help-27

[distant cypress]

you know that exponentials cannot be zero

[restive river]

it would be -inf

[distant cypress]

not, exponentials cannot be negative

[restive river]

oh

1/x

idk why am i guessing math

[distant cypress]

think of the domain of the function

its only between 0 and pi/2

between those

sin(x) is minimum at x=0 but then 0^0 is 1 choosing between 0 to 1 sin(x) is less than 1 so something less than 1 but > 0 is raised to something less than 1 but>0 would be less than 1 but>0

so the minimum value would somewhere when x is less than 1 but greater than 0

[restive river]

tending to 0?

[distant cypress]

it would not tend to 0 if its between the range 0<x<pi/2

[restive river]

i am sorry but i don't undertansd this

[narrow plank]

,w graph x^(sin x)

[woven radish]

Results provided by WolframAlpha

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Requested by shenny17#0

[restive river]

its not x^sin(x) its sin(x^sin(x))

[narrow plank]

oop

[distant cypress]

what its just say is that for function $a^{x}$ the range will be less than 1 if $0<a<1$

[woven radish]

Bring Back Beatrix

[restive river]

when the function is defined for 0 to pi/2?

[distant cypress]

you will find the minimum at x when you found where $x^{sin(x)}$ is at its minimum

[woven radish]

Bring Back Beatrix

[distant cypress]

in this case its only for 0 to 1

so the minimum x is between 0 and 1

[restive river]

bro i honestly don't understand anything, can we start again just for the minima part step by step.

apologies for wasting your time

please?

[distant cypress]

its alr, ok i start from the begginning for minima

[restive river]

thanks

[distant cypress]

the whole function is this $ln(sin(x^{sin(x)})+1)$

[woven radish]

Bring Back Beatrix

[distant cypress]

the ln function there would only approach minimum if

$sin(x^{sin(x)})$ is at its minimum

[woven radish]

Bring Back Beatrix

[restive river]

ye

[distant cypress]

and since $0<x<\frac{\pi}{2}$

[woven radish]

Bring Back Beatrix

[distant cypress]

$sin(x^{sin(x)})$ will not be negative

[woven radish]

Bring Back Beatrix

[distant cypress]

because $x^{sin(x)}$ will not be negative

[woven radish]

Bring Back Beatrix

[distant cypress]

so you can treat this function $x^{sin(x)}$ as like $a^x$

[woven radish]

Bring Back Beatrix

[distant cypress]

if $x=0$ then $x^{sin(x)}$ will be 1, but if you let $0<x<1$ the range will be $0<R<1$

[woven radish]

Bring Back Beatrix

[restive river]

$sin(x^{sin(x)})$ will not be negative, because $x^{sin(x)}$ will not be negative

[woven radish]

smartypants11062008

[restive river]

this

[distant cypress]

yes

we just eliminate the instance where the sine goes on to angle greater than 2pi

[restive river]

even if x^sin(x) is not negative, how does that signify that sin can't be negative

oh

[distant cypress]

because we only focus on $0<x<\frac{\pi}{2}$

[woven radish]

Bring Back Beatrix

[distant cypress]

so $sin(x)$ will not be negative

[woven radish]

Bring Back Beatrix

[distant cypress]

and so is $x^{sin(x)}$

[woven radish]

Bring Back Beatrix

[distant cypress]

now

[distant cypress]

we return here

if $x>1$ then $x^{sin(x)}$ will be greater than 1

[woven radish]

Bring Back Beatrix

[distant cypress]

because the base wil be greater than 1

so we eliminate that domain

and also 0

so the only remaining domain where we could find the minimum is $0<x<1$

[woven radish]

Bring Back Beatrix

[restive river]

yes ok

[distant cypress]

and if you want to get that x where its minimum you have to find where will the minimum of $x^{sin(x)}$ is

[woven radish]

Bring Back Beatrix

[restive river]

yea

[distant cypress]

know you could solve the derivative of that to find the minima and maxima but it would be too complicated

but this has some analogy

you could think of it this way

[restive river]

in the domain (0,1)?

[distant cypress]

yes

[restive river]

we can do it by derivating it right?

[distant cypress]

yes but it will be complicated

[restive river]

how can i find the minimum value of x^sin(x) then?

[distant cypress]

here

you could solve it numerically

or you could solve for derivative and then apply the newton method

to find the root of the derivative

if you solve it numerically you would find it somewhere like $\frac{\pi}{9}$

[woven radish]

Bring Back Beatrix

[distant cypress]

and i dont think that you could express the minima symbolically

because if you substitute x as $\frac{\pi}{9}$ the answer will not be symbolical

[woven radish]

Bring Back Beatrix

[distant cypress]

$ln(sin({\frac{\pi}{9}}^{sin(\frac{\pi}{9})})+1)$

[restive river]

hmm

[woven radish]

Bring Back Beatrix

[distant cypress]

but another way if you want the exact

.w

you get the derivative of $x^{sin(x)}$

[woven radish]

Bring Back Beatrix

[distant cypress]

which is

wait how do you prompt wolfram alpha

,w derivative x^{sin(x)}

[restive river]

sin(x).x^sin(x)-1

[woven radish]

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Requested by Bring Back Beatrix#7534

[distant cypress]

that

[restive river]

where does this log come from?

[distant cypress]

you could take the term in parenthesis

i solve for the derivative of $x^{sin(x)}$

[woven radish]

Bring Back Beatrix

[distant cypress]

you could take the term in parethensis to get the root

$sin(x)+xlog(x)cos(x)=0$

[woven radish]

Bring Back Beatrix

[restive river]

oh sorry bro, i have to go to school

can you just continue this? i will read it later

[distant cypress]

solve this equation for x and you will find the x where the minima is

alr

,w solve sin(x)+x*log(x)*cos(x)=0

yup even wolfram alpha could not solve it

[woven radish]

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

Requested by Bring Back Beatrix#7534

[distant cypress]

so i could not think of other way that to solve this equation numerically

[distant cypress]

this one

but i believe its somewhere like $x \approx \frac{\pi}{9}$

[woven radish]

Bring Back Beatrix

[distant cypress]

that will make the range to be $ln(sin((\frac{\pi}{9})^{sin(\frac{\pi}{9})})+1)<R<ln(2)$

[woven radish]

Bring Back Beatrix

[devout snow]

@restive river Has your question been resolved?

[distant cypress]

you could apply newton's method here to find the root

[devout snow]

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[sage socket]

There are three ways I can do this, but I think trying P(A|B complement) = P(A union B complement ) / B complement would be easier. Not sure how id approach P(A union B complement) = P(A) * P(B complement).

Now, I think I'm having some trouble setting this up because I haven't been given enough to start rewriting P( A union B complement).

[sage socket]

Just looking for hints.

It's not a given, but I know that B complement = 1 - B

[devout snow]

@sage socket Has your question been resolved?

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[restive river]

Why is it true that [
\dd \phi = \phi_x \dd x + \phi_y \dd y
]

[restive river]

Can someone link me an article for this I just genuinely forgot lol

[devout snow]

@restive river Has your question been resolved?

[tropic olive]

unless φ is some special function i have no idea of this is just the derivative of φ with respect to x

dφ/dx=φx +φy*dy/dx

it s also called the total derivative of φ

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

[restive river]

Thank you!

.close

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[jovial mauve]

Find the equation of the line that passes through the point (1,2,3,4) and is tangent to the function f(p,q,r,s)=p²+q²+r²+s²+pq+rs-2p-2q at the point (p,q,r,s)=(1,3,5,1).

[jovial mauve]

(Status 1)

[restive river]

Parametric equation of the line yes?

[jovial mauve]

Yes

@restive river

<@&286206848099549185>

[devout snow]

@jovial mauve Has your question been resolved?

[rose girder]

What does it mean to be tangent to the function f?

[jovial mauve]

Nvm.

.close

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[dark sparrow]

please explain h

i dont get how they did ti

<@&286206848099549185>

[small quartz]

Hi @dark sparrow, what don't you understand in this explanation?

You choose a pair out of the 13 possibilities. Then you choose 2 suits out of the 4 for this particular pair.

Then you need to choose the additional 3 more kinds out of the 12 remaining kinds

And for each of the chosen kinds you need to choose a suit out of the 4 possibilities

[dark sparrow]

The second half of the numerator

why isn’t it just50

[small quartz]

Give your explanation and we'll see where it's wrong

[dark sparrow]

cos you chose a pair

and you have to choose 3 mor

so isn’t it 50c3

[small quartz]

You want exactly 1 pair

50c3 includes more pairs

and includes the possibility of having a triple or quad

[dark sparrow]

so can we subtract those possibilities from 50c3

is that right

[small quartz]

Try

You can, but it would be more complicated

[dark sparrow]

ok thanks

[small quartz]

Wanna try?

You are welcome anyway

(:

[dark sparrow]

too tedious

.close

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[marsh agate]

Hey guys, i have a problem here which i dont understand the solution

[marsh agate]

I need to find AD here

I got the same answer as the solution but why PE = BE ?

[topaz beacon]

Law of equal tangents

[devout snow]

@marsh agate Has your question been resolved?

[marsh agate]

Oh thanks, i didnt see it

.close

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[odd elbow]

Why is this wrong...?

What I did was

I made a single object out of A,G,J

Which will be in the order of AGJ or GAJ

Then arranged it, so the number of arrangement will be 9! For 9 object which is B,C,D,E,F,H,I,K,(A,G,J)

[inland seal]

are they supposed to be in consecutive order

[odd elbow]

No, 8 obj can be in any order ... but for A,G,J they've restrictions....

[devout snow]

@odd elbow Has your question been resolved?

[wooden axle]

8!?

[odd elbow]

?

How?

.clos3

.close

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[vocal marsh]

Every way I break it down it doesn’t give me any of the answers provided, I’m usually getting -3, -6 or -11.849

[dreamy shore]

Lhopital all the way

[vocal marsh]

Daaaaamn we haven’t learned that yet

[dreamy shore]

everytime I see any limit question which does not solve when I just plug in the values, I just go for lhopital rule without even thinking twice (ik this is a very bad way though)

[finite briar]

@pseudo basin

[inland seal]

oh no

[dreamy shore]

It should be possible with some amount of factorisation I believe

[vocal marsh]

To cancel the t-1 in the bottom right

Once I get that I usually get this

[final laurel]

the best strat

[pseudo basin]

lhop basher (derogatory)

[inland seal]

that's not how algebra works

you can't cancel it out from only one term

[vocal marsh]

If I factor out from the beginning term?

[dreamy shore]

No way, remember you can only cancel terms in N and D if the term in N is multiplied by something and then isn't added to anything else

[vocal marsh]

I can’t factor -t^3 out from -t^4 + t^3?

To get t - 1?

[vast rain]

you can, but you can't cancel

[vocal marsh]

Gotcha, why not in this instance?

Because the plus sign on the right of the term?

[vast rain]

the real question you should ask is when can you cancel

and it basically comes down to only when you have [something] x A / A, where A is the thing you want to cancel

you don't have that here

[vocal marsh]

So because the t-1 in the numerator is bonded to other terms via addition right

In this instance

[vast rain]

the numerator just simply isn't of the form (t-1) * something

so how could you cancel out the (t-1) from the denominator and the numerator?

[vocal marsh]

Oh my wording is off

I’m asking for reassurance basically

[vast rain]

it really has very little to do with "bonding" or "addition" in particular

there are no special cases to remember

if you struggle with remember cancellation rules in general, then don't remember any of them; it's only a notational shortcut

cancellation is just removing a multiplication and division that "cancel" each other out or an addition and subtraction that do this

so if I were to multiply by (t-1) and then divide by (t-1), I could remove both of those as long as t isn't 1

[vocal marsh]

So because my t-1 in the numerator isn’t isolated I can’t cancel it, basically

[vast rain]

if by that you mean that the numerator doesn't look like (t-1) * [everything else], then yes

[vocal marsh]

Yes

Okay so now

(Preciate that btw)

Substituting in my value sends this to zero, and all my current ways of reducing this are getting me non listed answers

How should I go about approaching this

[vast rain]

so you're saying that if you substitute t = 1 into the numerator, you get 0?

[vocal marsh]

Denominator

[vast rain]

well yeah the denominator is obvious

but what about substituting into the numerator?

[vocal marsh]

If I substitute it into the numerator I have to do that to the denominator as well no?

[vast rain]

I'm aware

[vocal marsh]

I was asking

[vast rain]

but you should try to first figure out whether the limit is of an indeterminate form

[vocal marsh]

So I didn’t bother

We haven’t to my understanding dealt with the delineation yet

[vast rain]

for example, if you substitute t = 1 into the numerator and denominator, and you get something like 1 / 0, then you can immediately tell that it diverges

[vocal marsh]

Okay

Yeah we haven’t covered that concept in depth yet

[vast rain]

surely you have

the first types of limits you cover are the somewhat simple ones, like lim as x -> 4 of 2x + 3

[vocal marsh]

Ah yes

But we haven’t done derivatives yet

Right now if we get an undefined denom we just make a table of decimals just running up to where the limit is approaching

[vast rain]

there's no derivatives involved

you just need to figure out whether it's an indeterminate form, like 0/0

[vocal marsh]

Every problem I’ve gotten so far in my calculus unit has resulted in 0/0 upon first substitution of the limit value

So I’ve either had to get rid of the denominator through cancellation or using a table

[vast rain]

okay, that may be true for the particular problems you get, but learning math is partly about being able to use techniques to solve problems where you're not given such hints

[vocal marsh]

Yeah I understand that

[tight spire]

or u can also differentiate the numerator and denominator

[vast rain]

you can't do that until you know that it's in an indeterminate form

[tight spire]

to get rid of the variable for some of the questions thats is

[vocal marsh]

However just seeing as I’m starting out with this I just want to be able to get through this process repeatedly

[vast rain]

and in any case, because the numerator and denominator are polynomials

IF it is an indeterminate form

you can use polynomial long division

[vocal marsh]

Oh yeah

[vast rain]

try not to memorize the process

[vocal marsh]

I personally need to before I can understand the alternative paths

[vast rain]

it's not about the alternative paths

[vocal marsh]

If I can’t establish success I have no mental standing

[vast rain]

it's about coming up with your own plan of attack for a problem

using what you know

[vocal marsh]

Through a process

Yes

[vast rain]

no, not through a process that you memorize

but a process that makes sense to you

[vocal marsh]

Okay this is entering semantics

I appreciate what you’re saying

[vast rain]

it doesn't make any mathematical sense to just assume that the thing is in indeterminate form

that's why I told you to check it

but suppose it is in indeterminate form, that is you know both numerator and denominator are 0 when t = 1

since the numerator and denominator are polynomials, by the definition of "root," you know that they have roots at t = 1

what does that tell you about their factors?

the big problem with memorizing a process like you've done is that you can't distinguish what is mathematical nonsense and what is mathematical sense, because it's just all memorized

which also causes problems like what you tried with cancellation

[vocal marsh]

I understand

[vast rain]

that's basically the error-correction mechanism that mathematics people use when their memories get fuzzy

you just run everything through again and make sure that it all makes sense

[vocal marsh]

Yeah I get it

But this doesn’t help me at all, frankly

Yes, should I not memorize everything, yes

[vast rain]

what does that tell you about their factors?
can you answer that question?

[vocal marsh]

It tells me that t = 1

[vast rain]

given the context of the question, the thing you write is guaranteed to come out to 0 if you didn't mess it up

[vocal marsh]

Yes

[vast rain]

it tells you more than that

[vocal marsh]

Yes

[vast rain]

suppose I tell you that x^2 - 4x + 3 is 0 when x = -3

[vocal marsh]

It tells me this doesn’t diverge

[vast rain]

well, it actually doesn't tell you that at all

[vocal marsh]

0/0?

[vast rain]

what does this tell you about the factors of the polynomial?

[vast rain]

for example, (x-1)/(x^2 - 2x + 1) diverges as x -> 1, despite the fact it's 0/0

[vocal marsh]

X - 3 x - 1

[vast rain]

wdym

[vocal marsh]

That’s what I would factor that into if that’s what you’re asking me

[vast rain]

(x-3)(x-1)?

[vocal marsh]

With parentheses

(X-3)(x-1)

Set to zero

[vast rain]

that's not correct, either, but that's sorta missing the point

[vocal marsh]

Two roots

I don’t know what your point is

[vast rain]

I don't really care about what the factors are, but what the specific information that it's 0 when x = -3 means

you could imagine that I give a much harder example that you can't completely factor

that information still tells you something

[vocal marsh]

Yes

[vast rain]

what does that information specifically tell you

[vocal marsh]

It means it’s shifted 3 in the positive direction

Is that what you’re asking?

[vast rain]

no

suppose I give you this polynomial: $f(x) = x^{5} - 18 x^{4} + 98 x^{3} - 144 x^{2} - 99 x + 162$ and furthermore, I tell you that $f(1) = 0$. Can you tell me at least one of its factors?

[woven radish]

Saccharine

[vocal marsh]

162

[vast rain]

no

[vocal marsh]

Or 0

[vast rain]

0 is not a factor; otherwise it would be 0

what is the significance of the information that f(1) = 0?

[vocal marsh]

I don’t know what context this is

[vast rain]

there's no context beyond what I've told you

I'll just tell it to you

f(1) = 0 for a polynomial means that (x-1) is a factor

I think it's called the factor theorem or whatever

Can you see why if (x-1) is a factor of a polynomial f, then f(1) = 0?

[vocal marsh]

Yes

I’ve never had that concept fleshed out in that sense to be honest

[vast rain]

the other direction is a little harder to prove, so I'll omit it

if you take a look at your limit, do you see how the numerator is a polynomial, and so is the denominator?

[vocal marsh]

Yes

[vast rain]

and furthermore, they're both 0 at t=1, right?

[vocal marsh]

Yes

[vast rain]

so by what we just discussed, they should both have a factor of (t-1), right?

[vocal marsh]

Yes

That was what I was attempting to get from the numerator

[vast rain]

so it's pretty obvious how the denominator factors: (t-1)

[vocal marsh]

Yeah that’s what we have

[vast rain]

but now you're trying to find what times (t-1) gives the numerator, right?

[vocal marsh]

Yeah

[vast rain]

so in general, if we're trying to solve the question of "what times this gives that" what do we do?

[vocal marsh]

It depends right

[vast rain]

or in an actually more analogous context than is obvious, how do you answer the question "what times 105 gives 210?"

[vocal marsh]

I’d divide 210 by 105

And then multiply that by 105

Damn I really am going to have to do polynomial long division arent I

[vast rain]

so let's simplify your numerator to $-t^4 + t^3 + 3t^2 - 3$ (actually, I just noticed what you wrote on the paper has a mistake on it; this is another reason why you should do sanity checks like the numerator is 0 at t = 1)

[woven radish]

Saccharine

[vocal marsh]

Yeahhhhh

[vast rain]

it's not any different than the long division you learn in grade school

or you can literally just ask a calculator

[vocal marsh]

No I know I’m just thinking there’s a more efficient way

[vast rain]

efficient in what way?

[vocal marsh]

My calculators bad, gotta write everything

[vocal marsh]

Less steps and more clear

[vast rain]

there aren't many steps at a high level

[vocal marsh]

Instead of just number crunching

[vast rain]

it's just divide -t^4 + t^3 - 3t^2 - 3 by (t-1)

[vocal marsh]

Yeah

[vast rain]

ask a computer to do the number crunching then

,w divide -t^4 + t^3 - 3t^2 - 3 by (t-1)

[vocal marsh]

I can’t

But doesn’t that return something I have to process the limit through yea?

[vast rain]

,w simplify (-t^4 + t^3 + 3t^2 - 3) / (t-1)

wait I messed up

the numerator isn't 0

[vocal marsh]

It’s all good I’ll do it on paper

[woven radish]

Results provided by WolframAlpha

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

Requested by Saccharine#3780

[vast rain]

why do it on paper when you have technology

[vocal marsh]

Idk

I’m old and remember when I couldn’t use calculators for extensive problems like this in grade school

I want to use it though

[vast rain]

they're probably trying to make sure you actually know how to do it

[vocal marsh]

I also want to actually know how to do it tbh

[vast rain]

but there's nothing wrong with using a calculator for something that you are already expected to know

[vocal marsh]

True very true

[vast rain]

otherwise we would be stuck doing all arithmetic by hand

[vocal marsh]

Very true

So that’s what I’m doing right, I didn’t make any other mistakes getting to this point?

[vast rain]

it's - 3

in any case, you now have $\lim\limits_{t \to 1} \frac{-t^4 + t^3 + 3t^2 - 3}{t-1} = \lim\limits_{t\to 1} -t^3 + 3t + 3$

[woven radish]

Saccharine

[vocal marsh]

Wait wait wait

That’s where I f’d up

How is it -3

[vast rain]

the position function is called g(t)

[vocal marsh]

Yep

[vast rain]

so the numerator of your fraction should be g(t) - g(1)

g(1) = 5

[vocal marsh]

Yep

[vast rain]

therefore, g(t) - g(1) = g(t) - 5 = -t^4 + t^3 + 3t^2 - 3

[vocal marsh]

G(1) should equal 7, no?

[vast rain]

,w evaluate -t^4 + t^3 + 3t^2 + 2 at t = 1

[woven radish]

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[vocal marsh]

1+1+3+2?

[vast rain]

-1+1+3+2

[vocal marsh]

It’s 4 exponent

Inside the parentheses

[vast rain]

none of the exponents matter

t = 1

1 raised to any exponent is still 1

[vocal marsh]

That should make it positive if it was negative

[vast rain]

not sure why you think it's not -1

[vocal marsh]

(-1^4) = 1

[vast rain]

it's -1

,w (-1^4)

[woven radish]

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[vocal marsh]

That makes no sense

[vast rain]

oh you might be wrong on the order of operations

[vocal marsh]

-1 times -1 times -1 times -1 = 1

Okay

[vast rain]

yeah it's not -1 times -1 times -1 times -1

it's -1 times 1 times 1 times 1 times 1

[vocal marsh]

Omg

That’s so majorly frustrating

[vast rain]

(-1)^4 = 1

[vocal marsh]

How would in gods name would I delineate that

UGHHHHHHH

[vast rain]

that's what the parentheses mean

the parentheses mean "do this operation first"

[vocal marsh]

yeah but it’s all inside

[vast rain]

the stuff inside still obeys the order of the operations

[vocal marsh]

Yeah I get that

[vast rain]

the parentheses mean "evaluate this expression first and then substitute it back in"

[vocal marsh]

But there was no parentheses around that term

It’s just -t^4

[vast rain]

okay, so then everything obeys the usual order of operations

[vocal marsh]

So what should I assume is it either:

-(t^4) = -1

Or

(-t^4) = 1

[vast rain]

both of those things are -1

[vocal marsh]

No

That makes no sense

[vast rain]

both of those things convey the exact same order of operations

they do

[vocal marsh]

It’s not

You said so

[vast rain]

you've basically written -(1^4) = -1 and (-1^4) = -1

[vocal marsh]

This ^

[vast rain]

let's go over the order of operations of both so you can see they're the same

[vocal marsh]

But the actual problem has zero parentheses

I’ve added them

[vast rain]

both of those are the same as -(1^4)

[vocal marsh]

They can’t possibly be

[vast rain]

I'm not sure why you've written this when it's not true

[vocal marsh]

It is

-1 to the fourth is positive 1

[vast rain]

,w evaluate (-t^4) at t = 1

[woven radish]

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Requested by Saccharine#3780

[vast rain]

let's go over the order of operations for both of them then

[vocal marsh]

-1 times -1 times -1 times -1

[vast rain]

suppose we are asked to evaluate (-t^4) at t = 1

[vast rain]

except it doesn't mean that

[vocal marsh]

But how would one infer that

There’s zero information to assume that

[vast rain]

we recall the PEMDAS order, and we see that we should evaluate the expression in the parentheses first

well actually

let's substitute 1 in for t first

so we get (-(1)^4)

now the PEMDAS says that we should evaluate parentheses, inner parentheses to outer [I've added parentheses around the 1 just to indicate that's the whole number]

so we are required to evaluate -(1)^4

then PEMDAS says we should evaluate exponentiation next

so we compute (1)^4 = 1

and substitute it back in

[vocal marsh]

Yes

I get all this

[vast rain]

this gives us -1

[vocal marsh]

What my point is

[vast rain]

if you get all of this, then why do you write that (-1^4) = 1?

[vocal marsh]

There is no information to tell us if whether the negative is included in the exponentiation

[vast rain]

there is

that is what the order of operations tells you

[vocal marsh]

There isn’t

Look

[vast rain]

when you write -a, you are writing -1 * a

these mean the same thing

[vocal marsh]

You don’t actually know that though

[vast rain]

-t^4 is simply -1 * t^4

yes you do

this is the convention used by every mathematician

[vocal marsh]

That’s great

[vast rain]

that is what the order of operations is about

[vocal marsh]

I want consistency

[vast rain]

establishing a convention for what the order is

it is consistent

can you give an example where this is inconsistent?

[vocal marsh]

(-1^4)

-1 times itself 4 times

[vast rain]

I literally just explained to you why that comes out to -1

[vocal marsh]

No

You’re adding the negative after the fact

[vast rain]

that does NOT mean (-1) times (-1) times (-1) times (-1)

it just doesn't

[vocal marsh]

We’ll then a boat load of math problems I’m doing on a regular basis are just wrong then

[vast rain]

the order of operations says that you do multiplication (and therefore unary negation) AFTER exponentiation

[vocal marsh]

So naturally numbers aren’t negative

[vast rain]

no

[vocal marsh]

And every negative number is really just a positive multiplying by a negative 1

[vast rain]

that doesn't make any sense

there are negative numbers yeah

[vocal marsh]

It does

Not intrinsically

Is what that’s saying

[vast rain]

well, if you want to get into the weeds of it, the negative numbers are defined as additive inverses of the positive ones

[vocal marsh]

It’s saying that we never multiply -1 times -1 over and over

[vast rain]

that's not true

if I write (-1)^4, then that means -1 times itself 4 times

[vocal marsh]

Okay when then

YEA

OKAY THEN SO:

[vast rain]

,w (-1)^4

the parentheses indicate which is done

[woven radish]

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[vocal marsh]

That makes sense

I see it now

[vast rain]

do you see what the parentheses do here?

[vocal marsh]

Yeah

[vast rain]

they tell you that you have to do the -1 first

then raise it to the fourth

the usual order of operations would have you do 1^4 first and then multiply by -1

[vocal marsh]

That’s insane, this threw me off completely

So negative signs basically take last priority compared to exponents otherwise when noted by parentheses

[vast rain]

negative signs are just multiplication

[vocal marsh]

Is basically what is true and consistent?

Right so it does then

[vast rain]

unary negation is a shorthand way of writing -1 times the thing

[vocal marsh]

Yes

So yes is the answer to my question then right

[vast rain]

yes it comes after exponentiation

[vocal marsh]

Thank you

So for the future

I have to understand that that is what is actually being notated as

[vast rain]

I don't think this is a particularly special case of order of operations

it's just the standard order of operations

[vocal marsh]

It’s my general understanding of the notational context

And not implying (-1)^4

[vast rain]

I guess if you haven't seen things like (-3), maybe it's new but yeah

[vocal marsh]

I have

But tbh

99% of the problems I’ve been doing are (-1)^4 and not -1^4

Which is a little strange but I’m glad I understand why I messed that up now

No wonder it wasn’t making sense

[vast rain]

mathematics is often full of very subtle distinctions

which is why you have to state things precisely and come up with a consistent system of rules about it

[vocal marsh]

Yeah for sure

[vast rain]

an example from real analysis is the difference between continuous almost everywhere, continuous, uniformly continuous, Lipschitz continuous, absolutely continuous, differentiable, and continuously differentiable

those things do not mean the same thing

and even if 99% of the functions you deal with are differentiable, you can't just assume it elsewhere, and you have to be careful about details like these

[vocal marsh]

Yeah

We haven’t covered anything related to it concerning, just solving it to my knowledge, that’s why I was a bit fuzzy on that language

[vast rain]

so can you figure out the answer now?

[vocal marsh]

Yeah, I’m going to do long division

Is there anywhere you recommend for calculating it

For the future

[vast rain]

WolframAlpha does it fine

I personally use sympy for more complicated things

[vocal marsh]

Wolfram alpha?

[vast rain]

you can also get a very good approximation with just a scientific calculator

https://wolframalpha.com/

[vocal marsh]

You’re the best

Yeah I really appreciate it big time

.close

[devout snow]

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[scenic surge]

confused by this question

do i have to assume that there exists a 0 white color scheme and then show 2^2023 exists too?

idk what to do

[pseudo basin]

you have 2 different things to show the existence of

1. show there exists a 0-white-color scheme
2. show there exsits a (2^2023)-white-color scheme

a white-color scheme is an assignment of the colors black and white to subsets of A (i.e. in effect a function from P(A) to {black, white}) s.t. the union of any number of sets of the same color is also that color

the number of white sets under said assignment is the n in n-white-color scheme

@scenic surge does this make sense y/n

[scenic surge]

one sec lemme read

sorry but im confused by

"a white-color scheme is an assignment of the colors black and white to subsets of A"

so a white color scheme consists of black and white colors?

[pseudo basin]

could say that sure

if youre unhappy with the name i could come up with a different name for it

[scenic surge]

no i get it

[pseudo basin]

theres a better one i have in mind but i will not say it unless you ask me to

[scenic surge]

and if we take union of two same colors then the union is also the same color set?

so white_1 U white_2 = white_x

[scenic surge]

sure

[pseudo basin]

bad notation

[scenic surge]

oh

white_x U white_y = white_z?

[pseudo basin]

worse notation

[scenic surge]

shit my bad

[pseudo basin]

ok, in that case let's call a function from P(A) to {black,white} a powerset coloring.

this is not part of any argument but an establishment of terminology:
when considering a powerset coloring c : P(A) -> {black, white}, for a subset S ⊆ A, we say "S is white" to mean c(S) = white, and likewise for black.

[scenic surge]

wait sorry one second afk j got kicked feom the study room

[pseudo basin]

ok, ping me when you are ready

[devout snow]

@scenic surge Has your question been resolved?

[scenic surge]

alright back my bad

[pseudo basin]

ok

with one bit of terminology established, here's another one.
we say that a powerset coloring is good if for every S, T ⊆ A it is true that:

• if S is white and T is white, then S ∪ T is white
• if S is black and T is black, then S ∪ T is black
  [||there is a subtle difference between this and the definition you were given. but if you don't see it, do not worry about it -- take my word for it that mine is equivalent to theirs.||]

[scenic surge]

I see

[pseudo basin]

an n white color scheme, then, is a good powersest coloring under which exactly n subsets are assigned white, and all the others are assigned black.

are you good on that or do you need some time to process it

[scenic surge]

wait so

[scenic surge]

im so confused now

oh

so if we have 4 white color scheme then we also have 4 whites assigned to each?

[pseudo basin]

no

a 4 white color scheme is a good powerset-coloring under which exactly 4 subsets of A are white, and all other subsets of A are black.

[scenic surge]

alr

so if

we have a 0 white color scheme then

we have exactly 0 subsets of A which are white, and the remaining which should be 0 are subsetf of A which are black

[pseudo basin]

swing and a miss.

[scenic surge]

oh whats wrong

[pseudo basin]

[if we have a 0 white color scheme then] we have exactly 0 subsets of A which are white
this is correct
and the remaining which should be 0 are subsets of A which are black
and this is a trainwreck

i'm trying my best to guide you through the reasoning but you are getting caught up in the formal details.

[scenic surge]

alr mb

but like

idk how to approach q2 and how this question is related to induction

[pseudo basin]

it isn't

[scenic surge]

oh what

[pseudo basin]

at least, the solution i have in mind has nothing to do with induction at all

[scenic surge]

i have an induction test tmr n our prof suggsted this question

[pseudo basin]

in some sense

i mean like, unless you require an explicit reproof by induction that |P(A)| = 2^|A|

but that is it

[scenic surge]

mhm i guess its prolly the next 2 parts where I require induction

ok so back to the question

[pseudo basin]

the only possible powerset-coloring which assigns white to zero subsets, i.e. one in which no subsets are white, is the one in which all subsets are black.
it is exceedingly obvious that the "everything is black" coloring is good.
the (2^2023) white color scheme you seek is the other extreme.

idt i can hide this anymore

but it is just this glaring

[scenic surge]

oh because union of two empty set is still an empty set

[pseudo basin]

no

i mean, yes, you're right, but that's irrelevant to my point.

[scenic surge]

then for 2^2023 could I say

everything is white

[devout snow]

@scenic surge Has your question been resolved?

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[restive river]

Excuse me, I need help with this

[restive river]

I think I'd have to use sigma notation

but I don't know how to use it in this sense

also don't mind the 4

I don't have an answer for either

[devout snow]

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[devout snow]

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[austere trellis]

How should I solve this

[austere trellis]

I don’t even understand what the question is asking for

[stone stump]

largest positive integer
floor(-pi) = -4

anyway, ignoring that technicality

I would suggest to try some examples

[austere trellis]

It’s largest integer that is less then or equal to x

Wdym some examples

[stone stump]

if k=2, then you are looking for numbers n, n+1 with f(n)+f(n+1)=99

can you find some?

[austere trellis]

That would be 44 and 45

[stone stump]

are you sure?

I am not asking for n+(n+1)=99

what is f(44)

[austere trellis]

I don’t know?

[magic pine]

what is f(10)

[austere trellis]

I don’t know

Whenever I see questions with … in the middle I just don’t know how I should approach it

[magic pine]

we’re not asking anything wild though

[austere trellis]

Since it is infinite amount of n+1

[magic pine]

do you know what f(n) is?

like how it’s defined?

[austere trellis]

No

[magic pine]

reread your question

[austere trellis]

Yes?

Oh

So f(10) would be 1

F(44) would be 4

[magic pine]

answer this

[austere trellis]

So the total would have to be

From 990 to 999?

And that can be formed from f(n)+f(n+1)+f(n+2)+…+f(n+1)

But in …

How many f(n)s will be there

[magic pine]

you’re overthinking it

[austere trellis]

How can we solve if we don’t know what’s rly in that …

[magic pine]

it’s just saying

k consecutive integers

if k = 2

it’s 2 consecutive integers

so you do

f(n) + f(n+1)

[austere trellis]

449+450

448+449

447+448

446+447

[magic pine]

what is this

[austere trellis]

445+446

[magic pine]

your first pair was fine

idk what the rest of these are

[austere trellis]

Are the combinations that I thought could work

If they add up

They are still in range of 990 to 999

[magic pine]

but you’re doing f(n) + f(n+1)

plug in 445 and 446

and see what you get

[austere trellis]

Oh right

Wait

But to fit in

It has to be 49something + 50something

To make it 99

Like 499+50p

500

[magic pine]

yeah that should work

[austere trellis]

So there’s one pair in k=2

So do I continue for k=3

[magic pine]

yes

[austere trellis]

And if I do k=3

It would be f(n)+f(n+2)=99

[magic pine]

no

f(n) + f(n+1) + f(n+2)

i hope you can see the pattern

[austere trellis]

And I’m supposed to find

What exactly in the question tho

[magic pine]

reread it

if you’re not sure

[austere trellis]

So I’m looking for

A part

Where I can’t have consecutive pattern?

@magic pine

[magic pine]

i take it you’re still not sure what the question is asking, so read it over again and tell us what about it confuses you

[austere trellis]

Just confirm with me if I’m right or wrong real quick

I’m looking for k

Where I can not provide consecutive numbers for n that makes 99?

[magic pine]

not as your phrased it

it’s something similar

so again, please read it over again

and ask about anything that confuses you

[austere trellis]

I have to have k number of consecutive numbers

Is that what u missed

What I missed*

I’m not sure what I’m misunderstanding from the question

I’m looking for k that is a smallest integer

@magic pine

[devout snow]

@austere trellis Has your question been resolved?

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[restive river]

Just an arithmetic problem (trying to find the solutions of the system)

[restive river]

i did a series of subs starting from the 4th equation

but i messed up somewhere

i have been eyeballing it for like a good 2 minutes idk how i fucked up

[wooden veldt]

Use row operations to start killing the entries above the 1s if you want to make your life easier

[restive river]

honestly i find just doing subs after getting echelon form to be easier

imo

but okay sure maybe that will tell me how i fucked up

[wooden veldt]

Personal choice ig, but you seem to be struggling with it just in REF rather than RREF

[placid rover]

just rref it.

[restive river]

okay okay

[placid rover]

point of matrix reduction is that its meant to be quicker

because ur not writing useless symbols

[restive river]

i just hate writing it in matrix notation constantly because like

u have to write each element over and over

[wooden veldt]

I sentence you to solving a 10x10 system without matrices

[restive river]

just time consuming to write down 25 entries in a 5x5 matrix for every reduction yk?

[placid rover]

you can shortcut in rough

just write a big line over a row if it doesnt change

[restive river]

okay thats smort

[placid rover]

and omit 0's in the bottom left

...

[restive river]

okay lemme rref

[devout snow]

@restive river Has your question been resolved?

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[junior cairn]

I don't know what to do for problem d

[winter patrol]

d) is a cone stuck to a square prism

[junior cairn]

where do I start with?

[winter patrol]

volume of a square prism
volume of a cone

[junior cairn]

do I need to add them all up

[winter patrol]

same principle as parts a,b,c

[junior cairn]

I got the wrong answer though

[winter patrol]

show work

if you have work, show it. don't wait to be asked for it.

[junior cairn]

1/3 x π x (4.8/2)^2 x 5.2 + 4.8 x 4.8 x 1.5

[winter patrol]

why are you diviing 4.8 by 3

also 5.2 isn the height of the cone

[junior cairn]

by bad

I meant by divided by 2

[junior cairn]

wait, I thought it was?

[winter patrol]

5.2 is the height of the whole thing,

[junior cairn]

oh....

[junior cairn]

I got the correct answer

all I had to do was minus the 5.2 by 1.5

and that gives the height for the cone

okay so

for problem e

do I just halve the height of the whole thing?

.close

[devout snow]

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[radiant drift]

how do i solve this

[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'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above

[radiant drift]

1

[devout snow]

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

[radiant drift]

if i dont know where to begin, i have no work

when two conjugates subtract

a+bi-a+bi=2bi

=2 sqrt 3

bi=sqrt 3

[thin tapir]

No

[radiant drift]

oh

its the argument

[thin tapir]

|2bi| = 2 sqrt(3)

[radiant drift]

i see

maybe squaring both sides

yield

2bi* -2bi=4*3

[thin tapir]

Then with the fact that x/y^2 is real you can almost find the argument of x

Up to a finite number of possibilities

[radiant drift]

4b^2=12

b^2=3

b=sqrt 3

[thin tapir]

And then you should be able to find |x|

[radiant drift]

did i get the calculation right

[radiant drift]

how

[thin tapir]

Or b = -sqrt(3)

[radiant drift]

yeh right

[thin tapir]

Do you know the notation x = r·e^(it)?

[radiant drift]

yes

thats a form to represent complex number

[thin tapir]

Then y = r e^(-it) so x/y^2 = 1/r e^(3it)

And that's real

So π divides 3t

And if you know both b and t you can find a and therefore |x|

[radiant drift]

wait why is y, the i is negative?

[thin tapir]

Because it is the conjugate of x

[radiant drift]

ohh

so its

r·e^(it)/r·e^(-it)*2

?

[thin tapir]

r·e^(it)/(r·e^(-it)^2)

[radiant drift]

that simplifies to

1/r·e^(it)??

[thin tapir]

No

[radiant drift]

whats the 3 doing there

[thin tapir]

r·e^(it)/(r·e^(-it)^2) = r·e^(it)/(r^2·e^(-2it)) = r/r^2 · e^(it)/e^(-2it) = 1/r · e^(it - (-2it))

[radiant drift]

oh

ahh

im so slow at calc sorry

i see it now

[radiant drift]

so u just use the possible t that makes pi divides 3t, and then calculate for x?

[thin tapir]

That should work

Perhaps there is a more direct way

[half minnow]

tf

@radiant drift

[radiant drift]

?

[half minnow]

|a + bi - (a - bi)| = 2sqrt(3)

agree?

[radiant drift]

not with toby, but yeh

i already did that step

and found b

[half minnow]

sir lint..

what is b

[radiant drift]

+- sqrt 3

[half minnow]

idk man just

then just mult by complex conjugate

oops forgot the i

[radiant drift]

o maybe thats easier

OH

[half minnow]

stop this...

you cubed asshole...

[distant peak]

who did that colouring and how to do that lol(not related just wanna know its cool)

[half minnow]

snow

the command is named after him

[radiant drift]

mniip hates it, stop

[woven radish]

$\snow{orthogonal}$

[radiant drift]

mute u like how Ann was treated...

[half minnow]

its tyranny... is it not?

[radiant drift]

bro continue

[half minnow]

LOL nice

LINT

[radiant drift]

do u multiply up and bottom by a-sqrt3 i

[half minnow]

use Toby's law of conservaTION

NO

[radiant drift]

...

[half minnow]

u ass do u not know how to use complex conjugate

stop this clownery

[radiant drift]

...

[tropic olive]

@radiant drift x=a+bi you have found b right?

[restive river]

theres a nice geometric way I think

[radiant drift]

yes

just continue from leskinen

plz

[radiant drift]

this is nice

[half minnow]

ok

[tropic olive]

so from the first eq you have /a+bi-a-bi/=2sqrt3

/a/=sqrt3

[radiant drift]

let leskinen speka...

all hail

[half minnow]

conjugate of (a-sqrt(3)i)^2 is (a+sqrt(3)i)^2

[radiant drift]

yes linty...

[half minnow]

actually its not but it does the trick

now Watch...

[radiant drift]

sure

ok

wizard of ou

[half minnow]

AGREE?

[half minnow]

(yes toby im gonna make him solve a cubic)

[restive river]

[radiant drift]

yeh sure

|bro just give me

the easiest way

[half minnow]

lint

im trolling

if u actually expanded it

[radiant drift]

...

[half minnow]

its Trivial...

like DV Game...

[radiant drift]

5 year old?

[half minnow]

stop this...

[radiant drift]

what is the bottom expanded

[half minnow]

anyway can u solve it from here

LINT

DO IT URSELF

[radiant drift]

omg do u want me to expand everthing?

[half minnow]

no

[radiant drift]

in the denominator

[half minnow]

Bruh

u dont need to do that

[radiant drift]

toby ur gif didnt even load for me and it disappeared

[arctic field]

when will you realise ...

[half minnow]

...

[radiant drift]

ok thats nice

[half minnow]

lint...

[radiant drift]

fine

[half minnow]

why do we lose... commutativity... when we go into quataernions...

[radiant drift]

i get it now

[arctic field]

get real linty

[radiant drift]

lint plz give me some hint for this

[half minnow]

bruuv

[radiant drift]

@half minnow

wait

[half minnow]

is the denom real

[radiant drift]

its loading

[half minnow]

-_-

enough

[radiant drift]

no its not that

[arctic field]

cmon lets be real here ...

[radiant drift]

...^3/ (a^2+3)^2

[half minnow]

Be Real

Now

ok

did u ask ann

stop Asking in Homework Help...

[radiant drift]

wtf

[half minnow]

this is anti chris problem

you arent expected to solve it...

[radiant drift]

is this right

[half minnow]

yes...

u ann asker...