help-49

[carmine sigil]

The entire area in that circle?

That might be tricky coming from the side

Oh, this is a question about the central angle theorem

I take back what I said about this being tricky. This is pretty straightforward once you know the trick.

[queen bane]

Can you help me im struggling to solve this by myself 😭😭😭

[carmine sigil]

@queen bane do you recall the statement of the central angle theorem?

[queen bane]

I don’t really know what that is , because my teacher just said to find out how to solve it and we didn’t study that yet…

[carmine sigil]

Essentially, as you get closer to the pair of light houses the angle between them gets bigger

Exactly on that circle arc the angle reaches 30° (due to the central angle theorem)

So to keep out of the danger zone, you just need to make sure the lighthouses stay close enough together so that the angle between them is less than 30°

[midnight plank]

@queen bane Has your question been resolved?

[midnight plank]

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[solemn solar]

in this diagram, to prove similarity, my proof is

Right angle = right angle
corner is common
angles for corresponding angle

is this right?

[violet dune]

yes

[solemn solar]

k

so do i just assumed that the lines where the 14 and 4 are equal

cuz usually theres a marking or sum to indicate

[lyric charm]

what do you mean by equal

14 ≠ 4 so these segments are not equal in length...

[solemn solar]

i meant paralel

its alg tho

i alr solved the question i js forgot to close

.close

[midnight plank]

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[midnight plank]

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[steel rivet]

#calculus

[pearl hull]

What are you looking for?

[thin tree]

Is it the same question?

[pearl hull]

I’m closing the channel since there is no existing question in this channel, claim a new one once you get a question

.close

[midnight plank]

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[thin tree]

How can u close

[pearl hull]

He’s gone for 10 minutes without response

Claiming a new one doesn’t seem difficult

[thin tree]

But like can others close?

Like can I close a random person's help

[pearl hull]

You require a helpful role

[thin tree]

Oh

[midnight plank]

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[long lotus]

any ideas for this limit?

[midnight plank]

@long lotus Has your question been resolved?

[topaz urchin]

you can write it as exp[lim n-->inf (f(n)-1).n^2)]

f(n) is 1 + 1/n(n+1) btw

[long lotus]

hmmm

[solid iris]

$b_n=\log a_n=n^2\log(1+\frac{1}{n(n+1)})$

[grand pond]

ロケットジャンプ

[solid iris]

looks like lhopital or taylor log(1+x) works

[long lotus]

mmm if I just apply natural log on the thing it would get me to 0×infinity which is undefined isn't it

[solid iris]

do some algebra to see why lhopital works

[long lotus]

lhopital on which one

[long lotus]

this?

[solid iris]

yes

find me a form where lhopital works

[long lotus]

didn't it have to be infinity/infinity or 0/0 to do lhopital?

ohhhh

okay let me see

[solid iris]

im sleeping soon, do u see the algebra

[long lotus]

I'm not sure

getting that 1 in the same fraction?

inside the log

or is that not it

[solid iris]

get me this form

[long lotus]

uhhhh T-T

I swear I'm really trying i just see it?

[solid iris]

rewrite bn to get 0/0

@long lotus stuck?

[long lotus]

you don't want to see how my notebook looks like rn 💀

is it supposed to be obvious?

[solid iris]

$\frac{\log(…)}{1/n^2}$

[grand pond]

ロケットジャンプ

[solid iris]

make a note for future. its a good trick

[long lotus]

oh my god......

my teacher showed me that some time ago and I've already forgotten

[solid iris]

easy algebra but important trick

[long lotus]

okay so now I use lhopital and then it should go smoothly

[solid iris]

rest is up to u 🙂

gotta sleep. peace

[long lotus]

alrighty I think I can figure it out from here

many thanks for the patience

sleep well!

[solid iris]

np and ty

[long lotus]

.close

[midnight plank]

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[midnight plank]

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[untold oyster]

What is this form called again?

[rain wasp]

factored form?

[untold oyster]

Ohh i seee

How about this.

[untold oyster]

The second one

[winged maple]

also factored

[untold oyster]

Is there a specific factor term? Or just factor?

[violet storm]

factors = things being multiplied together

in the multiplication 2 * 3 = 6, 2 and 3 are factors

[winged maple]

I mean you could say it's a quadratic factorisation but the first one is also a quadratic

[violet storm]

in the multiplication x * (x+2) in your example, x and (x+2) are factors

[dusty portal]

Factored, or vertex form (really a techncality between the two).

[winged maple]

idk. I'd just say you are factorising them

[violet storm]

(x+1)^2 = (x+1) * (x+1) again factors being multiplied together

[untold oyster]

Yesss

[violet storm]

so "factoring (aka factorising) something" is taking a quantity that is multiplied together and seeing if you can break it up into the constituent parts that have to be multiplied together to obtain the original expression

factorise 6: write it as 2 * 3

factorise x^2 + 2x: write it as x(x+2)

[untold oyster]

Yess

[untold oyster]

Does 6 write it as 6*1 is a factor?

[violet storm]

yes

[untold oyster]

Okay thank you guys!! You are a big help!!

[violet storm]

for numbers, oftentimes it's useful to get a version with just prime factors being multiplied together

but at its simplest most general you can just think of factors as the pieces making up a multiplication

[midnight plank]

@untold oyster Has your question been resolved?

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[midnight plank]

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[wind fiber]

!status

[midnight plank]

What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above

[wind fiber]

1

2

[carmine sigil]

ok, you have proven that f1 is injective. What about surjective?

[wind fiber]

I’m a bit lost on how to actually prove it

I’m trying right now

[carmine sigil]

What is your opinion, is it surjective?

[wind fiber]

Yes, because I thought it’s binective because each a maps to exactly one element

[carmine sigil]

if it's surjective then every member of the codomain (the real numbers) has a number in the domain (the integers) which f1 sends to

So in order to prove that it's surjective, you need to prove that for any real number y, you can find an integer x, such that f1(x) = y

Alternatively, if you find a y such that there is no such x, then you have proven that f1 isn't surjective.

So now that I've clarified the above, do you still think that the function is surjective @wind fiber ?

[wind fiber]

Yes

[carmine sigil]

ok, can you find an x such that f1(x) = pi?

[wind fiber]

Okay maybe not

X not interfering

Integer

[carmine sigil]

yup!

and that's the key

while you can solve for x, the value you get isn't in the domain

So, you have proven by example that f1 isn't surjective.

[wind fiber]

Hmmm

What does Z arrow R mean, doesn’t that mean if I input integer, I get back real number

[carmine sigil]

yes, that means f1 is a function that takes an integer and gives you back a real number.

[wind fiber]

Pi is not integer

[carmine sigil]

that is correct, pi is a real number.

I think you're a little confused.

So surjective functions are functions that have the entire output space covered by the input space mapped by the function

in this case

we selected a member of the output space

and showed that it has no member of the input space which the function maps to it

so pi being a real number is not a problem, because the output space is the space of real numbers.

[wind fiber]

What is output space and input space

[carmine sigil]

domain and codomain

sorry

[wind fiber]

Input is the number we give x?
Output is the number we get from, y?

[carmine sigil]

codomain and domain respectively

output = codomain, input = domain

[wind fiber]

Ok

[carmine sigil]

and yes, in the example I gave above, x is a member of the domain, y of the codomain

[wind fiber]

Okay

Z is domain and R is codomain?

[carmine sigil]

yes

[wind fiber]

Okay

I got the whole thing understood

Thank you, I’ll do the question later, library is closing here

[carmine sigil]

🎉

[wind fiber]

Thanks

!close

.close

[midnight plank]

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[amber marlin]

Is it necessary to write $t \in \mathbb{R}$ ?

[grand pond]

SELVATOR

[amber marlin]

What if $t \in \mathbb{C}$

[grand pond]

SELVATOR

[subtle blaze]

Well I don’t know, do you have more context of this problem

[midnight plank]

@amber marlin Has your question been resolved?

[amber marlin]

Just coordinates

Of vector of a line

[dense harness]

Are you in a real or complex space

[subtle blaze]

Well if you want a curve

A curve is a function from some interval into your space

If you let t ∈ ℂ then you don’t have an interval anymore

And subsequently you don’t have a curve

[midnight plank]

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[ionic magnet]

Among the given pairs of functions, indicate a pair of mutually inverse functions.

[midnight plank]

@ionic magnet Has your question been resolved?

[blissful talon]

Do you know what the words mean? Can you test them?

You can test these on a few ways. You could switch the x and y in one and see if it's the same as the other. Or you can plug one function into its pair and see if it simplified to plain x

[midnight plank]

@ionic magnet Has your question been resolved?

[midnight plank]

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[uncut rapids]

95c

,rotate

[uncut rapids]

,rotate

[uncut rapids]

i do not know how to continue

should i change bases

into maybe 3 ?

which would give me above 1 everywhere

like 1/log(3)(x) ..

and so on?

but how do i continue after

[elder sierra]

Do you need to got from left to right?

@uncut rapids

[uncut rapids]

sorry could you clarify ?

i mean yes but any way is fine

[normal prawn]

Is x=16 the solution?

[elder sierra]

You need to show the expression in the image is true, right?

[uncut rapids]

i need to resolve it

[elder sierra]

Or just to solve?

Oh

[uncut rapids]

not that it is true

im sorry for not clarifying beforehand

[elder sierra]

Use a base change rule

[normal prawn]

what's the answer? 16?

Wait

[elder sierra]

Use base change rule in $log_3x$ so that is becomes an expression with $log_x$ in numerator and denominator

[grand pond]

Max Hetfield

[uncut rapids]

oh

[elder sierra]

$$\log_{B}(M)= \frac{\log_{A}(M)}{\log_{A}(B)}$$

[grand pond]

Max Hetfield

[uncut rapids]

so ill change base

as A is x

?

[elder sierra]

Yes

[uncut rapids]

for all 3 of the logarithms?

[elder sierra]

Yeah, why not

nO SORRY

Only for the ones that are not log_x

[uncut rapids]

wdym

[elder sierra]

log_3x and log_81x do a base change with those

[uncut rapids]

alr

N

,rotate

[uncut rapids]

do i move the first to the left and divide?

or

[elder sierra]

Checking

Good try to cancel terms

What do you get?

[uncut rapids]

i do not know which ones to cancel

move first () to the right

[elder sierra]

Yes

[uncut rapids]

and cancel the top and down?

[elder sierra]

Yes

[uncut rapids]

the logx 3

alr

wait a min

is this right?

[elder sierra]

No, log 81x shouldn't be there

The cancel is right, tho

[uncut rapids]

what should be there instead?

1/log..

?

[elder sierra]

Yes

[uncut rapids]

alr

can i put both of them at -1 so i can reverse the fractions ?

[elder sierra]

Give me a sec

Checking something

Use addition property of logarithms in the right side

[normal prawn]

is the answer nine 🥺

[elder sierra]

Do the addition property

[uncut rapids]

to get 3x and 27 ?

[elder sierra]

Sorry I meant the product property

$$\log_{B}(M \cdot N)= \log_{B}(M)+\log_{B}(N)$$

[grand pond]

Max Hetfield

[elder sierra]

This on the right side

[uncut rapids]

into 3x and 27 as m and n ?

[modest furnace]

what are we solving?

like what topic

[uncut rapids]

solving this

logarithms

[elder sierra]

Logarithms

[modest furnace]

ok thanks

for letting me know

[elder sierra]

Waiting on you

[elder sierra]

How did you end up with that denominator?

[uncut rapids]

3x*27 is 81x

what denominator did u get

[elder sierra]

Nah, don't worry just a different way of doing it

[uncut rapids]

then could've i done / ()^-1

to reverse the fraction

and get log x 3x / log x 3 = log x 81x

then use that formula

and simplify

?

[elder sierra]

Thinking...

[normal prawn]

im invisible

[modest furnace]

i think i got the ans

[normal prawn]

what is it

[elder sierra]

Let's see divyesh

[modest furnace]

x= 1.9,9

[normal prawn]

1.9?

how!?

[modest furnace]

hollup

ill show the steps

[elder sierra]

Try it

[uncut rapids]

alr

[modest furnace]

we are solving this right?

[elder sierra]

Yes

[modest furnace]

x= 1.9,9 (this is the ans in decimal form)

[elder sierra]

How

Show the procedure

[modest furnace]

SO SORRY

wait

0.1,9

typo

[normal prawn]

0.1?

[modest furnace]

yea

[normal prawn]

sussy

show us ur work

[modest furnace]

im sadly on my pc and cant take pics

[elder sierra]

@uncut rapids How you going?

[uncut rapids]

ill send a pic in a minute

[normal prawn]

umh alr

[modest furnace]

rlly sorry guys

[normal prawn]

no prob dude

[modest furnace]

what r u getting as ur ans

[uncut rapids]

now im not so sure if it was the right choice or not to use / ()^-1

[normal prawn]

Only 9

[elder sierra]

From here, send the sum to the numerator of the left side. And send the numerator of the left side to the denominator of the left

Then try to cancel terms

[modest furnace]

uhmm

[uncut rapids]

so i send the numerator of the left side to the denominator of the left ?

[modest furnace]

x1 = 1/9 , x2 =9

[uncut rapids]

sorry it is quite late rn

[elder sierra]

The denominator to the right

[modest furnace]

i think ur right

[elder sierra]

No issue, we're solving this together

[normal prawn]

Oh 1/9? I thought u said 0.1

[uncut rapids]

sorry which denominator

it is log 3 3x

[modest furnace]

it was a typo

my bad

my keyboards laggy

its 1/9

[modest furnace]

i mean in the decimal form it is 0.1

[elder sierra]

Let me grab a piece of paper and try to do it. I'll ping you

[uncut rapids]

alright take your time

[normal prawn]

Oh alr mb bro

[modest furnace]

@elder sierra what ans are u geting

[modest furnace]

nah issok

[elder sierra]

Trying rn, was doing it in my mind but this definitely needs pen and paper, so I grabbed them

[modest furnace]

yeah i did it on a pen and paper too

im just waiting to see if my answer is right or not

[midnight plank]

@uncut rapids Has your question been resolved?

[modest furnace]

@uncut rapids what are u getting as the answer

[uncut rapids]

i am not able to continue sadly

[modest furnace]

any guesses

or like

anything close

for x1 i got 1/9

and x2 = 9

[elder sierra]

I think I got an answer...

[modest furnace]

what'd u get?

[elder sierra]

But let me replace it and check it numerically

I got x = 3

[modest furnace]

hmm

@normal prawn he also got the same answer which i got

is there any way to verify

like any textbook solutions or smth

[uncut rapids]

i will check

[normal prawn]

YA

[modest furnace]

btw which grade question is this

[normal prawn]

idk

how old r u divyesh?

[modest furnace]

17

[normal prawn]

just asking if u don't mind :p

[uncut rapids]

1/9 and 9

are the solutions

[elder sierra]

Add me as friend, I will find the answer

[modest furnace]

yaaayy

[uncut rapids]

we note log 3 x = t

[normal prawn]

woww

[uncut rapids]

and we have 1/t * 1 / 1+t = 1 / 4+t

thats what the manual said

[modest furnace]

nice dude

[uncut rapids]

real straight forward

[modest furnace]

ohkk

[uncut rapids]

so we have to restart ig

[modest furnace]

hmm

[elder sierra]

I'm double checking but I'm sure 95% I'm wrong, I did some non-invertible operations and that's an almost sure way to get some strange answers

[normal prawn]

Just a small suggestion - when u apply the base change propertyy, change the base to 3 and not x

[uncut rapids]

i cant believe ive wasted 1 hour on this exercise lmao

[modest furnace]

DAMN!

[elder sierra]

Got it

[elder sierra]

He's right

That equation has 2 answers

9 and 1/9

[uncut rapids]

Can you send a picture with your solution?

[elder sierra]

That wouldn't be allowed in this server

But I can guide you quickly

The solution can be reached in less than 10 steps

Ready?

[uncut rapids]

yes

[elder sierra]

Start with the original euqation and do what @normal prawn said. Change to base 3 all logarithms

[uncut rapids]

a lr

[modest furnace]

woah

it took me a lot more than tha

t

[normal prawn]

Why?

[uncut rapids]

then

[elder sierra]

Don't worry mate, I used a lot of logarithm properties

[modest furnace]

hmm

[elder sierra]

We should help people get the answers, not straight out do their homework

[modest furnace]

ur right dude

[elder sierra]

@uncut rapids Waiting on you mate

[uncut rapids]

so the first ()() is 1/ log 3 x and 1 / log 3 3x

[elder sierra]

Can you show a photo?

That way it will be faster

[elder sierra]

Perfect

Do you know what log_3(3) is equal to?

Replace and show me what you got

[uncut rapids]

1

no?

[elder sierra]

Yes

[uncut rapids]

replace all with 1 i guess

[elder sierra]

Yes

[uncut rapids]

that i can skip

multiplying the first two

[elder sierra]

OK, then reorganize in such way that there are no denominators

[uncut rapids]

im not sure what ill get

[elder sierra]

Try and show me

[uncut rapids]

i am not good with multiplying fractions of logarithms

[elder sierra]

Don't worry, you don't need to simplify

Just replace the 1s and then reorganize following my instruction

[elder sierra]

Perfect, now reorganize

[uncut rapids]

how

[elder sierra]

Mate, send all terms to their respective other side

I'm sure you know what that entails

[normal prawn]

Ooooohh that's nicee

[elder sierra]

,rotate

[elder sierra]

Ok, that's a way of doing it

But less make it easier

Send the denominators as numerators

on their respective opposite sides

[uncut rapids]

alright

[elder sierra]

Waiting on you

[elder sierra]

No, mate

Delete the last 2 steps. What I asked you to do should end up in $\log_{3}81x = \log_{3}x \cdot \log_{3}3x$

[grand pond]

Max Hetfield

[elder sierra]

Take your time to understand and then tell me when you're ready for the next step

[uncut rapids]

sorry mate its really late rn

but

yes

i kind of got it

[elder sierra]

Don't worry, just write it

OK, then do you know the product property of the logarithms?

[uncut rapids]

yes

[elder sierra]

$$\log_{B}(M \cdot N)= \log_{B}(M)+\log_{B}(N)$$

[grand pond]

Max Hetfield

[elder sierra]

Take a look at this expression and tell me where you could use it

Hint: 2 of 3 logarithms can be treated with that property

[uncut rapids]

idk mate

[elder sierra]

Mate, make some effort, you can do it

[uncut rapids]

if we move

log 3 3x

to the other side

as a divide

[elder sierra]

You're halfway to the solution

[uncut rapids]

we get log 3 x = log 3 27

[elder sierra]

No, no}

[uncut rapids]

cant we do that?

nvm we get x = 3 i believe

[elder sierra]

Not yet

Mate Show me the picture of what you've got right now

I know it's late, but you can do it

You only have 5 steps left

[elder sierra]

Good. Now answeer this for the next step:

2 of the 3 logarithms in that final expression can be dealt with the product property

Which ones?

[uncut rapids]

the second and the one after =

cant they ?

[elder sierra]

Yes

[uncut rapids]

all could be but yes

[elder sierra]

Then do it

[uncut rapids]

ok

[elder sierra]

Use the property

Show me what you get

[uncut rapids]

,rotate

[elder sierra]

,rotate

[elder sierra]

Oh, you did it differently. Let me check

Right side has a mistake: 81 is not 3^3

Left side is good but use parenthesis to organize the expressions

[uncut rapids]

3^4 ?

did that

now what

[elder sierra]

What do you get?

[uncut rapids]

a organised expression

an*

[elder sierra]

Can you show me the picture?

So we can know we're on the same page?

[elder sierra]

,rotate

[uncut rapids]

i just use delta now

[elder sierra]

No. Left side is not like that. It's log_3x (log_3 x + 1)

[uncut rapids]

after i do t = log 3 x

oh

[elder sierra]

Correct it and show me again

[uncut rapids]

wait

[elder sierra]

Solve for t

2 steps left

[uncut rapids]

t = +- 2

[elder sierra]

Replace t

[uncut rapids]

log 3 x = +- 2

[elder sierra]

Do you know how to solve that?

[uncut rapids]

yes

I got like 8 more exercises

I'll be doing them tmre

Tmrw

Thanks a lot

it took a while

to solve this

[elder sierra]

That last step does not solve the equation tho 😅

$B^{\log_{B}(x)} = x$ Use this

[grand pond]

Max Hetfield

[uncut rapids]

Edgm

Wdym

[elder sierra]

The way you solve for x

May be interpreted as handwaving by your teacher

In order to solve log_{3} x = +- 2

[elder sierra]

You need to use this property

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[tiny gate]

Don’t understand how they differentiated 😭

[fallow scarab]

they found the derivative with respect to M then set M=0

[tiny gate]

Why v’(W) and not v’(M)

I’m tweaking

[fallow scarab]

When M=0, W + (theta - 1) * M = ?

you can think of it as chain rule

$v(W + (\th - 1) M) = v(f(M)), \ f(M) = W + (\th - 1) M$

just differentiate using chain rule then plug in $M=0$

[grand pond]

riemann

riemann

[midnight plank]

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[tiny gate]

Ahh ok thanks that helps

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[vocal briar]

for something like this am i expected to be able to draw this by hand

for b

like sinx is fairly simple

but do I have to find the intersection point and whatnot

incase the functions change which one is "on top"

or is there a way to do it thats better and without calc

[midnight plank]

@vocal briar Has your question been resolved?

[distant vigil]

if you find the intercept points then you can just pick one case between those two intercept points to substitute values in

and considering both functions should be continuous whichever one is larger should be larger for all points in that interval

what this means is once you show that the two points intercept, say at 0 and 4, you could substitute any value between 0 and 4 (let's say 1) and which ever value is larger should be larger across that interval

[vocal briar]

ok ty

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[vocal briar]

is there a quick and easy way to graph this

i set them equal to each other and js got x= 1

which means they change at that point

but I can't find the upper bound or lower bound easily

by substitution

[polar star]

you can apply standard transformations to square root functions

[vocal briar]

what if i dont have it memorized 😭

[polar star]

then solve it algebraically

[vocal briar]

how

i set them equal

[polar star]

you integrate wrt y

[vocal briar]

😡😡😡

ok

[polar star]

x = 5-y^2 and x = y^2/4

now do the analysis as you would normally (ie finding bounds and integrating wrt dy)

[vocal briar]

ohh

i see

ok

thank you

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[quasi stirrup]

hiya, i need help with this exercise. i'm stuck on a, haven't done b yet.

i know that 1/f(x) will have the same domain as f(x) since f(x) is always positive. i don't know how to calculate the range though

[quasi stirrup]

(please ping me once you respond)

[amber marlin]

Any more details?

I think f(x) mustn't be 0

$\frac15 < y < \frac12$

[grand pond]

SELVATOR

[amber marlin]

$D_f: -1 < x < 6$

[grand pond]

SELVATOR

[amber marlin]

@quasi stirrup

[midnight plank]

@quasi stirrup Has your question been resolved?

[quasi stirrup]

SORRY HERE

well so this is what it says @amber marlin

and it's confusing

[amber marlin]

Alr here

if a>b, 1/a < 1/b

$2 < 3$ but $\frac12 > \frac13$

[grand pond]

SELVATOR

[amber marlin]

Bc $\frac12 = 0.5 > \frac13 = 0.33$

[grand pond]

SELVATOR

[quasi stirrup]

i get it now

also i have another one

so above, it says that the local minima of y = f(x) will be the local maxima of y = 1/f(x)

but in the example, the local minimum of f(x) is (-3, -2) however the local maximum of 1/f(x) is (-3,-1/2). why? in other examples it's identical

[quasi stirrup]

<@&286206848099549185>

[solid iris]

do u know the def of local max?

[midnight plank]

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[prime garden]

If x is any positive real number, find in terms of x the closest perfect square greater than or equal to x.

[prime garden]

Note: One can use the floor function.

Let floor(x) be represented shortly as [x].
I got this:
([root(x)] + 1)^2

This is certainly correct for non-perfect squares.

It is obvious that the closest perfect closest greater than or equal to x in this case is x itself.

So adding 1 will give an incorrect answer.
How to do this using only the floor function?

I know it can be done using the ceiling function:
(ceiling(root(x)))^2

[brittle grotto]

it also uses the "root" function 💀

[prime garden]

The question did not say "One can ONLY use the floor function".

[brittle grotto]

well yeah, after you edited it just now 💀💀💀

[prime garden]

No, I did not.

[smoky walrus]

that doesn't give you the closest
using round would be closest

[fading osprey]

i think u need either ceil or piecewise function for this

[smoky walrus]

and you can make round(x) from floor(x)

[brittle grotto]

ok can a mod get the edit log here

[prime garden]

Okay, sure.

[brittle grotto]

boom u answered ur own question

[prime garden]

Call.

[brittle grotto]

because u removed the restriction

[prime garden]

Oh.

[brittle grotto]

💀💀💀

[prime garden]

Ceiling function is disallowed.

[prime garden]

How does it not, the ceiling function?

[smoky walrus]

there's more error

try x=4.000001

ceil(sqrt(x)) = 3

oh greater than or equal, didn't see that

thought it's just closest, mb

[prime garden]

So does the ceiling function always work?

[smoky walrus]

yeah, probably

[prime garden]

There are only two cases to check: when x is a perfect square, and when it's not.
I am quite confident the ceiling function completely works.

I want to use the floor function.

[smoky walrus]

I think ceil(x) = x + mod((1-frac(x)), 1)
frac(x) is x - floor(x)
and mod you can make as well I think

oh wait the mod here is used as frac

so ceil(x) = x + frac((1-frac(x))

[brittle grotto]

ok I don't mean to be a rule bender here but, if you haven't restricted piecewise,
f(x)= x when x is a perfect square and ([root(x)] + 1)^2 otherwise
should work fine, right?

[lyric charm]

ceil(x) = -floor(-x)

[prime garden]

Damn

[lyric charm]

so one can circumvent that

or are you gonna ban this too

[prime garden]

no

[brittle grotto]

i was thinking along the lines of this, but I didn't know how floor was defined for negatives lol

[lyric charm]

floor rounds down to the nearest integer

[prime garden]

no

[lyric charm]

where down means towards negative infinity

[brittle grotto]

so piecewise is not allowed?

[prime garden]

nah lol

[lyric charm]

e.g. floor(-0.4) = -1

[brittle grotto]

oh, I see

[prime garden]

"less than or equal to x"

[brittle grotto]

"round down" still sounds ambiguous 💀

[prime garden]

-1 < -0.4

thanks all

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[lyric charm]

where down means towards negative infinity

[brittle grotto]

oh oops i can't read

sorry about that. I'm awake purely on caffeine right now

actually i am also just so ass at arithmetic these days and idk why. I had to explain some SAT problems earlier this week and did so much incorrect arithmetic

[midnight plank]

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[tame summit]

Okay I can write the first bit, but how do I write the second bit?

[lyric charm]

equivalence relationship

[tame summit]

🥳

Hi Ann

[lyric charm]

hello

[tame summit]

Sorry I keep writing stuff wrong

[lyric charm]

i mean

they ask you to show that 1 R 3 and 1 R 5 and 3 R 5 and 2 R 4 but that not a R b for a in {1,3,5} and b in {2,4}.

to put it in the dumbest way possible.

bc that's all there is to it

[tame summit]

Yes I know but

How do I write the answer

[lyric charm]

wdym

[tame summit]

Like, for equivalence I have to prove that it's reflexive, symmetric and transitive in a systematic way

What do I write for the second bit 😭

[lyric charm]

you are told a R b <=> |a-b| is even

surely verifying that e.g. 1 R 3 is a matter of plug and chug

or do you want to spend 17 pages proving that 2 is an even number

[tame summit]

I don't have to do that much but I just gotta prove it for some reason or I won't get marks 😭

Like, it's basic logic reasoning, why do I need to prove it 😭

[lyric charm]

you need to prove that 2 is an even number?

do you also need to prove that -2 is negative and to prove that |-2| = 2 and all the other shit?

like come on its just basic arithmetic at the end of the day innit

[tame summit]

Okay ykw it's fine I'll like, deal with it 🙏 I'll write something

Okay wait, question

If I prove a set to be in an equivalence relation, that means its subsets are also in equivalence relation, right??

[lyric charm]

...

what does "a set is in an equivalence relation" even mean

[tame summit]

Idfk how to word stuff in this chapter okay 😭😭😭

[tame summit]

Okay, an equivalence relation with itself

[lyric charm]

...

that

still doesnt make any sense sorry

unfortunately your inability to word things is obstructing my understanding of what you are trying to say

[tame summit]

R is an equivalence relation, are subsets of R an equivalence relation?

(idk how else to rephrase that)

[lyric charm]

no, not necessarily.

[tame summit]

Okay

Thank you

[solid iris]

it may help you to try finding a counterexample

[lyric charm]

e.g. if you had an equivalence relation R under which 4, 20 and 69 were each related to one another, but then you decided to take R' := R \ {(4,20)} then R' won't be an equivalence relation anymore

[tame summit]

Okay

[tame summit]

... I'm pretty sure I can't find a counter example for the 2nd bit

[lyric charm]

what 2nd bit

what are you actually talking about rn

[tame summit]

What are they talking about???

[lyric charm]

who where

[solid iris]

im talking about “any subset of any eq rel is an eq rel”

[tame summit]

Ohhh

Okay okay

Yes counter examples 🙏🙏

I'm losing brain cells and people are prolly losing braincells cause of me 💀🙏

[solid iris]

think positive! everyone struggles at smth

u get better with practice

ann gave u a hint on counterexample

[lyric charm]

im still confused as to what problem op is doing

or what he actually wants to say or ask about

[solid iris]

(ii)?

[midnight plank]

@tame summit Has your question been resolved?

[tame summit]

Can I write the proof like this

[tame summit]

I was trying to ask how I should write the proof

[tame summit]

For the second part of the question

The thing that I asked about equivalence and subsets was just a general doubt that I had

[lyric charm]

i guess this is good enough

i don't really have anything to add there

[tame summit]

Okay

Thanks!

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[sick compass]

If we turn U=span((-2,1,0,0),(-3,0,1,0),(-4,0,0,-1)) intoba system of linear equations, what would we get? I got x1+2x2+3x3+4x4=0

[sick compass]

In R^4

[fresh crow]

I think you might have made a mistake at the end, I get 4x4 = x1 + 2x2 + 3x3

[sick compass]

Hold on

I meant

U=span((-2,1,0,0),(-3,0,1,0),(-4,0,0,1))

Sorry for the confusion

[fresh crow]

Oh okay then you're right

[sick compass]

Awesome! Thanks a lot

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[echo niche]

P(x) = x^3 +ax^2 +bx +2 has factors x+1 and x-2. Find a and b.

[echo niche]

Honestly i forgot how to do polynomials after not doing them for 4 days lol, so help would be appreciated from anyone available. Thanks.

[viral dagger]

a polynomial function f(x) having a factor x-a would imply f(a)=0

try using that

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[echo niche]

ok

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[ionic thicket]

how do i do this? ignore my working

[ionic thicket]

i know how to do suvat

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[dense ginkgo]

3 questions i dont know how to solve.
first one i was only able to prove the first statement correct.
second one basically all i could do is use the fact that AC=CE and thus <CAE = <CEA = 80 and then idk but the answer is C
third one i did something but probably in the wrong direction. i added no extra lines tho

[dense ginkgo]

the third question is also C

wait no i added lines DC i think

which is im just guessing parallel to EB

and assumed FD=FC so <FDC = <FCD = x

and i did something to reach <BFC = 2x iirc

for the first question's second statement im guessing it has something to do with the slopes divided by eachother but idk what that is

and looks like the third statement has something to do with subtracting the x intercepts from one another

but idk what to do with that

answer for the fisrt question is D btw

<@&286206848099549185>

is this that hard or is there just no one

looks like theres no one

[midnight plank]

@dense ginkgo Has your question been resolved?

[dense ginkgo]

i will be back in 14 hours

15

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[grave kernel]

f(x) = 2^x + 2^|x|

[grave kernel]

and f:R->R

i need to comment on if its one-one or onto

so well i graphed it on the positive side, broke down the mod, its 2^(x+1), for the negative part i kinda got stuck graphing it, so i just put some values and i checked from AM GM that its >= 2

and for x = -1 its 5/2
x = -2 its 17/4

so it should be many-one and into

right?

but the answer is given one-one

even checked the graph

in desmos

its many-one

or im missing something idk

[midnight plank]

@grave kernel Has your question been resolved?

[grave kernel]

<@&286206848099549185>

[fallow scarab]

Show the answer and question

[fallow scarab]

Why do you think that's enough to prove onto?

Do you know the definition of onto R?

[grave kernel]

well its not covering the entire set R

its minimum value is 2

[fallow scarab]

.

[grave kernel]

im on laptop,
The function f: R -> R defined by f(x) = 2^x + 2^|x| is
a)One-one and into b) Many-one and into
c)One-one and onto c) Many-one and onto

this is the question

and for the answer its just written option (a)

[grave kernel]

yeah

[fallow scarab]

Show

Screenshot or picture is best

[grave kernel]

i didnt differentiate and stuff
just drew the graph

[grave kernel]

.close

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[ashen herald]

any luck, my friends and i belueve its x=42 or 83

[lyric charm]

show work

[ashen herald]

.close

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[lyric charm]

have it your way i guess...

[ashen herald]

its all good we got it

[lyric charm]

ok

[ashen herald]

thanks 😭

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[dusty portal]

Filler

[grand pond]

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

[dusty portal]

Is this valid or am I hacking

Feel like I just pulled some strings that activated nukes

I’m scared

[carmine sigil]

My understanding is these sorts of manipulations are only dangerous if the series is not absolutely convergent

[grim sierra]

i think the first step is wrong

but im not sure gimme a sec

actually reindexing maybe right

[dusty portal]

Changing the summand from k to k+1 and the index from k=1 to k=0?

Lol I feel like I just hacked

I knew there was an even more clever way than my textbook presented

[grim sierra]

wait are u sure its wrong?

[dusty portal]

So it’s right?

[dusty portal]

Not now

I fixed a computation error and it seems to be right

,w sum from k=1 to infinity k/3^k

[grand pond]

Results provided by WolframAlpha

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

Requested by exiled_hype

[carmine sigil]

Seems that way

[dusty portal]

Lol

W revelation

[carmine sigil]

Gj

[grim sierra]

it loosk right yeah

[dusty portal]

Thanks you guys

.close

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[dusty portal]

.reopen

[midnight plank]

✅

[dusty portal]

Another question

$2^{1/3}4^{1/9}8^{1/27}\ldots$

[grand pond]

;(

[dusty portal]

Is this just $\prod_{k=1}^{\infty}(2^k)^{\frac{1}{3^k}}$?

[grand pond]

;(

[dusty portal]

Which would be $2^{\sum_{k=1}^{\infty}\frac{k}{3^k}}=2^{\frac34}=\sqrt[4]{8}$.

[grand pond]

;(

[dusty portal]

Augh

The sum feels so weird to write

[hearty rune]

seems alright

[dusty portal]

And in general $\prod_{k=1}^{\infty}a^{f(k)}=a^{\sum_{k=1}^{\infty}f(k)}$, right?

[grand pond]

;(

[dusty portal]

Obviously for a convergent sum f(k) and convergent product

Whooooo

This feels so fun

[hearty rune]

yeah, just follows from exponent laws

[dusty portal]

Yippee

Alright I might give up I have to use Cesaro sums for the next exercise

I’ll close it for now

.close

[midnight plank]

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[midnight plank]

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[grizzled gust]

Could someone explain to me how (a,b) = (d)

[grizzled gust]

like what does this notation mean

I dont get how a set could equal another

[polar star]

give more context

what are you reading

[grizzled gust]

it is a number theory book

[polar star]

read the definition before the lemma

ive never done number theory but clearly they defined what (a1,a2,a3...) means

seems like a linear combination

they refer to it as an ideal

[hard umbra]

two sets are equal if they contain the same elements

the statement says that the two ideals are equal

it is the fact that in Z, all Z-linear combinations xa + yb are actually multiples of d

[midnight plank]

@grizzled gust Has your question been resolved?

[midnight plank]

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[proper imp]

is this correct

[open yarrow]

No, concave down means the points of which the graph exhibits an 'n' shape and concave up is when its a 'u' shape

[hasty haven]

this is a graph of f''(x)

[open yarrow]

Oh right, mb didn;t see that

[proper imp]

if its f”(x) i read it differently?

[open yarrow]

I think its correct

[proper imp]

thats even worse bc idk how its correct

concave up is u and concave down is n but it doesnt look like that

[hard ore]

Uh at x=0 I wouldn't say it's concave up or down

But I'm not sure

[hard ore]

Oh true lol but you're describing a different graph

But I see how it's confusing

[open yarrow]

concave up is where 2nd derivative is positive and down is the opposite (I'm pretty sure)

[hasty haven]

its like that if ure looking at a graph of f(x)

this is f''(x) so >0 means conc up and <0 means conc down

[proper imp]

isn’t like that normally?

positive = concave up

[open yarrow]

and a point of inflection on 2nd derivative is when the direction changes and y=0

[proper imp]

neg = concave down

[open yarrow]

Only for 2nd derivative - when you do the 2nd derivative you're essentially doing a trend analysis

So you just need to check for direction

The 'n' and 'u' is when its f(x) you're looking at

[proper imp]

so POI is x=a , x=0, x=e

[open yarrow]

Not x=0, give me a minute to find you a video that can explain this to you much better than I can

So for points of inflection, you need to see if it is concave down before the point, and then up after the point (or vice versa) - which is typically easier to see when they cross the x axis

However, point 0 is concave down on both sides, so it's not one

https://www.youtube.com/watch?v=8ddkoXSmzqk

https://www.youtube.com/watch?v=OhqNbQi9QPk

These might be useful

[midnight plank]

@proper imp Has your question been resolved?

[amber marlin]

(b,d) not (a,e)

[midnight plank]

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[midnight plank]

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[last slate]

Would like my answer to be checked

$$\lim_{n \to \infty} \frac{196}{n} \cot{\frac{\pi}{n}}$$

[grand pond]

Edmund Cloudsley

[last slate]

$$ = \lim_{n \to \infty} \frac{196}{n \cdot \tan{\frac{\pi}{n}}}$$
as $n \to \infty , \frac{\pi}{n} \to 0$. For asmall values of $\theta$, we can approximate $\tan \theta \approx \theta$
$$\therefore \lim_{n \to \infty} \frac{196}{\cancel{n} \cdot \frac{\pi}{\cancel{n}}} = \frac{196}{\pi}$$

[spring wave]

this is correct

[grand pond]

Edmund Cloudsley

[last slate]

alrighty! thanks so much

.close

[midnight plank]

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[midnight plank]

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[molten bay]

https://youtu.be/PNrXPtaBd_M?si=R594YMIlsJdzwrA-

[molten bay]

When i draw it desmos it looks infinity

[lyric charm]

can you show the equation you're putting into desmos

[molten bay]

(cos(sinx)-cosx)/x^4

[lyric charm]

no like screenshot the desmos thing

i think it's an input error on your part

[molten bay]

Ohh my bad

You are right

[sonic valley]

try this https://www.desmos.com/calculator/dxuk4dktio

[molten bay]

Thank you guys

.close

[midnight plank]

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[molten bay]

limit ( x ) tends to ( 0 ) ( \frac{\cos(\sin x) - \cos x}{x^4} )

[grand pond]

Andy

[lyric charm]

$\lim_{x \to 0} \frac{\cos(\sin(x))-\cos(x)}{x^4}$

[grand pond]

Ann

[molten bay]

I tried to do it with expansion

I got -x^6/6

[lyric charm]

ok, show work