help-0

[rustic hazel]

ok

[oblique pier]

what do you think they're gonna equal to if we plus them?

[rustic hazel]

90

[oblique pier]

Nice

Now look at HGI and IGH

[rustic hazel]

also 90\

[oblique pier]

well

do see a connection here?

[rustic hazel]

yeah

[oblique pier]

what's that connection

[rustic hazel]

(probably wrong) its a 90 45 45 triangle

so its similar

[oblique pier]

nah

i want you to look at what i just tell you to look at

[rustic hazel]

their complimentary

[oblique pier]

Do you think they plus the same angle to give you 90 degree?

[rustic hazel]

for which

[oblique pier]

.

.

[rustic hazel]

ok

so

its aa?

[oblique pier]

a summarise

Do you see that they're both plusing the same angle to give you 90 degree?

shub?

[rustic hazel]

yes

[oblique pier]

We what can we conclude about FGI and IHG

i gotta take a shower srry

but i already give you a clue

[rustic hazel]

ok

[oblique pier]

You can do it

You got 2 similar triangle now

[rustic hazel]

ye

so

if their similar

would it mean

[oblique pier]

focus on the proportion lengths..

bye!

[rustic hazel]

their half

of each other

[rustic hazel]

ok cya

@oblique pier r u back

[lone heart]

@rustic hazel Has your question been resolved?

[rustic hazel]

um ima close ticket

.close

[lone heart]

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[still briar]

Good morning! Might someone be able to point out where I went wrong on this question? Correct answers haven’t been released yet, and it’s the final so I’m not sure that they will. First pic is the test question and second is my work. (Sorry, I’m not sure which subparts specifically are wrong either)

[still briar]

I should mention that the question should be worth 15 points total, so I think it’s most likely that just one of the parts is wrong as opposed to a little wrong in all

[lone heart]

@still briar Has your question been resolved?

[drifting hull]

2b) If n=1, then there are two 1-1 function (if i remember the definition of 1-1 correctly). And part d) is wrong. E.g. every function from c) is not onto ${0,1,2}$ and these are more than 3

[ocean seal]

Alexander42

[drifting hull]

@still briar

[still briar]

thank you very much for your response, i'll read through it now

i guess where i'm confused is how the logic from part d is different from part c

by "every function from c is not onto ..." do you mean it should look something like 3^n - 2^n ?

[native granite]

I think this is closer yes
I think you can use c to prove d

Like, for a function $f : A \to {0,1,2}$ not to be surjective, it is the same as choosing a function $f : A \to {0,1}$ or $f : A \to {0,2}$ or $f : A \to {1,2}$

[ocean seal]

Twenty

[still briar]

this is very interesting, thank you for this explanation

[native granite]

And there are $2^n$ choices for each of these three possibilities
So I guess that makes $3.2^n$ not surjective functions
I'm not 100% sure this is my first attempt too ^^ but this makes sense I think

[ocean seal]

Twenty

[drifting hull]

almost. You count the constant functions twice there

[native granite]

Oh right

Thanks

So that would make $3 \times 2^n - 3$ i.e $3(2^n-1)$ not surjective functions

[ocean seal]

Twenty

[native granite]

Hmm let me check for low values of n though

[still briar]

thank you, you two! i'm leaving to go on a walk right now but i'll definitely read back through this and work the rest of the problem out

[native granite]

Okay !

[still briar]

should i leave the ticket open for now?

[native granite]

Well i'll let you check for n=1,2,3 then ^^ (not that this certifies the result 😆 but at least check that it's not negative)

[native granite]

Hmm it will close automatically for inactivity so I guess you'll have to reopen one anyway

[still briar]

but the messages will still be here if closed?

[native granite]

Yes

[still briar]

excellent, i'll be back later then 🙂

[native granite]

Maybe you'll have to scroll up a bit though :p

[still briar]

@drifting hull thank you very much as well ❤️

[native granite]

You can use the search function of discord to look for your messages if you can't find them directly

[still briar]

.close

[lone heart]

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[fluid chasm]

Yo I have an issue with some math homework I have to do. I wanna figure out how to calculate the probability of atleast 3 times a 0.03 chance occurance over 25 instances. I looked some stuff up with dice as a comparison but I couldnt figure it out, since they all dealt with an atleast one occurance. Id love to get some help with this

[fluid chasm]

I thought of like 25nCr3 times 0.03^3 times 0.97^22, but because of the atleast I dont think thats correct

Its in dutch so I dont know how many people know of it, but this is the problem, specifically b

[blissful anchor]

you want to use the binomial distribution law

[fluid chasm]

How would I go about using it here?

[blissful anchor]

I don't fully understand your problem in your first message, what are your unknowns?

[fluid chasm]

The only unknown I guess is the end result. I need to calculate the probability of out of 25 occurances, atleast 3 have to be the desired outcome, where the chance of the desired outcome = 0.03. I dont know how I go about it

[blissful anchor]

So you have the formula:

the chance of a probability happening n times if the events are independant

is

the combination of n, k * p^k * (1-p)^(n-k)

with n being the total amount of tries

k being the amount of tries you want to succeed

and p being the probability of succeeding

[fluid chasm]

Ok, this takes into account the atleast thing? With having to be 3 or above?

[blissful anchor]

where is the atleast?

[fluid chasm]

Oh, I said it in the first message, forgot to add it in the second

[blissful anchor]

the easiest way to do this using the binomial formula is to calculate the chance of it happening 0, 1 and 2 times and then doing 1 - those probabilities

there may be another way to avoid brute forcing like this but I don't know it

[fluid chasm]

Ok, then I guess ill just take each of them individually and just do that, bit of a hassle

[blissful anchor]

yeah, there is a trick to do it with the ">" sign but it was kinda a thing that worked in a few cases not everytime, unsure you can do it here

[fluid chasm]

Wait, that doesnt work? The answer would be above 1

I did 1 - 25nCr0 x 0.97^25 - 25nCr1 x 0.03 x 0.97^24 - 25nCr2 x 0.3^2 x 0.97^23 and I got a negative number

[blissful anchor]

if i write it out it looks like this

[fluid chasm]

with p being the 0.03 correct?

[blissful anchor]

yea

there should be parenthesis after the 1-

otherwise you will get a number bigger than one my bad

[fluid chasm]

Ok, I figured it out, I put 0.3^2 instead of 0.03, always reread. I got an answer out of it, good to see.

[blissful anchor]

coolio

[fluid chasm]

If I have another problem ill open up another help chat, thanks

.close

[lone heart]

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[alpine sable]

f(x)=x³-5x²+ax
g(x)=f(x)-f(2)
number of root of g(k)=|g(x)| : h(k)
h(8)=4
g'(0)=?

[alpine sable]

g'(x)=3(x+1)(x-5)
=3x²-12x-15
but i think theres an error

it doesnt match with f'(x)

this time its prob my fault

so help

<@&286206848099549185>

[lone heart]

@alpine sable Has your question been resolved?

[alpine sable]

no

help imo

oh god

ok fixed the problem

<@&286206848099549185>

it was number of roots

not root

mb

[lone heart]

@alpine sable Has your question been resolved?

[remote heron]

im still confused by what the problem is

can you post the original prompt?

[lone heart]

@alpine sable Has your question been resolved?

[lone heart]

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[alpine sable]

hello
is {0,1} a field?

[alpine sable]

it doesnt satisfy closure under addition 1+1=2

[remote heron]

you could define it though right

use addition mod 2

does that work

[alpine sable]

sorry i dont understand the modular i didnt reach it yet

[remote heron]

https://en.wikipedia.org/wiki/GF(2)

depends on the operations

[alpine sable]

what does modulo mean

when they say like 1mod2

[remote heron]

you dont need to use modulo if you dont want to

you could use some isomorphic field

[alpine sable]

but i need to understand it

😦

[remote heron]

so mod means like rolling over

you know those like

odometers?

well lets say remainder

thats prolly easier to understand

modulo gives remainder of division

so mod 2 just means

divide a number by 2

discard everything but the remainder

[alpine sable]

you see i have 3 books of linear algebra
the second book talks about modular and etc..
im reading now the 1st book which is talking about infinite fields
now it started to talk about finite fields and i reached the section where it says is {1,0} a field by seeing the axioms

[remote heron]

afaik its common to define addition mod the largest element in these groups

theres nothing fancy about mod, you are just taking the remainder

mod 2 boils down to telling whether a number is even or odd

does that make sense?

[alpine sable]

Ima gona read that section

This is where i started btw about finite fields

Sorry its in hebrew

[remote heron]

but

[alpine sable]

But basically these diagrams

[remote heron]

you dont need modulo

you could define it another way like

say 0 is false

and 1 is true

then instead of

instead of addition use NAND i think?

and instead of multiplication use

[alpine sable]

But the bottom line is

[remote heron]

id have to think

[alpine sable]

its not a field right?

[remote heron]

{0,1} is just a set

you can pick operations on that set that build a field

or you can pick ones that dont

and okay

let the set be {FALSE, TRUE}

its a field under NAND and AND

no modulo needed

[alpine sable]

i dont understand i feel like im missing something here

• 0 1
  0 0 1
  1 1 0

how does that table count?

like why is 1+1=0

what im missing

[remote heron]

depends on how you want to think about it

1+1=2

wed define that were using addition mod 2

so first you add the 2 numbers

then divide them by 2

[alpine sable]

so with regular addition it doesnt work

[remote heron]

and take the remainder as your result

it doesnt work with regular addition

[vale crag]

also it's not like that table is invalid

[remote heron]

alternatively you can use TRUE (NAND) TRUE

[vale crag]

it's a valid operation to define on {0,1}

[remote heron]

or its only 2 elements in the set just

yea

just define it to be that way

[vale crag]

an operation is just something that takes two elements of your set and spits you out another element of the set

maybe it ends up being useless but it's still something you can define

[alpine sable]

basically its the additive inverse of 1 its either 1 or 0

cant be 0

so its 1

[vale crag]

well 1+1=0 here, so yeah 1 is the additive inverse of 1

[alpine sable]

can i ask on what logic did we assume that

like why did we say that

[vale crag]

it mimics the dynamics of even and odd numbers actually

0 representing the even numbers, 1 the odd numbers

if you add 2 even numbers, you get an even number

if you add an even and an odd number, you get an odd number

if you add two odd numbers, you get an even number

the operation pretty much reflects that

that's one way to see it at least

[alpine sable]

In the book it says a + b is the left over from addition a and b as intergers and devide them by n

Can i ask what they mean by "left over"

[vale crag]

the remainder when you divide a+b by n

oops I pretty much restated exactly what they wrote

[alpine sable]

so in z7 5+5=10 and left over is 3

10/7

[vale crag]

yeah

so in Z7, 5+5 = 3

[alpine sable]

so in z2 1+1=2 2/2=1

and the left over is 0

[alpine sable]

so thats why is 0 here

[vale crag]

yup

[alpine sable]

omg i finally understand it

😳

[vale crag]

it's pretty much the thing I said above, but using computations instead of me using words to describe it @alpine sable

[vale crag]

[alpine sable]

you guys are angels ngl

i started the semester this week
they got me scared and said the exam will be hard etc..

how hard is linear algebra?

like if i understand it quite good will i score more than 60

[vale crag]

that's from a linear algebra class ?

(i'm not surprised just wondering)

[alpine sable]

linear algebra 1

yea

[vale crag]

I mean depends how in depth your class will go

but you could probably blackbox all this finite field stuff when you'll have started the actual linear algebra of this class

[alpine sable]

wana know the topics?

[vale crag]

usually we just work in R or C (especially in a first class)

[alpine sable]

yes we are only in C and R

[vale crag]

but if you're in CS it's nice to pay attention to finite fields

[alpine sable]

we wont go in depth

[vale crag]

ends up in a lot of places

mostly cryptography tbh

[alpine sable]

Its for my major computer science

[vale crag]

but yeah I didn't find lin alg too hard personnally

did CS also

I self-studied a bit of material beforehand tho

it's quite refreshing after a fuckton of calculus in high school tbf

[alpine sable]

1. Linear system of equations
2. The space Fn
3. Matrices
4. Determinants
5. Finite fields
6. Linear spaces
7. Basics and Dimension Theory
8. Linear copying
9. Representation of Copies Using Matrices
10. Self-values
11. The scalar multiplication

its google translate sorry

[vale crag]

it's quite standard material yeah

you should be ok I mean idk you, so maybe you won't, but I hope you will

[alpine sable]

i'll study hard ngl

thanks alot man ❤️

[vale crag]

also don't forget to check out some other resources if you don't like how your prof teaches

like MIT opencourseware for example

there's also some book recommendations here #resources

if you want to see something else

[alpine sable]

my university gave me 3 books

they are quite usefull

[vale crag]

oops #books-old

[vale crag]

well nice^^

@alpine sable anyway let's close this

[alpine sable]

yea sorry i forgot

thanks again ❤️

.close

[lone heart]

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[alpine sable]

Hey I just wanted to ask quickly

[alpine sable]

what is divided by what in a Frequency table in order to get the mean

ik u have to divide by total frequency

tho wut value do u have to divide

anyone mind answering pls?

also wuts the formula for area of a circle

.close

[lone heart]

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[alpine sable]

.reopen

[lone heart]

✅

[alpine sable]

what is divided by what in a Frequency table in order to get the mean
ik u have to divide by total frequency
tho wut value do u have to divide
anyone mind answering pls?
also wuts the formula for area of a circle

[slender gull]

Pi r^2

For area if the circle.

[alpine sable]

thx

[slender gull]

About your frequency table.

[alpine sable]

?

[slender gull]

Do you have an actual question? I hardly think “what is divided by what in a frequency table”

Is enough context.

[alpine sable]

oki how do i get the mean

in a frequency table

[slender gull]

Oh.

[alpine sable]

U have got ur points frequency halfway and fx

those 4 normal things

[slender gull]

$\f{\sum{f_ix_i}}{\sum{f_i}}$

[alpine sable]

that you'll find in a frequency table

fx divided by f?

[ocean seal]

What the hell am I doing here?

[alpine sable]

Thxxx

sry to bug u for this small thing

i too forgetful for my own good

[slender gull]

In any case, I must tell you you CANT cancel f_i

[slender gull]

If you thought you could. And no issue if you did not.

[alpine sable]

oki?

[slender gull]

Yeah all good, have fun.

[alpine sable]

oi wait

one last question!

@slender gull

[slender gull]

Just post your question.

[alpine sable]

How do i tell the difference between alternate and opposite angles

[slender gull]

Depends on the problem.

Any example?

[alpine sable]

uh lets take ur average 2 parallel lines

with the Z and blah blah blah

[slender gull]

The blah blah blah is what decides the answer.

So I don't know how you expect to solve it without that.

[alpine sable]

oki let me draw a diagram

gimme a sec

just leve it

leave*

nice learning with u

.close

[lone heart]

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[alpine sable]

Reading the directions on this. Still unsure how the x² comes to be.. it says join x-1/x+5 together but I don't see how it makes the X squared

[high rapids]

How do you add fractions?

Like if I give you 2/7 + 3/5 what are the steps?

[alpine sable]

common denominator, so 35. and then multiply out

[high rapids]

Yes

[alpine sable]

hmm

[high rapids]

Do the same here

Show steps

[alpine sable]

So they all have in common X so multiply by X

[high rapids]

Also where does it say join x-1/x+5?

Are you doing the wrong question?

[alpine sable]

oh just typed it in wrong

[high rapids]

Yes

[alpine sable]

meant this part

ahh okay

okay okay interesting

so then the final result is this

then the domain is (x-1)(x+2) so x cannot equal 1 or -2?

[high rapids]

How did you get those domain?

[alpine sable]

sending

omgg i feel like this is so simply why am i makin these mistakes LOL but yeah thats what i did

even on the next one.. i feel like im doing nothing wrong for the domain

[high rapids]

I'm confused? How do you find domain of a function in general?

[alpine sable]

the denominator

[tardy kelp]

depends. in this case the dominator, if f(x) = x^a then x has to be > 0 . It depends on the function used to compose it. Like ur function is f(g(x)) where f is a rational function and g a polynomial function

and the polynomial function (ax+b for example) doesn't have any restrictions

but the rational one has one

the dominator has to be different from 0

[high rapids]

This shows nothing

[lone heart]

@alpine sable Has your question been resolved?

[alpine sable]

I factored

What times what is 2, well 1 x 2 is 2 and the 1 has to be negative to become 1

all other sites seem to confirm but it’s marking me wrong. odd

[lone heart]

@alpine sable Has your question been resolved?

[lone heart]

@alpine sable Has your question been resolved?

[lone heart]

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[frail rivet]

hello why does this NOT converge uniformly?

[frail rivet]

i get that $$||f_n-f|| = sup { | \frac{nx}{nx+1} - {1 ~for~ x \neq 0, 0 ~for ~x = 0 |, x \in ]0,1] } \cup {0} $$ which should be $$ sup { | 1 - 1 | } = 0 $$

but apparently the solution is 1

[ocean seal]

b3s4d

[tacit arch]

Where does it say it doesn't

[frail rivet]

solutions our prof gave us

sadly there is no proof it just goes from $$ ||f_n - f|| = sup{ | -1 | } = 1 \neq 0. $$

[ocean seal]

b3s4d

[tacit arch]

Did you compute your sup norm wrong

[frail rivet]

Well the pointwise convergence is correct

[tacit arch]

If x=0 then ||f_n - f|| = 1

[frail rivet]

yes but

i need to take the sup so x = 1 right?

the $$ \cup {0} $$ part i got from x = 0. bc then 1 -1 = 0

[ocean seal]

b3s4d

[tacit arch]

If ||f_n - f|| equals 1 at one point, how it can the sup norm be zero over the whole interval

[frail rivet]

oh

wait im dumb

setting x to the last possible interval number isnt always the sup

so i have to find a x where nx / nx+1 - 1 is the biggest right

ye probably, thank u

.close

[lone heart]

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[alpine sable]

if anyone could walk me through this expansion i'd be greatful

[lone heart]

@alpine sable Has your question been resolved?

[alpine sable]

<@&286206848099549185>

how are they doing this substitution?

looks like they are replacing q in some places and not others and adding c in some places and not others

[naive valley]

T(q) <= cq^2

T(n-q-1) <= c(n-q-1)^2

therefore T(q) + T(n-q-1) <= cq^2 + c(n-q-1)^2 <= max(cq^2 + c(n-q-1)^2)

and therefore max(T(q) + T(n-q-1) ) <= max(cq^2 + c(n-q-1)^2)

@alpine sable

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[mortal mantle]

Hi i need to apply string theroy to this but im not sure what it is exactly in relation to bridges

can anyone point me to some resources or tips ?

[naive valley]

jump over to the 10th dimension where there's a bridge, and drive across it?

[mortal mantle]

yes lol but i need this to work in 2d lol

or 3 technically

[naive valley]

srsly though, what does this have to do with string theory?

is there any context for this problem

[mortal mantle]

Its labelled string theroy

so it has something to do with it

this has been the case for previous levels

[naive valley]

is it possible to show an example of how you used string theory in a previous level?

do you mean "string theory" as physicists use it, the alternative to the standard particle model?

[mortal mantle]

i havent used string theroy previously

[mortal mantle]

if you are talking about physcicsts and bridges and stuff, then yes

its like civil enginereing i believe

[naive valley]

well for string theory as physicists use it, strings are tiny one-dimensional loops that are the fundamental building blocks of everything

but maybe here they want you to draw an actual macroscopic string?

[mortal mantle]

well the idea of this game is to get the car across

[mortal mantle]

so im not sure how we can fit this into that lol

[naive valley]

and like how do you solve this stuff in general, by drawing objects on the image?

i'm not familiar with whatever this app is

[mortal mantle]

its a videogame

you basically construct somethign that can get the car across

[naive valley]

which videogame

[mortal mantle]

polybridge 2

[naive valley]

ah so like a rope bridge maybe

"string" theory

[mortal mantle]

yeah possibly

but i cant figure out a rope bridge at all

[naive valley]

but how does this relate to math

like why did you choose this server

[mortal mantle]

I thought there was some mathematical answer to this in relation to strings ?

[naive valley]

could be

[mortal mantle]

maybe someone who is good at building bridges could help haha

[naive valley]

like if you have a rope anchored at two points then the shape of the curve it forms is a catenary

which is a math thing

[mortal mantle]

it should be something like this i think

[naive valley]

involving hyperbolic trig functions

[mortal mantle]

i could try this yeah

[naive valley]

oh, that's like a truss/suspension bridge or something?

i forget the words

[mortal mantle]

yeah somehting like that

i just need ideas of different bridge shapes to do with string

[naive valley]

got it

[mortal mantle]

maybe i should ask in a physics server ?

[naive valley]

well there's the classic suspension bridge

like the golden gate bridge

btw, this is just in two dimensions right?

[mortal mantle]

3D

[naive valley]

i suspect a physics server would be more suited, assuming one exists with enough traffic to be helpful

[mortal mantle]

yeah and it involves tensions of string so maybe

ill try in one of those

[naive valley]

mm yeah that's like ... rusty memories for me

good luck!

[mortal mantle]

thank you though! i will try some of your ideas

ty

close.

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[twilit cypress]

Not sure if I am asking in the right place since this is half math and half computer science, but imma try anyways....Anyone know how to prove a function is of a certain efficiency using Big-O? I think my brain is going to explode

[lone heart]

@twilit cypress Has your question been resolved?

[twilit cypress]

<@&286206848099549185> any comp sci folk in here that can help me with the math behind Big-O? I would be forever in debt to you

[twilit cypress]

😦 Thanks anyways, I still really appreciate this server

.close

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[cedar field]

Can you explain a list of when you can cancel out numbers (algebraically)?

[lone heart]

@cedar field Has your question been resolved?

[stone glen]

So we have $a+x=b+x$

[ocean seal]

Lelouch

[stone glen]

Then we can subtract both the sides by x

And we get $a=b$

[ocean seal]

Lelouch

[stone glen]

So for writing in short , usually a mark is put on them, and we say they cancel out

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[cedar field]

Sorry for posting again, My computer died. Can you explain a list of when you can cancel out numbers (algebraically)? And can you also make it so I can copy and paste it?

[lone heart]

@cedar field Has your question been resolved?

[cedar field]

<@&286206848099549185>

[tame willow]

hi

Is this like

every case?

i mean there’s really no way for me to tell you every single one but

The main one is in fractions you can cancel over multiplication

[cedar field]

most ones?

with examples

[cedar field]

or is that too much

@tame willow

[tame willow]

I mean

Well like

Your best bet is practicing with problems that have lots of cancelling

And figuring out what you can and can’t do

[cedar field]

but I need to know how to first though.

[tame willow]

Well if you asked me for a comprehensive list of every single human being on earth in this moment

[slender gull]

@cedar field

[tame willow]

I wouldn’t be able to provide every case

Same here but in a smaller scale

[slender gull]

I can't tell you all of those, like wumpus said.

I have one for you though.

Can you solve for x?

In

$\frac{(x+1)^2}{x+1} = 0$

This

[ocean seal]

What the hell am I doing here?

[slender gull]

Are you kangaroo?

[willow canopy]

i feel like this is a tricky question though

[slender gull]

Well whatever

[alpine sable]

did u intentionally make it

[tame willow]

Let kangaroo solve it.

[willow canopy]

and the person is asking for a general solution

[slender gull]

I made it intentionally, delete all your messages.

[willow canopy]

general tips i mean

[tame willow]

Kangaroo would you like me to provide you the rule here?

[cedar field]

yes please

[ocean seal]

What the hell am I doing here?

[tame willow]

If you have some quantity on top to a certain power, over the same quantity on the bottom, you can do subtraction of the exponents if the top is bigger

So here,

We have 2 $(x+1)$

[ocean seal]

wumpus

[tame willow]

On top

So we can divide one by the bottom

Which cancels

And gets us just one left.

Then we can say $x+1=0$

[ocean seal]

wumpus

[tame willow]

$x=-1$

[ocean seal]

wumpus

[slender gull]

But that's false.

Because there are no solutions to the problem I gave.

x = (-1) makes the denominator zero.

Which again, I told you, you can't have.

[tame willow]

I didn’t look at that

[slender gull]

Huh.

[tame willow]

Yeah I missed that

[slender gull]

You cancelled that.

While the goal here was to tell kangaroo you can cancel as long as you aren't cancelling zero by zero.

[native granite]

This function is defined at x=-1 though

[slender gull]

,w (x+1)^2/(x+1) = 0

[ocean seal]

Results provided by WolframAlpha

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

Requested by Yours truly#3785

[slender gull]

We're talking solutions, there are no solutions though, and I think what I meant here was the caution while cancelling.

Because that's what their query was, in the first place.

So I don't know what you're on about.

[cedar field]

How am I supposed to learn cancelling without someone telling me?

[native granite]

Ok fair enough. For me this function is equal to x+1 for all x (even x=-1) but I guess this is a question of notations here

What I meant is that $\frac{(x+1)^2}{x+1}$ is a continuous function on R (or at least it admits such continuous extension, which I guess is the difference here between what you had in mind and what I had in mind)

[ocean seal]

Twenty

[vale wigeon]

y'all i think this is veering way off from what OP asked

and this is all @slender gull's doing, i would wager

[slender gull]

👀

[native granite]

Yep sry, was just explaining my point

[slender gull]

Sorry, but the best way to illustrate what you can cancel is by showing what you should keep in mind while cancelling?

Or so I thought.

[native granite]

This is an interesting discussion, if you want to pursue it @slender gull feel free to ping me elsewhere

[slender gull]

I shall. Soon enough, not just yet.

[vale wigeon]

to attempt to answer OP's question,

'cancellation' in algebra refers to one of two things: replacing the sum of a number and its additive inverse with zero (a + (-a) = 0), or the product of a nonzero number and its multiplicative inverse with 1 (a * (1/a) = 1)
it can also sometimes refer to a third thing: replacing the composition of a function and its inverse

while both additive cancellation and multiplicative cancellation can have many different forms in practice,
attempting to catalogue all possible forms of either one is kind of pointless.

of course, the number that is cancelled need not look like a single letter. it may be a very long expression indeed.

[cedar field]

ok

[vale wigeon]

the most conceptually difficult context where cancellation requires care is with fractions

especially long ones which have a lot of stuff in the numerator, denominator or both.

if you want to do something to the numerator alone, or to the denominator alone, you can treat them as expressions in their own right, and accordingly all forms of cancellation are as fair-game as can be.

however, it is more often that the numerator and denominator are both modified, concurrently, by the cancellation of a common factor as it is often known, and in this way the fraction remains unchanged despite both its numerator and its denominator becoming different - but only if you do it properly

this kind of cancellation is multiplicative.

[cedar field]

So how do you think I should learn cancellation?

[vale wigeon]

pick up your favorite algebra book, or go to khanacademy, and practice. practice, practice and practice. get a few things wrong at first or further down the line.

no royal road to it, really.

[cedar field]

ok

thanks

everyone

https://tenor.com/view/thank-you-guys-dwayne-johnson-seven-bucks-the-rock-many-thanks-gif-20130341

bye

.close

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[untold tangle]

How do you solve this?

[slender gull]

What do you mean solve?

Rationalize the denominator?

[untold tangle]

Yes

[fervent timber]

https://www.mathsisfun.com/algebra/rationalize-denominator.html

[untold tangle]

THANK YOU!

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[nocturne mantle]

If a father is twice as old as his son. Write down an expression for their total ages.

[tacit arch]

Let b be the son's age and f be the father's age

[nocturne mantle]

yep

so what would it be?

[dry ridge]

That for you to figure out. Read the thing again. How is father and son age related?

[lone heart]

@nocturne mantle Has your question been resolved?

[tacit arch]

Try setting up an equation. No one's gonna give you the answer or do the problem for you

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[tepid drum]

Hi

[tepid drum]

I have a question

When we want to get the derivative if sinhx

We get (d/dx)((e^x-e^(-x))/2) = (e^x+e^(-x))/2

Why /2 not /4

If we use the quotient rule we should end up getting the result /4 instead of /2

[wary stream]

Because 1/2 can be pulled out as a constant

And you just take the derivative of the numerator

[tepid drum]

Yes but we can use the quotient rule by considering a function defined as g(x)=2

Why is this method wrong

[wary stream]

You can use the quotient rule, but it's not necessary if you can pull out 1/2 since it's a constant

You'd be doing extra work for no reason

[tepid drum]

But if u use quotient rule you will get a different result

My question is why am I getting a different result when using quotient rule

[wary stream]

And if you did quotient rule, I'm pretty sure stuff would cancel and the denominator be 2 still

[wary stream]

Because you messed up somewhere

[tepid drum]

Yes you are right

You will end up by getting the derivative of the numerator multiplied by 2 and divided by 4

2/4 will simplify to become 1/2

I am stupid bro

Thank you so much and sorry if I annoyed you

[nocturne mantle]

what grade r u in

[tepid drum]

10

Self study

I shouldn't do this mistake

[wary stream]

But like I said, it's much easier if you noticed that 1/2 is a constant and then you won't have to use quotient rule

[tepid drum]

I finished 10 this year

[tepid drum]

Yes you are right

Tysm again and have a nice day

.close

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[alpine sable]

is this 15?

[wide raven]

instead of asking that, just confirm what each variable is

full cat=10

cat only face=5

paws=2

[lone heart]

@alpine sable Has your question been resolved?

[alpine sable]

erm

sure

[lone heart]

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[stable night]

hi guys, my idea is that:
assuming the chess master plays once everyday, she would have played 7 games a week.
so there must exist a day where she played 2 or more times

then if she played 21 games consecutively everyday, this means that she played for at most 3 weeks.

[vale wigeon]

she*

[stable night]

oops

so the games are the pigeons and the days are holes?

[vale wigeon]

would say this requires a somewhat more refined approach.

[stable night]

hmm, can i get some hints

i cant see how pigeonhole principle ties into the last sentence

[vale wigeon]

imagine the chess master keeps a log of her games in the form of a 2 by 77 table in which she records how much she played at the end of each day, but in the following fashion:

the top row is labeled "Games played thus far", while the bottom row is labeled "Games played thus far + 21"

[stable night]

right

[vale wigeon]

since she plays at least one game daily, the numbers in the top row are all distinct from each other, and the same holds for the bottom row.

on the other hand, since she plays no more than 12 games per week, she plays no more than 12*11 = 132 games in total

[stable night]

right

[vale wigeon]

and thus her cumulative numbers of games played never go beyond that

[stable night]

can i ask why is the bottom row labeled "... + 21"?

[vale wigeon]

you'll see soon enough.

[stable night]

yesmam

[vale wigeon]

the numbers in the bottom row thus go no higher than 132+21 = 153.

so the table as a whole contains integers ranging from 1 to 153 inclusive.

that make sense?

[stable night]

right

yes

[vale wigeon]

right.

well here's the kicker:

there are a total of 2*77 = 154 entries in the table.

there are more of them than there are possible values.

therefore, there must be a value that shows up twice in the table.

[stable night]

erm wait why is it 2 * 77?

[vale wigeon]

it's a 2 by 77 table

2 rows (games played; same+21) by 77 columns (one for each day)

[stable night]

oh oh right

[vale wigeon]

these duplicates, by the argument i laid out above, cannot occur in the same row.

[stable night]

yepp

[vale wigeon]

so one of them will be in the totals row and the other in the totals+21 row.

[stable night]

wait sorry but are the "possible values" just values i can put inside any box

[stable night]

yes

[vale wigeon]

the possible values are whole numbers from 1 to 153 inclusive

[stable night]

oh right

[vale wigeon]

sketch up an example table and think about the meaning of this duplicate.

[stable night]

Okay one moment

[vale wigeon]

||this duplicate will lead to two numbers in the "Games played thus far" row that differ by exactly 21. and this is exactly what we want!||

[stable night]

Wait, i understand

But i still dont get the purpose of the bottom row

[vale wigeon]

it's to force a duplicate via pigeonhole

\begin{tabular}{c|ccccc}
Day & $\dots$ & $i$ & $\dots$ & $j$ & $\dots$ \
\hline
Games thus far & $\dots$ & $n-21$ & $\dots$ & $n$ & $\dots$ \
\hline
Games thus far + 21 & $\dots$ & $n$ & $\dots$ & $[...]$ & $\dots$
\end{tabular}

[ocean seal]

Ann

[vale wigeon]

these n's are the duplicate whose existence pigeonhole guarantees

[stable night]

Right

So if theres an n on top and n on the bottom

Wait

If theres n on the top and n+21 on the bottom then she would have played it for 21 days consecutively?

Er wait no

[vale wigeon]

no

she played n-21 games from day 1 to day i inclusive, and n games from day 1 to day j inclusive

this gives us that she played n-(n-21) games from day i+1 to day j

[stable night]

Wait so there are 1 to 153 inclusive possible values and 154 slots

Then, by pigeonhole principle this means that there must be at least 1 duplicate in both rows

So she played 21 days from day I to J
Aka she played n-(n-21) games from day i to day j

Wait why is it i+1?

Okay so

1. we create the bottom row to force out pigeonhole principle
2. the duplicate causes 2 numbers to be the same, and the numbers cant occur in the same row cuz she plays at least 1 game everyday
3. since the 2 numbers of different rows are duplicated, there will be 2 values in "games thus far" row with a difference of 21

[vale wigeon]

yes

[vale wigeon]

{1, 2, ..., j} \ {1, 2, ..., i} = {i+1, i+2, ..., j}

[stable night]

oh right

how would u advice me on developing this level of intuition next time

like I didnt think of thinking in terms of a table

even if i did i wouldn think of using pigeonhole principle on that

[vale wigeon]

dunno

[lone heart]

@stable night Has your question been resolved?

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[stable night]

ill continue trying

thanks alot

.close

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[cerulean garnet]

so f(p) would be 3-2p

im not sure about f(p+1)

is it just 3-2p+1

[gray ingot]

no

3-2(p+1)

[cerulean garnet]

oh okay

so

its f(p) = 3-2(p+1) - 3-2p

[gray ingot]

no again

f(p) isn't equal to that

f(p+1)- f(p) = 3-2(p+1) - (3-2p)

[cerulean garnet]

oh right yeah

.close

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[iron swallow]

Hi

[iron swallow]

Hey

<@&286206848099549185>

I need

@keen pasture can you help

Or not

[keen pasture]

Nice pattern.
Maybe you could describe the steps you did to reach this. For example flipping, rotating...

[iron swallow]

Yes

Ik that is the top question which I like know how to do but this is the question I am stuck on can you help

[iron swallow]

Ya I going to be like I moved shape a in the vector 5,0 to make shape b etc

[iron swallow]

But this is so confusing

Do you get it

[keen pasture]

Also look at the symmetries: is each individual building Block symmetrical? Is the whole thingy you created symmetric? You should also focus on the middle part, which has great proberties

[iron swallow]

Ya

[keen pasture]

Maybe draw some lines

[iron swallow]

Like the is symmetry at 16.5

The invariance is the hard bit

[iron swallow]

Yup

[keen pasture]

You could draw some more lines, but ok

[iron swallow]

Where

How

[iron swallow]

Oh liek for each shape

But how do I say that

@keen pasture

Please help man I am so screwed I have a test tomorrow on this

[keen pasture]

With those lines

[iron swallow]

Kk

Can you help with inverance

I don’t even know what it is

@keen pasture you there

[keen pasture]

Don't know what it means

[iron swallow]

Ya I don’t

[keen pasture]

That may help for tomorrow

[iron swallow]

Ya ik this

Not inverance

@keen pasture

My teacher honestly suck never though us anything

Can you help me please

[lone heart]

@iron swallow Has your question been resolved?

[iron swallow]

Ya

[keen pasture]

You should ask Google. He knows a lot.

(That's a deliberate personification)

[keen pasture]

Take a look at these. If you know them by heart tomorrow, you will have an significant advantage

[lone heart]

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[keen pasture]

https://en.m.wikipedia.org/wiki/Invariant_(mathematics)

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[wanton junco]

Let f(x) be an integrable function on [a,b]
prove that $e^{f(x)}$ is also integrable on [a,b]

[ocean seal]

Deus_Vult

[wanton junco]

First observation is that f(x) is bounded by some real number M, therefore g(f(x))=e^(f(x)) is also bounded by e^M

now can I use Cauchy's mean value theorem and say that at every interval [c,d], sub interval of [a,b]
e^(f(d))-e^(f(c))=e^x_0(f(d)-f(c)), x_0 being some element in [c,d]?

That would allow me to easily show integrability using darboux sums
however, Cauchy mean value theorem requires both g(f(x)) and f(x) be continous on the interval
is it enough that g(x)=e^x is continous on the interval ?

[willow canopy]

any composition of continuous functions are continuous themselves

so yes

(both has to be cts on the same interval of course)

[wanton junco]

of course
but I f(x) in this case isn't continuous necessarily

[willow canopy]

oh yea true

[wanton junco]

so g(f(x)) isn't necessarily continuous .. (?)?

would the theorem still apply ?

[willow canopy]

my intuition says no

try something like

[wanton junco]

like, does x_0 exist ?

[willow canopy]

f(x) = 0 when x<0

and f(x) = 1 when x>0

such an x_0... would still exist huh?

[wanton junco]

in this case yea actually

[tawdry portal]

I don't think the statement is true if $f $ is not continuous, take $f(x) = \frac{1}{\sqrt{x}}$ for $x>0$ and $f(0) =0$. Then $f$ is integrable on $[0,1]$ but $e^{f}$ is not

[ocean seal]

Maldor

[willow canopy]

oh yea true

ty for the better counterexample lol

[pliant dune]

depending on the type of integration I'm not sure f would be integrable

[wanton junco]

the counterexample is right
but in our course material we defined integrability using riemann and darboux integrals

therefore an unbounded function is not considered in the definition of integrability
and f(x) is necessarily bounded

should've mentioned it, sorry ..

[tawdry portal]

I see, that makes sense

[pliant dune]

One thing I notice is e^x is increasing so you have something like inf(exp(f(x))) = exp(inf(f(x))) and sup(exp(f(x)))=exp(sup(f(x))) but idk if that would help the calculations

[wanton junco]

yea definitely
that would be the next step if the mean value theorem holds

hmm
what if I consider g(x)=e^x
and consider the interval [f(c),f(d)], where f(c)<f(d)
then the mean value theorem will hold
e^(f(c))-e^f(d)/(f(c)-f(d)) =e^b, s.t. b is in [f(c),f(d)]

and since f is bounded by M, it follows that for every such interval, [f(c),f(d)] and every element a in such interval, a<M
and then e^b<=e^M

that would give me the desired result

[willow canopy]

that is actually very smart holy

yea nothing is stopping you from doing that i think

especially since f is bounded

[lone heart]

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@wanton junco Has your question been resolved?

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[fossil shoal]

Hi, I think I just went full on stupid here. Why is the normal vector in the direction of the arrows in this comic given by (cos(beta), -sin(beta)) and not (-cos(beta), -sin(beta))? If you think about it on the unit circle, we are in the 3th quadrant, no?

[pliant dune]

because sin and cos already account for the signs

the - sign in front of the sin is because beta is measured clockwise from positive x axis instead of anti-clockwise

[fossil shoal]

If you draw an vector for (cos(pi/3), -sin(pi/3)) you get a vector in the 4t hquadrant.

not in the third.

so you have to take -cos(beta), leading to what I said.

ah beta would be >pi/2? ^ ^

[pliant dune]

-pi/3 ends up in the 4th quadrant, but beta is drawn in the third

[fossil shoal]

yeah thanks I just saw my mistake.

Have a nice day.

.close

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[native cloud]

Can anyone help me with this question?

[native cloud]

Not sure what to do after this

[rigid smelt]

plug the point into the equation to evaluate

[gray isle]

your notation is very questionable

[native cloud]

wdym?

[native cloud]

which one?

[gray isle]

what's with those d^2y/dx^2 and dy/dx on the side
and those evaluation lines

[native cloud]

Ah that just for (at point (-2, 1))

One of my teachers used to write it that way

[gray isle]

you're not at a point where you can evaluate yet

so you shouldn't be writing that stuff at this stage

[native cloud]

Oh okay

[gray isle]

currently all you have is the first derivative,
differentiate and apply chain rule again to get the second derivative

[native cloud]

Where should I be applying chain rule?

[gray isle]

when differentiating stuff with y

[native cloud]

Like this?

[gray isle]

no

derivative of a quotient is NOT the same as the quotient of the derivatives of the numerator and denominator

[native cloud]

I should use quotient rule then, right?

[gray isle]

that would be something that is appropriate

[native cloud]

What do I do next?

[lilac nest]

substitute dy/dx with the expression you found before and then substitute x and y

[native cloud]

Wdym by substitute dy/dx?

[gray isle]

you had an equation telling you what dy/dx was

[native cloud]

Substitute this in dy/dx?

[gray isle]

repalce the dy/dx in what you just got with "that"

[lilac nest]

you found before what dy/dx is when calculating the first derivative, that's what you have to substitute

[native cloud]

So like this?

[lilac nest]

yes

[native cloud]

.close

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[willow tinsel]

someone help me with this

[willow tinsel]

pls

compound/annunities interest question

[lone heart]

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[willow tinsel]

I have simplified an equation to this:

11.390625 = (3/2)^x
or
11.390625 = 1.5^x

How do I get x from this. I know the answer is 6 but I would like to know the steps involved

[willow tinsel]

<@&286206848099549185>

[willow canopy]

take log base 1.5 on both sides

[willow tinsel]

.close

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[lone heart]

@trail spruce Has your question been resolved?

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@trail spruce Has your question been resolved?

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[graceful tulip]

How does this step work?

[earnest current]

have you tried using $\sum (a + b) = \sum a + \sum b$?

[ocean seal]

texaspb

[earnest current]

looks like they also used some kind of identity for binomials

I'm not sure though.

[graceful tulip]

I think your answer is correct makes sense

They basically just excluded it

Okay thanks

[earnest current]

no problem

[graceful tulip]

.close

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[long roost]

Hi, I am very confused on this topic, I got help yesterday but still to no avail. My thought process is a(area) = 1/2 x a(apothem) x p(perimeter). In this problem I got 12tan45 equals a(apothem) and then multplied it by the permimeter and then times 1/2 but was told this was incorrect. any assistance please

[vale wigeon]

the apothem is not equal to 12 tan(45°)

[long roost]

But isnt 360/8 45 meaning this would be an isosiles triangle?

so the angles would be 45,45,90?

[vale wigeon]

the triangle you're thinking of is isosceles, but its angles are not 45°, 45° and 90°.

45° is the angle at its apex, not one of the angles at the base.

[long roost]

Ok, I think I'm overcomplicating it a little bit. Simply put I am confused on why I am getting the angles of the triangle incorrectly if that makes sense.

Like when I divide 360/8 and get 45

is that the correct thought process?

[vale wigeon]

let me draw a picture

[long roost]

Thank you

[gray isle]

it would be a good idea to clearly mark where your angles are

[long roost]

I am drawing the triangle out on paper

[gray isle]

instead if doing "stuff" just because you reached a nice value

[vale wigeon]

these two triangles are not the same

the triangle on the left is the one that your octagon is made from 8 copies of.

the triangle on the right is what you confused it with.

[long roost]

Ok the triangle on the left, is it an equilateral triangle?

[gray isle]

it has a 45° angle...

[vale wigeon]

no, it is not an equilateral triangle.

[long roost]

Ok in my problem am I using right triangle trig?

I would think so?

[vale wigeon]

if you want to use right-triangle trigonometry then sure

but you will not be able to do it without actually sitting down and drawing a diagram that accurately represents your problem

[long roost]

I am drawing the diagram

I wrote it out

multiple times

anyway, I would like to know where my thought process/solving method is incorrect?

If I divide 360/8

I am putting the 45 at the apex of the pyrimad

[vale wigeon]

that's not a pyramid that's an isosceles triangle

[long roost]

Im spitballing

[vale wigeon]

...

[long roost]

There are two triangles

[vale wigeon]

no.

[gray isle]

draw your diagram and show us

[long roost]

ok

thats a 45 at the top

[vale wigeon]

what does this diagram have to do with your problem?

[long roost]

oh wait hold on

[gray isle]

instead if doing "stuff" just because you reached a nice value

[vale wigeon]

again, what does this diagram have to do with your problem?

[gray isle]

I'd recommend starting with the full octogon

[long roost]

Ok in polygons your figure is made up of equal triangles

[vale wigeon]

also MS Paint has a straight line tool

[long roost]

correct?

[vale wigeon]

too vague.

[gray isle]

and the consider what 360/8 determines and gives you

and how it would be represented graphically in that octogon

[vale wigeon]

to find the area of a regular polygon with n sides,
the first step is to take the polygon's center, and draw from it straight lines connecting the center to every vertex.
this breaks the regular polygon into n copies of the same triangle.
this triangle is isosceles because two of its sides are lines connecting the center to a vertex, which all have the same length,
and the angle at the apex of this isosceles triangle is 360°/n, because there are n such angles at the center, they are all equal and they add up to a full 360°.

[long roost]

let me read this hold on

[vale wigeon]

that is a yikes-level diagram

[long roost]

not drawing a da vinci

[gray isle]

and based on the way you're splitting, you don't get triangles

[long roost]

Ok what do you think I am not getting so I can fully understand

[gray isle]

first draw an octogon properly

and/or harvest one from Google images

[long roost]

good?

[vale wigeon]

seems good enough if you're careful enough about what comes next

[long roost]

Ok now I draw 8 lines from the center to the vertex with 45 degree angles in each of the apexs