help-13

[stray crypt]

since i do know the book has it

[sacred anchor]

gng just say binomial theorem 😭

[native heath]

its not just binom tho its generalised

falling factorial and all that

[stray crypt]

@cobalt iron

i get -5/2

[cobalt iron]

u rationalise wrong

numerator will be 5x-6

[stray crypt]

right my bad

thx

[native heath]

calling this rationalisation is insane

[cobalt iron]

ahh man this is how calculus is done

yes u can use multiple expansions but if i give u some ques or that kind u can also wont able to do by expanions

[stray crypt]

u indian?

[cobalt iron]

yes

[stray crypt]

cool

[cobalt iron]

😂

[stray crypt]

im just a newbie learning for a test soon

both ideas @native heath @cobalt iron are cool but ill go with the more simple tricks for now

anyways thx yall ❤️

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[sleek agate]

Can someone help me get past this point and actually understand it? My teacher had to correct me like 8 times before this

[sleek agate]

???

[sullen cipher]

The 2nd term is missing an h

[sleek agate]

Ohh

[sullen cipher]

It's (z+h)z
Not z+h(z)

[sleek agate]

Ok

Now i have - zh/h

[sullen cipher]

$\frac{-h}{z(z+h)}$

[wraith dagger]

Victimizer

[sullen cipher]

U hv this now?

[sleek agate]

No

[sullen cipher]

U r still missing the z after h

[sleek agate]

I have now

I see where i messed up

[sullen cipher]

And when u do the subtraction
The numerator os supposed to be z-(z+h)=z-z-h

[sleek agate]

Ye

[sullen cipher]

Write it in this way

[sleek agate]

What to do now?

Ok i will fix it

[sullen cipher]

Didnt they ask u to js express it in common denominator form

Oh I see it

[sleek agate]

Well it says find common denominator and simplify so i assume that’s it but my professor wasn’t clear

[sullen cipher]

U had am initial 1/h

[sleek agate]

Yea

I can multiply

[sullen cipher]

Yes

What do u get

[sleek agate]

-h/(z^2+hz)h

Do i distribute?

[sullen cipher]

The h cancels out

[sullen cipher]

Not necessarily

[sleek agate]

How cancel out. One os positive the other os negative

It isnt the same h

[sullen cipher]

Js cancel out the h

The negative sign remains

[sleek agate]

Oh so like -1

-1/z^2+hz

[sullen cipher]

Yes to repesent it neatly, u can express it as
$\frac{-1}{z(z+h)}$

[wraith dagger]

Victimizer

[sleek agate]

Oka

Hms

It looks good to me

[sullen cipher]

Yea simplified

[sleek agate]

Ya

Ty!

Imma leave this open because I’m working on math still

[sullen cipher]

Np

[sleek agate]

Can someone check my paper to make sure im right? Btw its a worksheet

[cedar kiln]

@sleek agate Has your question been resolved?

[sleek agate]

<@&286206848099549185>

[cedar kiln]

@sleek agate Has your question been resolved?

[ancient lodge]

1. right answer but this is very all over the place

Continuity requires $f(0)=\lim_{x \to 0} f(x) \implies b=\lim_{x \to 0} (x+3)=0+3=3$

[wraith dagger]

Civil Service Pigeon

[ancient lodge]

rounded answers are fine I assume?

you have the right idea but this can be written a lot more nicely

We have that $f(1)=2$ and $f(2)=10$. Because $f$ is a polynomial, it is continuous for all real numbers, and is thus also continuous on $[1,2]$. Therefore, by the intermediate value theorem, $f$ takes on every value between $f(1)=2$ and $f(2)=10$ at least once on the interval $[1,2]$. Since $9$ is between $2$ and $10$, the claim is proven.

[wraith dagger]

Civil Service Pigeon

[ancient lodge]

no

coterminal angles does not apply here

honestly just leave it all in radians

-pi/2 was fine

[sleek agate]

Oh

[sleek agate]

I dont understands

[ancient lodge]

coterminal angles represent the same position on the unit circle, but they are not the same number.

here, you care about the number

so just keep it as such

[sleek agate]

Oh

Thanks!

[ancient lodge]

[cedar kiln]

If you are done with this channel, please mark your problem as solved by typing .close

[sleek agate]

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[pearl glen]

Hi! I'm new here, I hope I got the question protocol right.
I have a pedantic notation question about "similar triangles" . Included is a screenshot of the answer key to a homework question.
Does the order of letters designating an angle matter? I am aware the three letters indicate what part of a figure make up the angle, but is $\angle CAB$ equivalent to/equally correct as $\angle BAC$ ?

[native heath]

yes

[wraith dagger]

Raymanovich

[pearl glen]

Cool, Thanks! Could I reasonably expect both to be scored the same on a test? If that's a fair question to ask

[native heath]

yes there's literally no difference

it's just 'the angle between sides ca and ba'

[pearl glen]

I guess I was confused, because the textbook stated the order matters specifically when stating similarity, and this was one of the steps getting there

but that's a separate case than

[native heath]

thats a triangle!

[pearl glen]

right, different case

Thank you!

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[fallen hollow]

How to do 57

I got a diff ans

[tropic oxide]

show work and answer

[fallen hollow]

I did it very rough

1/33 and -1

😭

[ember heath]

🤔

what did u do

[ember heath]

if u cant show atleast tell us

[fallen hollow]

Um I tried getting the product of solutions which is basically inverse of a:c from the two eqn

[ember heath]

yes

how did u try

[fallen hollow]

Got 33 and -1

Tho

No option matches

[ember heath]

how did u try

[digital bane]

do you have the full working? just show that

easier than trying to explain your thought process

[ember heath]

they did it in "rough"

which is kinda fair cause i do that sometimes too

[fallen hollow]

Used binomial to expand (alpha + beta)^4

And simplified till I only had

the first eqn LHS (power 4 one) and product of solutions

So I solved for product by supposing it as x but got two solutions: 33 and -1

Is the question's options wrong or did I do an error 😭

@ember heath

[ember heath]

u have alpha + beta = 4

square that first

[fallen hollow]

Yes.

I got two values of alpha * beta from the eqns: 33 and -1

Can you check if this is correct?

[ember heath]

its wrong

[ember heath]

do this

[fallen hollow]

Ye

Then?

[ember heath]

then tell me what u get

[fallen hollow]

Bruh

its too long to typee mann, this is easy step

Go on

[ember heath]

ok

we want to square again but squaring with 3 terms is tough

so take 2ab to the otherside and then square

[fallen hollow]

Bruh, I alr did all of this

I straight up

Used binomial theorem

Its the same thing

😭

What value(s) did you get of alpha * beta?

[fallen hollow]

.

<@&286206848099549185>

[ember heath]

then if u didnt make a calculation mistake ur ans should be correct

[fallen hollow]

can u check?

.close

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[storm inlet]

hello

[storm inlet]

help needed

[cedar kiln]

No need to ask “Can I ask…?” or “Does anyone know about…?”—it’s faster for everyone if you just ask your question! See https://dontasktoask.com/

[storm inlet]

Description
Create, graph and post a polynomial function with the following features:

a minimum of Degree 6,
a minimum of 3 different x-intercepts, and
at least 1 factor with each type of multiplicity (1, 2, and 3)
In a few sentences, explain your mathematical process and include a detailed sketch of your graph. Your graph must be hand sketched. No online graphing tools will be accepted. Marks will not given for computer generated sketches.

i have this so far f(x)=x6−x5−19x4−7x3−2x2−12x+360

i need someones help making the graph

[iron wren]

Does each type of multiplicity mean there has to be a root repeated 3 times, a root repeated twice and a root by itself

[tropic oxide]

first off use ^ for exponents, so x^6 - x^5 etc.

[digital bane]

try desmos or geogebra?
ignore this

[tropic oxide]

and second, can you show us what your thing looks like in factored form

cause surely you cooked it up in factored form, yes?

[digital bane]

oh wait hand-sketched, sorry

[tropic oxide]

@storm inlet

[storm inlet]

.close

[cedar kiln]

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[tropic oxide]

eh?

i mean, ok.

[storm inlet]

sorry for wasting yalls time

i just work better with 1 on 1

i need help grpahing thhis

can only one person help me

@tropic oxide

.reopen

[cedar kiln]

✅

[tropic oxide]

well my question's gonna remain the same

can you show the factorized form of this?

[storm inlet]

1 sec

[tropic oxide]

given the requirements, i think it would be quite natural for you to have started with it.

[storm inlet]

f(x) = (x - 2)(x - 5)(x + 3)^2(x^2 + 4)

[tropic oxide]

at least 1 factor with each type of multiplicity (1, 2, and 3)
seems like you're missing a factor of mult 3.

so yours doesnt really fit the defn.

err

rather your thing doesn't fit that requirement\

keeping the min degree as 6, you'll want to do something like $$f(x) = (x-a)(x-b)^2(x-c)^3,$$ where $a$, $b$ and $c$ are free for you to pick (but must be different from each other).

[wraith dagger]

Ann

[tropic oxide]

then for graphing, following the form i just suggested, you mark off the x-ints at x=a, x=b and x=c, and sketch out a graph crossing the x-axis for a, touching for b, and crossing-but-like-a-cubic for c.

[storm inlet]

this better f(x) = (x + 2)(x - 1)^2(x - 3)^3

[tropic oxide]

and then try to join the pieces up as best you can. perhaps also find the turning points (though i would not envy anyone who would have to solve the quintic eq f'(x)=0, so maybe take some computer assistance just for that... or wing it)

[tropic oxide]

yeah wonderful

[storm inlet]

can u grpah on desmos for me

i do not know how to use demos at all

never used it

[tropic oxide]

you can literally copy-paste

f(x) = (x + 2)(x - 1)^2(x - 3)^3
into the desmos calculator

[storm inlet]

oh ok

[tropic oxide]

graphing an equation in desmos is as easy as simply typing it in

[storm inlet]

this shows up

[tropic oxide]

well don't put []

or erase those

[storm inlet]

i copied exactly what u said

[tropic oxide]

and then write: f(x) = (x + 2)(x - 1)^2(x - 3)^3

f(x) = (x + 2)(x - 1)^2(x - 3)^3

i really dont know how else to explain it to you, you know how to copy-paste right?

[storm inlet]

ye

i dont understand

what am i doing wrong

this right

[tropic oxide]

that looks like a fine graph.

try shift+clicking while your cursor is near the y-axis to adjust the axis scale

you can get a graph something like this

it dips down quite a lot and the gradient at its x-intercept of -2 is quite steep

[storm inlet]

so is this good?

[tropic oxide]

i mean it's computer-generated so you can't put it in your submission

[storm inlet]

i know that

im saying i can draw his right

this*

[tropic oxide]

sure you can.

i would maybe make it purposefully not to scale a bit

[storm inlet]

can u make the final one and sned it to me

[tropic oxide]

no

!noans

[cedar kiln]

The purpose of this server is to help you learn, not to hand out answers. Do not ask someone to give you the answer directly.

[tropic oxide]

what i mean though is that you should make this bit look bigger for illustration purposes

[storm inlet]

.close

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[neon hinge]

so.. i got an old exam paper from my senior as well as his solution and im wondering for this question can i just do what i did instead of using squeeze theorem?

[tropic oxide]

you can't speak of $\sin(\infty)$ as if it's a thing that exists, no.

[wraith dagger]

Ann

[tropic oxide]

you are still essentially attempting to apply the squeeze theorem (under the most charitable interpretation of your work) but you are being super mega wonky about it.

[neon hinge]

so even if ik that sin(inf) is within [-1,1] and ik that no matter what value it is within that set when i multiply it by 0 it will return 0 i still cant assume that 0*sin(inf) = 0?

[tropic oxide]

sin(∞) does not exist

stop talking about a thing that does not exist as if it does

[neon hinge]

mmmm alr thx a lot for answering

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[exotic zenith]

How do you do d

[iron wren]

bring y-3 to the other side by multiplying both sides

[exotic zenith]

Im at this step

@iron wren

[iron wren]

do you need to get y by itself

[exotic zenith]

Yea

Im trying todo that

Y=

[iron wren]

its a quadratic

[exotic zenith]

So what am i supposed to do

[iron wren]

id just use the quadratic formula on y^2-6y-t^2/2+C=0

[exotic zenith]

So whats C

Since quafractic is abc

Not abcd

Theres 4 of them

[iron wren]

t^2/2+C

is the constant

i mean -t^2/2 + C

[exotic zenith]

So thats one whole thing?

[subtle harbor]

it's a quadratic in terms of y, you want to solve for y, so you treat anything that doesn't involve y as a constant

alternatively you could complete the square

probably much easier

i.e. you have y^2-6y=t^2/2+C, you can add 9 to both sides, 9 gets absorbed into C

y^2-6y+9=(t^2/2)+C

then you can easily factor LHS

[cedar kiln]

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[neon hinge]

Is this correct?

[neon hinge]

Ye mb

I accidently send him into the underworld

Anyways other than that point is there any other problem? (Mainly the chain rule)

[tropic oxide]

your execution is correct (besides the x that's on charon's boat for no good reason)

though imo your planning is a little bit wonky

maybe you could share the problem exactly as given to you originally?

i wanna see if there was any like, special instruction to do something in a specific way.

[neon hinge]

I just think of this question myself since i wasnt sure if i really understand the chain rule

[tropic oxide]

right...

[neon hinge]

To like.. test myself and such

[tropic oxide]

well there is somewhat of an issue in that differentiating f(x) itself requires the chain rule.

so you're kinda signing yourself up for extra work

i would at least work out $f(g(x))$ first and found it to be $\sqrt{3x^2+16}$, then i would go on to differentiate that. (with inner function as $3x^2+16$)

[wraith dagger]

Ann

[neon hinge]

Ye that will def make this easier

[tropic oxide]

yeah so my advice is dont make your life difficult on purpose.

[neon hinge]

Anyways thx a lot

Appreciate it

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[languid ice]

Find the smallest possible value of the expression [(a+b+c)/d] + ... in which a, b, c and d vary over the set of positive integers.

[split ice]

what does the ... represent, sorry?

[chrome quail]

!original

[cedar kiln]

Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.

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[rocky torrent]

Consider f (w) = (w − 3)^2. Starting from w0 = 0 with learning rate η = 0.1, perform two
steps of gradient descent. Recall: update wt+1 = wt − ηf ′(wt). Can someone show me what I need to do to start or a similar problem?

[tropic oxide]

$w_{t+1} = w_t - \eta f'(w_t)$

[wraith dagger]

Ann

[tropic oxide]

did you mean this?

@rocky torrent

[rocky torrent]

yes

thats what it should be thanks

[tropic oxide]

ok, do you know what the notation f' means?

[rocky torrent]

probably not. f prime?

or the derivative

<@&286206848099549185>

[jovial zealot]

hello hello

Yes im helper

[rocky torrent]

howdy

so is the w the gradient

[jovial zealot]

my english is not good, so i try my best

[rocky torrent]

and maybe we need to solve that formula twice

[jovial zealot]

I generally dont know im not that smart

but i still understand

[rocky torrent]

ok maybe find someone else

[jovial zealot]

let me atleast try

[rocky torrent]

ok thanks

[jovial zealot]

Lemme ask something

Can you please claify on what you mean by the gradient

Like the gradient of scalar function

[rocky torrent]

I don't know myself

[jovial zealot]

No worries its alright

[hasty dove]

To start, you should find the derivative using the power rule and the chain rule.

Then just perform the updates

[rocky torrent]

ok ill try that and show my work

[cedar kiln]

@rocky torrent Has your question been resolved?

[rocky torrent]

here

I think this is what you wanted me to do

does w mean the gradient

@hasty dove

<@&286206848099549185> anyone? I think im almost done

[steel crest]

f'(x) in general means derivative of f at x

[rocky torrent]

did I mess up?

[frail citrus]

here, the two steps of gradient descent mostly means the second derivative of the provided function

[rocky torrent]

but what i did was i used the function twice

wouldnt that be a different answer

[steel crest]

two steps means use the formula they gave twice

[frail citrus]

prolly a new convention was bought into play here

[rocky torrent]

does w usually mean the gradient

[frail citrus]

we differentiate the func twice as gradient, in short, means slope/derivative

[frail citrus]

i never saw/heard such a abstract thing

maybe for you people it maybe true in this specific problem

[steel crest]

w is just the value of your guess

w_0, w_1, w_2, ... is your series of guesses

they are asking for w_2, or your guess after second update

[frail citrus]

it kinda relates to recurrence relations

maybe manipulating it might give us some insight?

or maybe that step is foolishness

[rocky torrent]

but why is w0 relevant

[steel crest]

the qn is literrally asking us to apply it twice

[steel crest]

because you calculate w1 using it

[frail citrus]

to find higher values of w_1, w_2and so on

[steel crest]

they gave you a formula for making the next guess given an older guess

[frail citrus]

idts, as they related/mingled it with slopes and gradients too

[rocky torrent]

so the final awnser should be a number not a formula

[steel crest]

yes

[frail citrus]

but this is an interesting question

[steel crest]

you use the slope to update the guess

[rocky torrent]

ok ill try again soon

[steel crest]

||pls dont overcomplicate this, there are no double derivatives and all in this, they are just following the slope||

[rocky torrent]

i got 42.24

[steel crest]

i think you made some calcn error

[frail citrus]

👍 mb then

their terminology ditched me

and caught me off guard

[steel crest]

w1 = w0 - 0.1 * f'(w_0)

why are you calculating f(w_0) ?

[rocky torrent]

oh

so its not nesscarry

s0 it should be 1.08

is that for sure correct

[steel crest]

i think so

[rocky torrent]

ok

[cedar kiln]

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[wicked hazel]

This is for an introductory Linear Algebra class.

So, I'm working through the textbook and just finished Binary Vectors. It continues on to do the following as shown in the photos.

The circle diagram/clock analogy doesn't make sense to me... is there another way to explain it?... I'm trying to understand where they got the numbers in the table, mainly the 2+2=1

[split ice]

another way to think about modularity is "sorting" all the integers into boxes. in this case, you are sorting integers into 3 boxes: integers of the form 3k, of the form 3k + 1, and of 3k + 2 (where k is allowed to be any integer).
Why only those? Because if we do, say, 3k + 3, that's the same as 3(k+1), so the cycle starts anew. Similarly, 3k-1 = 3(k-1) + 2, and the cycle proceeds backwards.

This logic is formalized as an equivalence partition on the integers (commonly referred to as a set of equivalence classes). Every number in a box is treated equal: ..., -5, -2, 1, 4, 7, 10, 13, ... are all the "same" number modulo 3. So are -4, -1, 2, 5, 8, 11, 14, ... - i'll let you figure out what the last "box" is

So when we talk about 2 + 2, that's of course 4, but 4 goes in the "3k+1 box." We represent this box by its smallest positive member, aka 1. So as far as modulo 3 is concerned, 2 + 2 = 1 because it is in the same box as it.

[livid hound]

btw there's a typo in figure 1.21,
bottom left should have
-1,2,5

[wicked hazel]

Ok, this makes a lot more sense. Thanks!

[split ice]

np! the clock messes me up too >.>

[cedar kiln]

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[weak granite]

As an person that learns math in arabic, I'm terrible when it comes to math terms in english. I wanna ask a question about this, does anyone recgonize this so I can explain my problem properly? these are all vectors and its relating a vector to its magnitude and another vector (of magnitude 1) thats in the same direction

[spiral orbit]

go ahead with the problem

[weak granite]

i wanna know where this came from

how did we conclude it

[crisp halo]

what does your note say, it looks like you are finding the basis vector pointing in the same direction as P

but i am not sure what the right hand side says

[weak granite]

is that hwat its called? basis vector

[floral arrow]

$\frac{\vec{v}}{||\vec{v}||} = \vec{u_v}$

[wraith dagger]

Nel

[weak granite]

yes that

[floral arrow]

Something like that?

[weak granite]

exactly that

[floral arrow]

That's just a unit vector

[weak granite]

and the magnitude of u is 1

[floral arrow]

Yeah

[weak granite]

ok, where\did this come from

[crisp halo]

ah wait are unit and basis vectors not the same thing

[floral arrow]

Wdym where it comes from?

[crisp halo]

ah wait the basis ones are always in i, j or k direction?

[weak granite]

whats the proof? how did we conclude this?

[spiral orbit]

suppose you have a unit vector u, multiplying it by a scale p gives you a new vector with same direction but now with scale p

[floral arrow]

So ||v|| is the magnitude of v

[crisp halo]

you can think of it as the length

[floral arrow]

If you divide v by its magnitude, you get a vector in the same direction but of magnitude 1

[crisp halo]

of vector v, no direction

[weak granite]

thats quite literally what the rule states, I just- cant really imagine it

[karmic field]

you can just put || on both sides

[floral arrow]

Get the definition of the magnitude of a vector I guess

[surreal cave]

the vector <0,2> has magnitude 2, scaling it by 1/2 gives <0,1> which is a unit vector, and has magnidtude 1

[karmic field]

and you have that the magnitude of the vector is 1 on the left side

[floral arrow]

If v = (3, 4), its magnitude is sqrt(3^2 + 4^2) = 5
v/||v|| = (3/5, 4/5) and if you check its magnitude, you get sqrt((3/5)^2 + (4/5)^2) = 1

[weak granite]

OH

yes. i understand now

[floral arrow]

The process is also called "normalization"

The result, u, would be a normalized vector

(kind of a synonym with unit vector)

[weak granite]

everything makes hella sense now

ty everyone

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[merry flame]

Hi there, is anyone able to help with this problem, I have tried expanding the left hand side and doing simultaneous equations with a and b but Im unsure it’s right

[spiral orbit]

What did you get?

[merry flame]

11/13 for a and 35/13 for b

[spiral orbit]

,w 3a+5b=16, 2a-b=-1

[wraith dagger]

Results provided by WolframAlpha

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Requested by peacegiant

[spiral orbit]

yep

[merry flame]

Thanks

.close

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[royal loom]

I have a question

[royal loom]

Say that you have a TK_(3,3)

I am wondering how many different ways up to automorphism there are to have 2 paths between inner vertices of the TK_(3,3)

I tried approaching it first by just placing the 4 inner vertices

but it was more complicated than I thought

[royal loom]

Also, restrict to the case that the paths are from different sides of the TK_(3,3). So no path from its own side to its own side.

I actually am so lost in the sauce its hard to even describe what I am trying to count

so if someone can help me even think of a way to say it that would be nice too

[cedar kiln]

@royal loom Has your question been resolved?

[muted bear]

whats tk3,3

@royal loom

[royal loom]

a subdivision of K_(3,3)

[muted bear]

do you have a picture of it?

ossie

my guy

work with me here

[cedar kiln]

@royal loom Has your question been resolved?

[muted bear]

i cant find a pic of tk3,3

,w tk(3,3)

[wraith dagger]

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Requested by garlicbredfries

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@royal loom Has your question been resolved?

[cedar kiln]

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[magic solar]

Question was what value of x minimises the integral

<@&286206848099549185>

[livid hound]

!15m

[cedar kiln]

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.

[livid hound]

also, the worked solution is there

what's your issue with it

[magic solar]

Oh wow didn’t see the worked solutions there

I don’t understand it

[livid hound]

,tex .FTC2

[wraith dagger]

ℝαμOmeganato5

[livid hound]

FTC was applied, with a(x) being constant here, the term being subtracted is just 0
leaving just the first term
which is what they have there

[magic solar]

I don’t understand how to diff the integral if the bounds is not x, but instead a(x)

[livid hound]

you multiply by its derivative

[magic solar]

Why?

[livid hound]

,tex .FTC1

chain rule

,tex .FTC1

[wraith dagger]

ℝαμOmeganato5

[magic solar]

I don’t see how chain rule can be applied here

[livid hound]

plugging in your bounds,
= F(b(x)) - F(a(x))

how would you differentiate that. (applying chain rule)

[magic solar]

F(b(x)) would become b’(x) x f(b(x))

[livid hound]

not quite

careful of capitalisation

F'

[magic solar]

yeah which is f right

[livid hound]

yes

[magic solar]

So ok

I’m not connecting the dots here

[livid hound]

now lets consider what happens when your bound is a constant,

if a(x) = k
what's its derivative?

[magic solar]

Derivative of?

[livid hound]

a(x), k

[magic solar]

0

[livid hound]

yes

[livid hound]

now note, that this iss the first term in FTC posted above
and the second term here will be 0 because derivative of the lower bound is 0

[magic solar]

yeah

oh right ic

[livid hound]

so comparing what you have to FTC,
your f(t) = e^(t^2)
and b(x) = x^2 - 3x
and a(x) = -3

[magic solar]

yep thanks

Geometrically, I thought that the area was minimised when the bounds are equal

Ie 0

But would it be negative area here

How would f’(x)=0 give a minimum integral

It could be a maximum as well no?

Or even a inflexion

[livid hound]

from first derivative alone, you'd know its a stationary point
to determine what type you'd need more,
like second derivative test or consider slopes before and after x=3/2

[magic solar]

Oh I just realised that the FTC works regardless of the constant bottom bound because it just becomes 0

Ok then thanks

.close

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[cedar kiln]

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[opal hinge]

ykw

This sounds familiar

I think I have done it in a help channel

what have you tried?

[cedar kiln]

@wide gorge Has your question been resolved?

Channel closed

Closed by @wide gorge

Use .reopen if this was a mistake.

[wide gorge]

.reopen

[wide gorge]

I used trig and found ac and bd

[opal hinge]

.reopen

hmmm

[wide gorge]

.reopen

.reopen

i kinda dont understand if the lines can intersect or cant

[native heath]

bot broken?

.reopen

why did it get closd anyway

[wide gorge]

I was eating

[native heath]

bot is fine.

.reopen

your channels fucked

[wide gorge]

.reopen

🙁

[native heath]

just get a new channel

[wide gorge]

ok

.close

[cedar kiln]

Channel closed

Closed by @wide gorge

Use .reopen if this was a mistake.

[opal hinge]

what

[wide gorge]

wait

.reopen

[cedar kiln]

✅

[wide gorge]

yes

[opal hinge]

wth

[wide gorge]

?

[opal hinge]

nah nothing, could you show your work

[native heath]

💀

[wide gorge]

I dont got my phone w/ me rn

[native heath]

this is closed too??

.reopen

amazing

[wide gorge]

its open

[native heath]

@wide gorge youre cursed

[wide gorge]

^^

[native heath]

ah ok

great

[prime plover]

Can anyone tell me how to do l hospital rule sum

[opal hinge]

I will make a diagram gimme a min

[digital bane]

!occupied i believe

[cedar kiln]

Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).

[native heath]

dont

also occupied

[opal hinge]

what did you get for AC and BD

oh wait

[wide gorge]

30 adn 10sqrt3

[opal hinge]

hmm I remember we have to express AD in term of BC

[wide gorge]

how?

simmilar triangles?

cod and aob?

[opal hinge]

hmm

using area

[wide gorge]

how?
s

im dumb sorry

[opal hinge]

this's different from the one i did

Everything the saem

except BC and AD are perpendicular

[wide gorge]

cant we use pythogoras

i mean trig

for bc and ad

and calculate it from,t heir

we know dist b/w the liens

[opal hinge]

wait hold on

okay so the fact that we don't know position of A,B,C and D

[opal hinge]

let say it look like this then I think we have to do some trig here

[wide gorge]

idk

can you help

[opal hinge]

idk my brain is not braining

[wide gorge]

my math is not mathing

you know some website that can explain this to me

<@&286206848099549185>

wait

can we fidn co-ordinates of each point

[crimson sedge]

do you know derivatives?

[opal hinge]

I believe he do

[wide gorge]

no

i think i do

for physics

[opal hinge]

oh

okay so I have to find another way lo;

[wide gorge]

I found the co-ordinates

can I just use distance formula and completign sqaure

[opal hinge]

I'm not sure, I haven't done it that way

[opal hinge]

I mean what tools do you have to solve the problem?

only inequality?

[cedar kiln]

@wide gorge Has your question been resolved?

[cedar kiln]

Channel closed

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• Show your work, and if possible, explain where you are stuck.
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[exotic orchid]

Hey there , i cant seem to understand how something can divide n^2 being less than n and not divide n , how does this work , isnt a factor of n also a factor of n^2

[crimson sedge]

4|6² but 4 doesn't divide 6

[exotic orchid]

whats that line dividing 4 and 6 in the middle

[crimson sedge]

it's true that a factor of n is also a factor of n² but the opposite isn't true

"divides"

[exotic orchid]

ah

[exotic orchid]

but n is

given to us

[crimson sedge]

I showed a simpler example

[exotic orchid]

oh wait no i see what you mean

how do we approach such problems

i tried to apply euclid's lemma but got nothing

yk the n=ab+r thing

[crimson sedge]

n²=2^38 *3^24

[exotic orchid]

yep

[crimson sedge]

a divisor of n² must be of the form 2^a 3^b with 0≤a≤38, 0≤b≤24

[wide gorge]

Helpp

WIt srry

[crimson sedge]

not here

Lol

[wide gorge]

Ok remember the geo etry question wayyy above

[exotic orchid]

alright

[crimson sedge]

use another channel

[wide gorge]

This

[exotic orchid]

but that divisor will also divide n

[wide gorge]

Ok

Srry

[crimson sedge]

not necessarily

n=2^19 * 3^12 so a divisor of n must be 2^a 3^b with a≤19, b≤12

So if you have a=20, b=12, then your number only divides n² but not n

[exotic orchid]

ohhh

i see

[crimson sedge]

in this example the divisor is actually bigger than n, but you can take for example a=20, b=1 and the now your number is less than n but doesn't divide n

[haughty osprey]

I just did this half an hour ago lol, ioqm 2024

[exotic orchid]

lol you too?

[haughty osprey]

Yeah

[exotic orchid]

thats pretty cool

[haughty osprey]

Number of divisors for n and n^2 will help

Should be your first step

[exotic orchid]

for n it should be 12+19+19*12

and for n^2 it should just be these values multiplied by 2

[haughty osprey]

You sure?

[exotic orchid]

i think so

for n^2 it will be like 24*38+24+38

[spiral fog]

Umm

Do you know the formula for number of divisors?

[exotic orchid]

i dont actually

im eyeballing it by the exponent here

[haughty osprey]

Better

But not exactly

[spiral fog]

You are off by one

[exotic orchid]

like

the number 1

we include that aswell?

[spiral fog]

Yes...

[exotic orchid]

oh

[exotic orchid]

what does alpha denote here

[spiral fog]

Its + 1 because it can be zero

[spiral fog]

The exponent

[haughty osprey]

It's just the exponent

[exotic orchid]

okay

[haughty osprey]

So can you get the number of divisors for n and n^2 now?

[spiral fog]

Btw what is ioqm

[haughty osprey]

Indian Olympiad Qualifier in Mathematics

[spiral fog]

Oh ok

[exotic orchid]

yeah

oh and one more thing whats the p1 p2 thing

[spiral fog]

Not so pretty problems but ig india does very well on the IMO so doesnt matter

[spiral fog]

Prime numbers

[exotic orchid]

okay so in our case its just 2 and 3

[spiral fog]

Yes

[haughty osprey]

Yeah

[exotic orchid]

oh so its just (12+1)(20)

for n

[spiral fog]

Like fundamental theory of arithmetic..

[haughty osprey]

Yes

[exotic orchid]

and for n^2 its 25*39

[haughty osprey]

Yeah

[exotic orchid]

ah , mb they dont teach us these things in school

atleast in grade 10 which is my age

[spiral fog]

They dont. But if you are solving problems from competitions, you should probably self study that

[exotic orchid]

fair enough

[spiral fog]

Most of my knowledge is from self studying

[haughty osprey]

Btw

Next thing for lesser than n

[exotic orchid]

we will make an inequality?

with a and b?

[haughty osprey]

You should know if d is a factor of n^2 , then so is n^2/d

[spiral fog]

Not really

[haughty osprey]

Eh

[spiral fog]

You should know how factors of squares look like

[haughty osprey]

Symmetric about d=n btw

So can you find out the number of diviosrs of n^2 lesser than n?

[exotic orchid]

yea lemme try

[spiral fog]

Like look at 36 for example

1 , 2 , 3 , 4 , 6 , 9 , 12, 18 , 36

[exotic orchid]

487

[haughty osprey]

Yeah

For n, just subtract 1

Should be easy after

[exotic orchid]

👍

[spiral fog]

I would personally use 488 (including n) as its goona get subtracted anyway

[exotic orchid]

but our requirement is for less than n

[spiral fog]

Yes but n divides n

So you are goona subtract it later on

[exotic orchid]

ah

[haughty osprey]

That works but then you don't have to subtract 1 from the number of divisors of n

[spiral fog]

Well yes exactly

I mean both work

[haughty osprey]

Yeah

[exotic orchid]

alright

i seem to get a little hold of this

now

thanks vm

[spiral fog]

Just dont remove the divisors of n including n if using 487 as you will get an off by one error

[exotic orchid]

alr

[spiral fog]

Also the numbers here are so small that this couldve been brute forced in like 10 mins prob

[exotic orchid]

😔 💔

[spiral fog]

Like idk if i would be motivated to find a smarter solution to just count them manually doing casework.

[exotic orchid]

fair enough 😭

its like 30 questions

with 3 hours

we have plenty of time

to bruteforce

[spiral fog]

But if its like $n = 2^{100} \cdot 3^{251}$ then you would have no other options

[wraith dagger]

casework

[exotic orchid]

i see

this would have a lot of permutations

so fair enough

we would have to use the proper method

[spiral fog]

Oh thats like AMC format or what else

[exotic orchid]

yeah

30 ques , no negative marking

after IOQM its RMO

[spiral fog]

Regional Math olympiad?

I mean india has a big population so i get why so many exams

I just dont like the AMC/AIME format

[exotic orchid]

yeah

ioqm>rmo>inmo>imo

between those all the training camps included

[spiral fog]

Oh they call everything an olympiad

I kind of divide olympiad and competition in my head

Like
Olympiad math problems
Competitive math problems

[exotic orchid]

are they not the same

[spiral fog]

Its just me ig

[crimson sedge]

lol for a moment I thought the > meant harder

[exotic orchid]

😭

[crimson sedge]

surely there's tst tho right

[exotic orchid]

tst?

[spiral fog]

Like olympiad math problems should be harder and be like proof-type

While competitive math problems just use like 1 trick and its mostly calculate something or such

[exotic orchid]

yeah no im not old enough to understand the difference here

[spiral fog]

That should be inmo if i got it right

[crimson sedge]

no

[exotic orchid]

indian national maths olympiad

thats inmo

[spiral fog]

Tst means team selection test

[exotic orchid]

oh yeah

[crimson sedge]

yeah

[exotic orchid]

yeah

after inmo

[spiral fog]

Inmo selects for the imo?

[exotic orchid]

yes

[crimson sedge]

no

[exotic orchid]

and even after that

there is a selection camp

[spiral fog]

Oh so you have the tst on camps

[exotic orchid]

yep

but those are cracked kids

thanks for the help everyone

hey can anyone like uhh unoccupy this channel

[manic kelp]

if this is your channel you have to type .close

[exotic orchid]

.close

[cedar kiln]

Channel closed

Closed by @exotic orchid

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[cedar kiln]

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[cedar kiln]

Please don't occupy multiple help channels.

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[mental quail]

learning summation notation for the first time.

can someone with a writing pad or ability to like scribble how i should be answering this show me?

first time btw meaning like...i only know this is an adding problem because of the E....and k is the starting point, the 6 is the ending, 3k is what you need to do with those two numbers 1 all the way up to 6........

[stiff brook]

youre idea of the notation is on the right track!
it essentially means 3(1) + 3(2) +... + 3(6)

[chrome elk]

Just a small correction: 1 is the starting point, not k

But yea you seem to have the right idea

[mental quail]

how would i write it though, like i have a piece of paper with that initial problem written down, so i need an = sign somewhere

how do i show me steps

what does the final answer look like

[stiff brook]

∑3k = 3(1) + 3(2) +... + 3(6)

[mental quail]

do i just write 3(1) + 3(2) +... + 3(6) below it and = xyz

[pulsar garden]

yes

[stiff brook]

and then u can add them up

[pulsar garden]

can someone help me with grade 11 maths

[stiff brook]

this channel is occupied so you should open a new channel

[cedar kiln]

Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).

[mental quail]

so my final answer was 63.

beside the initial problem i wrote everything down like ∑3k = 3(1) + 3(2) +... + 3(6)
then just expanded it to be like = 3 + 6 + 9 ... + 18 one line lower
and then on the last line i just have = 63

if i were asked this on a test do i need to like rerwite the ∑3k = 63?

[stiff brook]

what u have written is perfect!

if u directly write ∑3k = 63, u may not get marks for working

[mental quail]

ahhh, so they usually want steps with all that stuff...

[stiff brook]

yeah in math u should ideally write down ur steps

[mental quail]

for big sums is it okay to use the " ... " ?

[stiff brook]

yeah u can write the first 2 terms, use ..., then the last term for example

[mental quail]

im asking because that might come up then i'd have to learn how to get my calculator to do it after entering parameters or something

[stiff brook]

in calculator, u can directly solve sums depending on your calculator

[mental quail]

that was the first practice problem, in the initial image

this is the second Lol

they expect me to use a calculator right?

[stiff brook]

yeah a calculator seems reasonable

[mental quail]

cause 27 steps basically otherwise...

[stiff brook]

later on, u will encounter sum identities of i^2 and i which will make ur life easier and u will no longer need to use a calculator

[chrome elk]

I mean you can use the summation properties if you've seen them

[mental quail]

no idea what this means 🥲 i already see i^2 in that second image lol

[mental quail]

ive seen nothing, this is like my first class ever thats asked me about summations. the only time i've seen the E otherwise was just in passing or briefly

[chrome elk]

I see, in that case use calculators if you're allowed

[mental quail]

oh, so i, i^2, and i^3 represent the formula or something like that

is that for like compound interest and idk, im just throwing words out, annuities etc

bond pricing

[stiff brook]

those are some useful summation properties to compute up to third degree polynomial sums

[stiff brook]

i dont do finance math so i dont know if there is a connection

[mental quail]

alright xD

but the i just represents whats on the right of the = sign, yeah?

or maybe ill just learn about it later lol

[stiff brook]

it is just the summation variabe
like 3k was the summation variable in your question

[mental quail]

thanks again

.close

[cedar kiln]

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[mental quail]

.reopen

[cedar kiln]

✅

[stiff brook]

do you need help with something else?

[mental quail]

sort of

they gave this as an example

and then now im thinking there are a ton of summation rules out there, and i was just searching for stuff now

because they rewrite the given sum/example by moving the 11 to the left

[mental quail]

i forgot the word but the summation now just looks like a...function? and we are just doing some operation or something to it now, 11 times the sum of i^2

plus 12 times i

if im understanding what the 11 and 12 mean

[stiff brook]

in mathematics, we call this linearity.
we have that ∑(ka+cb) = k∑a + c∑b
where k and c are arbitrary constants and a,b are the variable of summation

[mental quail]

they give another example
i think they intend for 605 to be the final answer.

but relating back to the first ever summation image....shouldn't it also have a substitution of 2, 3, 4, 5? here they just skip to 5 and solve?

[stiff brook]

They are using the third property

∑i^2 has a closed form expression which is as shown in the image

you can just plug in the value of the upper bound into that equation and get the answer

[mental quail]

ooh....so the n is just that upper bound...and theres no need to start with n = 1 then add with replacing n = 2 ... until n = 5

[stiff brook]

yes

[mental quail]

feels kinda confusing when the first example was with k, and that was adding a series of numbers together

but i guess the i function is just it's own thing

[stiff brook]

u can do the same here by doing 11(1^2 + 2^2 +... + 5^2)

u will get the same answer

it is just easier to use the properties

[mental quail]

oooh

[mental quail]

so for this image they would use the i^2 property on the left, and the single " i " on the right?

[stiff brook]

yes exactly! we have a property for the single i as well

[stiff brook]

the 2nd one

[mental quail]

im referring back to your img, yeah

[stiff brook]

yeah

[mental quail]

😄 thanks, gonna read the textbook more

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[cedar kiln]

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[steep steeple]

im ngl my brain aint braining

[dull oxide]

What does the problem want you to do

[steep steeple]

apply linear ode to 4 and 5

working on 4 rn

got it to dy/dx + 2(y/x) = x(^-4)

[dull oxide]

Oh there are equal signs in there

[steep steeple]

oh yeah

[dull oxide]

They look like minus

[steep steeple]

yeah i should've mentioend that

mb

for whatever resaon the eual signs are very washed out

[steep steeple]

once i got here i kinda forgot what i'm supposed to do

[stiff brook]

you multiply both sides by the integrating factor and recognize it as d/dx (y*I(x)) = x^(-4) * I(x) where I(x) is your integrating factor

[cedar kiln]

@steep steeple Has your question been resolved?

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[teal folio]

Can someone explain to me the answer? i get where the ⅓tan³x is from but not the -tanx and +x,

[void sand]

the integral of sec^2x is tan x, and the integral of 1 is x

[dusk thorn]

+c

[void sand]

there's an implicit step between the second last and last where they did [\int (\tan^2 x \sec^2 x - \sec^2 x + 1) \ dx = \int \tan^2 x \sec^2 x \ dx - \int \sec^2 x \ dx + \int 1 \ dx ]

[wraith dagger]

higher!

[void sand]

and then they used this

[teal folio]

Ohhh okay okay

Thank you!

[teal folio]

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[winter falcon]

26 and 27

[cedar kiln]

@winter falcon Has your question been resolved?

[winter falcon]

<@&286206848099549185>

[tough finch]

Quick question: is this a pathfinder book?

[wary elbow]

try cubing the the second equation given in the question, you should get clues

[tough finch]

Because I swear to god I've seen this book before somewhere

[wary elbow]

same bro

giving me flashbacks

[winter falcon]

whats pathfinder

[tough finch]

Nvm 🥀

Pathfinders are as far as I know some of the hardest books of their respective subjects used to prepare for competitive exams like JEE and Olympiads

[winter falcon]

64x cube + 1/x cube + 48x + 12/x = 27

how does this help me

[tough finch]

Ok so you got the 1st 2 terms

[wary elbow]

27 right?

[winter falcon]

yes

[tough finch]

Now how do you think you can substitute the original equation into the last 2 terms?

48x+ 12/x?

[winter falcon]

yes

[tough finch]

Because you have the value of 4x+1/x

So maybe try taking sm in common to get that?

[winter falcon]

OHH

i didnt notice that

[tough finch]

Yeah

Similar process in previous question

[winter falcon]

i got 64 x cube + 1/ x cube = - 9

[tough finch]

Seems right

So what option do you think is correct?

[winter falcon]

option a

but it doesnt explain the f(alpha) = f(beta)

[wary elbow]

are you aware of the quadratic equation?

[winter falcon]

yes

[wary elbow]

you could multiply the second equation by x

form a quadratic equation

[tough finch]

Wouldn't that just complicate it unnecessarily?

[winter falcon]

wait f(x

nvm

i thought i got it but i did not

[wary elbow]

find alpha and beta

they are both unequal

[tough finch]

Imaginary roots no?

4x²-3x+1?

[wary elbow]

3x sqared -12 x +1

that's the quadratic equation

[tough finch]

I think you're getting confused

We're on 27

[wary elbow]

ohoh

my bad

I was talking about 26

😅

[tough finch]

Still imo this method seems way too long

Calculating roots is completely unnecessary

[wary elbow]

you don't have to

you just find out that they are irrational

so option a would be correct

[winter falcon]

😭

brother

[wary elbow]

i'm sorry, did I confuse you?

I just tried it and it was pretty short