help-27

[viral lynx]

Which other way sry?

Ah wait

I get you

[vital edge]

Ah thank god

I was gonna have to go hunting for a video

[viral lynx]

So if it doesnt then it cant have an extrema , cause it will just be increasing or decreasing, right?

[vital edge]

Actually I just realized

None of what I said is relevant

You want to check for global extremas

My bad

Also hi cloud, all yours cuz I've clearly fucked up lol

[viral lynx]

No but I think I get it right?

[viral lynx]

Can someone just confirm

[viral lynx]

The diagram seems to match what I had in mind

[acoustic leaf]

so knowing the derivative is greater than or equal to 0 (with zeros only at isolated points), we can sketch what the points where the derivative is equal to 0 look like. they have to look like places where it briefly goes flat but is pointing upward on both sides. so it's actually increasing the entire time

[surreal sequoia]

can someone help with this

[drifting sierra]

The point here is just that the function is strictly increasing except on some points where the derivative is 0

[surreal sequoia]

nel can you help me

with this i dont get this

[acoustic leaf]

please get your own help channel #❓how-to-get-help

[surreal sequoia]

i did nobody respond

[vital edge]

Dude be patient

[viral lynx]

But never decreasing, therefore there's no extrema, right?

[viral lynx]

Or even, but not decreasing

[drifting sierra]

A constant function is never decreasing but has an extremum (the constant value)

[viral lynx]

Oh

So the values where the graph is constant ARE extrema?

[acoustic leaf]

well if it was zero for some interval then it would be even. but if the derivative is only zero at single points then it's actually increasing the entire time

[drifting sierra]

Locally, sure

Not necessarily globally

[viral lynx]

Ahhhhhh

I get it now

Thank you, everyone explained it really well!!

❤️

.close

[devout snow]

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[drifting sierra]

Tbh for this question I would just think of the limit

sin(2x) is only between -1 and 1 so it's irrelevant, and the limit of 2x at either infinity is either infinity, so...

[devout snow]

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[north hound]

What is the comparision between the problems 3, 4 and the problems 6,7? (This is asked at the last of the question 7.)

[glass canopy]

Look at what you need to prove, and there is a similarity between 3 & 6, 4 & 7

[north hound]

Is the similarity that both the pair of problems require the user to proof relations of the form A + B = C and A - B = C, where the As, Bs and Cs are symbolically equal in each pair of problems?

[glass canopy]

Is it coincidence?

[devout snow]

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[keen sundial]

What is this saying

What’s e

[pseudo basin]

e?

[keen sundial]

I mean r

[pseudo basin]

ok right

let's keep going with the "20 athletes, 3 medals" example

unfortunately though it will be extremely tough if you claim not to know nor have reference for the multiplication rule

but anyway, here is the pic i showed again

for 2 medals (gold and silver)

there are 20*19 possible winner lineups

[keen sundial]

What is the multiplication rule it might be later in the slides or next few lectures

[pseudo basin]

phrased in combinatorial terms it'll be something like this:
if one choice can be made in m ways, and another choice can be made in n ways independently of the first,
then the number of ways to make both choices at once is m*n

for a simpler example, if you need to dress for work and can pick from 6 shirts and 4 pairs of pants, and have to wear one of each,

then that's 6*4 = 24 possible outfits you can make

or e.g. if you are looking at lunch options in the canteen and there are:

• 4 different first courses
• 3 different second courses
• 5 different desserts
  (and you have to pick one first course, one second course and one dessert)
  then the number of ways that you could assemble your tray is 4 * 3 * 5

[keen sundial]

I’m not sure what’s combinatorial terms

[pseudo basin]

you're overthinking it

"phrased in a combinatorial way" perhaps

if you want a wording that's less prone to get stuck on

[keen sundial]

Ok yeah I understand this

Continue with the athlete example

[pseudo basin]

ok

so you understand now why the number of lineups with a gold and silver medal is going to be 20 × 19, yes?

[devout snow]

@keen sundial Has your question been resolved?

[keen sundial]

yes

[pseudo basin]

ok

let's try to add bronze to the lineup now

if we have already fixed a gold and silver medalist, how many options do we have for bronze?

[keen sundial]

wdym fixed a gold and silver medalist?

[crystal dawn]

if you have already chosen a gold/silver guy

[pseudo basin]

did you suddenly stop understanding the words "gold medalist" and "silver medalist"

or maybe "fix"

i have no idea what part of "fix a gold and silver medalist" is unclear

[keen sundial]

17

[pseudo basin]

why 17

there's 20 athletes in the pool

[keen sundial]

yea first was gold so thats 19, second was silver which is 18, third is bronze which has to be 17

[pseudo basin]

no

gold has 20 options

silver has 19 options

bronze in fact has 18 candidates not 17

[keen sundial]

then bronze has 18

[pseudo basin]

yes exactly

so then

using the same logic by which there are 20 × 19 lineups if we just give out gold and silver,

if we give out gold, silver and bronze, how many lineups will there be?

@keen sundial you get 5 minutes

[keen sundial]

20 * 19 * 18

[pseudo basin]

ok

great.

do you see how this logic could continue if we wanted to have more places in the lineup than just 3?
e.g. if we wanted to give out prizes down to and including 5th place, how many possible winner lineups would we have?

@keen sundial

[keen sundial]

5!

could you clarify this before we proceed

Im not sure whats happening here

[pseudo basin]

incorrect

logic not applied...

[pseudo basin]

you're overfocusing on a piece of the picture that doesnt really form a logical part of it

the picture is most naturally broken down into rows

the first row contains all the 2-place lineups where #01 gets the gold medal. there are 19 of these.

the second row contains all the 2-place lineups where #02 gets the gold medal. there are also 19 of these.

the third row..

do i need to continue?

[keen sundial]

yea I get it now

continue

[devout snow]

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[deft flower]

what can I further do to solve this limit

[native jay]

the power is 1/x ?

[deft flower]

yes

the solution should be 1/e

[native jay]

can u give me the function

that ur studying

[deft flower]

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

nvm I found the solution

[native jay]

oh ok

good

[deft flower]

.close

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[devout snow]

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[forest palm]

p(a' ∩ b) how would you draw this as a venn diagram?

[pure kelp]

you could do this step by step. first start out with two venn diagrams. side by side. in one, shade a' and in the other shade b

now look at the area that would be common to both shaded areas

with that draw a 3rd one

[forest palm]

i understand the concept that it means P(B) - (A ∩ B) but i just cant visualize it on a venn diagram

So only the area in B that doesnt intersect with A?

[proud perch]

Yea

[forest palm]

.close

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[vapid socket]

?

[pseudo basin]

!

[vapid socket]

?

[pseudo basin]

be more specific in what you're ?ing at

[vapid socket]

Im confused on how to approach the quest

[pseudo basin]

that limit of a sum looks awfully like some kinda integral

[vapid socket]

Figured that but its the geo part

Thats bugging me

[pseudo basin]

ok so maybe shift everything down one unit so that B is now the origin and the equation of the curve is 4(y-1) = x^2

express this in polar coordinates

then integrate the resulting r(θ) function from -pi/2 to 0

[vapid socket]

Polar

How do u do that for a parabola

I only know parametric

[pseudo basin]

x = r cos(θ), y = r sin(θ) -> put that into 4(y-1) = x^2

ig it's gonna look ugly as hell but still

[vapid socket]

[pseudo basin]

... i do not know what that emoji is supposed to mean

[vapid socket]

Looks cooked

How am i supposed to form the r/n

To int

[pseudo basin]

yeah honestly i dont fucking know

its gonna take some serious trig crunch no matter what i think

[vapid socket]

[pseudo basin]

unless theres some stupid trick specifically for this question

[vapid socket]

How am i supposed to think of this during the exam

[vapid socket]

Idek

I wrote like 400 pages of notes

On parabola

Circles

[languid jasper]

I wonder if B being the focus does anything

[vapid socket]

Ellipse

And hyperbola

[devout snow]

@vapid socket Has your question been resolved?

[rare kernel]

is there a solution given 💀

[languid jasper]

@vapid socket I got 2/(1-sinx) intgeral from 0 to pi/2

Its undefined tho

The issue is that at when r=n you are making an angle of pi/2 but that point is at infinity

[devout snow]

@vapid socket Has your question been resolved?

[vapid socket]

No

Wtf is this question

[rare kernel]

just leave it no point on wasting so much time on 1 question

[devout snow]

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[rocky gate]

Hi! Theres this proof ive done. But i think it isnt a proper proof and I’ve listed the flaws as well. Could someone help me verify it? Thank you!

[stone stump]

the whole point of the notation of S and T like that is to say that they have n or m elements

[rocky gate]

So i could use this proof even if its short?

Also my book uses a proof using contrapositive and contradiction for that one. I was wondering if thats a proof technique for proving double implications

[stone stump]

how did your book define what that notation even means?

cause it should mean something like:

• there is a bijection f:{1,...,n}->S
• we denote f(1) by s_1, f(2) by s_2 and so on

and from there |S|=n should be obvious

[rocky gate]

The first line is whats mentioned in the book, its part of a preposition

[devout snow]

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[devout snow]

@rocky gate Has your question been resolved?

[devout snow]

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[molten wagon]

For example x = y, is it a straight line on the coordinate system

?

[vast drift]

yeah

any

ay+bx+c=0 equations are straight lines

[molten wagon]

So is a, b, c here all zero?

[hard ocean]

They are parameters

[lyric hornet]

in the y=x case, a=1, b=-1

[polar chasm]

x = y can be rearranged to
x - y = 0
so it would be a = 1, b = -1

[faint hearth]

c=0

[lyric hornet]

or a=-1 and b=1

[polar chasm]

there are other options as well, but this one is prolly the most natural one

or a = -pi, b = pi

[molten wagon]

And to find zero points…

Delta eight?

Right*

,w 1 + 410

[woven radish]

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[molten wagon]

,calc 2/2

[woven radish]

Result:

1

[molten wagon]

,calc 0/2

[woven radish]

Result:

0

[molten wagon]

Zero points?

???

,w (1,0)(0,0)

[woven radish]

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[molten wagon]

And c is at 0?

<@&286206848099549185>

[supple knot]

Your math is all over the place. Show the actual problem you're working on

[supple knot]

This Delta came out of nowhere

[molten wagon]

I said I use “delta right?”

[molten wagon]

I’m just trying to figure out what x = y equals to…

It has actually zero numbers…

[supple knot]

What is Delta then?

What does a "zero number" mean?

[molten wagon]

1

[supple knot]

I mean what are you calling Delta

In x = y, there's no Delta

[molten wagon]

https://en.wikipedia.org/wiki/Zero_of_a_function

[molten wagon]

How

[supple knot]

Just answer this

How did you conclude Delta = 1

[molten wagon]

Because

[supple knot]

What does Delta even represent

[molten wagon]

A is 1 b is 1 c is 0

[molten wagon]

It’s used to calculate zero numbers

[supple knot]

Read this again

[supple knot]

No idea where you get Delta from

You're confusing different things together

[molten wagon]

Okay then

A 1 b -1 c 0

[supple knot]

x=y is zero only when (x, y) = (0,0)

,w plot y = x

[woven radish]

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[molten wagon]

It’s x = 0 and y =1

[supple knot]

If x=0 and y=1, does x=y?

[molten wagon]

No…

But I used the method

[supple knot]

.

[supple knot]

.

[molten wagon]

But how do you interpret that equation

I don’t have any numbers

[supple knot]

x is the independent variable so it can take on any numbers

[molten wagon]

Why can’t the line be for example here?

[supple knot]

y follows what x equals by the relationship y = x

[supple knot]

Because that's not x = y

,w plot y=-x

[woven radish]

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[molten wagon]

Why?

How would I even know that this looks like that

[supple knot]

Pick numbers for x, then take the negative of them and plot them

For lines you only need to do that for two points

[molten wagon]

Okay x is 5

5 = y

,w 5 = y

[woven radish]

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[supple knot]

(5,5) is a point on x=y yes

[molten wagon]

,but can you explain ?

[supple knot]

I did here

[molten wagon]

Okay y = 5

It’s like this

,w y=5

[woven radish]

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[molten wagon]

??? It’s not the same line

[supple knot]

y = 5 is a line

[molten wagon]

But not directed like that …

Where’s this?

[supple knot]

You're just using Wolfram wrong

[supple knot]

This is how Wolfram interprets y=5

[molten wagon]

So y=5 can be this?

[supple knot]

No

[molten wagon]

Then what do you mean

[supple knot]

You're leaving out this information

You're thinking wolfram knows what you're trying to do

When all you give it is y=5

[molten wagon]

But you just said that y=5 isn’t this

[supple knot]

Correct

Where do think I'm contradicting myself

[molten wagon]

Dude you also said that this isn’t x=y

Why can’t x=y be for example this? @supple knot

[supple knot]

Right. I already said your black line was y=-x

[supple knot]

.

[supple knot]

The red line doesn't go through (1,1)

[molten wagon]

It does

It doesn’t matter

[molten wagon]

It goes through 0,0

And you said that x = y is 0,0 0,0

<@&1280681159495385169>

[supple knot]

Wrong

[supple knot]

Read this again

(0,0) is one point, (1,1) is another point on x=y

[molten wagon]

Show them

…

[supple knot]

You can see it yourself in desmos

https://www.desmos.com/calculator

[devout snow]

@molten wagon Has your question been resolved?

[molten wagon]

How do you know it is 1,1

[supple knot]

When x = 1, y=x implies y=1

[molten wagon]

But dude

OK.

But for example

x = y + 3

@supple knot

How would you do that

,w x = y +3

[woven radish]

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[supple knot]

what is "do"

plot?

[supple knot]

you should just learn this. connect two points and they make a unique line

[molten wagon]

….

• 3…

so 1 for x

(1,4)?

@supple knot

[supple knot]

you can stop pinging me

[molten wagon]

<@&286206848099549185> okay anyone else?

.close

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[viral lynx]

test

[fringe garnet]

Hello, I would like someone to check if I did well on my work.

[supple knot]

[supple knot]

those are just questions. where's your work

[fringe garnet]

With this

[fringe garnet]

Here it is:

[fringe garnet]

You here?

[jaunty mantle]

Did you even read your instructions

Why do your solutions look so hand wavey when that is precisely what they want to avoid

[fringe garnet]

Well, I don't really know what to do.

[jaunty mantle]

Perhaps you should look at some lecture materials from your class

[fringe garnet]

Okay. I'll redo it, then.

[fringe garnet]

But wait, aren't lecture materials like not helpful in certain cases?

[jaunty mantle]

Literally look at your own answers

None of them actually answer what they want you to do

I doubt this assignment was given with no help at all, it’s a pretty big and in depth assignment that you seem to have no idea how to do

I’m sure they must’ve covered some of this in class

[fringe garnet]

Yeah.

[jaunty mantle]

As to reconstruction of classic theorem proofs there are many online you can find

[fringe garnet]

I'll try reviewing them.

[jaunty mantle]

I doubt you’re allowed to just quote Markov’s inequality without proof either

[fringe garnet]

Ok, Ok.

.close

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[woven radish]

Renato

[devout snow]

@spring oasis Has your question been resolved?

[faint gorge]

Is there anything you tried?

[spring oasis]

f(x,y) - P3(x,y) = R(x,y)

[devout snow]

@spring oasis Has your question been resolved?

[spring oasis]

i think the big idea is to find out the second order taylor pol

out of the third order taylor pol

[spring oasis]

Forget this

[faint gorge]

i did

so we dont know f but its taylor polynomial, looks like it's ought to be related to the theorem from before

[spring oasis]

Yes

This exercise is hard

[spring oasis]

How do I find which part of the third order Taylor poly is the second order poly

[faint gorge]

my thoughts would be to either substitute x=u+1 and y=v-2 and then grouping P_3 should be easier, I guess

[spring oasis]

They give the Taylor poly because that's the point

[spring oasis]

Grouping p3?

[faint gorge]

ya

because before you would have to factor terms like (x-1) and (y+2) instead

which i mean is harder

[spring oasis]

Just

Help me find p2 please

Trust me the calculations get simple if we figure out p2

[faint gorge]

it's given lol

[spring oasis]

If we find p2 exercise simplifies

Trust

[spring oasis]

Well

[faint gorge]

that's some insane algebra, personally i would substitute then you group x and y not (x-1) and (y+2)

[spring oasis]

F and P are equal at the point

(1,-2)

The idea is simple @faint gorge

f and P3 are equal at the point (1,-2)

[faint gorge]

sure

[spring oasis]

What I mean is, f(1,-2) = P3(1,-2)

[faint gorge]

i understood that

[spring oasis]

If we differentiate P3 wrt x and evaluate (1,-2) we get fx(1,-2)

[faint gorge]

WHOA

that's smart

it's 3 am is my excuse

[spring oasis]

No worries

[faint gorge]

almost thought it was my help channel and u were my helper

[spring oasis]

I have been scratching my head a lot with this one

[faint gorge]

dont overscratch tho

[zenith spoke]

didnt you solve this before tho?

[faint gorge]

illusions

[zenith spoke]

was there something unclear at that time

[zenith spoke]

lmao

[spring oasis]

The method

[zenith spoke]

like it seemed to come out of nowhere or something like that?

[spring oasis]

There is a simpler way

[zenith spoke]

hmmm thats interesting

yea i didnt necessarily give the simplest one, just the one that came to mind

tho whats this simpler way

[spring oasis]

Use this

Find second order Taylor poly

Then use Taylor theorem

It's hard to explain

@faint gorge

[zenith spoke]

you have the 3rd order taylor polynomial but not the original function so how would you find the 2nd order taylor poly

also how wdym by use taylor's theorem

[spring oasis]

There is a corollary

@faint gorge

[faint gorge]

@spring oasis i think @zenith spoke is willing to help as well

[spring oasis]

Yes

[spring oasis]

You have the third Taylor poli, using that and this you can figure out the second Taylor polynomial

This

If we differentiate P3 wrt x and evaluate (1,-2) we get fx(1,-2)

[spring oasis]

This is what I mean by Taylor theorem corollary

The remainder

[zenith spoke]

I mean suppose that you have successfully found the 2nd order polynomial

what next

[zenith spoke]

what does this tell you

[spring oasis]

f(x,y) = P2(x,y) + R2(x,y)

f(x,y) - P2(x,y) = R2(x,y)

[spring oasis]

We have that (f(x,y) - P2)/norm{(x,y)-(1,-2)}^2 = 0

[zenith spoke]

so you are doing all that in order to substitute P_2 in place of f(x,y) in the limit?

[spring oasis]

Trust

I can explain

But you have to trust my ass

[zenith spoke]

or maybe to find out that this limit is the same as the limit of the quotient you are talking about +4?

oops mb sorry universe i thought that i was replying to renato

i didnt mean to ping (ik its late for you thats why i am saying this)

[spring oasis]

Focus

[zenith spoke]

I mean the way I told you about before is most probably easier than this ngl

the thing is that what you are doing here isnt exactly meaningful

you are trying to find P_2 and substitute it for f(x,y) in the given limit expression

[spring oasis]

Last time we solved this we ignored P3

[zenith spoke]

but you can already substitute P_3

[zenith spoke]

what

[spring oasis]

No

[zenith spoke]

how did we do that

[spring oasis]

Sorry

[zenith spoke]

we literally substituted P_3

[spring oasis]

Yes I remember now

[zenith spoke]

and worked things out from there

[spring oasis]

No

[zenith spoke]

I mean thats the purpose of the question if you ask me tbh

(at least thats what i am seeing lol)

[spring oasis]

I am saying something else

[zenith spoke]

hmmm then whats the conclusion you want to reach

[spring oasis]

f(x,y) = P2(x,y) + R2(x,y)
f(x,y) - P2(x,y) = R2(x,y)

[zenith spoke]

right all good till now

[spring oasis]

Trust my ass

You gotta trust dude

[zenith spoke]

I am trusting you but what is the next step is all i am asking

[spring oasis]

Ok

First we find p2

After we find p2, we use it this way f(x,y) = P2(x,y) + R2(x,y)
f(x,y) - P2(x,y) = R2(x,y) using Taylor lemma

I think it's better if we go ahead and do it, you will see

We will arrive same place in less steps

[zenith spoke]

ok so this is the taylor series in multivariables

at the end is the 2nd order polynomial for a function of 2 variables

ie P_2

[zenith spoke]

this works for single variable functions

here we have a function of 2 variables

[spring oasis]

Works aswell for multivariate

[zenith spoke]

you need partial derivatives etc..

[spring oasis]

Yes

[zenith spoke]

which is this

a better version since 2 variables so slightly less horrible notation

[spring oasis]

You gotta trust me

[zenith spoke]

we have $P_3$ which is the horrible\ $P_3(1,-2)=P_2+\frac 1{3!}(f_{xxx}(1,-2)+3f_{xxy}(1,-2)+3f_{xyy}(1,-2)+f_{yyy}(1,-2))$

[woven radish]

ali yassine

[spring oasis]

yes, but we dont care about that, we want to find P2

[zenith spoke]

but we have P_3

and we want to get P_2 from there

to find P_2 we need to know the partial derivatives of f

[spring oasis]

no

I already show you the theorem

look at the example

[zenith spoke]

again, this is for functions of one variable

ok so how do you apply this to multivariable functions

[spring oasis]

so, there is this formula for P2

,, P_2 = f(x_0, y_0) + f_x(x_0,y_0)(x-x_0) + f_y(x_0,y_0)(y-y_0) \ + \frac{1}{2!} {f_{xx}(x_0, y_0)(x-x_0)^2 + 2f_{xy}(x_0,y_0)(x-x_0)(y-y_0) + f_{yy}(x_0, y_0)(y-y_0)^2 }

[woven radish]

Renato

[spring oasis]

this is the formula for P2

and we know that f(x0, y0) = P3(x0,y0)

we know that fx(x0, y0) = P3x(x0,y0)

we know that fy(x0,y0) = P3y(x0,y0)

we know that fxx(x0,y0) = P3xx(x0,y0)

we know that fxy(x0,y0) = P3xy(x0,y0)

we know that fyy(x0,y0) = P3yy(x0,y0)

[zenith spoke]

ah ig this is right

i am sleepy so i might say stupid stuff

[zenith spoke]

here you probably meant P_yy right?

[spring oasis]

yes

[zenith spoke]

alright for now i will just listen

i wont interrupt your work so go ahead

[spring oasis]

f(1,-2) = 3
fx(1,-2) = 2
fy(1,-2) = 0
fxx(1,-2) = 0
fxy(1,-2) = -1
fyy(1,-2) = 8

you still here?

@zenith spoke

[zenith spoke]

yes i am

[spring oasis]

,, P_2 = 3 + 2(x-1) + 0 + \frac{1}{2!}{0 -2(x-1)(y+2) + 8(y+2)^2} \ P_2 = 3 + 2(x-1) + \frac{1}{2} {-2(x-1)(y+2) + 8(y+2)^2} \ P_2 = 3 + 2(x-1) -(x-1)(y+2) + 4(y+2)^2 \ P_2 = 3 + 2(x-1) + (1-x)(y+2) + 4(y+2)^2 \ f(x,y) = P_2(x,y) + R_2(x,y) \ R_2(x,y) = f(x,y) - P_2(x,y)

[zenith spoke]

alright

[spring oasis]

now we use taylors lemma

[zenith spoke]

note that i didnt check any calculations but i am trusting you on that

[spring oasis]

trust

unless I made a typo everything should be correct

because most of the stuff was calculated by wolfram

this is the only part I did by myself

we use taylors theorem

,, P_2 = 3 + 2(x-1) + 0 + \frac{1}{2!}{0 -2(x-1)(y+2) + 8(y+2)^2} \ P_2 = 3 + 2(x-1) + \frac{1}{2} {-2(x-1)(y+2) + 8(y+2)^2} \ P_2 = 3 + 2(x-1) -(x-1)(y+2) + 4(y+2)^2 \ P_2 = 3 + 2(x-1) + (1-x)(y+2) + 4(y+2)^2 \ f(x,y) = P_2(x,y) + R_2(x,y) \ R_2(x,y) = f(x,y) - P_2(x,y) \ \lim_{(x,y) \to (1,-2)} \frac{f(x,y) - P_2(x,y)}{\norm{(x,y) - (1,-2)}^2} = 0

[woven radish]

Renato

[spring oasis]

but we have another limit

we have this

[zenith spoke]

right

so now the moment of truth

what now?

[spring oasis]

,, \lim_{(x,y) \to (1,-2)} \frac{f(x,y) - P_2(x,y) + P_2(x,y) + xy -y -3 + 4(x-1)^2}{\norm{(x,y) - (1,-2)}^2} \ \lim_{(x,y) \to (1,-2)} \frac{f(x,y) - P_2(x,y) }{\norm{(x,y) - (1,-2)}^2} + \frac{P_2(x,y) + xy -y -3 + 4(x-1)^2}{\norm{(x,y) - (1,-2)}^2} \ \lim_{(x,y) \to (1,-2)} \frac{P_2(x,y) + xy -y -3 + 4(x-1)^2}{\norm{(x,y) - (1,-2)}^2} \ P_2 = 3 + 2(x-1) + (1-x)(y+2) + 4(y+2)^2 \ \lim_{(x,y) \to (1,-2)} \frac{3 + 2(x-1) + (1-x)(y+2) + 4(y+2)^2 + xy -y -3 + 4(x-1)^2}{\norm{(x,y) - (1,-2)}^2} \ \lim_{(x,y) \to (1,-2)} \frac{-xy + 4y^2 + 9y + 19+ xy -y -3 + 4(x-1)^2}{(x-1)^2 + (y+2)^2}

[zenith spoke]

right

stuff simplify nicely here

[spring oasis]

3 + 2(x-1) + (1-x)(y+2) + 4(y+2)^2
3 + (2x - 2) + (y+2-xy-2x) + 4(y^2 + 2y + 4)
3 + (2x - 2) + (y+2-xy-2x) + (4y^2 + 8y + 16)
3 + (y-xy) + (4y^2 + 8y + 16)
3 -xy + 4y^2 + 9y + 16
-xy + 4y^2 + 9y + 19

[zenith spoke]

no why did you do that

you were on the right track

do not expand 4(y+2)^2

[zenith spoke]

keep it as 4(y+2)^2

then take 4 as a common factor and simplify

[spring oasis]

can I get some help

we are getting there

,, \lim_{(x,y) \to (1,-2)} \frac{f(x,y) - P_2(x,y) + P_2(x,y) + xy -y -3 + 4(x-1)^2}{\norm{(x,y) - (1,-2)}^2} \ \lim_{(x,y) \to (1,-2)} \frac{f(x,y) - P_2(x,y) }{\norm{(x,y) - (1,-2)}^2} + \frac{P_2(x,y) + xy -y -3 + 4(x-1)^2}{\norm{(x,y) - (1,-2)}^2} \ \lim_{(x,y) \to (1,-2)} \frac{P_2(x,y) + xy -y -3 + 4(x-1)^2}{\norm{(x,y) - (1,-2)}^2} \ P_2 = 3 + 2(x-1) + (1-x)(y+2) + 4(y+2)^2 \ \lim_{(x,y) \to (1,-2)} \frac{3 + 2(x-1) -(x-1)(y+2) + 4(y+2)^2 + xy -y -3 + 4(x-1)^2}{(x-1)^2 + (y+2)^2}

[zenith spoke]

yea sure

[woven radish]

Renato

[zenith spoke]

from here simplify just like you did before editting

but at the end do not expand the term 4(y+2)^2 this time

[spring oasis]

3 + 2(x-1) + (1-x)(y+2) + 4(y+2)^2
3 + (2x - 2) + (y+2-xy-2x) + 4(y+2)^2
3 + [(2x - 2) + (-2x + 2)] + (y-xy) + 4(y+2)^2
3 + (y-xy) + 4(y+2)^2
3 + y -xy + 4(y+2)^2

,, \lim_{(x,y) \to (1,-2)} \frac{f(x,y) - P_2(x,y) + P_2(x,y) + xy -y -3 + 4(x-1)^2}{\norm{(x,y) - (1,-2)}^2} \ \lim_{(x,y) \to (1,-2)} \frac{f(x,y) - P_2(x,y) }{\norm{(x,y) - (1,-2)}^2} + \frac{P_2(x,y) + xy -y -3 + 4(x-1)^2}{\norm{(x,y) - (1,-2)}^2} \ \lim_{(x,y) \to (1,-2)} \frac{P_2(x,y) + xy -y -3 + 4(x-1)^2}{\norm{(x,y) - (1,-2)}^2} \ P_2 = 3 + 2(x-1) + (1-x)(y+2) + 4(y+2)^2 \ \lim_{(x,y) \to (1,-2)} \frac{3 + 2(x-1) -(x-1)(y+2) + 4(y+2)^2 + xy -y -3 + 4(x-1)^2}{(x-1)^2 + (y+2)^2} \ \lim_{(x,y) \to (1,-2)} \frac{3 + y -xy + 4(y+2)^2 + xy -y -3 + 4(x-1)^2}{(x-1)^2 + (y+2)^2} \ \lim_{(x,y) \to (1,-2)} \frac{4(y+2)^2 + 4(x-1)^2}{(x-1)^2 + (y+2)^2} = 4 \cdot \lim_{(x,y) \to (1,-2)} \frac{(x-1)^2 + (y+2)^2}{(x-1)^2 + (y+2)^2} = 4 \cdot 1

[zenith spoke]

nice

so now what do you notice about the numerator and denominator?

[woven radish]

Renato

[spring oasis]

broo

[zenith spoke]

great!

thats you rn

[spring oasis]

Literally idk if it was more or less work

At the end it was pretty similar

I do prefer doing partial derivatives rather than complete the square tho

[zenith spoke]

yea well you shouldve expected that the work would be similar from some point onwards

you will always have to pass by the algebraic sinplification/manipulation step with this kind of questions

[zenith spoke]

yea go for the way that you see fit

[spring oasis]

Well I do remember we needed difference of cubes at one point or another

@zenith spoke

Exercise was hard tbh

[spring oasis]

Like, if you recall in the other solution dude

We inserted the P3 directly in the limit

[zenith spoke]

yes we did

[zenith spoke]

well you bombed it 🔥

[spring oasis]

But, in here instead of inserting P3 we inserted P2

Which was a little shorter

[spring oasis]

Like, the first time we did this, we inserted P3 but that's why at some point or another we had to use difference of cubes

Using P2 was a little shorter but required to compute the partial derivatives of P3

[spring oasis]

Well I just wanted to show you this method

[zenith spoke]

this was a nice way to tackle the problem ngl

[spring oasis]

Using Taylor's theorem and using the fact that f(x0) = P(x0) and f'(x0) = P''(x0)

And so on and so forth

[spring oasis]

Anyways, have a good one

I learnt a bunch today

[zenith spoke]

have a great day/night too

i enjoyed our conversation!

when you are done please type .close

[spring oasis]

.close

[devout snow]

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[pure stone]

use power of point A

also notice something about A, P, D, O and Q?

[rain summit]

those 5 points are cyclic

[pure stone]

indeed

use that to get AHP + AHQ = 180

[devout snow]

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[rain summit]

.close

[devout snow]

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[urban aurora]

I don't understand this reasoning. My textbook doesn't mention F having to be simply connected anywhere

[urban aurora]

My textbook just says f needs to be continuous on a domain

[devout snow]

@urban aurora Has your question been resolved?

[devout snow]

@urban aurora Has your question been resolved?

[supple knot]

Not simply connected implies discontinuous

[devout snow]

@urban aurora Has your question been resolved?

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[sturdy fiber]

,rccw

[sturdy fiber]

What did I do wrong?

(please mind the handwriting)

[supple knot]

How did you get the antiderivative on like 4

[sturdy fiber]

I applied integral by parts for xlnsinxdx taking x as u and lnsinx as v

[supple knot]

Should show the full step

[sturdy fiber]

sorry

I was too shocked seeing π=2

[supple knot]

That's how you know you made a mistake

[supple knot]

And why you shouldn't skip steps

[sturdy fiber]

I see now

[devout snow]

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[grave delta]

I’m stuck, this question is guiding me through l’hospitals rule step by step. And I have just finished deriving, this is the second step, but I am now lost at the third step.

[grave delta]

I’m confused because I don’t understand what they are asking of me

[lunar harbor]

Differentiate, not derive. But they’re really just asking you to rewrite $\frac{2\sqrt{5x}}{5x}$ in an equivalent form with only $2$ in the numerator.

[woven radish]

Civil Service Pigeon

[lunar harbor]

aka ||divide both the numerator and denominator by sqrt(5x)||

[devout snow]

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[devout snow]

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[languid jungle]

beginner with inductions could anyone help

[terse orchid]

For induction you want to show that the statement is true for n=0 (in this case)

Then show that if the statement P(n) is true for any n, then it's P(n+1) is true for n+1

[languid jungle]

yes I'm familiar with the step, not so much on how to act on it

[terse orchid]

Are you familiar with proofs?

[languid jungle]

I've done it before

[terse orchid]

Prove $P(0)$ and $(P(n) \implies P(n+1))$

[woven radish]

flynger

[iron fiber]

Have you written induction proofs before?

[languid jungle]

oh sorry not familiar with those kind of proofs, I've just done a unit in highschool that had formulas and different than this

[languid jungle]

no

I do know base case is usually 0, 1 sometimes and we prove for that, then we try to prove for every number after which is n + 1

and so on

[terse orchid]

In order to prove $p \implies q$ you are basically showing that if you assume $p$ to be true then $q$ must be true

So you usually write. "Suppose $p$." and then use logic to arrive at $q$.

[woven radish]

flynger

[iron fiber]

Ok but have you read induction proofs before and are familiar with how they are structured ?

[languid jungle]

ohh that's what you were asking about to prove p0 and pn

[languid jungle]

I'm familiar with the steps I need to take, I have tried doing proofs I kind of struggle on how to turn it into my hypnosis

I'm trying to learn the structure so I could get a better hang of inductions

[iron fiber]

Ok so in this case what would your induction hypothesis look like ?

[terse orchid]

Example of a proof

Prove that $x=1 \implies x^2-1=0$.

$\textit{Proof. }$Suppose $x=1$. Then $x^2=1$, so subtracting 1 on both sides gives $x^2-1=0$.

Hence, if $x=1$, then $x^2-1=0$.

[woven radish]

flynger

[terse orchid]

Does that make sense?

[languid jungle]

sorry using paint

[languid jungle]

yes

[ocean gale]

Wait I’ve done this proof before

[iron fiber]

You Are just rewriting the problem statement

[terse orchid]

For the inductive case in induction you have to show

$P(n) \implies P(n+1)$ for any $n \in \mathbb{N}$.

[woven radish]

flynger

[languid jungle]

if it was already n + 1 before would it be n + 2 now? like ths

[languid jungle]

yes I got what we have to be proving

[iron fiber]

Now what is your Next step ?

[languid jungle]

inductive step, prove for k + 1

[iron fiber]

Ok and what would that look like on paper ?

[languid jungle]

like how I would do that?

I'm not really sure, I get stuck here, the hypnosis is pretty straightforward

[iron fiber]

Yes this is a little more tricky you need to break up the sum on the left. I will do something quickly

[languid jungle]

okay thank you!

[iron fiber]

Ultimately this is what you want to arrive at when I say „break up the sum“

Are you familiar with this ?

The hypothesis that you stated above is ultimately where you want to arrive at

Now would you know how to move forward from here ?

[devout snow]

@languid jungle Has your question been resolved?

[languid jungle]

how did you make this?

[languid jungle]

yesyes

[iron fiber]

Ok so using that Knowledge what is the next step. Look at other induction proofs and try to solve this one

[languid jungle]

this is the first one from the textbooks I'm not famliar with others to look at yet

[iron fiber]

You need to apply your induction hypothesis to the formula now to end up here

Do you understand why ?

[languid jungle]

ohh I see, the right side shown here is when you break the sum like you said but since we can't put all the sums we just do 2^k+1

[devout snow]

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[iron fiber]

Sort of yes. We apply $2^{k+1}$ because we are breaking off the $k+1th$ part of the sum. Does that make sense ?

[devout snow]

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[languid jungle]

???

[languid jungle]

yes

this breaknig off the sum part, does it apply with all sigma induction?

[iron fiber]

Yes

[woven radish]

Baker

[languid jungle]

ah I see that makes other equation similar

I see the structer of such now

.close

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[lilac crescent]

Modular of z=(1-z)=1/z

[lilac crescent]

Solve for z in C

Idk how to start

,,z \in \mathbb{C}, |z| = |1-z|=|\frac{1}{z}|

[terse orchid]

Is it true that $\bigg| \frac{1}{z}\bigg|=\frac{1}{|z|}$?

[woven radish]

flynger

[lilac crescent]

Yep

[terse orchid]

Try seeing what you can get out of $|z|=\frac{1}{|z|}$ which only has one unknown

[woven radish]

flynger

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[zealous nest]

So, for this, I don't understand how the co-ordinate of P is -0.5,2.5 because everytime I do it, I keep getting -0.5,1.5

[zealous nest]

because I diffrentiated it, to get 6x^2 + 6x

and then cuz it equal -1.5

i got the equation

6x^2 + 6x + 1.5 = 0

and then I solved the quadratic to get

-0.5

and then subbed it back into the first equation

and I keep getting

1.5

[cinder bobcat]

well i plugged -0.5 into it and got 2.5

[zealous nest]

ok, maybe I just need to redo it again..

hm

maybe I'm just a bit stupid lmao I just redid it again 😭

I need to take a break (I have my gcse math mock tomm and I'm stressing, clearly)

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[devout snow]

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[devout snow]

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[dull thicket]

I just got to angle-bisector and triangle proportion theorem, how do i find both x and y?

[devout snow]

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

[next saddle]

what does the angle bisector thm state?

@dull thicket still here?

[devout snow]

@dull thicket Has your question been resolved?

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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.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• 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.

[cunning valley]

what percentage do I need on my finals which are worth 30% of my mark to pass the course? (50%+)

[cunning valley]

I'm dead serious btw

[vital edge]

If your finals are worth 30%

And you need 50% to pass

I'm sorry, but you're screwed mate

[faint sage]

don't think its possible

most you're getting is ~48%

[cunning valley]

...

[vital edge]

Even if you get full on the final, you'll only be at 48.6

[cunning valley]

I have another assignment worth ~15%?

[vital edge]

Try and get full on that

[cunning valley]

or two assignments

like two assignemnts worth 15% in total

[vital edge]

Try to get full marks on the assignments

[misty crest]

you’re cooked

[cunning valley]

my teacher marks so hard tho

[vital edge]

Then you would need ~60% on the final

[misty crest]

what class is it

[cunning valley]

english

[errant harbor]

That's bad

[misty crest]

oh who cares then

[faint sage]

you're going to need pretty much full marks for everything left

[misty crest]

you need the credit to graduate?

[cunning valley]

I dont need the credits

but I need the course

do i js take it next year

[misty crest]

🤔

[cunning valley]

or next semester i meant

I wasn't locked in

[misty crest]

sure

[cunning valley]

I literally did not do a single thing in english

I slept through 99% of it

[misty crest]

it’s not the end of the world tbh

just english

what’s your major

[cunning valley]

its just english

I'm in high school

[misty crest]

oh

lol

dawg

how do you take it next semester??

[cunning valley]

i got 95%+ in all my other math/science shit

[misty crest]

don’t classes run year round

[cunning valley]

nah its 1 semester for english

[misty crest]

🤔🤔🤔

[cunning valley]

how did I even end up here

18.60%:(

[misty crest]

only time i had alternating courses waa for bullshit electives

[misty crest]

yea this is pretty bad

[slate kite]

depends on the school, at my school its per semester.... only some IB courses are full sem

[slate kite]

do you procrastinate

[misty crest]

in my senior year after getting into uni i don’t think i did more than two assignments for my english class in the fourth quarter and i still had like a 50

how do you get an 18

[slate kite]

tbh its probably their time management

[cunning valley]

nah not for anything except english

[slate kite]

aaa

how

[cunning valley]

i dont do jack shit in english

LMAOO

wait actually

[slate kite]

is it like too boring

[misty crest]

i don’t blame you tbh

[supple knot]

Try asking in #study-discussion

[faint sage]

that might be the issue

[cunning valley]

it might be

idk

[misty crest]

do you do nothing in class?

[cunning valley]

idk I don't hand in anything on time

for english

FOR ENGLISH

[slate kite]

not in other classes, pretty sure its only english

[cunning valley]

yeah its just english

[slate kite]

aaa

[cunning valley]

i mean my english isnt even bad

my reading is like

[slate kite]

is it just too boring for you

[cunning valley]

idk my teacher is weird too

[slate kite]

or like do you just dont have the motivation for assignments

[cunning valley]

i got the 18% from a test

I think

reading comprehension

[misty crest]

what grade is this

[cunning valley]

grade 11

[slate kite]

holy

[misty crest]

damn

bro

💀

[slate kite]

no like

[misty crest]

that’s the year to lock in

[cunning valley]

its only november

[slate kite]

i suggest at least getting a BOOK

[cunning valley]

maybe I can come back from this

[cunning valley]

I've never read a book outside of english class

[slate kite]

that you are interested in

[cunning valley]

except for textbooks

[misty crest]

you probably read some things as a kid

[slate kite]

no like if you are super zoned out of english, just pull out the book and start reading it, at least that'll kinda help

[cunning valley]

I'm to lazy to do that bruh 😭 😭

[slate kite]

also teacher will probably not get mad at you if you are literally reading in an english class lol

[cunning valley]

I just go to sleep

whenever teacher tells us to read

[misty crest]

*too

[cunning valley]

to*

[misty crest]

no

[slate kite]

1. has the teacher ever stopped you
2. have you considered getting an engaging book

[faint sage]

mmm should probably stop doing that

[cunning valley]

once I think

[misty crest]

what do you want to do after school

[cunning valley]

uni

[misty crest]

damn

[cunning valley]

is it geniunely over for me

[slate kite]

holy shit, is the teacher just unresponsible 😮‍💨 .....

[misty crest]

maybe not over over

[slate kite]

i mean...

[misty crest]

like you can get into unis still

[slate kite]

yeah

[cunning valley]

i got

[slate kite]

if your sciences/maths are above 90s...? maybe they'll consider english as an outlier

[cunning valley]

ap physics 1, 2, calc bc, biology this year

[misty crest]

i knew kids with garbage gpas that went to uni but like definitely not prestigious ones

[cunning valley]

for ap

I'm self studying physics 2/calc bc

[misty crest]

how are you in your other classes?

grade wise

[slate kite]

pretty sure he says its in the high 95s?

idk though

[cunning valley]

yeah

almost 100

[misty crest]

that’s good i guess

[cunning valley]

math is a snooze fest too

[misty crest]

in what sense

like it’s easy and you get 100s or like you don’t do shit

and get 20s

[cunning valley]

I don't do shit and get 100s

[slate kite]

wait...

[misty crest]

are you in regular english

[cunning valley]

yeh

[slate kite]

shi

i was gonna say you could have used AP mark

😮‍💨

[cunning valley]

I'd get like a 3 on

an ap english test

my writings nothing special

[misty crest]

damn isn’t there like nothing to do in regular english

[cunning valley]

kinda

I think its still comebackable to 75 or smt

[slate kite]

uhhh

[cunning valley]

i may have slightly exagerrated

[misty crest]

stop playing video games or some shit

[cunning valley]

I literally don't do shit except math and physics at home

[slate kite]

hmm

[cunning valley]

and I just do math in english class

[misty crest]

doubt it

[slate kite]

do you do math or physics in school

[cunning valley]

math/sleep

[cunning valley]

im fr bruh

[cunning valley]

exaggerated*

[slate kite]

i mean...

[cunning valley]

I still got

1 essay

1 reading comprehension test

2 assignments

and final exam

[misty crest]

i mean i was a similar way in high school like i’d pull out math books in english class but my teacher was ok with it because i handed my shit in and had a good grade

[cunning valley]

maybe comebackable

defo passable

[cunning valley]

what was ur grade in english

[slate kite]

maybe comebackable

[misty crest]

depends on the year but it was always around 97

[cunning valley]

thats insane

wtf

[slate kite]

some universities are kinda lenient on a low mark if the rest of the marks are high...? depends on things tbh

[misty crest]

fuck my senior year english teacher though

had low 90s in that class

[cunning valley]

does writing just come to you naturally

or maybe it's because I don't read books

idk

[slate kite]

probably the books thing

[cunning valley]

I write like a fucking AI

[misty crest]

maybe don’t use gpt

[slate kite]

uhh...

[cunning valley]

I don't use chatgpt

for my essays

[slate kite]

maybe read more books

and like

learn how they phrase

[cunning valley]

is it even possible though

its november

how fast can my writing skills improve

[slate kite]

uhh

[misty crest]

i mean you don’t even need to read that many books like if you just read anything even online you’ll develop the skill

[slate kite]

depends on how much you want to put in

actually

audiobooks work too

[cunning valley]

I think I get 80s on my essays if I try

[slate kite]

?

[cunning valley]

80s on my essays if I try a little

cuz my past 5 essays have been 80%

[slate kite]

what about just audiobooks

[cunning valley]

like exactly 80%

like

I kinda understand how to improve my writing

first of all I make 5-6 careless mistakes in each essay

and then the other thing is

I think I should go deeper in my analysis ig

and rephrase them so they flow better

[misty crest]

maybe

[cunning valley]

and for reading its usually 90%+ that shits kinda free

[misty crest]

my 10th grade english teacher was begging me to major in english

tried looking for occupations that combined english skills with stem

i was like no sorry

[slate kite]

my math teacher was begging me to major in the maths since i was doing some bs in grade 10

like above calc and beyond lmfao

[cunning valley]

u did calc in grade 10?

how long will it take me to self study entire BC

[misty crest]

like 2 weeks

[cunning valley]

no like fr

[misty crest]

you could do it in 2 weeks

[cunning valley]

I cracked open a textbook

yesterday

I think limits are kinda easy

[misty crest]

i did physics c e&m + physics 2 in like 3 weeks

[cunning valley]

continuity too

I haven't got to derivatives

[misty crest]

spring break was a nightmare

[cunning valley]

wtf?

I gotta self study physics 2 aswell

[misty crest]

physics 2 was the easy part tbh

[cunning valley]

how about physics 1

[misty crest]

uhh idk i did that in school in 11th

[cunning valley]

phsyics 1 is kinda free ngl

[misty crest]

without any physics experience it might be a bit more difficult

[cunning valley]

I learned

unit 3 in 3 hours