square on right side and you'll get similar form, you can get the vertex from dat

[grave locust]

oh ok

oh ok

its 2 and 4

i meant

2 and -4

yes

thanx

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[twilit field]

!status

[midnight plank]

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

[twilit field]

4

D

as sin is an increasing function

[brittle rampart]

sin is not increasing everywhere

[twilit field]

oh yeah

nvm

[brittle rampart]

but you are on the right track id say

[twilit field]

hmm, x^3 is increasing everywhere

so I would think that A is right then

[brittle rampart]

yes. you can also prove it using inequalities

and the proof makes use of the fact that x^3 is an increasing function

[twilit field]

hmm

[violet storm]

you can also come up with counterexamples to the others pretty easily:
B) -2, 0, 1, (-2)^2 is largest, not min
C) similar idea with negatives: -2, 0, 1 |-2| is not the min
D) in the 2nd/3rd quadrants, theta is increasing but sin(theta) is decreasing (y coordinate of unit circle is going down)

[twilit field]

Yeah, got it thanks!

I'm trying to think of a proof involving inequlaities

let r<s<t

then as n^3 is increasing everywhere

r^3<s^3<t^3

[violet storm]

you can kinda just say:
r < s < t assume this without loss of generality, in other cases you can just rename the variables
r^3 < s^3 < t^3 then just cube everything 😄
ya exactly what you did

[twilit field]

this can similarly be proved for any permutation of r,s,t

[twilit field]

thanks!

can I close this now?

[violet storm]

i guess technically <=

[twilit field]

$\geq$?

[grand pond]

Why am. I here

[violet storm]

since they arent necessarily distinct 😄

but yeah

[twilit field]

Ok, thanks!

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[cloud lintel]

for Q2

how does the curve increase with downwards concavity

the second derivative is always positive so doesn’t that mean it’s always concave up?

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[cloud lintel]

<@&286206848099549185>

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[last slate]

u, v are vectors (n x 1)
A is a matrix (n x n)
is this true:
u(Av) = (Au)v

[last slate]

⚠️ you cannot write the left hand side. Your matrices are of size:
n × 1, n × n and n × 1

u(Av)
isnt Av going to be processed first, giving a n x 1 vector
which is going to be multiplied (inner product) with u which is also a n x 1 vector

You should then take the transpose of u

ohhhh right

i see

i forgot abt this

thank you!

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[last slate]

a bit confused about the last part. Why is it not 1-1 on the boundary?

[last slate]

oh for context's sake, the domain R is taken to be a rectangle with bounds

a < u < b and c < v < d

I suppose the value of your function on the boundaries must be equal at some points

can u explain further please?

find the value on r on the boundaries of its domain and solve the corresponding equation. If it is not a 1 to 1 correspondance the equation r(φ,θ) = m doesn't have a unique solution

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[last slate]

Hi

[last slate]

How do I find the y-intercept?

[unique stream]

what did you attempt so far

[last slate]

Im not sure where to start

[unique stream]

hm, can you determine the gradient of the line?

You know that the line goes through (1,6) and (5,0)

and to get the gradient of any line you can always use
Difference in Y

Difference in X

So (0 - 6) / (5 - 1)

Since from the point (1,6) to (5,0) the difference in Y is -6

and the difference in X is 4

[last slate]

I got the gradient

Rise/run

[unique stream]

k, now if you know that the gradient is -3/2

can you find the point (0,???)

on the line

its equivalent to finding any point on the line

I could also ask for what's the Y value of the line if I give you X = 17 instead

you have (1,6) as a point on the line. If we decrease the X value by 1, how does the Y value change?

[last slate]

Is it (0 , 7.5)?

[unique stream]

yop

if you decrease X by 1

then Y decreases by the gradient

so -(-3/2) = +3/2

can you also find where the line intersects at X=17?

it's the same principle

[last slate]

Sorry im not understanding this. Can you explain it in another way pls?

Like this

What steps would i tale to fin the y int

Gradient is 2

[unique stream]

well you have the point (-1, -5)

and you seek (0, ???)

just like before

since the gradient is 2 the line must go through

(0, -3)

(1, -1)

(2, 1)

(3, 3)

(4, 5)

etc

You can determine any intersection

it's not just about the x-axis and y-axis intersections

[last slate]

Ohhh so i just add (or subtract) the gradient from the y intercepts?

[unique stream]

for whole integers yes

that's less or more the meaning of the gradient :D

[last slate]

😆 thank you

[unique stream]

So if you take the above line

and I ask you where does the line intersect for X = 0.5?

so (0.5, ???)

[last slate]

-2?

[unique stream]

yop

and where does it intersect for X = 0.1? 🐣

[last slate]

-0.4?

[unique stream]

hm not quite, but I'd be intrigued how you tried to solve it

[last slate]

Is it -0.5?

[unique stream]

nop

how do you try to calculate it

[last slate]

For the first one, I did -2÷5

I got -0.4

[unique stream]

hm, well we know that the line intersects (0, -3)

now what happens if we increase X by 0.1?

[last slate]

-0.3?

[unique stream]

then Y increases by 0.1 * gradient

[last slate]

So for example:
Coordinate is (2,4)
Gradient is 2
X increases by 0.5
Y is now y+0.5x2?

[unique stream]

Exactly!

you can do so for an arbitrary point

I could also give you some point
(14, 20) and a gradient of 6
if X increases by 85.33
then Y increase by 85.33 * 6

[last slate]

Are you an actual teacher? You teach very well

[unique stream]

or if Y decreases by -2
then Y decreases by -2 * 6

[unique stream]

nup :]

[last slate]

What grade level is the math we r doing rn

[unique stream]

hm am not sure in the American system, maybe 7th?

[last slate]

Oh where r u from

Im from Australia

[unique stream]

germany

[last slate]

7th graders learn this stuff in usa???

How old r they

[unique stream]

idk as said, somewhere between grade 6 and 9 probably come functions and their properties

[last slate]

Im in yr 10 and im learning this from you rn the school didnt even teach us this

[unique stream]

oh ok

[last slate]

Well thank you very much for your help today 😊

[unique stream]

but the power of the above principle allows you to find any point on a line, x-axis and y-axis are just special cases where Y=0 or X=0

[unique stream]

np 🐛

have you had functions before?

In the form of polynomials for instance:
f(x) = 7x + 3

[last slate]

No

Thats some yr 11/12 stuff

[unique stream]

yr 11/12 = grade 11/12 or in terms of age 11/12

[last slate]

age 17/18

[unique stream]

kk

maybe to give a brief rundown, I presume almost everything you have encountered so far in math is rather static, numbers have fixed values, geometric objects have fixed properties etc

idk what other aspects of math you had so far but at least I'd presume so

[last slate]

This is correct

[unique stream]

but then at some point one may wonder why we don't also define some new type of "object" or "thing" that isn't static

[last slate]

The most weve done where someone can mean more than two numbers is something like a=+-4

[unique stream]

ah k, maybe PQ-formula?

at least that's where we were introduced to ±

nywys

[last slate]

Like this?

Ive done this before i think I’ve forgotten

[unique stream]

kk, but that'd suggest you have done functions before

in the form of f(x) = ...

[last slate]

Idk im getting confused 😭

[unique stream]

regardless, consider that we'd try to invent some new type of math object that isn't actually static, it doesn't have a fixed value for example

but rather, we want something that can have multiple values perhaps

preferably like a machine, we throw a value in and the machine spits out a new value

that machine wouldn't be static because its output depends on what we throw in

so in specific we would e.g. give that machine a 3 as input and it spits out 5

if we ignore what the machine does for now

it's just about this new concept of a machine that has input & output

and in essence that's how you can picture what a function is meant to do in math

You give the function an input, for instance a number (but it doesn't have to be, I'll get to that in a sec) and it might spit out another number

so just like how you could write variables before:
x = 5
x would be a static variable with the value 5

we might now also define this new type, a function:

which for the sake of notation is written as
FunctionName(Input) = Output

@last slate understandable so far? :)

[last slate]

👍

[unique stream]

so maybe to make a super simple function to start with, let's say we have a function that we call Useless

[last slate]

That looks like code

[unique stream]

in a way it is like code yeah

and that function Useless spits out the number 0, regardless of what we throw in

then we'd that function this way:

Useless(Input) = 0

a function can have any name, just like a variable

usually you'll find that most mathematicians just use the name f for functions because it denotes "f"unction

but it's really just the name of it

ok so above I defined that new function called Useless

if I want to work with it now, I could throw something in, for example the number 5

which you'd write as
Useless(5)

and we can calculate what comes out, since we know what the function does

it always spits out 0, so we know:
Useless(5) = 0

or I could throw something else in, idk, maybe your name Shinémabu

Useless("Shinémabu") = 0

it doesn't matter what I throw in, the function will simply spit out 0, that's how we defined it above

ok that function didn't really seem that useful

maybe let's try one with a better output

eg I could define a function called Increase:

Increase(number) = number + 1

and besides this definition I additionally state that the function Increase only allows numbers as input

picture it like a machine that only allows specific types of input

[last slate]

Functions is how we tell our calc what to do?

[unique stream]

so with that new function, I can at least do something a bit more interesting, eg
Increase(6) = 6 + 1 = 7

[unique stream]

it doesn't have to be about the calculator, it's about a fully new concept of mathematical things

before you only had variables, geometric objects like lines/cubes/triangles/...

this new thing, functions, allow for so many new possibilities

now I might create another little function, which I'll call Square:
Square(number) = number * number

Also straightforward

Square(4) = 4 * 4 = 16

or I could even use a distinction, with a new function I'll just call, idk, Blub:

Blub(number) = 0 if the number is even else 1

so Blub(5) = 1

Blub(2) = 0