help-36

[pliant shore]

not (3 + 1)(4)

[amber portal]

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[pliant shore]

the rest is correct

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[tranquil pine]

if a/b = c/d
Then is (a+b)/(ax+by) = (c+d)/(cx+dy)?

[soft zealot]

Wolfram Alpha doesn't understand your query!
Perhaps try rephrasing your question?
Click here to refine your query online

Requested by anjaliji_74295

[willow stone]

a=bc/d from the ratio

substitute it in the eq n you'll find out

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@jaunty mauve Has your question been resolved?

[rain ember]

mhm

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[smoky oxide]

can someone help me with this?

so we have this formula for derangement. how do i prove that Dn is even if and only if n is odd?

[whole halo]

When n is 1, 2, 3, 4, 5…
See whether n! is odd or not
Then move n! into the sum and rewrite as a sum of integers

[vital crag]

possibly also find a recursive formula

[smoky oxide]

how?

[amber holly]

Or you can also consider the parity of each n!/i!

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[copper sorrel]

how do I find out if the following is convergent or divergent

[vital crag]

log(n) < sqrt(n) for all n > c for some c

c =3 or 4 or something

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[clever solstice]

i wonder if you guys are familiar with these

[vital crag]

this is hardly math

try looking for a sql server

[clever solstice]

Ah ok i thought it was math cuz of "algebra" but cs kinda has its own math/languages sometimes

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[tame notch]

help

[tame notch]

can someone quickly tell me how 1/10 becomes 1/50

I get everything else

[vital crag]

u sub

[tame notch]

i wut

also I have a side question

why is the integral taken for (30+5r) and not just r^1/2 by itself

[vital crag]

https://tutorial.math.lamar.edu/classes/calcI/SubstitutionRuleIndefinite.aspx

[tame notch]

from that

I'm unclear how 4/5 prompts

oh it's

1 / 1+ 1/4

4/5

duh

but then

1 / 1+ 1/2 = 2/3

which again I get

I still don't get how the 10 turns to a 50

[vital crag]

did you try using u sub

[tame notch]

wait before that

[tame notch]

why isn't 18x^2 integrated

i got it by mathway

u = 50+r5,
du = 5dr, isolate for dr
dr= 1/5 du
sub back in

but I don't get why 18x^2 drops out in the example you sent

unless it's because

18x^2
dx = du / 18x^2
and then the top bottom 18x^2 canel out

leaving you with integral (6x^2 + 5)^(1/4) dx

which is just 1/(1+1/4) or 4/5 (6x^2+5) ^(5/4)

is that right?

u= whatever
du = whatever dx
and then dx = du / whatever

is that the general idea?

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@tame notch Has your question been resolved?

[tame notch]

Is it du = (whatever du is) *dx
And then isolate for dx ?
@rie.mann

Is that the general ideaaaaaaaaaa

That seems to be the general ideaaaaaaaa

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@tame notch Has your question been resolved?

[tame notch]

Can you please just verify

[tame notch]

I'd this is it

I believe so

That that's it

I'd just like confirmation

[vital crag]

Just use Wolfram if you want to verify definite integrals

[tame notch]

But is that the way to do it

The sub with u

It's

du= (result) dx
dx= du/(result) and then plug back in

@vital crag

So du is always 1/du when plugged back in

Unless there's a negative exponent

[tame notch]

.close

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[sage river]

How do I solve this without the graph bruh

like

-X^2 + 4X <= 0

hi knief

im having the same problem again

[rocky tusk]

hi

[sage river]

idk if u remmeber

LOL

so denom is 3, numurator is 0 and 4

[sage river]

and after this i did:

-1 (x - 4) > = 0

[rocky tusk]

$x^2 - 4x \geq 0$

[sage river]

so. my intervals are (- infinity, 0) (0,3) (3,4) (4, infinity)

[soft zealot]

knief

[rocky tusk]

$x(x-4) \geq 0$

[soft zealot]

knief

[sage river]

negative at the front

[rocky tusk]

for equality, x = 0 or x = 4

[rocky tusk]

i flipped the sign

[sage river]

uh

bruhhh

[rocky tusk]

bruhhh

i like positive x^2

[sage river]

ya but even if uflip the sign its suppose to be

-x(x-4) > = 0

[rocky tusk]

^?

$x\leq x^2-3x$

[soft zealot]

knief

[rocky tusk]

$0\leq x^2 -4x \iff x^2-4x \geq 0$

[soft zealot]

knief

[sage river]

one sec im just gonna show a picture of what i did

Ignore the answer at top left

I’m just trying to see why I can’t get it with my graph

So as u can see, my graph says it’s positive at two spots which is opposite from the answer

[rocky tusk]

well you had $\frac{-x^2+4x}{x-3} \leq 0$

[soft zealot]

knief

[sage river]

Ya isn’t hat right

[rocky tusk]

but you ignored the x-3 it looked like

[sage river]

No I know that’s the denominator I was just solving the numurator

[rocky tusk]

$\frac{-x(x-4)}{x-3} \leq 0$

[soft zealot]

knief

[rocky tusk]

hence

[sage river]

Yes

[rocky tusk]

$\frac{x(x-4)}{x-3} \geq 0$

[soft zealot]

knief

[rocky tusk]

for equality, x = 0 or x = 4

note that x ≠ 3 so we should also consider that point

[sage river]

How do I know when to flip the sig.

[rocky tusk]

when dividing/multiplying by a negative

[rocky tusk]

you can also say $0\leq \frac{x(x-4)}{x-3}$

[soft zealot]

knief

[rocky tusk]

just from adding it to both sides

[rocky tusk]

this is the same as

[rocky tusk]

^

[sage river]

Ok I see

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[thorn stump]

i should just use that projection formula thing right?
(x dot u)/(|u|)^2 and multiply it with the vector u

cool cool thank you

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[timber rivet]

how can I prove that this series is bounded?

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@timber rivet Has your question been resolved?

[wicked robin]

Idk if it might be helpful, but notice that 2sqrt(n) = 2n/sqrt(n) = 2/sqrt(n) + … + 2/sqrt(n)

So you can rewrite the partial sum $x_n$ as $\sum_{k=1}^n \left(\frac{1}{\sqrt{k}} - \frac{2}{\sqrt{n}}\right)$

[soft zealot]

Aslan

[wicked robin]

But this is still sort of useless I think

[timber rivet]

😭

[wicked robin]

Hm maybe it’s possible to show that that first sum term u got in $x_n$ is bounded above by $2\sqrt{n} -1$, hence $x_n$ must be bounded above by $ -1$

[soft zealot]

Aslan

[wicked robin]

Then u want to somehow show that x_n is bounded below aswell which might be the hard part

[timber rivet]

first sum term u?

[wicked robin]

Like the sum u had in x_n

“u” as in “you” lmao

I’m just being lazy by not saying the sum 1 + 1/sqrt(2) +…+ 1/sqrt(n)

[timber rivet]

yea i see now but I guess that it might be impossible

[wicked robin]

Impossible?

Oh do you even know if it is bounded or not?

[timber rivet]

aah yes it's bounded and we have to prove it, What I mean is that I think it might be a bit more complicated lol

But nvm, I don't really know

I think this is the only way we can work now anyway

[timber rivet]

Is this bc that the first term 1 + 1/sqrt(2) +…+ 1/sqrt(n) is easier to evaluate while the last term is more difficult to handle?

[wicked robin]

It seems easy to bound above by 2*sqrt(n) - 1 is all

Like there isn’t really a question of x_n being bounded above of course

The obvious problem is to bound it below

[wicked robin]

So can you bound this sum below somehow?

And it still being useable of course

Like we want 2sqrt(n) to cancel out and leave a constant or something which is bounded below

But this is just one mode of thinking

[wicked robin]

I attempted here to convert into a more concrete series but that didn’t really work

[timber rivet]

I just realized I misused a word

we want to prove that {xn} this sequence is bounded

not the sum

my fault

[wicked robin]

Yeah I know

I figured

[timber rivet]

but still how can we think of it

[wicked robin]

I mean yeah I tried thinking about as x_n being a partial sum of some series

But that didn’t work

Then I tried to just argue about normal sequences being bounded like above

So you want to show x_n is somehow bounded below

Then a way to do that

Is to the bound the sum in your sequence from below

The sequence is not a sum

But it’s equal to a sum plus some other term

That is what I was talking about when I said sum btw

In any way I have to go so good luck!*

[timber rivet]

Oohk ty!!

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[jagged flare]

if EF is the angle bisector of CED then how would i find the length of EF?

[jagged flare]

CE=21 DE=28

[odd seal]

Dumb pythagoras method: Drop a perpendicular EG from E to make a similar triangle to the DEF. Then calculate distance between FG, and use pythagoras' thm

[normal dirge]

you could also brute force using sine rule.. CED is given to be a right angle?

[jagged flare]

yeah

waitt stewarts theorem exists

[tranquil pine]

use area

[jagged flare]

is this correct?

.close

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[jagged flare]

eugh wrong image

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[arctic slate]

bro how do u open a help channel

[arctic slate]

oh

[raven marsh]

💀

[arctic slate]

what's the answer

[tranquil pine]

we can't tell you the answer directly

[arctic slate]

oh

[tranquil pine]

but keep in mind the characteristics of the graph

[arctic slate]

what's the process

to solve it

[tranquil pine]

walk through each option 1 by one
and just check if the graph corresponds to the option

[arctic slate]

bet

[tranquil pine]

do you want me to walk you through the options?

[arctic slate]

yes please

🙏

@tranquil pine

so is garrett the correct one?

[tranquil pine]

wai mate

[arctic slate]

mb

[tranquil pine]

mate allison is right btw @arctic slate

since the gradient of the curve is 3/4

aka 0.75

[arctic slate]

oh

so

how do I do something like this

[slender rampart]

find the slope of MO and OQ

[arctic slate]

how

[slender rampart]

delta y over delta x

so change in y over change in x

[fleet flame]

this is linear so you can just bound the phenogram

just do it bruh

[slender rampart]

lets use (0,3) and (6,6)

for line MO

the equation for calculating the slope is $\frac{y_1-y_2}{x_1-x_2}$

[soft zealot]

pppoopoo

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✅

[violet stream]

D,A,B,C,1/2,75

[slender rampart]

equation for calculating slope is $\frac{y_1-y_2}{x_1-x_2}$

[soft zealot]

pppoopoo

[slender rampart]

!noans

[final saddle]

The purpose of this server is to help you learn, not to hand out answers. Do not ask someone to give you the answer directly.

[violet stream]

fourth one is D

But do you know how I got these answers?

You want to know?

[fleet flame]

bro just give him the wrong answers

nvm he already doing it

[violet stream]

You know how the equation of the line looks like?

Linear function

y = kx+ n

have u seen this anywhere

ok well that equation represents a line. You are given some k which is the slope and n which is the value where the line crosses y-axis

then whenever you look at the graph and you want to know the value of that line at the point x, you plug in kx+n and that is the value

You understand what I just said?

https://www.desmos.com/calculator/adowej4zoc

ok click this link and try moving the values k and n around

[unreal sigil]

I am curious. Are these assignments you got from school? And if so wouldnt you have gotten an explanation from school?

[violet stream]

put n = 0

yeah just write n = 0 or move the slider

[fleet flame]

desmos looking ahh

type shi

[violet stream]

yes

and then put k = 1

broo

it was 1/2 for previous answer

and 75 for this

0.5 = 1/2

fractions

you leanre that?

good, so write 1/2

no man

YEAH 1/2 IS THIS

.

75

But listen i gtg, you can do it, understand linear function

Go try to look some yt tutorials if my explaintion is shit im sure there are plenty it will only take u 10-20mins

[violet stream]

and u will know to solve this problems on your own

[lime crest]

what are you asking

its all right

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[strange shoal]

ok so i have the solution

the solution is x = a+(b.a/(1-b.c))c

its the 2nd part

thats struggling w

[gritty solar]

Well what happens when b.c = 1?

[strange shoal]

well ofc u get a divide by 0 error but thats not the point of the question

its talking abt

whats the significance

@gritty solar

[gritty solar]

I think all you can conclude is that a and c are perpendicular

[strange shoal]

how have u come to that

[gritty solar]

dot product b in the original equation

[strange shoal]

yhyhh

alr

thanks bro

i acc appreciate that

i missed that

but thanks

.close

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[obtuse knoll]

uhm, in this line, is this answer wrong?

[obtuse knoll]

P is false and Q is true, P->Q should be true as well? and then true->R(false) will give us 0/false

just kinda confused since this is the solution we were given and I think it's wrong

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@obtuse knoll Has your question been resolved?

[paper oxide]

yes it looks like a mistake

[obtuse knoll]

thank you

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[fair gulch]

I have the first derivative here.

How do I find the local maximums and minimums of 1 and 3?

Am I doing it wrong?

[scarlet igloo]

you did the derivate correctly

to the find the critical points you find every possible way the function can = 0

then plug numbers left and right of the critical numbers into the original function (before you did the derivative) and the signs of those numbers will tell you if it is going up or down

so if the left of the critical number it’s positive and the right of the critical number is negative then its a local max and vice versa

[fair gulch]

What I got was 2x^3-12x^2+18x but I don't know how to find the CN for those.

As my first derivative.

[scarlet igloo]

the critical numbers are all the value of x where the function is zero

so if x were 0 the whole function is 0

[fair gulch]

Right.

[scarlet igloo]

so that’s a critical number

[fair gulch]

So what would the other ones be then?

[scarlet igloo]

0, 2, 3

[fair gulch]

How did you find those?

[scarlet igloo]

if you plug those numbers into for x the function = 0

it’s easier to do if you factor out the function a bit more

[fair gulch]

Right but I'm trying to find them on paper.

[scarlet igloo]

2x(x−3)(x−3)=0

[fair gulch]

Yeah I got the same result as you, but I don't get how you got 2.

[scarlet igloo]

i recommend watching professor leonard on youtube

i learned all i know in math from him

[fair gulch]

Should I ping @ Helpers for this?

[fair gulch]

@spring depot Could you help with my original problem?

[scarlet igloo]

i messed up it’s not 2

don’t worry

i was grazing through it

[soft zealot]

JD

[fair gulch]

Why is f'(x) saying there's a point at 1 though?

[scarlet igloo]

it’s the inflection points

using the second derivative

[fair gulch]

So are these, like, mismatched or something?

[scarlet igloo]

the second derivative is how you find concavity

it’s the third part of the problem

[fair gulch]

I see.

I think I figured out the issue lol.

Ty for helping.

[scarlet igloo]

np

if there’s anything just @ the helpers i have to go

if not you can close the convo

[fair gulch]

Okay.

[final saddle]

@fair gulch Has your question been resolved?

[fair gulch]

How do I find the slant asymptote of this TBH?

[patent chasm]

hey @fair gulch

[fair gulch]

Hi.

[patent chasm]

wht is -16,32

[fair gulch]

Not that.

The equation atop it.

[patent chasm]

the symptote is x= -8

[fair gulch]

Slant asymptote.

[fair gulch]

<@&286206848099549185> Would you be able to help me with this or nah?

[patent chasm]

if u show me figure of symptote i wud try to find eution

[fair gulch]

.close

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[mortal berry]

$\int \frac{x^2}{1+x^2}$

[soft zealot]

crazytime

[raven marsh]

hello

[mortal berry]

Hello

I'm in trouble

[gritty solar]

• 1 - 1 in the numerator

[tranquil pine]

Yup!

[raven marsh]

note that the integrand is 1 - 1/(1+x^2)

is there anything special about the fraction?

[tranquil pine]

I remember asking this exact question 3 months ago when I had just started with calculus 😅

[mortal berry]

$\int 1 -\frac{1}{1+x^2}$

[soft zealot]

crazytime

[raven marsh]

yes

[mortal berry]

Oh

[raven marsh]

what can we do with the fraction $\frac{1}{1+x^{2}}$

[soft zealot]

Death.cv

[mortal berry]

$\arctan(x)$

[soft zealot]

crazytime

[tranquil pine]

$$\int \frac{x^2 + 1 -1}{1 + x^2} dx$$

$$=\int \frac{x^2 + 1}{x^2+1}dx - \int \frac{1}{1+x^2}dx$$

[soft zealot]

Edmund Cloudsley

[tranquil pine]

That’s correct

[mortal berry]

Thanks very much

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[tranquil pine]

I think I also saw a solution that uses U-sub for this question

I can show that as well if you’d like

[raven marsh]

np

[tranquil pine]

Bit of an overkill but yea good practice if you’re just starting out

[raven marsh]

yeah i think you could use that

[mortal berry]

.reopen

[final saddle]

✅

[raven marsh]

but theres no need to

[mortal berry]

Ok!

[mortal berry]

.

[tranquil pine]

Yeah I’m typing that just sec

[gritty solar]

wat

[tranquil pine]

Using latex

[gritty solar]

u sub or trig sub

[tranquil pine]

Yeah trig sub sorry

Trig I meant trig

[raven marsh]

oh

[tranquil pine]

Ok the latex worked lol

Here

Took ages to type 🥹

[soft zealot]

Edmund Cloudsley

[tranquil pine]

This is a much more 🤮 way of doing it @raven marsh mentioned

Using +-1 is much more elegant

[raven marsh]

yeah but nice proof tho

[raven marsh]

thats so familiar

[tranquil pine]

I literally had to find my book for this

Given how stupidly and unnecessarily complex it is

[raven marsh]

lol

[raven marsh]

yeah

[mortal berry]

👍

Thanks very much again

[tranquil pine]

Happy to help

Have a great day

[mortal berry]

Nice solution

[tranquil pine]

Is it really nice? 😓

[mortal berry]

Yes!

It's alternative

[tranquil pine]

Okay

If u say so

But don’t ever use it in an exam

It’s far too long

[mortal berry]

Okay!

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[warm python]

$b=a^2k_1$
\
$c = b^3k_2$
\
$\implies c=(a^2k_1)^3k_2$
\
$\implies c=a^6k_1^3k_2$
\
from this we can conclude that $a^6 \mid c$

[soft zealot]

A dense set

[amber holly]

Looks good

[warm python]

Cool

thanks

I apologise for asking y'all to verify such simple proofs just revising for my end terms

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[warm python]

.reopen

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✅

[warm python]

We here use the lemma that $a \mid b \implies a \mid k_1a+k_2b$
\
From this lemma it follows that $a \mid 3b^3 -b^2$.
\
Applying this lemma once more we get $a \mid 3b^3-b^2+5b$
\
This concludes our proof

[soft zealot]

A dense set

[wary copper]

factorising b out works directly might be a litlte more intuitive

[warm python]

Oh yeah

doing this as a direct proof is going to be fun to say th least

If $5 \mid 2a \implies 2a=5q_1$

[wary copper]

can you use euclid's lemma

[soft zealot]

A dense set

[wary copper]

ie, if prime | ab, then either prime | a or prime | b

[warm python]

Prime factorisation that is?

no

[wary copper]

https://en.wikipedia.org/wiki/Euclid's_lemma#:~:text=In algebra and number theory,those integers a or b.

[warm python]

Ah, I'll have to prove that first then

[wary copper]

kind of is a lemma of that ig

[scarlet sequoia]

alternatively, multiply by -2 and add 5a

[warm python]

Right, ann had mentioned that

[scarlet sequoia]

but yeah euclid's lemma comes from bézout theorem

[warm python]

hmm

[warm python]

I don't follow

[final saddle]

@warm python Has your question been resolved?

[scarlet sequoia]

2a = 5k

-4a = ...

-4a + 5a

[warm python]

ah

got it

thanks

so $-4a+5a=5(k+a)$
\
so $a=5(k+a)$

[soft zealot]

A dense set

[warm python]

ah

cool

thanks

[scarlet sequoia]

5 * (-2k+a) no?

[warm python]

ah yes

my bad

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[dire sky]

hello

[dire sky]

Can anyone please help me understand something in this solution? From what I see we substituted y(1)=1 in the second branch of the piecewise function But it only accepts x>1. So how did we subsitute x=1 here ??? I don't get it

[final saddle]

@dire sky Has your question been resolved?

[dire sky]

<@&286206848099549185>

[dire sky]

:/

<@&286206848099549185>

[long flame]

Hello @dire sky

It's a limiting value

We are just putting x as x tends to 1+

That's all

Hope this helps

[dire sky]

Can u explain more

Like why do we do that

[long flame]

Because we the function to be continous

For it be differentiable

So at x =1

Both definitions should be equal

[long flame]

Both definitions should be equal at x =1

@dire sky

[dire sky]

Ommm...

[dire sky]

Why tho?

[long flame]

Because if they aren't equal

[dire sky]

This is a piecewise fn right? They should be different

[long flame]

Function wouldn't be continuous

[long flame]

Basically like where the first function ends

The second starts

[dire sky]

Where did u know that it should be continuous?

[long flame]

Like this

1st definition is yellow

Which is continous from 0 to that green point

And 2nd definition is red

Which is continous afrom the green point and beyond that

The place where they needed to be continuous the green point itself

Which in your question is x=1

[dire sky]

I see I see

But

[long flame]

See here they aren't continous

[dire sky]

Why do we need a continuous equation in 1st place

I don't get it tbh

[long flame]

Because for non continous function the Differential doesn't exist

[long flame]

Like here

Differential is the slope of the function right?

At that point

What would be the slope here

Not defined

[dire sky]

Ohhh right

It's way more clear now

Thank you very much

[long flame]

Ur welcome

[dire sky]

Have a good day/night my friend

[long flame]

You too

Kindly .close the chat

If you don't have anything else to ask

@dire sky

@dire sky

Kindly close

The chat

So somebody else can get help

[dire sky]

Oh sorry I forgot

.close

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[wispy tiger]

b⋅v = a⋅v = -0.1736
solve for v given a and b
|V|=|a|=|b|=1
lets assume
vector b = (0.66590456, 0.66053921, 0.34678389)
vector a = (0.06358067, 0.78442331, -0.61695833)
then how????
or is it just impossible

[inland kettle]

this basically just means if you have the unit n-sphere

v will be on the line segment that perfectly separates a and b, in a sense

[long flame]

@inland kettle I think what you are trying to say is that v will be an angle bisector for a and b

Did I get that right?

[inland kettle]

in 2D, yes

in 3D, sort of, yeah

very apt summary @long flame, thanks!

[long flame]

Welcome

Please continue

[night spruce]

Is it true that Arccotg(tg(x)) = 1/x and tg(arccotg(x)) = 1/x? How would you prove it

[final saddle]

@wispy tiger Has your question been resolved?

[wispy tiger]

hmmm

[final saddle]

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[hoary skiff]

$$(-x^{2})^{2} = x^4$$\
or\
$$(-x^{2})^{2} = -x^4$$\
and why?

[soft zealot]

𝕿𝖆𝖘𝖐

[subtle path]

ok

first one

[wet warren]

the first one because for any negative number squaring it give a positive number

[subtle path]

the difference comes with the parenthesis

-x^4 = -(x^2)^2

but x^4=(-x^2)^2

just as

(-x)^2= x^2 but -(x^2)= -x^2

[hoary skiff]

$$(-4x^{2})^{2} = (-4)^{2} \times x^{4}$$

[soft zealot]

𝕿𝖆𝖘𝖐

[hoary skiff]

right?

[subtle path]

yes

it equals 16x^4

[hoary skiff]

so it's like
$$(-x^{2})^{2} = (-1)^2 \times x^{4}$$

[soft zealot]

𝕿𝖆𝖘𝖐

[hoary skiff]

isn't it

[subtle path]

yes !

thats one way to explain it

[hoary skiff]

$$(-x^{2})^{3} = (-1)^{3} \times x^{6} = -x^{6}$$

[soft zealot]

𝕿𝖆𝖘𝖐

[hoary skiff]

So if I understand correctly

[subtle path]

yes

[hoary skiff]

this is true too

[subtle path]

yes

[hoary skiff]

ok thanks!

[subtle path]

it basically means (-(x^2)^3

[hoary skiff]

$$(-(x^2)^3)$$
?

[soft zealot]

𝕿𝖆𝖘𝖐

[hoary skiff]

or $(-(x^2))^3$ is what you mean?

[soft zealot]

𝕿𝖆𝖘𝖐

[subtle path]

yes

the second

[hoary skiff]

$$(-1)^3 \times x^6 = -x^6$$

[subtle path]

i'm sorry i'm a bigggg noob at latex so i cant write it like at

[soft zealot]

𝕿𝖆𝖘𝖐

[subtle path]

yh thats it

[hoary skiff]

np

though it ends up being the same anyways $((-1) \times x^6) = -x^6$

[soft zealot]

𝕿𝖆𝖘𝖐

[hoary skiff]

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[cunning shadow]

would these values be correct?

[fleet flame]

just check the eigen values

[cunning shadow]

im not sure what that is

[fleet flame]

no i just wanted to say random buzzword

[cunning shadow]

uhhh

[urban burrow]

complete the square

[cunning shadow]

yes

so

take out coefficient

which is -3

so a=-3

[fleet flame]

ok it isnt the right values (i checkedc)

[cunning shadow]

aha

can you explain how you got the right values

[fleet flame]

,tex .cts

[soft zealot]

Steel

[fleet flame]

i just converted the quadratic to vertex form

by completing the square

[cunning shadow]

yes i know the general idea of it but

you see when

so you take out -3

so u divide 12

by 3

giving you 4

then you half 4 which is 2

and square 2

to get 4 again

dont u then multiply 4 by -3

and add that to 8

to get-4

[urban burrow]

,tex .cts

[soft zealot]

kaue

[urban burrow]

how do you do that

[urban burrow]

you mean -3

[cunning shadow]

ou yes

-3

[formal trail]

google "riemann math discord preamble"

[cunning shadow]

so u get -4

[urban burrow]

yesh

[cunning shadow]

so value for b = -2

then c would be

-2 squared

multiplied by -3

add 8

?

[urban burrow]

when you factor out -3, 8 becomes -8/3

[cunning shadow]

you have to divide the 8 by -3 aswell?

[urban burrow]

no hold on

you have $-3(x^2 - 4x) + 8$

[soft zealot]

kaue

[urban burrow]

remeber $(x + k)^2 = x^2 + 2kx + k^2$

[soft zealot]

kaue

[urban burrow]

you need to write $x^2 - 4x$ in that form

[soft zealot]

kaue

[urban burrow]

notice it can be written as $x^2 + 2(-2)x$

[soft zealot]

kaue

[urban burrow]

so k = -2, now you need to add (and subtract) k² (which is (-2)² = 4) so you can factor it

[cunning shadow]

im so confused

[final saddle]

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[fair ingot]

How can I find the horizontal asymptote of -7x / (4x + 2) ? Or really any rational function in general.

[fair ingot]

<@&286206848099549185>

[final saddle]

@fair ingot Has your question been resolved?

[tranquil pine]

[i read the question wrong]

[fair ingot]

oh ok lol

[tranquil pine]

you can set x to infinity if you want

i would also suggest partial fractions

[fair ingot]

so it will be -7/4 ?

[tranquil pine]

i think so

[fair ingot]

ok chatgpt said im correct so i must be correct lmao

sorry can i ask one more question?

[tranquil pine]

that is not ideal reasoning

but you are correct

[tranquil pine]

sure

[fair ingot]

with a function like (x-1) / 3(x-1), the hole would be at x=1, but wouldnt the vertical asymptote also be at x=1?

[runic chasm]

wouldn't that function graph be y=1/3?

with a hole at x=1

[fair ingot]

wow im actually dumb

yes ur right

how did i actually not realize that

[tranquil pine]

no vertical asymptote here

you may check this by finding the left and right limits at x=1 (if you really want to)

[fair ingot]

yeah

thanks a lot for answering my question

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[terse crypt]

this what i've written correct

[terse crypt]

theres no solution in the textbook

[final saddle]

@terse crypt Has your question been resolved?

[terse crypt]

<@&286206848099549185>

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[hoary skiff]

Given: \[1em]
$\bigtriangleup ABC$ is isosceles ($AB=AC$) \[1em]
$\overline{AD}$ is a median to $\overline{BC}$ \[1em]
$E$ is laid on top of the median $\overline{AD}$ \[1em]

Prove:\[1em]
A. $\bigtriangleup BEC$ is isosceles\[1em]
B. $\bigtriangleup AEC \cong \bigtriangleup AEB$\[1em]

I understand how to prove A, I have an idea on how to prove B but I am struggling a bit.

[soft zealot]

𝕿𝖆𝖘𝖐

[hoary skiff]

I proved △BEC is isosceles by proving △CED is congruent to △BED, then I thought of proving m∠AEC = m∠AEB somehow and then use SAS to prove B.

sup it's u again

mhm

ah stupid me

SSS

AAAAAAAAAAAAAAAAAAA

thanks

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[hoary skiff]

well that's what's happening when you study for so long

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[warm python]

Now this is a good question

We first prove that if $a>1 \implies x<\frac{x+y}{a}<y$

[soft zealot]

A dense set

[warm python]

wait

this feels wrong

yup def wrong

Well, I can still work with this

[scarlet sequoia]

only true for a = 2

[warm python]

yeah

$x< \frac{x+y}{2}<y$

[soft zealot]

A dense set

[warm python]

But

$\sqrt{2}{x}<x<\frac{x+y}{\sqrt{2}}< y<\sqrt{2}y$

[soft zealot]

A dense set

[scarlet sequoia]

yeah but that won't advance you anywhere

oh now what you wrote is def wrong

[onyx peak]

the edited version is false now

[scarlet sequoia]

just consider any irrational number (positive)

say sqrt(2)

[warm python]

ah

right

makes sense

[scarlet sequoia]

and transform it into an irrational number that is as small as you'd like

[warm python]

yeah, got it

oh, continue

sorry

[scarlet sequoia]

that way, x + (that irrational number as smol as you want)

is between x and y

now, how can you create a smaller irrational number from sqrt2 for example

[warm python]

we have $100x<50(x+y)<100s$

[scarlet sequoia]

?

[soft zealot]

A dense set

[warm python]

We then have $\frac{100x}{\sqrt{2} < \frac{50(x+y)}{\sqrt{2}}$ < \frac{100y}{\sqrt{2}}$

[soft zealot]

A dense set
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[warm python]

$\frac{100x}{\sqrt{2}} < \frac{50(x+y)}{\sqrt{2}} < \frac{100y}{\sqrt{2}}$

[soft zealot]

A dense set

[warm python]

We now take the greatest integer of both

[scarlet sequoia]

I feel like this isn't going where you want

[warm python]

the extremes

this gives us $70(x)<\frac{50(x+y)}{\sqrt{2}}<70 y$

[soft zealot]

A dense set

[warm python]

so $x<\frac{5(x+y)}{7\sqrt{2}}<y$

[soft zealot]

A dense set

[warm python]

,w is $x<\frac{5(x+y)}{7\sqrt{2}}<y$

[soft zealot]

Results provided by WolframAlpha

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

Requested by physicsrocks

[warm python]

okay, this won't work

[scarlet sequoia]

how does this tie back to x < irrational < y tho?

[warm python]

I'm trying to blow the numbers apart, insert an irrational between them, and then shrink it back down

Like I have 100x <100y

I know (x+y)/{2} is between them

[onyx peak]

Have you made any progress with this specific approach of manipulating this yet? If not, maybe it's time to try another approach...

the first idea is not always the right idea

[warm python]

Another approach would be to start with an irrational, construct two rationals, from it

and the flip my work around

[onyx peak]

or just trying to insert an irrational number between rational x, y but in a different way than you tried till now

[warm python]

somethhing like the bisection formula

ooh

$\frac{x \sqrt{2}+y}{\sqrt{2}+1}$

I'll have to derive this formula if I'm to use it though

[soft zealot]

A dense set

[onyx peak]

idt this will work

[warm python]

one minute

forgot the name of that theorm

[onyx peak]

maybe look at what rafi suggested

[warm python]

Ah, I can take the weighed mean of the points

[onyx peak]

Let me propose a simpler problem to solve first:
||Can you find an irrational number between 0 and y, where y is positive?||

[onyx peak]

but this is also interesting, and could work, im worried that you might have a hard time proving it works tho...

[warm python]

It will work

I know that

But proving it though

[warm python]

y/sqrt{2}

[onyx peak]

okay, cool

can you use that to solve the og problem?

you just found an irrational number r, s.t.
0 < r < z

you need something such that x < r < y

if you want to, you can look at rafi's hint

[warm python]

hmm

[warm python]

Let $x,y$ be real numbers . if one wants to divide the line segment between them in the ratio $m:n$ , the that number is given by $\frac{nx+my}{n+m}$
\
Let $x<\frac{nx+my}{n+m}$

[onyx peak]

yeah, good luck proving that that number is gonna be irrational

[soft zealot]

A dense set

[onyx peak]

hmm

[warm python]

Well, I can set n to be 1, and m to \sqrt {2} ,and with x,y, being rational , I'm done

[onyx peak]

provei t

prove that the result is irrational

[onyx peak]

but actually, if you chose better m and n, it wouldnt be that hard to prove

$\frac{\sqrt{2}x+1y}{1+\sqrt{2}}$

[soft zealot]

MæthIsAlwaysRight

[onyx peak]

you'd need to prove this thing is irrational

[warm python]

We first prove that $x< \frac{\sqrt{2}x+ y}{1+ \sqrt{2}}<y$

[soft zealot]

A dense set

[onyx peak]

yeah, this is the obvious part

[warm python]

$x+ \sqrt{2}x<\sqrt{2x}+y<y+\sqrt{2}y$

[onyx peak]

actually, you can even work with this

[soft zealot]

A dense set

[onyx peak]

you just need to carefully prove it's irrational

it's not that hard as i thought it would be

[warm python]

I need to prove this is irrational, yes?

[onyx peak]

yeah

[onyx peak]

the fact that this holds follows just by the way you constructed it

[onyx peak]

by this

[warm python]

I didn't prove that holds true though

[onyx peak]

alright, then it follows by this instead

[warm python]

We wish to prove $\frac{\sqrt{2}x+1y}{1+\sqrt{2}}$ is irrational
\
We invoke the fact that the product of a rational and irrational is irrational to solve this.As is their sum. This tells us $\sqrt{2}x+y$ is irrational, and that $1+\sqrt{2}$ is irrational
\
We now multiply and divide by $\sqrt{2}-1$

[onyx peak]

which gets you to irrational / irrational situation

[warm python]

yup

[onyx peak]

and this is inconclusive, right?

okay, that looks good

[soft zealot]

A dense set

[warm python]

We wish to prove $\frac{\sqrt{2}x+1y}{1+\sqrt{2}}$ is irrational
\
We invoke the fact that the product of a rational and irrational is irrational to solve this.As is their sum. This tells us $\sqrt{2}x+y$ is irrational, and that $1+\sqrt{2}$ is irrational
\
We now multiply and divide by $\sqrt{2}-1$
\
This gives us $\sqrt{2}x+2x - \sqrt{2}y+y$
\
\
$\sqrt{2}(x-y)+2x+y$

this feels off

[onyx peak]

it's fine

try grouping everything rational and everything irrational

,w x < sqrt(2)x + 2x - sqrt(2)y + y < y

[soft zealot]

Results provided by WolframAlpha

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

Requested by methisalwaysright

[onyx peak]

huh

[soft zealot]

A dense set

[scarlet sequoia]

my we're still here

[warm python]

,w rationalise $\frac{\sqrt{2}x+1y}{1+\sqrt{2}}$

[scarlet sequoia]

or is it a different question

[soft zealot]

Wolfram Alpha doesn't understand your query!
Perhaps try rephrasing your question?
Click here to refine your query online

Requested by physicsrocks

[onyx peak]

okay, yeah

that works

[onyx peak]

yep... wai tried a different approach

[warm python]

same question, different appraoch

using weighed means

[onyx peak]

its longer, less obvious but worked

I actually think it's the same number as the one with your approach

[scarlet sequoia]

wait how is sqrt2 x + y/(1 + sqrt2) irrational

lemme check it's

[warm python]

It isn' always irrational

[inland kettle]

physics did you solve the ratl. points one

[warm python]

no

sorry

I'll do that tomorrow, want to finsih this and head to bed

[inland kettle]

sed

[soft zealot]

MæthIsAlwaysRight

[scarlet sequoia]

(sqrt2 x + y)(sqrt2 - 1) = 2x - sqrt2x + sqrt2 y - 1

oh it is irrational if x < y are rational

ok

[onyx peak]

it's literally just a different form of a possible number you'd arrive at using your method

[scarlet sequoia]

I was thinking about x + (y-x)/sqrt2

[onyx peak]

yeah, me too

this is the same but sqrt(2) + 1 instead of sqrt(2)

longer approach, but an original one i gotta admit

[scarlet sequoia]

how did you spoiler texit bot btw?

[onyx peak]

,texsp

[scarlet sequoia]

,texsp $2$

[soft zealot]

rafilou is not not born in 2003

[scarlet sequoia]

oh and you delete your message afterwards?

[onyx peak]

anyway, @warm python , 2x + y is rational, (x-y) is rational as well

and as long as x - y is non-zero, (x-y) * sqrt(2) is gonna be irrational

since it's rational * irrational

and then irrational + rational is irrational

[warm python]

and x=y is absurd

so yeah

[onyx peak]

yep

[warm python]

that's what I though

thanks

[onyx peak]

also check this and also

y = x + (y-x) / sqrt(2)

the y = x + (y-x) / sqrt(2) is probably the most obvious one

[warm python]

We wish to prove $\frac{\sqrt{2}x+1y}{1+\sqrt{2}}$ is irrational
\
We invoke the fact that the product of a rational and irrational is irrational to solve this.As is their sum. This tells us $\sqrt{2}x+y$ is irrational, and that $1+\sqrt{2}$ is irrational
\
We now multiply and divide by $\sqrt{2}-1$
\
This gives us $\sqrt{2}x+2x - \sqrt{2}y+y$
\
\
$\sqrt{2}(x-y)+2x+y$
\
Which is always irrational

[soft zealot]

A dense set

[warm python]

Will do

[inland kettle]

physics why ru still awake

[warm python]

$x+(y-x)(\sqrt{2}-1)$

[soft zealot]

A dense set

[warm python]

Idk

[inland kettle]

its fucking 12

😭

[warm python]

I should have slept 2 hours ago

[inland kettle]

yes

now go to sleep

this isn't the time for you to be awake

[warm python]

@onyx peak , mind if I turn in?

[onyx peak]

sure, good night

[warm python]

Thanks!

gn!

.close

[inland kettle]

no close?

[final saddle]

Channel closed

Closed by @warm python

Use .reopen if this was a mistake.

[inland kettle]

oh

timing

[final saddle]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[subtle urchin]

Calculate the exact value of this

tg = tan

i figured it out

.close

[final saddle]

Channel closed

Closed by @subtle urchin

Use .reopen if this was a mistake.

[final saddle]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[torpid zenith]

hi I have this ε-NFA and I want to turn it into NFA

[torpid zenith]

is this correct, only NFA?

(I forgot the start arrow for 1, my bad)

[crisp gate]

no it's nor right

because in your nfa aaa is accepted whereas in ε-NFA it isnt

[dull ravine]

It isn’t?

[crisp gate]

oh

it is

my bad

so yes his nfa is right

[dull ravine]

The input ‘a’ doesn’t work though i think

[crisp gate]

what do you mean it doesnt work?

[dull ravine]

For the e-NFA it is accepted through eae

[crisp gate]

oh

[dull ravine]

For the nfa it gets stuck at 3

[crisp gate]

true

[dull ravine]

It’s a very confusing e-nfa ngl

[crisp gate]

that's why i dont like e-nfa

[dull ravine]

But i think that a-loop in the nfa should be for 1 instead of 2

[crisp gate]

could work yes

[dull ravine]

@torpid zenith

[final saddle]

@torpid zenith Has your question been resolved?

[torpid zenith]

Hi sorry for the late answer