help-33

[severe owl]

How did you make that square?

[hallow plover]

i used my brain

its got insane computing powers

[severe owl]

no x²-2x +1 = (x-1)²

[hallow plover]

my brain running on hopes and prayers actually

[severe owl]

x²-x+1 = (x+1)²-3x

[hallow plover]

intel pentium type shit

[severe owl]

Bruh c'mon !

I believe in you

[hallow plover]

so you wrote it like that?

[severe owl]

Yes I want you to write it like this

Where x = tg²x

3% battery left ! The phone may die anytime!

[hallow plover]

PLUG IT IN

IVE GOT MORE EXERCISES TO ASK

also

what do i do after i write it

the way you said

[severe owl]

Did you write ?

[hallow plover]

i wrote exactly what u said

[severe owl]

So (1+tan²x) = sec²x

[hallow plover]

and nothing more

[severe owl]

Can you do this ?

[hallow plover]

wow, insane

what next

[severe owl]

In deno

[hallow plover]

no

[severe owl]

We get sec⁴x + tan²x

[hallow plover]

sec^2 - 3tg^2?

[severe owl]

(Sec²x)²

Now we need to do what we have always done

[hallow plover]

cry

and scream

[severe owl]

divide by sec⁴x 😭

[hallow plover]

ah

[severe owl]

Like you get

[hallow plover]

ok

[severe owl]

Then

(sin²xcos²x)/(1-3sin²xcos²x)

So imma do another numerator adjustment here

[hallow plover]

couldnt we also just write tg^2 x as sec^2 u -1?

and then separate the thing again?

[severe owl]

From where ?

[hallow plover]

so you said that sec^2 u - 3tg^2u = sec^4?

si that what you said?

[severe owl]

No

We have (1+tan²x)² - 3 tan²x

Right ?

[hallow plover]

okk

whatever

[hallow plover]

what do we do from here

[severe owl]

So it's sec⁴x - 3tan²x

[severe owl]

Have you reached here ?

[hallow plover]

yea

[severe owl]

Woahh

[hallow plover]

i swear to god

its on sight when we meet

on sight

[severe owl]

Now sin²xcos²x -1/3 + 1/3

[severe owl]

In numerator

So we're left to integrate

[hallow plover]

this exercise should not be so complicated

i feel like there should be more normal way of solving this exercise

[severe owl]

1/3 × 1/( 1-3sin²xcos²x)

[hallow plover]

we somehow chose the longest solution known to man for this exercise

[severe owl]

😭

[severe owl]

Let's reach the answer

[hallow plover]

here you go

thats how they did it

the ones ho made this problem

[severe owl]

See

[hallow plover]

no no

you see

big bro, i wrote well over 3 pages with your solution and we still aint done yet

[severe owl]

How can someone think of multiplying that factor, without extensively solving the problem?

[hallow plover]

yea

i know

thats what i said

[severe owl]

It's better to know how to solve

[hallow plover]

i dont like their solution either

its crap

but yours is so loooooong

[severe owl]

It's ofcourse the best , I won't disagree

[hallow plover]

yea but you wont know it unless you're the one who made the problem

[severe owl]

Truly !

[hallow plover]

so while i like your solution

[severe owl]

Else someone exceptional can surely

Unfortunately I ain't !

[hallow plover]

no no

ur solution is definetely something

its just that

[severe owl]

Ok lets just complete it

[hallow plover]

the point is to find the shortest solution possible

this exam has 20 exercises like this

[severe owl]

1% remaining

[hallow plover]

and you got 3 hoursto do it

so yea

[severe owl]

So this

[hallow plover]

rest in piece dude

[severe owl]

Next we know

[hallow plover]

aha

[severe owl]

Sin²xcos²x = (sin²2x)/4

So it's just

1/ (1+ k sin²2x )

Now you know how to do them by making the deno quadratic and dividing by cos²x and that tanx = z sub

Or just by visualisation

[hallow plover]

sure

[severe owl]

It's solvedddd !

I will surely look for some easy methods ! Bye ! Cya !

[hallow plover]

ok byeee

add me on discord back

and lets talk in private

.close

[marsh citrus]

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[eternal fog]

Hiii

[eternal fog]

I need some help with this question

I'll walk through my steps

I drew this triangle

I got AB to be a+b

[elfin berry]

Arya

Arya

[tulip idol]

Similarly work on others.

[eternal fog]

We need to find Ab first tho

Is my AB vector correct?

[zenith needle]

i got something of a question like that but it's basic one idk it can work here I used herons in it

In that one of mine the values were given of CB

It was to find AC and AB and CB was given where O as perpendicular

Nevermind 💀 i haven't saw vector

[marsh citrus]

@eternal fog Has your question been resolved?

[vale venture]

If AB was equal to a+b P would need to be somewhere on AB

[marsh citrus]

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[smoky elm]

hello, im working on this problem, and I was wondering if my work looks correct thus far, if not, how can i fix it?

[grizzled linden]

lol wish i could help but i think im abt to learn sin, cos and tan next year

[spark mason]

use sin^2+cos^2=1

[sacred idol]

slight error in 2nd step

$cos 2x - (1-cos^2 x) = cos 2x - 1 + cos^2 x$

[sacred idol]

you have changed -1 to +1

[smoky elm]

okay!

now what?

[sacred idol]

i can see the first block modified

but cant understand how its still 2cos^2 x = 3cos x after that

[smoky elm]

my apologies, i wrote the corrected solution on the bottom

[marsh citrus]

@smoky elm Has your question been resolved?

[fringe tulip]

Can u get it?

[marsh citrus]

@smoky elm Has your question been resolved?

[undone hemlock]

let y = cos(x) or something like that then solve it like youd solve any other quadratic equation

once you get solutions for y replace y with cos(x) and then you use inverse cos to find x

[sacred idol]

correct, now substitute cos x = n and youll get a quadratic

like said above

[marsh citrus]

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[manic stirrup]

how does partial derivitives work on ln(xy^2z^3) for example?

[leaden monolith]

Partial derivatives are with respect to a particular variable

You just pretend everything else is a constant

[manic stirrup]

yeah im just confused to why partial derivitive for x isnt 1/y^2z^3 and is instead 1/x

maybe i forgot some rule

[stoic slate]

Log properties

[leaden monolith]

Log laws

[manic stirrup]

since its all times eachother idk how you can seperate it like that

oh?

[stoic slate]

Log(abc) = loga+logb+logc

[leaden monolith]

Well, what’s the derivative of ln(2x)

You don’t really need log laws if you don’t want to

[manic stirrup]

yeahhhh that was it

[stoic slate]

“Need” is not important

Just use what you know

[leaden monolith]

My point is that if you fail to derive this

Perhaps you would fail at deriving ln(2x) as well

Which means your issue is not with partial derivatives

Instead your misunderstanding lies just in normal differentiation

[manic stirrup]

1/2x * 2? = 1/x

[stoic slate]

(2x)

[leaden monolith]

Now replace the 2 with all the other stuff

And you end up with still just 1/x

[manic stirrup]

Yeahhh okay i understand it now. Thanks ☝️

.close

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[eternal fog]

can someone help me visualise this question

I dont understand where p is

[rotund violet]

P can be anywhere

it can be inside the triangle it can be outside the triangle, it can be on the triangle

wherever they want

[eternal fog]

oh alright

.close

[marsh citrus]

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[rotund violet]

.reopen

[marsh citrus]

✅

[rotund violet]

im not convinced this is correct

oh wait it is nevermind

.close

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[lime bison]

Hi I need to show check whether this holds $\lim\limits_{n \to \infty} \frac{n-1}{n+1} = 1$

[elfin berry]

p1za

[lime bison]

using the following:

Let $b_n$ be a null sequence and $\vert a_n - a | \leq b_n, then \lim\limits_{n \to \infty} a_n = a$

[elfin berry]

p1za

[lime bison]

Im not sure how i should approach this problem

i cant really see the connection yet between the sequence and the rule I have given

[rotund violet]

i mean what is the definition of a null sequence

[lime bison]

the limit of the sequence is 0

[tulip idol]

Let $b_n = \frac{2}{n}$, is a null sequence. Now can you work around $a_n - a$?

[elfin berry]

Arya

[lime bison]

I dont think I understand

[tulip idol]

Firstly, do you get that the b_n I gave is a null sequence?

[quiet anvil]

write out (n-1)/(n+1) - 1, and simplify this expression, what do you get?

now do you see the connection with the value given by Arya?

[lime bison]

- 2/(n+1) ?

[tulip idol]

Yes, |a_n - a| = 2/(n+1) ≤ 2/n = b_n, which is a null sequence

[elfin berry]

Arya

[lime bison]

okay so I got that this is a null sequence

but where do I get that the limit is 1?

[quiet anvil]

you used the limit = 1 when selecting a.

[lime bison]

ohhh Im so stupid

thank you, wow that really was obvious lol

[quiet anvil]

Not stupid

just learning

[lime bison]

One more thing if you dont mind:
if I need to prove that the product of a bounded sequence a_n and a null sequence b_n is another null sequence: can I do it like this?
| a_n | ≤ C
which is equivalent to
| a_n | * | b_n | ≤ C * | b_n |
and that is equivalent to
| a_n * b_n | ≤ C * | b_n |
which is just
| a_n * b_n - 0 | ≤ C * | b_n |
And now applying the theorem above
the limit of a_n * b_n is 0?

sorry for no latex

[quiet anvil]

looks good to me.

[lime bison]

Alright thanks a lot everyone, i appreciate it

going to close this now

.close

[marsh citrus]

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[slim escarp]

Hi

[slim escarp]

Do you have any shortcut for this?

My attempt

Can we ignore the root?

I mean mod

[tulip idol]

That looks like too much work. Instead, use a number line and mark points -2/3, 2/3 as points A, B.

Before A, the inequality goes: -6x ≤ 8
Between A n B, it's 4 ≤ 8
After B, it's 6x ≤ 8

Club the solutions (take intersection)

[tulip idol]

[marsh citrus]

@slim escarp Has your question been resolved?

[marsh citrus]

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[weak wind]

Can someone please help me with this question:

[weak wind]

<@&286206848099549185>

[marsh citrus]

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[true parcel]

time at faster speed + 0.5 hr = time at normal speed

[weak wind]

okay

[quaint elm]

it might be helpful to let x be the usual speed

and start writing down equations

[weak wind]

t (time) = 120/x

x = 120/t

<@&286206848099549185>

[amber birch]

great, can you now set up the other equation?

wait

yeah okay you're using x for speed I see

[weak wind]

(x+8) = (120/s) - 1/2hr

i don't know how to write the 30 minutes in corresponding to the variables, so i'll just write it like that

<@&286206848099549185>

:

<@&286206848099549185>

[marsh citrus]

Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.

[weak wind]

.close

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[amber birch]

you're super close

it's just (x + 8) = 120/(t - 1/2)

you just substitute in the new speed and the new time

new speed = old speed + 8 (mph) = x + 8
new time = old time - 0.5 (hours) = t - 0.5

[marsh citrus]

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[novel juniper]

Trying to determine if this converges.
\
\
I was thinking of the ratio test, which gives $\lim_{n \to \infty} \frac{sin(1/n+1)}{\sqrt{n+1}} \cdot \frac{\sqrt{n}}{\sin(1/n)} = 1$. It thus doesn't converge

[elfin berry]

math rocks(wai)

[severe owl]

Series expansion

Then compare with

1/(n√n)

So you get, convergent

[novel juniper]

I don't know series expansion 😭

That's the next topic

[severe owl]

sinx = x - x³/3! + x⁵/5! ...

[static quarry]

or try multiplying and dividing by 1/n and then using the comparison test

[novel juniper]

hmm

okay

that works

thanks

both of you

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[clever sigil]

are both these results true?

[fervent rampart]

yes

[clever sigil]

I know about x limit infinity=0, but not sure about x limit 0=1

[fervent rampart]

https://www.khanacademy.org/math/differential-calculus/dc-limits/dc-squeeze-theorem/v/sinx-over-x-as-x-approaches-0

[marsh citrus]

@clever sigil Has your question been resolved?

[marsh citrus]

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[orchid oracle]

Can someone explain how this integral is transformed?

[orchid oracle]

when i complete the square of the expression under the root i get
this

but now im stuck

[elfin berry]

rainy

[calm vortex]

make a substitution to put it in the form of sort u^2+1

you can think about transforming 2(x - 1/4)^2 to 7/8 u^2 then factoring out the 7/8

[orchid oracle]

hmm

can i not just factor out the 2

[calm vortex]

you could

[orchid oracle]

and multiply it by someting to cancel it out

[calm vortex]

if you do that you'd then have (x - 1/4)^2 + 7/16

[orchid oracle]

yeah

so

$\sqrt{2((x-\frac{1}{4})^2+\frac{7}{16}})$

[elfin berry]

rainy

[orchid oracle]

and now just get that 2 out of the sqrt

[calm vortex]

and then you could just ignore the sqrt(2)

then you can do x - 1/4 = sqrt(7/16) u

so that itll be sqrt(7/16 u^2 + 7/16)

[orchid oracle]

yepp!

thankss

[marsh citrus]

@orchid oracle Has your question been resolved?

[marsh citrus]

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[marsh citrus]

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[regal kite]

Why won't the sine rule give the correct answer in this example (which uses the cosine rule)

[regal kite]

I tried solving it this way:

^^ btw FA is 25

[amber birch]

angle FBA is not 24 degrees

only angle FBC is 24 degrees

[regal kite]

AAAA

but

oh wait

yea

[regal kite]

yea now it makes sense lol, thank you!!!

[amber birch]

np!

[regal kite]

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[paper hornet]

Given that the perimeter of a triangle is 2p find the lenght of the sides, that arise when we turn the triangle around one of the sides

[paper hornet]

So I have some ideas but they don't seem to work

First of all the we have to maximaze the r^2*l

Where we can transfer r*l into a Heron formula

Due to that we have a function of only 2 variable

But it doesn't look like it is working

[marsh citrus]

@paper hornet Has your question been resolved?

[marsh citrus]

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[amber spade]

I need some help with improper integrals

[amber spade]

I have this steps for solving the problem

but what do I do in this case?

if the domain was something like (a, b) I would go
lim x->b- F(x) - lim x->a+ F(x)

but I am not really sure how to approach this one

[robust wigeon]

could u explain me demoivres theorem

hlo

[lucid zenith]

why do you think the same thing doesn't work?

[amber spade]

I mean I think it works, but I am not really sure

is it something like this:

[lucid zenith]

limits at infinity are well defined

[amber spade]

lim x->inf F(x) - F(1)?

[lucid zenith]

if you want to be more precise it would be $\lim_{b\to\infty}F(b)-\lim_{a\to 1^+}F(a)$

[elfin berry]

kheerii

[amber spade]

because 1 is included in the domain is it necessarily to have the second limit?

ohh okay

[lucid zenith]

what do you mean?

[cedar stone]

+infinity is important ig

[amber spade]

so the domain of the funciton is something like [a, inf),
and I thought because of the closed side we don't really need to use the limit when applying L-N

[lucid zenith]

integrals are always defined as limits

the only reason you can write F(1) like you did above is that F(x) is a differentiable function (by definition) and so by extension it is also continuous

you usually don't have to worry about open vs closed intervals for integration

[amber spade]

ohh I see

thank you very much, that cleared my troubles

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[hallow plover]

so its not continuous

for sure

[novel juniper]

first what are the conditions needed for a function to be integrable

[hallow plover]

continuous and differentiable

and its not continuous

so its not integratable

so now, is it strictly monotone? it looks like it to me honestly

[novel juniper]

technically only continuity is needed for integrability

[hallow plover]

what?

really?

[novel juniper]

recall the definition of a Riemann integral

[hallow plover]

oh yea i dont know those

thats the only thing im left to study

about integrals

its volumes, and Riemann integrals

[severe owl]

f is subjective right ?

[hallow plover]

volumes are easy tho

[hallow plover]

why do you think that?

[severe owl]

Draw the curve of the function

[hallow plover]

? for volumes?

i dont wanna study those rn

[severe owl]

Nah

[hallow plover]

?

[severe owl]

F(x) in 2d

[hallow plover]

you are replying to the wrong messages

ok sure

you might be right

but, why isnt it strictly monotone?

when you do the derivative for each branch you get something bigger than 0, which means its strictly increasing

[severe owl]

At x = 1.1

y = 0.2

At x = 1, y = 1

[hallow plover]

why are you checking like that?

[severe owl]

Doesn't that hamper nature ?

[hallow plover]

can you not prove it ?

to check the monotony you do the derivative, dont you?

[severe owl]

,w graph f(x) = x for x in [0,1] and f(x) = 2x-2 for x in (1,2]

[elfin berry]

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

Requested by xor_11

[severe owl]

Monotonicity is checked for continuous functions

[hallow plover]

i checked it witht he variation table

and, it looks nonotonous to me

[severe owl]

f(x)'s continuity is compromised after x = 1, it remains a monotonically increasing function but you see there x's value decreases

[hallow plover]

i dont see the graph

but damn

i didnt know that a function isnt monotonous if its not continuous

makes sense tho

i see

[severe owl]

Yes !

[hallow plover]

so now, we are only left with surjectivity and injectivity

so how would you check for that analitically

without using the graph?

to be surjective means that its Im f = [0,2]

but since it not continuous at 1, would it not mean that its not surjective?

[severe owl]

what's the definition of injectivity ?

[hallow plover]

that the function is strictly increasing or decreasing

[severe owl]

f(x1) = f(x2), implies x1 = x2

[hallow plover]

you can check like that too

[hallow plover]

but wouldnt it work my way too?

[severe owl]

That's the way of checking 😭

[hallow plover]

theres 2 ways of checking for injectivity and surjectivity

[severe owl]

I just founded y = 0.2 for x = 1.1

[hallow plover]

you can check it with or without derivatives

[severe owl]

And also y = 0.2 for x = 0.2

[hallow plover]

or looking at the graph i guess, but the point is to do it by myself

[severe owl]

It's clearly not bijective !

[hallow plover]

and im not drawing graphs

okk so wait

its injective right?

[severe owl]

Graphs are the easiest way to visualise

[severe owl]

It's not injective !

Injective means one to one mapping

[hallow plover]

ooh, so it would mean that x1= 2x2 - 2?

and thats not true

[hallow plover]

so you do it this way, right?

[severe owl]

For a single y you will get a single x !

[hallow plover]

i see, so like this?

[severe owl]

A counter example works in most of the cases, but you can generalise !

[hallow plover]

ok, so im right or wrong?

do you check like that or nah?

[severe owl]

I don't !

[hallow plover]

ok, you dont, but could you do it like that anyway?

like, is it correct?

[severe owl]

The overlapping region of y, for the graph would provide, why it isn't injective !

[hallow plover]

ok bud, i agree, its not injective, is this the correct way of checking that its not injective?

[severe owl]

You can write statements like for x € (1,1.5] we get range € (0,1) which is also achieved for x € (0,1]

[severe owl]

You can't say this for a discontinuous function! You will need to define the domain of x in each case because that changes the definition. An analysis is best

[hallow plover]

right

ok

.close

[marsh citrus]

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[marsh citrus]

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[lime bison]

If I need to show that the sequence n-1 has no upper bound is it ok if I assume that it has an upper bound C that is a natural number and then contradict my initial statement by showing that (C+2)-1 >= C. Im asking because usually the upper bound is defined as a real number

[devout mauve]

no you cant immediately assume the upper bound is a natural number

however, if you have an upper bound which is non-natural, can you find a natural number which is even bigger?

[lime bison]

well yes but im not sure how to express that mathematically, i have seen these ceil symbols but we never used them in uni so Im not sure if we are allowed to use them

[devout mauve]

do you know the archimedean property?

[lime bison]

yes, natural numbers are unbounded

i thought about this too, not sure how to apply it tho

[lime bison]

hmm i also kinda dont understand why i cant do this if we are not talking about the minimal upper bound

[devout mauve]

ok

so you do know that for every real number there is a bigger natural number

so if your upper bound is some real number, you can take a bigger natural number

which will again be an upper bound

and then you can do the rest of the proof

[lime bison]

yeah so why cant I immediately assume that the upper bound is a natural number?

since there always is a bigger natural number

like if it has an upper bound that is a real number it also has an upper bound that is a natural number no?

[devout mauve]

if you have to reason why you can assume it then thats a good indicator for why you cant immediately assume it, no?

at this level you do have to make this argument

[marsh citrus]

@lime bison Has your question been resolved?

[lime bison]

Yeah i guess thats true

is there any formal way of saying "take the next bigger natural number"?

[marsh citrus]

Channel closed

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[lime bison]

.reopen

[marsh citrus]

✅

@lime bison Has your question been resolved?

[devout mauve]

showing that the "next" bigger natural number exists needs a slightly longer argument

if x is given, then the set {n in N: n > x} is nonempty by the archimedian property

so then by the well ordering principle it has a least element

you can take that

although for your proof you really only need some natural number bigger than x

[marsh citrus]

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[lime bison]

.reopen how do I express that then

[marsh citrus]

✅

[lime bison]

and do I also need to prove that?

can we really not use the archimedean property to my advantage here?

[devout mauve]

wdym

I used it

I gave you the proof

[lime bison]

well we did none of that in class the thing is all we did is we said that the archimedean property tells us that N is unbounded

like is there no easier way? I dont think this type of proof is expected

or maybe a different idea to prove that n-1 is unbounded?

[marsh citrus]

@lime bison Has your question been resolved?

[devout mauve]

n-1 is just N but in a trenchcoat

but I assume you arent allowed to just say "obviously"

you started this channel with asking whether you can assume that your upper bound is a natural number

its not like you actually need that

[lime bison]

why tho?

my approach was to use C+2 as an index for the sequence

but the index for a sequence needs to be a natural number obv

[devout mauve]

assume the sequence is bounded, so there is some upper bound C

then there exists a natural number bigger than it because N is unbounded

and then use that

I mean thats not a crazy proof

there is no way around using that N is unbounded

(my point is that you dont explicitly need that C itself is natural)

[lime bison]

yeah i guess thats pretty straight forward

thank you for time, i really appreciate it!

going to close this now

.close

[marsh citrus]

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[marsh citrus]

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[stark trail]

To do this you can just create a counter-example

What have you tried

[summer grove]

!status

[marsh citrus]

What step are you on?
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3. I got an answer but I was told that it's wrong.
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7. None of the above

[split vigil]

This is where in at rn

I tried proof by contradiction

[stark trail]

contradiction is probably not the best way to do this

if you can simply find 3 vectors a, b, c where the equality is not true, you're done

[split vigil]

uh

[stark trail]

For example if I said

Prove that span(a,b) != span(a)

you could say

let a=(1,0), b=(0,1) then span(a)=span((1,0)) clearly does not equal span(a,b)=R^2

and hence it is disproven

[devout mauve]

they have to prove that it never holds

[stark trail]

oh

[devout mauve]

the first point of your argument actually showed the opposite of what you wrote down

it showed that a+b, b-c, c+a are in span(a,b,c), so that span(a+b,...) is a subset of span(a,b,c)

now you need to show its a proper subset

[split vigil]

isnt this my second point

[devout mauve]

no

you bungled the argument

[marsh citrus]

@split vigil Has your question been resolved?

[split vigil]

Can you enlighten me

[devout mauve]

well ok I chose bad words

both of your points show the same thing

namely that span(a+b,...) is a subset of span(a,b,c)

[split vigil]

oh i see

do i have to get rid of 1 or 2

[devout mauve]

whichever you want

[split vigil]

1 was simpler

ill get rid of 2

how can i continue?

[devout mauve]

you need to show that the other subset relation doesnt hold

for example by finding a vector which is in one of the spans but not the other

austin mentioned doing an example

that might help with finding which vector to choose

[split vigil]

uh

can i say for the scalars lambda1 = -1

lambda2 = 1

and lambda3 = 1

the vector [-1, 1, 1]T adds up to 0

therefore there exists a non trivial solution

[marsh citrus]

@split vigil Has your question been resolved?

[marsh citrus]

@split vigil Has your question been resolved?

[marsh citrus]

@split vigil Has your question been resolved?

[split vigil]

.close

[marsh citrus]

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[marsh citrus]

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[buoyant jetty]

Ivan and Adeline are in a classroom with a chalkboard. They are standing on different halves of the board, and on each half, the number $2$ is written. When Ivan's teacher gives a signal, Ivan multiplies the number on his side of the board by $-2$ and writes the answer on the board, erasing the number he started with. Adeline does the same on each signal, except that she multiplies by $2$. The teacher gives $10$ signals in total. How many times (including the initial number) do Ivan and Adeline have the same number written on the board?

[elfin berry]

938c2cc0dcc05f2b68c4287040cfcf71

[plush jolt]

Hint: ||the absolute value of both numbers are always equal.||
additional hint: ||since the multipliers (-2 and 2) are opposite, you can think of this as an alternating sequence||

[buoyant jetty]

wdym? can you elaborate the idea?

[tribal tree]

the fact that they both have 2 is actually irrelevant

but think about it like this

how about we go through each time the teacher gives the signal

what would Ivan and Adelines number be on the first signal

[buoyant jetty]

4

-4

[tribal tree]

yes

how about on the second signal

[buoyant jetty]

8 -8

[tribal tree]

no

multiply 4 by 2 and -4 by -2

[buoyant jetty]

4x2 = 8
-4 x -2 = 8

[tribal tree]

yep

and now they have the same number

alright how about on the third signal

[buoyant jetty]

16
-16

[tribal tree]

and the fourth?

[buoyant jetty]

32
32

[tribal tree]

yep

do you see the pattern?

[buoyant jetty]

nope

[tribal tree]

well look

every other signal

they have the same number

so initially they had the same

the first signal they didnt

the second signal they did

third they didnt

fourth they did

so every "even" signal they will have the same value

[buoyant jetty]

1 - no
2 yes
3 no
4 yes

[tribal tree]

yep

and itll continue like this

[buoyant jetty]

10 is even

[tribal tree]

so how many times (including the beginning) did they have the same value

[buoyant jetty]

5

[tribal tree]

close

[buoyant jetty]

2 4 6 8 10

[tribal tree]

yes but count the beginning too

count 0

0 2 4 6 8 10

[buoyant jetty]

dammit

[tribal tree]

its ok

it makes sense tho doesnt it

[buoyant jetty]

ye

[tribal tree]

we need to wait for the negatives to cancel for them to have the same number on the board

[buoyant jetty]

I appreciate it

yes

thank you very much

. solved

.solved

[marsh citrus]

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[marsh citrus]

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[opal valve]

Is it possible to do this question with the box method?

[opal valve]

Or do i HAVE to do long division

cs that TLmaths guy on yt said the box method works for everything

[main idol]

Show what the box method is

[opal valve]

Okay

1 sec

[neat quiver]

(x^2-x+2)(4x^2+4x-3)

i think

im in bed rn so i just thoguht abt it

no need to divide u can just figure out the coefficients by thinking abt it

[opal valve]

Wdym

[neat quiver]

ok so you have (x^2-x+2)(?x^2 + ?x + ?) = x^4+ax^2+11x+b

they ? r the coefficient u need to figure out

so u know the first one is 4 since x^2 * 4x^2 =4x^4

however for the next one u notice there is no x^3 term

we are at (x^2-x+2)(4x^2 + ?x + ?) = x^4+ax^2+11x+b and only an x^2 * x can prduce an x^3 term

so u have -4x^3 which we know, and then x^2*?x = 4x^3 since they x^3 term need to cancel out

so now u are at (x^2-x+2)(4x^2 + 4x + ?)

and u need 11x

we have 2*4x = 8x and we need 3x

so -x * ?=3x, ? = -3

so the factorization is (x^2-x+2)(4x^2+4x-3)

then expand to find a and b

can someone verify

[opal valve]

Why did the 4x^3 turn negative here

[neat quiver]

(x^2-x+2)(4x^2 + ?x + ?) cuz u multiply -x*4x^2, 2nd term of first polynomial multiply 1st term of second polynomial

[opal valve]

OHHHH SHITTTT THATS SO SMART BRU

The train of thought felt almost identical to doing the box method except there was no box

Thats so cool

[neat quiver]

do u get it now

[opal valve]

Yeahhh

[neat quiver]

ok im gonna go back to gooning

bye

[opal valve]

goodluck on ur gooning

Ty

.close

[neat quiver]

yw

[marsh citrus]

Channel closed

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[neat quiver]

oh yeah it basaically is box method

just use that

i couldnt do it since i was in bed

but box method makes it simpler and more organized

[marsh citrus]

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[buoyant jetty]

How many two-digit prime numbers can be formed by choosing two different digits from the set ${2, 7, 8, 9}$ to be used as the tens digit and units digit?

[elfin berry]

938c2cc0dcc05f2b68c4287040cfcf71

[novel juniper]

I don't think there's any nice way to do this

you have to list out the two digit primes and then compare

[twilit geyser]

I think you can avoid some bad cases

if the digits are the same, it won't work - why?

the last digit can't be 2 or 8 - why?

[buoyant jetty]

if the digits are the same is div 11?

[twilit geyser]

yup

[buoyant jetty]

unit digit can't be 2, 8 because then is div 2

29
89
97
79

okay that was awkward

[rotund violet]

yeah those are the 4

[buoyant jetty]

yeah i keep wondering wtf is this exercise

[rotund violet]

if you are allowed to flip numbers upside down you can get 67 also

it just said you had to use them

it didn't say that you couldn't do something to the digits

[buoyant jetty]

LOL

.close

[marsh citrus]

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[marsh citrus]

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[glass locust]

Function is continuous and differentiable everywhere. F'(x)>=8 . F(3) is -2. What can we say about f(5)

[vernal mantle]

Did you mean to capitalise F

[glass locust]

f(x)

[vernal mantle]

Ah alright

Welp I'd say then that f(5) would be positive

I mean it's not a very specific question

[meager badger]

Wpuls it also be at a higher y value?

Cause the function is always increasing

[vernal mantle]

I mean yea

Uh so the derivative is always greater than or equal to 8

So the lowest f(5) could be is if the function was a straight line with gradient 8 from x=3 to x=5

From there you can find the bounds of the value f(5) can actually be

Which I think is that they want

@glass locust

[marsh citrus]

@glass locust Has your question been resolved?

[bleak thistle]

apply MVT?

[glass locust]

I didn't understand MVT :((

[glass locust]

Oh alright cool thanks!

Is this what MVT is?

[vernal mantle]

Mean value theorem

[glass locust]

Yeah what you just said

Is that mean value theorem

Or is that just common sense

[vernal mantle]

I just use common sense I don't really know what mean value theorem is

[glass locust]

Oh alright, this was quite comfortable of a logic. Thanks!

[glass locust]

Could you just introduce MVT please

[marsh citrus]

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[restive loom]

hey trignometry doubt here:
we r given that
tanΘ + cotΘ = (3)^1/2
to find:
tan^6(Θ)+cot^6(Θ)

[fathom ridge]

I think this is more algebra than trig

[restive loom]

Idk

I tried cubing and squaring

[fathom ridge]

that should work

[restive loom]

I somehow ended up with tan theta + cot theta = 0

[severe owl]

x+1/x = √3

[amber birch]

yeah it's basically algebra because you have x + 1/x = sqrt3 and you want to find x^6 + 1/x^6

[amber birch]

if you don't mind finding x^4 + 1/x^4 first

[fathom ridge]

you don't need to though

[amber birch]

then (x^4 + 1/x^4)(x^2 + 1/x^2) = x^6 + 1/x^6 + x^2 + 1/x^2

if you want to avoid cubing

[fathom ridge]

yeah fair

[severe owl]

x²+1/x² = 1

[restive loom]

Guys what abt the root 3=0

[fathom ridge]

,w solve x + 1/x = 1

[elfin berry]

Results provided by WolframAlpha

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

Requested by neonperseus

[fathom ridge]

well tan and cot here take imaginary values

the trig is irrelevant they just didn't want to stir your brain by using x and 1/x

[amber birch]

if you know complex numbers gg

[restive loom]

K thanks

[marsh citrus]

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[final flume]

!status

[marsh citrus]

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.
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7. None of the above

[final flume]

2

im almost there but can't figure it out

this would be the figure

the rhombus side is the radius which is 8

as in the question

so one diagonal would be 8

wait bruhhhh

i think i found it

!status

[marsh citrus]

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

[final flume]

4

this would lead into 2 equilaterals

[final flume]

and altitude of them is root 3/2 * side

and that would be root 3/2*8 which is 4root 3

2 of those

so 8 root3

?

@severe owl

[severe owl]

Pythagoras?

[final flume]

no

i mean u could do that

but this is easier

[severe owl]

Surely !

[final flume]

lemme go to the answer of this question once

to confirm

[severe owl]

Anything works in geometry problems !

[final flume]

yess it's 8 root 3

i solved my own question silly me

thanks anyways

.close

[marsh citrus]

Channel closed

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[final flume]

.reopen

[marsh citrus]

✅

[final flume]

i have another question but it's algebra

thought it'd be a waste of tickets

so posting it here

@severe owl can u help in this one

!status

[marsh citrus]

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

[final flume]

1

[severe owl]

Try multiplying with the conjugate of deno

[final flume]

so rationalize

ok

[severe owl]

49-48 = 1 in deno

[final flume]

yeah 49 - 16*3

in the top that becomes

lemme calculate

[severe owl]

Surely

[final flume]

229+ 132 root 3?

lemme plug that in

cos it matches a+b root 3

[severe owl]

Is this what you get in the numerator?

[final flume]

ye

a+b should be 361 lemme check

[severe owl]

This is under √

[final flume]

oh wait

yeah

[severe owl]

132

So what are the factors of 132 ?

[final flume]

2, 3, 4, 6, 11, 12, 22, 33, 44, 66

[severe owl]

Ok so 2×66, 11×12, 6*22

Oh wait

Huhh !

,calc 132/2

[elfin berry]

Result:

66

[severe owl]

We need factors of 66

[final flume]

66*2

11

,22

33

2, 3, 4, 6, 11, 12, 22, 33, 44, 66

[severe owl]

Are you sure it was 229 ?

[final flume]

ye

check on wolfram as well

[severe owl]

,calc 179 + 323

[elfin berry]

Result:

249

[severe owl]

This gives me 249 though !!

[final flume]

323??

[severe owl]

32*3

[final flume]

expansion of (19+8 root 3)(7+4 root 3) becomes 133 + 76 root 3 + 56 root 3 + 96

<@&286206848099549185>

[severe owl]

It's 153

Oh soryy

[final flume]

19*7 is 133

[severe owl]

Wth am I doing

[final flume]

happens

[severe owl]

17*9 !

So 66 = 6*11 (121 + 108) (11² + 6√3²)

[final flume]

sure

[severe owl]

I'm extremely sorry, I really messed up 2 grade calculations !

So done ?

[final flume]

no

idk the answer

nor am i close to it

[severe owl]

You re

11² + 2×11×6√3 + (6√3)²

[severe owl]

a²+2ab+b²

[final flume]

ohh

so 11 + 6 root 3

answer's 17 in the ak

and 11+6 is 17

nice

ty

[severe owl]

I messed up real bad

[final flume]

it's fine

[severe owl]

Sorry again !

[final flume]

i've done worse

np

ty

.close

[marsh citrus]

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[marsh citrus]

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[rugged pendant]

Hello

[rugged pendant]

can someone help me with this (integration with variable shift)

I've seen many examples but I still can't understand it fully

\

[red nimbus]

can u highlight or explain the part you dont understand?

[marsh citrus]

@rugged pendant Has your question been resolved?

[rugged pendant]

so, we write t = \sqrt{u}

but then i don't understand the du part

because we multiply by 2t instead of the derivative of sqrt u

[red nimbus]

Because sqrt(u) = t

So dt = du/2sqrt(u) = du/2t

[rugged pendant]

but why do we multiply by 2t and not 1/2t in this case?

[red nimbus]

you wanna solve for du

you wanna rewrite the same integral in terms of t

thats the whole point

[rugged pendant]

sorry i still don't get vit.

$\frac{dt}{du} = \frac{1}{2\sqrt{u}} \iff dt = \frac{du}{2\sqrt{u}}$

[elfin berry]

Badr

[rugged pendant]

how to go from this to 2t? what am i doing wrong?

[elfin berry]

𝔸dωn𝓲²s

[red nimbus]

remember that t = sqrt(u) in the first place so they substituted it right away thus multiplying by 2t insteaf 2sqrt(u) but they are both the same literally

[rugged pendant]

ooh

so we need to get du not dt

thanks!

[red nimbus]

yes

[amber birch]

.close

[marsh citrus]

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[wary yew]

Here to derive the equation (1.5) they have taken u as dependent variable on x and y right?

[hazy lion]

yes

[marsh citrus]

@wary yew Has your question been resolved?

[wary yew]

so if I do partial derivative of phi_1 with respect to x, we get 1+ u_x right?

@hazy lion

[marsh citrus]

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[wary yew]

.reopen

[marsh citrus]

✅

[wary yew]

,rotate

,rotate

,rotate

,rotate

[wary yew]

I don’t understand how they got the partial derivatives as 2x, -2y

because u depends on x and y

[marsh citrus]

@wary yew Has your question been resolved?

[hazy lion]

yea what the heck

are they suddenly using u as a variable again

[marsh citrus]

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[wary yew]

.reopen

[marsh citrus]

✅

[wary yew]

I don’t really know what is going on,

[main idol]

Can also try #odes-and-pdes

Or even #multivariable-calculus

[wary yew]

Sure

[marsh citrus]

@wary yew Has your question been resolved?

[marsh citrus]

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[upbeat hound]

$x^2(a-1)-9a+9$

again, can someone please explain how we factor from here

[elfin berry]

Simon James B

[spark otter]

maybe factor -9a + 9

and then it'll become more obvious

[crystal lagoon]

Other channel closed lol

[upbeat hound]

what sing do we consider

[crystal lagoon]

you factor out a-1

[upbeat hound]

sign

when we factor 9 do we say -9 or +9

[spark otter]

if you factor by -9 you gotta remember to swap the sign of every thing inside

[crystal lagoon]

in this case -9 because then the inside is (a-1)

[upbeat hound]

and if we do +9 they remain the same

[spark otter]

try -9 to see if you get it

[upbeat hound]

$x^2(a-1)-9(a-1)$

[elfin berry]

Simon James B

[upbeat hound]

$(a-1)(x^2-9)$?

[elfin berry]

Simon James B

[crystal lagoon]

yes but you're not done yet

[upbeat hound]

i see it

[spark otter]

yep

[upbeat hound]

the difference of squares

right?

[spark otter]

yep

[upbeat hound]

$(a-1)(x-3)(x+3)$

[elfin berry]

Simon James B

[crystal lagoon]

✅

[upbeat hound]

it's not hard at all but i need to practice to see them well

<33

.close

[marsh citrus]

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[earnest yarrow]

How do I go about solving this?

[fervent rampart]

https://tutorial.math.lamar.edu/Classes/CalcII/SurfaceArea.aspx

[marsh citrus]

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[marsh citrus]

@calm robin Has your question been resolved?

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[calm robin]

Hi, so I'm stuck figuring out something with the variables I've gotten