help-23

[shut flare]

I could have the x be the adjacent side?

for arccos

I really wanna understand it

[lyric vine]

you want to find $\cos(\frac{\pi}{2}-\theta)$

[flat frigate]

A Fellow Human

[lyric vine]

which angle is $\frac{\pi}{2}-\theta$?

[flat frigate]

A Fellow Human

[broken yew]

nah

I was referring to the equation directly

You want to find angles such that sin and cos are both x

then relate them.

[lyric vine]

i mean both approaches work

[shut flare]

I think I see why thats the same as sin(pi/2), because if you shift the graphs by pi/2 they match

but I can't tell how I should show my work

for this problem

[shut flare]

How do I do this

[broken yew]

.

[shut flare]

is it on the 4th quadrant ?

Am I going all over the place lol

[lyric vine]

(i mean im not sure if you even need to prove the identity)

right so you marked out theta, one angle is pi/2 (right angle), what's the third angle?

and what would the ratio between the adjacent and hypotenuse be for that angle?

[shut flare]

oh

I'm thinking rn

does the total have to be pi?

180 degrees

[lyric vine]

(for euclidean space) yes

[shut flare]

oh so then

pi/2 - angle

damn

I just realized why that works, well, you made me realize

[lyric vine]

yw

[shut flare]

but I wanna see the end of this problem

so what can we conclude now

[lyric vine]

so you've proven the identity

[shut flare]

yeah

do I just write it out like that

my book doesn't provide solutions or explanatiosn for this particular problem

[lyric vine]

so if $x=\sin\theta=\cos(\frac{\pi}{2}-\theta)$, what does that tell you about $\arcsin x$ and $\arccos x$?

[flat frigate]

A Fellow Human

[shut flare]

arcsin(x) = arccos(pi/2-theta)

wait

I mixed something up

that doesnt seem right lol

[lyric vine]

no

you have two equations

\begin{align}
x&=\sin\theta \
x&=\cos(\frac{\pi}{2}-\theta)
\end{align}

apply $\arcsin$ for (1) and $\arccos$ for (2)

[flat frigate]

A Fellow Human

[shut flare]

arcsin(x) = theta
arccos(x) = (pi/2) - theta

and theta cancels out

I SEE IT

Hot damn

[lyric vine]

👍

[shut flare]

This was a very educational experience

Thank you everyone

[lyric vine]

yw

[shut flare]

I appreciate the help

I hate how they just throw facts and formulas at me in my school

Because when they teach me how it works I don't forget it

Though I guess it would take too long if they went into every detail

.close

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[crisp wren]

hi

[crisp wren]

f(x,y,z) = 4x^2 -y^4 + x^4 + z^5
fx= 8x +4x^3
fy=-4y^3
fz=5z^4

am I correct

<@&286206848099549185>

I waited over 15 minutes

[pulsar condor]

what's preventing you from using a calculator

and no you didnt

you waited a whole 2

[crisp wren]

nah

I sent it on another channel

noone answered

no one

so I made a new one

[pulsar condor]

doesn't matter

anyway, you can just use a calculator to check for yourself

[shut flare]

maybe https://www.symbolab.com can solve it

or something similar

I use it to check my work

[crisp wren]

It doesn't work

idk

it doesn't do multivariable calculus

[pulsar condor]

Given I instantly found the partial derivative sign...

yes, symbolab does in fact have multivariable calculus

[crisp wren]

alright let me see

again

found it

lmao

[pulsar condor]

Yeah.

Could have also used wolfram

[crisp wren]

I'm correct

thanks

bye

[pulsar condor]

Yeah you're correct, you just didnt need to open a channel

[crisp wren]

alright sorry about that

cya

[pulsar condor]

.close the channel.

[crisp wren]

.close

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[civic dove]

how do i do 5 and 6

[paper jolt]

For 5 we only need one senior to have their name called so we can add the probability to itself (probability sum rule) 4 times.

[civic dove]

so nCr(40,1)+nCr(40,1)+nCr(40,1)+nCr(40,1)

?

[paper jolt]

Oh I see, I thought this was probability practice not combinatorics.

Sorry

[civic dove]

<@&286206848099549185>

[safe radish]

@civic dove Has your question been resolved?

[civic dove]

<@&286206848099549185>

this one too

[safe radish]

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[wet spade]

https://www.math.nyu.edu/~avellane/HighFrequencyTrading.pdf
anyone please look this paper and tell me what are the prerequisites for understanding that paper ?

[wet spade]

understanding of prob

stochastic processes and stochastic calc

then ?

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[wispy prism]

Can someone explain

[iron tulip]

which question?

[safe radish]

@wispy prism Has your question been resolved?

[iron tulip]

9 or 10?

[wispy prism]

both

[iron tulip]

do you know how the highest common factor is defined?

[wispy prism]

yes

[iron tulip]

so what have you tried computing it?

[wispy prism]

I forgot how to compute it

[iron tulip]

one way is to look at the factors of 90 and 48, then the common ones, then the highest one

have you tried that?

[wispy prism]

yes

how to find the factors

[iron tulip]

$a$ is a factor of $b$ if and only if there exists an integer $k$ such that $ak = b$

[flat frigate]

Lance

[iron tulip]

try using that definition and look for shortcuts along the way

[wispy prism]

ok

what about question 10

[iron tulip]

what do you know about scales?

[wispy prism]

There are actual and map scales with ratio and to find the actual image of something you will use the map scale ratio

[iron tulip]

yes, so have you tried doing that?

i mean, how :P

[wispy prism]

Yes but i got the wrong answer

[safe radish]

@wispy prism Has your question been resolved?

[safe radish]

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[cedar crystal]

need to get area of it

but i really dont know how to get the altitude

to find height

or the base

[ancient escarp]

you're already given the height

[cedar crystal]

oh ok

then how do i get the base

?

[ancient escarp]

what theorem relates the sides of a right triangle

[cedar crystal]

trig?

or

wdym

wait no pythagorean

right?

[ancient escarp]

yes

[cedar crystal]

wait lemme see

but hold up

you cant use

pythagorean theorem

@ancient escarp

cause

this aint a right triangle

[ancient escarp]

that sucks

use trig then

[cedar crystal]

how

i dont have

[ancient escarp]

law of sines?

[cedar crystal]

another angle

[cedar crystal]

cant

[ancient escarp]

cos*(

**

[cedar crystal]

i dont have another angle

[cedar crystal]

i dont have 3 sides

[ancient escarp]

but you have 2 sides and an angle

that is opposite the unknown side

[cedar crystal]

o yea

true

lemme see

i got 3.57

ohh i got it

thanks dude

also can you help me

wit this

?

[ancient escarp]

what are you trying to find

[cedar crystal]

area

but i dont have altitude

and im not sure how to get it

cause vertical altitude isnt possible

[ancient escarp]

why not

[cedar crystal]

because you dont have any angles

if you make a mini triangle to solve for the altitude using sin

the other longer side is unknown

[ancient escarp]

if you make a mini triangle you know every angle

• a side

[cedar crystal]

how

your only given 36

[ancient escarp]

making a right triangle = creating a 90 degree angle

[cedar crystal]

oh kk

thx

got it

thx man

.close

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[gloomy stump]

Hi all, trying to figure out how to approach this: I started out by calculating expected value for each row. Then multiplied result by hit chance. Not sure if I'm even approaching this correctly. Any help is appreciated.

[safe radish]

@gloomy stump Has your question been resolved?

[gloomy stump]

<@&286206848099549185>

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@gloomy stump Has your question been resolved?

[olive wave]

hi I might be wrong, but wouldn't it just be 14/53/52/51/5 ?

oops the typing doesn't work, 1 times 4/5 times 3/5 times 2/5 times 1/5

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[steep mural]

I have a summation question. There is a sequence that goes n, 9n/8, 73n/8, 585n/512... I'm trying to figure out a general equation for it

[frigid geyser]

What exactly is the question?

Please post a pic or screenshot

[steep mural]

ok

this is what I'm currently working with. I think I need to turn into a summation and then find a general pattern for it

I'm having some trouble setting it up

[frigid geyser]

okay it seems your denominators multiply by eight each time?

is that right?

[steep mural]

yes

n/8^i

i think

[frigid geyser]

so we could factor out n and write $$n\sum_{i=0}^k \frac{1}{8^i}$$

[flat frigate]

Camilleone

[frigid geyser]

yes?

[steep mural]

yes that makes sense

[frigid geyser]

and now, do you know what a geometric sum is?

[steep mural]

The way that I have always done it is write out a few examples and find a general pattern

[frigid geyser]

well, good time to learn more things

[steep mural]

yes!!!

[frigid geyser]

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

what you have is something we call a geometric progression (in i)

strongly suggest reading this page for many details

[steep mural]

so the common ratio is 1/8

[frigid geyser]

yes

[steep mural]

the general equation would be (1/8)^(i-1)

Thank you for your help

that wikipedia page really helped

.close

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[rose terrace]

hello. how to simplify (2m)^-3?

[rose terrace]

.close

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[half elk]

What must be written on the blank so that the expression is a perfect square trinomial: 16x^2 - ____+25y^2

[grave ferry]

$(4x)^2-40xy+(5y)^2$

@half elk

[half elk]

I think the blank should be filled

[grave ferry]

in the blank

write it in the form of (a+b)^2

[half elk]

20xy?

[grave ferry]

no

40xy

[half elk]

I'm pretty sure it's a (coefficient)xy

[grave ferry]

yes

[half elk]

It's a perfect trinomial

[grave ferry]

40xy

yes

it should be 40xy

[half elk]

Oh, how did I get -40 then lol?

Thanks I get it now

I didnt factor it right :))

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[flat frigate]

Iron Man

[half elk]

.reopen

[safe radish]

✅

[grave ferry]

like this

[half elk]

Thank you!!!

Oh yeah I forgot there's alr a negative sign there

Thanks, Iron man

[grave ferry]

It's Ok!!

[half elk]

.close

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[ancient escarp]

don't do people's work for them

[grave ferry]

I just taught him

[ancient escarp]

you did the work for them

don't do that

[grave ferry]

hmm

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[snow pivot]

how do i easily get multiples of i? for example i^123

[finite wasp]

Powers of i? It cycles every 4, since i^4 = 1. So you can take away any multiple of 4 from 123 without changing the answer. What's the biggest multiple of 4 that's less than 123?

[snow pivot]

120

ah i see

thank you

[zinc crown]

use .close if no more questions

[snow pivot]

.close

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[marsh walrus]

.

[ancient escarp]

.:)

[marsh walrus]

.wow

.thats pretty cool

[young orbit]

Anyone able to dumb down the wording for b?

I don't actually understand what it's asking

[marsh walrus]

create a cost function

well maybe more like this

Profits = Revenue - Cost

I'll show the first one since its the trivial case i guess

we know revenue is 70 dollars per unit right?

and cost is 50 dollars a unit

part a is saying, we guessed exactly right

we sold every unit we made and have none left over

so we sold 445 at 70 bucks a pop

and each one cost 50 bucks to make

for a clean profit of 20 bucks a unit on 445 units

make sense?

do you see how the formula is going to change if you have some left over (that you didnt sell)?

@young orbit

[digital sky]

In the first part you wrote down an expected value of how much demand the company expects which the company bases its monthly orders on

So the company decides to buy 445 items / units

The second part tells you how much each item costs and how much they sell it for

And how much they actually sell

You can use this information to find out whether the company gains money or loses money and how much

e.g. If I buy 10 usb cables, they cost 1 dollar each. I manage to sell only 5 in a month at 2 dollars each. Then I made no profit or gain, 0 dollars in that month.

[young orbit]

So like, 50*300=15000 (total cost), then the revenue (70) - the total cost = -14930?

[digital sky]

no

If you think you are going to have 445 orders of an item and you buy enough to fulfill those orders, but you end up only getting 300 orders

You still paid upfront for more than 300

In this question part a suggests the company thought it was going to get 445 orders and that is how many it planned to order

Part b tells you to use that information to find out how much the company made, so you should think it bought to prepare 445* items

So the total cost should be 50 * 445

The question says it sold only 300 units at 70 dollars each (sale price / revenue per unit)

sorry, had some typos. Do you know how to calculate the the resulting profit / loss?

[young orbit]

I'm just trying to keep up

[marsh walrus]

you need to try writing some stuff

👀

[digital sky]

Do this again, this time correct it to show that the company was ordering 445 items which costed it 50 dollars each

[young orbit]

50*445=22,250, then 70-cost = -22,180?

The last question before this was a negative number

But it just feels wrong being that huge

[digital sky]

So that calculation for the total cost 50*445= 22,250 is fine

Now what is the total money that the company received (revenue) from selling these items? You would need to know how many items were sold and at what price

The question says ‘each unit demanded generates $70’ this means that the selling price is $70 each

Then it says ‘actual demand for the item is 300 units’, which means 300 items were sold

[young orbit]

So I take that 50*300=15,000

22,250-15,000=7250?

[digital sky]

no

‘Assume that each unit demanded generates $70 in revenue and that each unit ordered costs $50’

This means that when the company sells an item, they charge $70, and that same item costs the company $50 to get/make

So when you calculate revenue, you should use the selling price, not the item cost

since revenue is the the sum of the money that is received by the company

[young orbit]

70*300?

[digital sky]

yes

[young orbit]

1250?

It was

😢

Jesus

I don't think this was covered in the online lectures

Which didn't help

Sorry I had to make you type entire essays

[digital sky]

no problem

[safe radish]

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[modest cloak]

on a graphing calculator

[modest cloak]

how can i see the far right side? because i need to see where 2 graphs intersect

i see where they intersect on the left but not on the right

ive changed the window and still cant see

[winter pivot]

Zoom

Fit

[oblique warren]

Or you can adjust the graph. By going to window.

[modest cloak]

.close

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[safe radish]

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[errant prairie]

So I was wondering if it is called anything special when you take the digit sum of a number all the way down until it only has one digit. For example with 293 where it first becomes 14 and then 5

[errant prairie]

And also if there is a mathematical function for it, like the function for digit sum once is $\sum_{i=0}^{\lfloor \log_{10}(n) \rfloor} \frac{n \mod 10^{i+1} - n \mod 10^{i}}{10^{i}}$

[flat frigate]

Mikkel-T

[safe radish]

@errant prairie Has your question been resolved?

[quasi bison]

well it just so happens that taking the sum of the digits of a number repeatedly until you get down to 1 digit produces the remainder of the original number mod 9

except multiples of 9, which yield 9 rather than 0

[errant prairie]

Oh, thats really cool. Thanks! Do you know how I could write this out as a formula?

[safe radish]

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[errant prairie]

.reopen

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✅

[final loom]

Lol

indeed, yes it has a term to it

called the digital root

Look it up on wiki(!?) ig

[errant prairie]

Thank you both so much!

.close

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[trim maple]

A spinner with 6 equal sides has 3 red, 2 blue, and 1 yellow edge. A second spinner with 7 equal
sides has 4 purple and 3 green edges. Both spinners are twirled simultaneously. Find the probability of
getting: a red and a green

[trim maple]

nvm was looking at wrong asnwer

.close

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[quartz estuary]

I am going to show that the limit of f(x,y)=(1-exp(x^2+y^2))/(x^2+y^2) when (x,y) tends to (0,0) is -1

[quartz estuary]

So I stuck at here

Normally I can factorize the term

But I hv no idea about it now

[safe radish]

@quartz estuary Has your question been resolved?

[quartz estuary]

<@&286206848099549185>

[finite wasp]

To be clear, you can't use l'hopital on this?

[thin token]

just use expansion series

[finite wasp]

That works too: I'm just wondering how much we're allowed to assume here if we're at the level of epsilon-delta proofs.

If you don't want to use l'hopital or Taylor expansion, the best way is probably to make the substitution r = x^2 + y^2, giving you the limit (1 - e^r) / r as r approaches 0.

You can then use Bernoulli's inequality, along with the knowledge that e^r is the limit as n approaches infinity of (1 + r/n)^n to put bounds on e^r (for r close to 0) which you can use to evaluate the limit with the squeeze theorem.

https://en.wikipedia.org/wiki/Bernoulli's_inequality in case you need a refresher.

[safe radish]

@quartz estuary Has your question been resolved?

[quartz estuary]

Umm I think we can't? I am showing the limit existence.
The main reason is most likely my course haven't teach multi-variable calculus. We just talked about the existence and the non existence of the limit in R^n.

[quartz estuary]

You mean the binomial?

[quartz estuary]

Ok let me try this

[finite wasp]

l'hopital actually doesn't require the existence of lim f(x)/g(x) to be known as a prerequisite. You only have to know that lim f'(x)/g'(x) exists, which will be easy if you've already worked out what its limit is.

Although that's a moot point if you haven't covered it yet.

[quartz estuary]

Ok ic

[quartz estuary]

I apply the inequality and get $\exp{x^2+y^2}>1+x^2+y^2$, then $\abs{1+x^2+y^2-(1+x^2+y^2)}>\abs{1+x^2+y^2-(\exp{x^2+y^2})}$

[flat frigate]

parameters

[safe radish]

@quartz estuary Has your question been resolved?

[feral linden]

Told you the answer before I believe?

You already have that the limit of (1-e^u)/u=-1 using hopital rule.

So any ε>0 there exists δ>0 such that any |u|<δ, |(1-e^u)/u -(-1)|<ε

Now you just replace u with x^2+y^2…

[quartz estuary]

Yes but I am a bit confused about how it works in vector.
My prof lecture notes say that I have to prove if |f(x,y)-L|<epsilon then there exists delta such that 0<|(x,y)-(0,0)|<delta

Also by the |...| sign denotes the magnitude of the vector

[feral linden]

😂 |(x,y)-(0,0)|<ε is just x^2+y^2<ε^2 dude

Any ε, therefore any ε^2

[quartz estuary]

Oh

Omg

[feral linden]

😂

[safe radish]

@quartz estuary Has your question been resolved?

[final loom]

Let $t = x^2 + y^2$ and $t \to 0$

[final loom]

wait wha-

oml

Da heq

$0 < \sqrt{x^2 + y^2} < \delta$

[final loom]

$\abs{ \frac{e^t -t - 1}{t}}$

[safe radish]

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[tawny stone]

I was wondering what this could be?

[lone arch]

off the final amount so I think you first have to get the price, then subtract the coupon from it

[tawny stone]

31.5

Is the total + discount

How would I write an expression

[lone arch]

yeah

[lone arch]

can you write it as a single calculation without using variables?
Using 5 pounds and $7.25 per pound, then the 4.75$ discount?

[tawny stone]

I think they are asking for a variable?

I’m not sure

Lol

[lone arch]

yeah, but I think its easier to first do it without variables, just a single calculation

and then replacing the 5 with p

[tawny stone]

What would that look like

[lone arch]

final cost of 5 pounds = 7.25 * 5 - 4.75

thats how u get 31.5

now replace 5 with p

final cost of p pounds = 7.25 * p - 4.75

that should be final cost of p pounds

[tawny stone]

Oh p is pounds

Thanks man

31.5
7.25 * p - 4.75

Didnt work :/

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@tawny stone Has your question been resolved?

[safe radish]

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[lean chasm]

how do I solve $$\int e^{W(\frac{\frac{1}{4}x^2(2 \ln{x} - 1)}{e})}$$

[plucky elk]

What's W?

[lean chasm]

productlog

[lean otter]

is texbot down?

$\int e^{W(\frac{\frac{1}{4}x^2(2 \ln{x} - 1)}{e})}$

sheesh ok ;-;

[plucky elk]

yea texit bot is down

[lean chasm]

=tex \int e^{W(\frac{\frac{1}{4}x^2(2 \ln{x} - 1)}{e})}

smh

[lean otter]

I assume you've already tried using e^(W(x)) = x/W(x) and some sort of substitution?

looks painful either way

[lean chasm]

there we go

[lean chasm]

no idea what I'm supposed to substitute lol

wfa is not being helpful either

[lean otter]

try stuff until you get something or determine that it's probably not a good route to take

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@lean chasm Has your question been resolved?

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[lean otter]

I got -7 for the top one

but -3 is not equal to or greater than 2. I am not sure if I should write anything for the 2nd equation.

[untold canopy]

Just sub the value of -3 into the piecewise function and see what condition you met

Seems like you already have a trip on what to do

[safe radish]

@lean otter Has your question been resolved?

[lean otter]

maybe 😳

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[lean otter]

hm

[queen parcel]

What have you tried?

[queen parcel]

^^

[lean otter]

@queen parcel

uh After 1998, A = 1/2G

DOB(A) + DOB(G) = 3846

After 2002, how old is A

1998 - DOB(A) = (1998- DOB(G))/2

[fiery merlin]

Close.

[solid shell]

I'm guessing because I haven't written it out but could you solve for A given (1998 - 2A) + (1998 - A) = 3846?

[fiery merlin]

A is the age of Alex in 1998, so Alex was born in 1998 - A.

[lean otter]

yes

[lean otter]

dont know where that is crom

wait nvm I do but like dont know if it works

[solid shell]

Alex's year of birth was 1998 - A

And his grandmothers was 1998 - 2A

[lean otter]

cause 1998 - 2A is grandmas age

[fiery merlin]

No, it isn't.

1998 - 2A is when grandma was born.

[solid shell]

If you add those together you get 3846, then I believe you can solve for Alex's age

[fiery merlin]

Just like 2022 - your age is the year you were born.

2A is grandma's age.

[lean otter]

so A = 50

....

so what will I do now

[fiery merlin]

Well, what is the question asking?

[lean otter]

I have 54

1998 - 50

[solid shell]

👍

[lean otter]

2002 minus that

ok thanks

.close

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[winter pivot]

i got about 4 or 5 of these right just guessing

but i'm not sure which ones

i'm just gonna go through these in order

wait no shit lol

the definition of the directional derivative is the dot product of the direction vector with the gradient vector. if the two are perpendicular then it's 0

[zinc crown]

this would be the same reasoning as the last right? if the direction is perpendicular to the gradient, the dot product would be 0

[winter pivot]

This also has to be true since there are infinitely many vectors perpendicular to a given vector in R3

i'll be very honest

i have no fucking clue what a contour is

[zinc crown]

not sure what contour means

LOL

i mean its probably right cuz it seems like the definition of the graident

[obtuse jackal]

@zinc crown Does it still hold for non-continuous derivatives ?

[zinc crown]

oh wait i actually have notes on this

[zinc crown]

not sure

[obtuse jackal]

Yeah don't think so intuitively

If you just take the graph of |x| and rotate it into a 3D it seems wrong

Because the gradient doesn't exist

Bad counter-example

[winter pivot]

what is the contour

is that just like the surface of it

[obtuse jackal]

I believe it's a set of points where f has the same value. But I never took calc so check Wikipedia

[winter pivot]

no that's a level curve/surface

[zinc crown]

think it might just be any line that lies on the surface that passes through the point

so yeah

[obtuse jackal]

Wasn't sure if the names were for the same thing

[winter pivot]

wait are those the same thing?

a level curve is the contour of the surface for a given value?

i'm confused

wait ok those are the same thing

so is the gradient vector tangent to the contour?

wait yeah it has to be

https://samjshah.wordpress.com/files/2009/01/tangent3.gif

proof by visual

[obtuse jackal]

I thought it was always perpendicular

[winter pivot]

this is correct

i thought that the gradient vector was always tangent to the contour?

also 1 was false lol. directional derivatives are scalars not vectors haha

[obtuse jackal]

This also solves 6 and 7 btw

[winter pivot]

i figured out 3

correct me if i'm completely wrong on this

but the gradient vector itself is just a "velocity" if you will

and a velocity can't be tangent to a position

[obtuse jackal]

Kind of

[winter pivot]

you would need to take the gradient vector and turn it into the normal line to find the tangent to the contour

[obtuse jackal]

Gradient != velocity and contour != position though

[nova creek]

The gradient vector is always perpendicular to the contours

[winter pivot]

but it's not tangent to the contour

which is what i don't get

[nova creek]

It's not tangent to the contour because it's perpendicular.

I don't quite get what you're saying

[winter pivot]

isn't the tangent line to the contour perpendicular to it though?

[nova creek]

I don't see how that could be the case

If its perpendicular, then it crosses at a right angle to the curve. I don't see how that could be a tangent

[winter pivot]

it doesn't cross though

it meets the curve at the point at a right angle

i'm honestly not too sure though

[nova creek]

If it doesn't cross, how could it meet at a right angle?

[nova creek]

Moreover, I don't see how this image is a visual proof that the gradient is tangent to the contour. There is no contour on this graph

[winter pivot]

idk maybe something like this?

the line doesn't cross the curve but it forms a right angle

[nova creek]

But it's not perpendicular

When I say it crosses at a right angle, I mean the angle between the line and the tangent to the curve is 90 degrees

In this case, it's 0 degrees

[winter pivot]

wait ok

so the tangent is actually parallel to the curve at the point

as in it's the line with the same slope as the curve at that point

[nova creek]

Yes. I think. I'm not sure what the rigorous definition is, but considering we say that the gradient is perp. to the contour, I suppose the tangent would be parallel to the curve

I mean, consider the graph z = x² + y². The contours are a series of concentric circles centered at the origin, while the gradient is a bunch of vectors pointing directly away from the origin. It's clear the gradient is perpendicular to the contour everywhere

[winter pivot]

OH

yes

perfect

i get it now

yes. the gradient is always perpendicular to the contour so it can't be tangent

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[spare girder]

x³-8x+6=0, x is a rational number

[spare girder]

cant find the correct answet using horners scheme

didnt succeed in using viete's

[iron tulip]

what do you know about rational roots of integer polynomials?

[spare girder]

well i know that there are three of them

[bronze prism]

Are you, though?

You know the polynomial has three complex roots

Which may or may not be distinct, real, let alone rational

In this case there are three real roots (i just graphed it)

[spare girder]

yeah so how do i find those roots?

what do i use

[plucky elk]

Rational root theorem

[spare girder]

thanks

.close

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[final loom]

Are you doing a test rn?

[valid grove]

No

Lol

Sir gave us hw

Advance questions

[final loom]

🤯

And you're running late on hw? why so many questions bombarding all at once lol

[valid grove]

No

I have time rn

Cause tomorrow’s sunday

Gonna enjoy

Lol

[final loom]

Anyways, just calculate $$\lim_{x \to 2^+} (x - a)^{\frac{1}{2 - x}}$$ and $$\lim_{x \to 2^-} (x - a)^{\frac{1}{2 - x}}$$

[final loom]

make them both be equal

and the limit value = your f(2)

[valid grove]

But

[final loom]

but?

[valid grove]

If f is not defined at 2

[final loom]

But?

Yes

[valid grove]

Then how is it continuous

[final loom]

f is not defined at 2

YOU are supposed to define f(2)

to make it continuous

[valid grove]

Oh

Thanks @final loom

.close

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[calm spoke]

Asking this question again because last time my channel got timed out, sorry!

What would I need to know to find the surface area and volume of the shape highlighted in red? The size of every side and the angle of the sloped roof? Or would I use pythagoras theorem?

[safe radish]

@calm spoke Has your question been resolved?

[lost patrol]

Hi...

[calm spoke]

Hi

[lost patrol]

So what level are you in Maths

[calm spoke]

Year 10 high school maths

[lost patrol]

Ohhkie

[calm spoke]

We just started this topic and I'm struggling a bit to understand it

[lost patrol]

Ohh

Come in a VC and lets see if we can do it

[calm spoke]

Well my headphone mic doesnt workkkkkkkkkkk

[lost patrol]

Ohh

Np

You'll be probably using the side lengths

[calm spoke]

yea

[lost patrol]

Yeah

Also

Youll need the formulae for volumes of triangular pyramid and prism

[calm spoke]

Oh yeaa okay

thank you

is the entire roof a triangular pyramid?

[calm spoke]

<@&286206848099549185> 🙂

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[calm spoke]

.close

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[steel imp]

How many ways to role a sum of 8 with 2 5 sided fair dices

[plucky elk]

Can you write out the table for the possibilities

[steel imp]

Is there an algebraic approach

Like 4 and 4 is a possibility

[lyric ingot]

Count the amount of ways you can make 8, with [1,...,5] + [1,...,5]

[steel imp]

*1…5

[lyric ingot]

Oh 5 sided?

[steel imp]

Is the only way case by case

Ye

[lyric ingot]

There

[steel imp]

I can’t think of one

[lyric ingot]

What do you mean?

You have two of those 5 sided dice, so 1-5 + 1-5

There is definitely at least one

[plucky elk]

It's more work but yea

[steel imp]

Can u explain

[lyric ingot]

Where do you need explaining?

Understanding the problem?

[steel imp]

On how to solve this algreaincally

[lyric ingot]

@plucky elk He wanna know

[plucky elk]

https://www.quora.com/The-density-function-of-the-sum-of-two-random-variables-is-the-convolution-of-their-respective-densities-What-is-the-intuition-behind-this

It's a convolution of the two probability mass functions

[steel imp]

Thank you

[steel imp]

Yo what if I knew that each must of 3 then I could use stars and bars right

Originally it would be 1 bar ans 8

But since each as 3 we could subtract 6

Since the lowest number in a sum of 8 is 3 )3+5

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@steel imp Has your question been resolved?

[safe radish]

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[wooden merlin]

how can i formally say if n is divided by four, the remainder is 1
i think i can use mod somewhere but i forgot how to use it

[wooden merlin]

is this correct?

[olive wave]

the mod is correct, I so think yes

[wooden merlin]

okok ty

.clseo

[olive wave]

welcome 🙂

[wooden merlin]

.close

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[idle gazelle]

hi

[idle gazelle]

how do i do this

[safe radish]

@idle gazelle Has your question been resolved?

[idle grove]

greg recovers his glucose level after 4 for the lunch and 2.5 hours for breakfast (from where they start to where they end in Y axis) and 4 = 2.5+1.5, which means greg took 1.5 hour more to recover his glucose level

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@idle gazelle Has your question been resolved?

[safe radish]

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[safe radish]

@sudden merlin Has your question been resolved?

[sudden merlin]

<@&286206848099549185>

[broken yew]

write out what 378 is

[flat frigate]

Result:

100

[final loom]

mb, sry

[sudden merlin]

378 is

3x^2 + 7x + 8

[broken yew]

good.

And you want this to be a square

any ideas?

[sudden merlin]

well i thought of making it equate a n^2, but thats not helpful is it?

[broken yew]

i mean... i think thats the most natural thing to do

If you want it to be a square, set it equal to n^2

$$n^2 = 3x^2 + 7x + 8$$

[flat frigate]

Shuri2060

[broken yew]

What next 🤔

So ideas such as:

• modulo arithmetic
• difference of squares
• completing the square

come to mind

If you haven't already experimented with them

[sudden merlin]

oh difference of squares that might be interesting

i dont think i would need modulo arithmetic

and idk how to do properly anyway

but this quadratic has imaginary solutions

[broken yew]

thats a good point

this is a quadratic

and no it doesn't.

[sudden merlin]

well i should be more specific, the quadratic 3x^2 + 7x + 8 doesn't intersect with the x-axis

[broken yew]

well yh sure.

Notice this is a quadratic in x.

that you can solve.

[sudden merlin]

how....?

[final loom]

$$(6x + 7)^2 + 47 = 12n^2$$

[final loom]

1. Notice, this implies n ≥ 46

2. set m = 6x + 7 > 157, and rewrite:

$$12n^2 - m^2 = 47$$

[final loom]

where, you're asked only for the smallest value of n > 45, s.t. this equation is satisfied

or, you could notice: $$12(n^2 - 4) = m^2 - 1$$

[final loom]

and work your way out of there

[broken yew]

I was referring to using quadratic formula on the original thing

Does that not get you there

[final loom]

$$x = \frac{\sqrt{12n^2 - 47} - 7}{6}$$

[final loom]

that's the same thing moreoless, you'll be hunting after the discriminant

[broken yew]

$$x = \frac{-7\pm\sqrt{49 - 12(8 - n^2)}}{6}$$

[flat frigate]

Shuri2060

[final loom]

that's why I initially suggested D = perfect square, if you remember

[sudden merlin]

ok, i have so many questions lol

[final loom]

Yes?

[sudden merlin]

firstly how did u get to the 6x+7

[sudden merlin]

here

[final loom]

applying some brains?

I mean..

[broken yew]

• completing the square
  ===

that was one of my suggestions

To do that, the coefficient on x^2 must also be a perfect square

and you want to work with integers

thats why they multiplied through by 12

[final loom]

I believe you're capable enough to figure the how?

can we proceed to your next "questions"

[sudden merlin]

ok then wdym by pell equation, i'll search it up later but hows that relevant rn

[final loom]

I just deleted the part about pell equation

please ignore,

Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x² - ny² = 1 where n is a given positive nonsquare integer, and integer solutions are sought for x and y.

[broken yew]

I get the feeling plugging into quadratic formula is most straightforward method (if it works)

[sudden merlin]

ok well i said i had a lot of questions but idk how to formulate them lol, umm how about how did you notice this and also what did you use quadratic formula on?

[final loom]

I, did not use the quadratic formula

[broken yew]

I used it on our original equation.

$$n^2 = 3x^2 + 7x + 8$$

[final loom]

Also,

[flat frigate]

Shuri2060

[final loom]

did you read the other messages that I sent before the "noticing" part

[final loom]

.

that should be self-explanatory

About how I plan to proceed, however, is: $$12(n+2)(n-2) = (m+1)(m-1)$$

[final loom]

but how to proceed from here, is entirely up to you. That's all the lead I can provide tbh

from here, you're free to plug and check values into $$m^2 = 12n^2 - 47$$ should be a perfect square

[final loom]

it's pretty trivial to build up more and more conditions from here though, so you can handpick with much ease

Also, I think n ≥ 46 would also help you :D

[broken yew]

You're going to end up with the same math either way

But the quadratic is used as if this was a quadratic equation in x

n is a constant

$$0 = 3x^2 + 7x + 8 - n^2$$

[flat frigate]

Shuri2060

[broken yew]

In case you didn't see

[sudden merlin]

yeah i saw that

[broken yew]

This will lead to the same math as above done by Ansh.

[sudden merlin]

hmmm, so at this point i would just have try values out

and just figure it out from there?

[final loom]

not really, depends on your capability though

for example,

you see this: $$12(n^2 - 4) = (m^2 - 1)$$

[final loom]

And you can conclude that n must be even

then you set n = 2k, and rewrite $$48(k^2 - 1) = (m^2 - 1)$$

[final loom]

, k ≥ 23

[sudden merlin]

ok im still stuck on how u determined the (n^2-4) thing, can u explain

sry about the spam if u saw

but i still dont get how u arrived at that conclusion

[final loom]

Actually, it'd be convenient if you could read on it as well

[final loom]

btw, if you're manually calculating..

forget it

n = 166 is way beyond your reach 💀

which is still, only partly the answer..

[final loom]

If you do, it'd be interesting to find out that you can actually generate a sequence of solution: $$n = a_k, , , a_{k+2} = 14a_{k+1} - a_k$$

[final loom]

where, you can see that a_1 = 2, with m = 1 and a_2 = 16 with m = 55 satisfy the equation

this quickly gives a_3 = 222

and on checking,

,calc (sqrt(22222212-47) - 7)/6

[flat frigate]

Result:

127

[final loom]

is actually an integer

you get your x

[safe radish]

@sudden merlin Has your question been resolved?

[sudden merlin]

would this problem be considered an algebra I problem

and able to be done in 2.5 minutes?

just curious there isnt another way of doing it analytically?

[final loom]

well, if you're familiar with pell equation

yes

but idk abt the algebra 1 part

I suppose your initial assumption was wrong

$378_x$ is probably meant to be square of an integer in base 'x'

[sudden merlin]

then...

how would u do that lol

i forgot how to do bases stuff

[final loom]

well, then what

then you equate $$3x^2 + 7x + 8 = (ax + b)^2$$

[final loom]

notice a = 1 is possible only

so

$$x^2 + 7x + 8 = 2bx + b^2$$

[final loom]

solve for x > 25

[sudden merlin]

oits 2x^2

i think

[final loom]

Yes

tysm

$$2x^2 + 7x + 8 = 2bx + b^2$$

[final loom]

ooo

LoL same recurrence

mwahaha

[frigid geyser]

what the heck are you doing that needs that kind of recurrence solution

[final loom]

Camieeee

Pell Equation

but

the guy says it's algebra 1

[sudden merlin]

yes

[final loom]

[sudden merlin]

our teacher said that u would just use a calculator

[frigid geyser]

what's the question?

[sudden merlin]

and brute force it

[final loom]

Da heck does the question mean

378_x is square of integer in base x? in base 10?

[frigid geyser]

doesn't matter, a square's a square

[final loom]

is that how bases work

[frigid geyser]

yes, that is how bases work

[final loom]

then,

is this the correct interpretation?

[frigid geyser]

sure

[safe radish]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

[final loom]

and this?

@sudden merlin

.reopen

[sudden merlin]

.reopen

[safe radish]

✅

[final loom]

ty

Camie deep think

[frigid geyser]

so why multiply by 12?

[final loom]

cuz perfect square!?

[frigid geyser]

that LHS is no longer a perfect square

[final loom]

Dude omg

[sudden merlin]

🤣 there has to be an easier way

[frigid geyser]

either way there's going to be a stray factor of 3

[final loom]

$$12n^2 - m^2 = 47$$

[final loom]

now that's where I thought of pell equation

[sudden merlin]

i was thinking there was some property of perfect squares i didnt know about or smth

[frigid geyser]

has to be an easier way

for instance

[final loom]

I noticed a way

[frigid geyser]

$3x^2 + 7x + 8 = 3(x+1)^2 + (x + 4)$

[flat frigate]

Camilleone

[final loom]

$$12(n^2 - 4) = m^2 - 1$$

[frigid geyser]

replacing $x+1$ with $y$ because notation is ugly, $3y^2 + (y+3) = n^2$

[flat frigate]

Camilleone

[final loom]

or, $$(n^2 - 4) = (3x + 4)(x+1)$$

[final loom]

but what's there to conclude from this

I mean, one could possibly bound by (2y + 1)^2

?

[frigid geyser]

tbh at this point brute forcing might be faster HAHA

[final loom]

how'd you brute force though

[frigid geyser]

start with x = 26, work up

[final loom]

what's your guess

like, after how many iterations

do you think one might get the answer

[frigid geyser]

,w sqrt(3n^2 + 7n + 8) for n=26 to 85

[final loom]

none of them

would work

go higher

Higher

[flat frigate]

Results provided by WolframAlpha

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Requested by Camilleone#1596

[final loom]

meanwhile amazed that Camie actually remembers all those perfect squares