help-42

[dreamy wyvern]

and they do

alpha = 82.69 - 60
= 22.69

therefore the airplane is flying at a ground velocity of about 496 km/h on a bearing of about 023 degrees

BUT i don’t get the part with the alpha

this is my attempt

idk why i can’t send photos of my textbook tho

can someone explain

[calm coral]

@dreamy wyvern Has your question been resolved?

[calm coral]

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[blazing wadi]

Is f(n) from Naturals to Reals such that f(n) = 1/n a continuous function ?

[glass heart]

what do you think

[velvet osprey]

what topology does N come with in this case?

without that we cannot hope to discuss the continuity of any function with domain N

[velvet osprey]

!xy

[calm coral]

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.

[blazing wadi]

well nothing I was just confused my book stated that functions can be continuous at isolated points of the domain so I got curious and thought of this

[rigid kettle]

You are doing topology?

[blazing wadi]

Analysis

[rigid kettle]

hmm

well you can just think for now about the continuity statement

I believe the "forall" statement within it becomes vacuous

[blazing wadi]

could you explain that a bit more

[warm warren]

i mean if u pick the discrete topology on N

any function will be continuous

[blazing wadi]

Also could anyone explain what does continuous on a isolated point in domain means

[glass heart]

I am assuming this is just at the start of analysis and words like "topology" are new to you? I assume you have probably only seen the eps delta definition of continuous?

[velvet osprey]

yeah exactly

[glass heart]

or maybe the limit definition?

[blazing wadi]

i guess
I also have seen neighbourhood definition and sequential definition

[glass heart]

why dont you write out the definitions you have seen

[blazing wadi]

can I send u a picture of all definitions given

[glass heart]

yes

[warm warren]

whats the V thing

[blazing wadi]

neighbourhood of a point

[glass heart]

the first definition is the clearest here imo

we have A=N

lets say c=1

what happens if we choose delta = 0.5

[blazing wadi]

There is no f(x) within that range of x

umm hello ?

[glass heart]

wdym no f(x)

which x in A satisfy |x-c| < delta ?

[blazing wadi]

1

[blazing wadi]

mb got confused again 😔

[glass heart]

ok so only x=1 satisfies the condition

does x=1 also satisfy |f(x)-f(c)| < eps ?

[blazing wadi]

yes

[glass heart]

so that means that (i) is satisfied

[blazing wadi]

ok

But didn't the others say that it isn't continuous

[glass heart]

no

[blazing wadi]

also still what does continuity on isolated points of domain mean

[glass heart]

the point c=1 is isolated

[blazing wadi]

so what exactly were they saying

[glass heart]

there are no "nearby" points

they were asking for more information

[blazing wadi]

Oo

[glass heart]

on what "nearby" means. you could put another topology on N and then c=1 might not be isolated

anyway

[blazing wadi]

put another topology?

[glass heart]

the point here was that by choosing delta=1/2, the only x that satisfied |x-c| < delta was x=c itself

that wont be the case if c is not isolated

[glass heart]

just ignore it, you'll encounter it later maybe

[blazing wadi]

ic

[blazing wadi]

so we could generalize this and say that every function is continuous on isolated points of domain ?

[glass heart]

well if the function isnt defined at that point then the point isnt in the domain

[blazing wadi]

oh yes mb 😭

[blazing wadi]

doesn't this like contradict the def of continuity told to us in calc " u can draw the graph without lifting the pen " or lim f(x) = f(c)

Anyways Thanks alot @glass heart

.close

[calm coral]

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[glass heart]

its not like that is a rigorous definition

the limit definition also still works

[blazing wadi]

Hm

[glass heart]

but limits at isolated points are weird

thats why I chose (i)

[blazing wadi]

Huh? Isn't the limit not defined for isolated points

[glass heart]

well yes

[blazing wadi]

soo how does the def hold

[glass heart]

so you are kinda talking about nothing

[blazing wadi]

wdym ?

[glass heart]

well ok, works was maybe bad wording

.reopen

[calm coral]

✅

[blazing wadi]

So what's the right wording

[glass heart]

like your image said, the limit def is just for limit points

for isolated points you just say that the limit is not defined, so in that sense its not unequal to f(c)

or something like that

the point is, isolated points are weird

[blazing wadi]

y ?

[glass heart]

cause we just found out that any function f:N->R is continuous

in what world is that not weird

I would call it weird

[rigid kettle]

truly, what it really is, is 'point with no neighbourhood to speak of is not a discontinuity"

[glass heart]

the whole point of continuity in the end is to say something how the function behaves at nearby points

[blazing wadi]

I mean that u visualize a smooth graph when u talk about continuous so ok it's kinda weird

[glass heart]

if there are no nearby points then what are you talking about

[blazing wadi]

Hm

[rigid kettle]

You only have point being not continuous if you can find a neighbourhood around it which contradicts the continuity statement.
sortof. i think

Its just the nature of the definition

[blazing wadi]

I mean wouldnt the creators of the def thought of this and what exactly were they thinking making isolated points continuous

[rigid kettle]

yes its consistent

its not consistent with the natural language perhaps

but its far more convenient to define it this way

and the same goes for vacuous statements in general

[glass heart]

in the end isolated points dont matter

its much more of a funny quirk

[rigid kettle]

(look up what "vacuous" means)

[glass heart]

no function that people care about has isolated points in its domain

in the context of analysis

so why put extra effort into rewording the definition so that it excludes those points

[rigid kettle]

It also still fits in with "not lifting pen off paper" definition of continuity if you consider that you teleport whenever you have discontinuities in the domain

So f : R - {0} -> R with f(x) = 1/x is continuous

At 0, you teleport from minus to plus infinity, casually speaking.

(small and negative x, large and negative y) going to (small and positive x, large and positive y)

[blazing wadi]

teleport 😭

[rigid kettle]

but this is just kinda very casual cus i think i can think of some silly examples where this might break down

typical analysis

[rigid kettle]

A function (that jumps everywhere) defined on only Q isn't continuous over the euclidean topology is it? or is it?

[glass heart]

I hope not

[rigid kettle]

eg. f(x) = rand(x)

for some fixed random generator

yeah crazy shit

[blazing wadi]

euclidean topology?
I'm out

[rigid kettle]

euclidean topology just means the usual thing on R

[glass heart]

"normal way we measure distance"

[rigid kettle]

which is your usual definition

[rigid kettle]

So take exactly this definition and you will find f : Q -> R with f(x) = denominator when x is in simplest form
And f is not continuous

[blazing wadi]

Isn't that the thomae function

[rigid kettle]

maybe

idk, was just thinking of example that jumps around

[blazing wadi]

It's continuous on irrationals and discontinuous on all rationals

[rigid kettle]

i dont think ive written that down then

[naive meadow]

It’s the Thomas function restricted to Q

[blazing wadi]

ya that

[rigid kettle]

ah 0 if x irrational

yeah

[blazing wadi]

Yup

anyways good time talking to you all
have to go now

.close

[calm coral]

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[unkempt drift]

you want help with part b right?

it's easier if we focus on one part at a time

[gentle kelp]

b c an d yes

for b i tried doing it and got an answer -1 + root(2)

[unkempt drift]

cool let me check that using Desmos

https://www.desmos.com/calculator/warum5gdnr

unfortunately that's not correct

your work that you crossed out is on the right track

[unkempt drift]

you should get x - y = 0 or x = y
not x + y = 0

[gentle kelp]

did i do some small error somewhere

[unkempt drift]

yes it has to be

[gentle kelp]

lemme check quickly

[unkempt drift]

yeah should be k = -1 + sqrt(2), k = -1 - sqrt(2) also

[gentle kelp]

idk what i did i still get x+y = 0

[unkempt drift]

yeah -2kax-2ay=2ax+2kay indeed gives that

that's why

wait

[gentle kelp]

i think the equations should be correct

[unkempt drift]

okay they are correct

and yeah okay now I get x + y = 0

[unkempt drift]

setting the discriminant equal to 0 after subbing in y = x into one of the equations

and k can't be negative so k = -1 + sqrt(2) only

[gentle kelp]

so b should be correct

how do we do c

[gentle kelp]

here

[unkempt drift]

you can test a case for example, so at k = 0 the two circles are indeed intersecting

and then as k increases, the 1st circle moves to the right

the 2nd circle moves downwards

since the radius of both circles is fixed

that means as k increases, the circles move farther and farther apart and don't intersect after the critical value k = -1 + sqrt(2)

[gentle kelp]

so what do we write as set of values

[unkempt drift]

$k \in [0, \sqrt{2} - 1)$

[potent lotus]

south

[unkempt drift]

I'm counting tangent as not intersecting

[gentle kelp]

hmmm

now how do we do d

[unkempt drift]

you should think about what I just said

because it follows that the area is continuously decreasing as k increases

[gentle kelp]

so at max k we will have max overlapping area

[unkempt drift]

well, that's when the circles aren't intersecting at all!

for all k > -1 + sqrt(2)

[gentle kelp]

at 0 k we will have most overlapping area?

[unkempt drift]

yep

[gentle kelp]

how to find area tho

im confused

[unkempt drift]

have you tried doing circular sector - triangle

draw a sketch first if you haven't already

[gentle kelp]

i drew one

[unkempt drift]

a sketch specifically for k = 0 though

[gentle kelp]

yea i did that

so we can subtract area of triangle from area of sector

and that x2 should be the answer

[unkempt drift]

correct!

[gentle kelp]

but i dont understand how we will

get the area of sector

oh wait its in terms of a

ahhhh

[unkempt drift]

yeah

in fact you can use area similarity, so you can let a = 1 first

then multiply that answer by a^2

[gentle kelp]

i dont have that in my syllabus tho

[unkempt drift]

really?

I know this is a particularly clever way of doing it, but

[gentle kelp]

i got maths boards tmrw lol

[unkempt drift]

you don't have something like this?

[gentle kelp]

no

[unkempt drift]

oh good luck!

[gentle kelp]

ty man

[unkempt drift]

it's actually not obvious that the figure is similar after changing the value of a

the important part is the centres, (0, a) and (-a, 0)

and the radius being a

so changing a just is an enlargement/dilation centred at the origin

[gentle kelp]

is the answer

a^2/2(pi-root(3))

[calm coral]

@gentle kelp Has your question been resolved?

[unkempt drift]

nope

the triangle is right-angled

right-angled isosceles, in fact

[gentle kelp]

oh i mistekenly used the equilateral formula

[calm coral]

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[tough star]

is there a fast way to check if a rational function like this has even/odd symmetry?

like for a function like this

the exponents alternate between both even and odd, so it doesn't

is there a similar trick we can use with rationals?

[mortal orbit]

ig even/even = odd/odd = even, even/odd = odd/even = odd

[tough star]

oh

i see

nice

ty

[calm coral]

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[spiral knoll]

Hello, I need help with this

[spiral knoll]

I'm so lost

[naive meadow]

Start by simplifying the numerator and denominator individually

[spiral knoll]

So I have to find least common denomanators right?

so 4(x+4)/x+4 + 12/x+4

[naive meadow]

Right, that's the numerator

There's one more immediate simplification here

[spiral knoll]

x+4?

[naive meadow]

Well we have $\frac{4(x+4)}{x+4}+\frac{12}{x+4}$

[potent lotus]

Buzzing Hornet

[naive meadow]

What's the really obvious way to simplify this expression

[spiral knoll]

cross canceling?

[naive meadow]

Not quite

How do you simplify $\frac{a}{b}+\frac{c}{b}$

[potent lotus]

Buzzing Hornet

[spiral knoll]

add it

[naive meadow]

Yeah exactly

Which gets you...?

[spiral knoll]

so 12+4(X+4)/x+4

[naive meadow]

Yeah exactly

[spiral knoll]

then cancel x+4

?

or no

[naive meadow]

No, we can't cancel the the x+4 since our expression is $\frac{12+4(x+4)}{x+4}$

[potent lotus]

Buzzing Hornet

[naive meadow]

That's all we're going to do with the numerator for now

What about the denominator?

[spiral knoll]

27/x+4 and 9(x+4)/x+4

[naive meadow]

Right

So how can we simplify that?

[spiral knoll]

and combine

so 27 +9 (x+4) / x+4

[naive meadow]

Right

So now we have $\frac{\frac{12+4(x+4)}{x+4}}{\frac{27+9(x+4)}{x+4}}$

[potent lotus]

Buzzing Hornet

[naive meadow]

What's it look like the obvious thing to do here is?

[spiral knoll]

ya

cancel?

[naive meadow]

Exactly

Which gets us?

[spiral knoll]

16/36

4/9

[naive meadow]

Uh... I'm not sure how you got this

[spiral knoll]

oh

[naive meadow]

We have something that looks like $\frac{\left(\frac{a}{b}\right)}{\left(\frac{c}{b}\right)}$

What's the simplification we should make here?

[potent lotus]

Buzzing Hornet

[spiral knoll]

I honestly forget I remember learning it but

[naive meadow]

Well, do you remember how to divide by a fraction?

[spiral knoll]

no

thats so bad that idk how to do that

[naive meadow]

Ok so I will remind you then that $\frac{a}{\left(\frac{b}{c}\right)}=a\cdot\frac{c}{b}$

[potent lotus]

Buzzing Hornet

[spiral knoll]

oh like the keep change flip thing

I remember that now from when we were multiplying and dividing

[naive meadow]

Great

[naive meadow]

So how can we apply this

[naive meadow]

To this?

[spiral knoll]

ya

[naive meadow]

It was a question

How can we use it

[spiral knoll]

so do 12+4(x+4)/x+4 times x+4/27+9(x+4)

[naive meadow]

Right

And how can we simplify that?

[spiral knoll]

cross cancel

[naive meadow]

Yes

Which gets us?

[spiral knoll]

12+4/27+9

[naive meadow]

Uhh

Where did the x+4s multiplied by the 4 and 9 go

[spiral knoll]

dont you just cancel them?

[naive meadow]

I will point out that $\frac{a+bc}{c}$ is not necessarily equal to $\frac{a+b}{c}$

[potent lotus]

Buzzing Hornet

[spiral knoll]

oh

[naive meadow]

Or I guess that's not quite the error you've made

Regardless, if we're adding things in the numerator and denominator, you can't just cancel them

We can only cancel things if we have $\frac{ab}{ac}$

[potent lotus]

Buzzing Hornet

[spiral knoll]

oh

ok

So is it 12+4(x+4)/27+9(x+4)

[naive meadow]

Should be, yeah

[spiral knoll]

so what do I do after that

[naive meadow]

Well is there a factor you can pull out of 12+4(x+4)

[spiral knoll]

12 +4x +16

[spiral knoll]

4x+28

[naive meadow]

Uhhh

I was more thinking something along the lines of $ca+cb=c(a+b)$

[potent lotus]

Buzzing Hornet

[spiral knoll]

well cant it just be 4x+28/9x+63

then gcf

4(x+7)/9(x+7)

[naive meadow]

Well this is what I was trying to get at yes

[spiral knoll]

so 4/9

[naive meadow]

Right

[spiral knoll]

okk thanks sorry for taking so long

very helpful

.close

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[vital chasm]

i need help on part b

i mean its 1

but how do i formally prove it

[drifting seal]

is it supposed to be floor(sqrt(x))

[vital chasm]

yes

[drifting seal]

you mean sup right

[vital chasm]

yes

[drifting seal]

ok so have you shown it’s an upper bound

[vital chasm]

yes i can show that

wait

yes sure

its a upper bound but idk how to show its the supremum

[drifting seal]

what properties of the floor function do you know

[vital chasm]

all

its real analysis ofc i know most

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[drifting seal]

🤔

if you know all of them then what’s the confusion

[calm coral]

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[rapid beacon]

I want help verifying this proof

oh one mistake, please ignore the (N+k)/(N+k+1)

and assume its all just less than |a_(n+k)|

<@&286206848099549185>

right now I think its wrong but im not sure how wrong... like theres just one mistake to fix

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@rapid beacon Has your question been resolved?

[calm coral]

@rapid beacon Has your question been resolved?

[rapid beacon]

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[hearty tusk]

lowkey vague question though how do i convert integral to sum or sum to integral? ive never understood how to do it

[sleek basin]

like a taylor series?

[drifting seal]

$\int_a^b f(x) \dd{x} = \lim_{n \to \infty} \sum_{k = 1}^{n} f\left(a + \frac{b-a}{n}k\right) \cdot \frac{b-a}{n}$

[potent lotus]

knief

[sleek basin]

oh yeah there's that

[hearty tusk]

yea stuff like this one

[sleek basin]

so you're just confused on what's going on here?

i can help if so

[hearty tusk]

like when the sum is lowkey complicated i js cant convert it to integral

i have an ex

[velvet osprey]

you should share your exercise

[drifting seal]

send it

[hearty tusk]

this is like one of them

[calm coral]

@hearty tusk Has your question been resolved?

[drifting seal]

so this is an annoying style i guess because it doesn’t precisely match the definition you want

you can see clearly that b-a = pi/4 and match

but they have the shift of pi/6

the pi/4 k/n is precisely the x term

wherever you see the delta k that’s your x

then you’d think a = pi/6 here but we know b-a = pi/4 so it can’t be

which means a = 0 works though

and the pi/6 is just an annoying shift to throw you off

then the 4/pi comes from the fact that our 1/n in the sum is missing the pi/4

most of these questions are just about process of elimination tbh

[hearty tusk]

i lowkey get it its js matching with the standard equation right

[drifting seal]

there are multiple equivalent representations

[drifting seal]

yes

[hearty tusk]

gotta remember that thing

thanks 🙂

[drifting seal]

this is the thing to remember

[hearty tusk]

yea

[drifting seal]

it comes straight from the riemann sum

remember how you do finite approximations

you add increments from the left endpoint and evaluate there

[hearty tusk]

left endpoints?

[drifting seal]

like in the interval we’re integrating

[hearty tusk]

why left though

[drifting seal]

https://youtu.be/eidemLqiPyQ?si=w3GTCrtpyZxrHo0F

watch this

[hearty tusk]

kk imma review it

thanks!

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[granite dust]

is my reasoning right?

[glass heart]

no

bases dont contain all lin independent subsets

take for example S1={(1,0),(0,1)} in R^2 and B={(1,0),(1,1)}

[granite dust]

oh

[calm coral]

@granite dust Has your question been resolved?

[granite dust]

is this sufficient?

[quick talon]

Perfect !

Concise, clear and very true

[granite dust]

ty

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[viscid cobalt]

I forgot the steps

To solving this

Help

[velvet osprey]

step 1 is to subtract 2 from both sides

then make the lhs into a single fraction

[viscid cobalt]

No

I can't subtract

2 from both sides

It's not how we do it

In class

[velvet osprey]

deviating from in class methods is forbidden?

[viscid cobalt]

In this case yes

[velvet osprey]

what a stupid restriction...

[viscid cobalt]

There's a simpler method

That I forgot

[velvet osprey]

this is the simple method.

but if we are not allowed to use it then i guess you have to break it down into two cases: one where x-3>0 and the other where x-3<0

and in each case multiply both sides by x-3 but mind the inequality symbol

[dull wagon]

you could multiply both sides by (x-3)^2 and use wavy curve

[velvet osprey]

or that

OP does not remember which method is the one and only that's allowed in his class

we cannot really help with that

@viscid cobalt you have now been offered 3 different starting points of which you have rejected 1

[viscid cobalt]

Thanks

Ig

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[eternal shard]

looks okay, maybe elaborate 4) what exactly you demand

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[worn hedge]

What happened to the parts I highlighted

When they differentiated

[worn hedge]

Nvm

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[nimble harbor]

I have interesting integral problems that I can't solve at all 😭 . Plz help

[eternal shard]

post

[nimble harbor]

one sec

Define ${f_n (x) = x + \frac{x^2}{2} + ... + \frac{x^n}{n}, x \in [0,1]}$. Solve for
[\int_0^{\frac{1}{2}} \frac{(1+f_{n-1}(x))^{n+1}}{(1 + f_{n}(x))^n} \dd x]

[potent lotus]

k

[nimble harbor]

one person suggested that it is the absolute version of ln(1-x) expansion

[sharp narwhal]

i think you can simplify f_n(x)

[nimble harbor]

i find it interesting that ${\frac{\dd}{\dd x} (x + \frac{x^2}{2} + ... + \frac{x^n}{n}) = 1 + x + x^2 + ... + x^{n-1}}$

[potent lotus]

k

[sharp narwhal]

yes

and the rhs is geometric series

then integrate both sides

[nimble harbor]

?

[sharp narwhal]

$$ f'_n(x) = \frac{x^n - 1}{x - 1} $$

[potent lotus]

mumford

[eternal shard]

-ln(1-x)

[sharp narwhal]

what is absolute version?

[nimble harbor]

oh wait

😭

[sharp narwhal]

ah alr

[nimble harbor]

still

what do i do with this

ln(1-x) only holds if it is till infty, no?

[sharp narwhal]

yes

[nimble harbor]

but the question is to n

🥹

[nimble harbor]

i tried it, it gives another geometric sum and 1/x-1

[sweet stag]

whats the question?

[sweet stag]

so we need to evaluate the integral?

[nimble harbor]

ye

[sweet stag]

okii

[nimble harbor]

so..

any idea

on what do i do next

😭

@sharp narwhal

[sharp narwhal]

not sure

could try u = 1/2-x

or do long division

after all it's just polynomial/polynomial

could also test different values of n and find a pattern

[nimble harbor]

ok

i cant find

😭

@eternal shard , any idea?

[eternal shard]

i took some notes but i dont think i found something

[nimble harbor]

damn

thanks for helping tho, everyone

👍

i give up

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[nimble harbor]

question is

?

[jaunty monolith]

how do i write it in vertex form]

like a(x-h)2+k

[nimble harbor]

ok

do u know how to complete teh square

[jaunty monolith]

not really

i forgot

[nimble harbor]

so u add another number that makes it a square

in this case, we compare it to (x+2)^2

[jaunty monolith]

so like (x+2)(x+2)?

[nimble harbor]

I think u should review completing the square 😭

https://youtu.be/C206SNAXDGE?si=jm3F2I6YavOFvF5v

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@jaunty monolith Has your question been resolved?

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[plucky walrus]

??

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[eternal shard]

hi, any questions you have?

[young olive]

• Split the square into 4 sides
• Parametrize each side
• compute the integral for each (annoying but doable)

[noble narwhal]

which one is the first side

[noble narwhal]

what is the second method

[noble narwhal]

which one is the first side or i doesnt change anything ok

??£

which one is it?

[young olive]

Or you can just rewrite (z-1)/z as 1-1/z
So inside the square z = 0 and residue -1
And BOOM Integral = 2pii * (-1) = -2pi i

[noble narwhal]

ok thank you

[young olive]

You sure you understood

[noble narwhal]

and which one is the first side?

[noble narwhal]

not sure but imma do the first method first

[young olive]

Here order doesn’t matter (closed loop) but if you need a starting point:

• top side z(t) = t + i [FROM -1 to 1]
• right, bottom and left [just follow clockwise]

[noble narwhal]

ok

[young olive]

just know all integrals will combine to give -2πi

[noble narwhal]

and this, i dont know what residue is

we've only talked about index

winding number

[young olive]

residue is just fancy math for "what's left at the special point z=0

[noble narwhal]

oh ok, so i rewrite as 1 - 1/z and then what

i dont get the next step

@young olive

[young olive]

integrate

The integral of "1" part is simply 0

The -1/z part is the spicy bit

Its residue at z=0 is just the coefficient (-1)
Winding number around 0 is 1 (since we loop once)

integral = 2πi × (residue) × (winding number)

• residue -1
• winding number 1

use this formula to work out the final answer!

that's literally it
the residue is just the coefficient when your function looks like (something)/z at the special point
the rest is just applying what you know about winding numbers!

[noble narwhal]

oh ok thank youuu

👍

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[young olive]

Glad to help 😄

https://tenor.com/view/hmm-thinking-adding-up-calculating-cute-gif-14475367

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[plucky walrus]

Use the identity sin(x+y) = sin(x) cosine(y) + cosine(x) sin(y)

And cos(x - y) = sin(x) sin(y) + cos(x) cos(y)

[ancient grotto]

!nosols

[calm coral]

As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.

@delicate quest Has your question been resolved?

[delicate quest]

Am confused about the 2nd last step

Where I divided through by cosx

It's like dividing one side of an equation by 2 which completely changes it right

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[plucky walrus]

.wdym

[viral isle]

Am I doing those correct?

[viral isle]

<@&286206848099549185>

[rigid shell]

you should wait a little longer before asking helpers

in 5 and 6 you put local maxima and minima in disjoint points of the function

[viral isle]

I see

At 0 and 6 for 5 and 7 for 6?

[rigid shell]

it is not a mistake, but for example the point in 5. at x=1 is a local minima as all points of the function sourrounding it are greater

[calm coral]

@viral isle Has your question been resolved?

[viral isle]

I see

So we just leave it alone?

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[blissful field]

I just started antiderivatives and got slapped with "integral of (ax+b)^n = 1/a * ( (ax+b)^(n+1) ) / (n+1) )

[blissful field]

so i kinda get how they like

assumed that r = ax+b

then integrated r^n

but where did the 1/a come from

[tawdry flame]

It’d be useful to write the integral with the dx at the end

Because as you integrate r^n you need to replace dx by dr, a function of r

That you get by differentiating r wrt x

[blissful field]

mhm yeah

[tawdry flame]

What is dr/dx?

[blissful field]

a?

[tawdry flame]

Yes

dr/dx=a means dx=dr/a

So the integral becomes

(1/a)(r^n) dr

1/a is a constant so you can separate that and integrate r^n as usual

Then substitute r=ax+b

If that makes sense

[blissful field]

but like

werent we taught that we shouldnt consider dx/dy or anything like that as a ratio

as something divided by something

[tawdry flame]

I suppose you could since in many cases it behaves like a fraction

Or ratio if you like

[blissful field]

yeah fraction

okay hmm

[tawdry flame]

It is not a fraction but it behaves like one

[blissful field]

alright

[blissful field]

im writing this on paper trying to figure out what you did

[tawdry flame]

$\int{(ax+b)^n dx}=\int{r^n (\frac{1}{a}) dr}$

[potent lotus]

pianist

[blissful field]

yeaaaah i got it

yeah makes sense

theres another rule

but i didnt really try to figure it out by myself yet

but its the

[tawdry flame]

Yes?

[blissful field]

idk its weird but it says the integral of f(x)^(n) * f'(x) dx is like- integral of f(x)^n

which is f(x)^(n+1) divided by (n+1)

[tawdry flame]

Yes, and the integral you provided earlier is an example

[blissful field]

isnt that kinda hard to see

[tawdry flame]

$\int{f(x)^{n}f’(x)dx}$

[potent lotus]

pianist

[tawdry flame]

Make the substitution $r=f(x)$

[potent lotus]

pianist

[tawdry flame]

Then tell me if you can, what the integral becomes (with no x’s but only r’s)

[blissful field]

gimmie a sec, my pen disappeared

[tawdry flame]

(And dr)

[blissful field]

OOOH it turns into integral of r * (r with a dash ontop of it) * dx which is dr/f'(x) which is (r with a dash ontop of it) so they cancel out! so we're left with integral of r^n dr

the latex-less of me is

annoying

ik

[tawdry flame]

Dw

I suppose you do a lot of latex since you’re an undergraduate

But thats okay

[blissful field]

what

wait no what does that mean 😭

[tawdry flame]

You are in uni are you not

[blissful field]

no- im in second year of hs

out of three

[tawdry flame]

Oh i see

I’m in the first

👍

[blissful field]

of uni?

[tawdry flame]

Hs

[blissful field]

oh

[tawdry flame]

$\int{f(g(x)) dx}$

[blissful field]

is that- is that a test

whats that

[tawdry flame]

If you let f(x) = x^n

Wait

[blissful field]

oh you're giving me the general formula of how we integrate stuff like that

yes take your time

[tawdry flame]

Yes but im having a brain fart

I think

$\int{f(g(x))g’(x)dx}=\int{f(g) dg}$

[potent lotus]

pianist

[tawdry flame]

I think

But to be honest

No one uses these general formulas

It’s easier to do the substitution

[blissful field]

well im glad

bc-

wtf is that

oh i got it

[tawdry flame]

If you fix $f(y)=y^n$

[potent lotus]

pianist

[tawdry flame]

Then the statement becomes $\int{g(x)^ng’(x)dx}=\int{g(x)^n dg}=\frac{1}{n+1}g(x)^{n+1}$

[blissful field]

no no this is the same thing as be- wait a minute... if we have g(x) then- why do we even integrate to begin with- isnt that the point of this type of integration? bring the function back to un-differentiated form

[tawdry flame]

Some functions are not integrable, but yes

[blissful field]

i mean i see how this doesnt work

[blissful field]

also yeah

this is a bit hard to see

[potent lotus]

pianist

[blissful field]

like noticing the differntiated and normal forms of a function yk

[tawdry flame]

To answer your question then, if we have some power of g(x) in an integral multiplied by g’(x)

Then the integral is easy because of that formiula/substitution

You dont have to physically integrate g(x)

If you let the whole thing, $g(x)^ng’(x)$ be say $S(x)$ then we are just saying that $\int{S(x)dx}$ turns out to depend only on g(x) and n

[potent lotus]

pianist

[blissful field]

WHT

Oh.

[tawdry flame]

That you don’t need to find out the integral of g(x)

But only g(x) itself

[blissful field]

ooooooh

yeah

but wait so

the not so easy questions

is going to look like a function raised to n times the derivative of the function but- like multiplied

not in seperate parenthesees

[tawdry flame]

Uh

[blissful field]

yk what nvm

yeah yeah

ik

tysm for helping me

[tawdry flame]

Not sure if i understand

[blissful field]

i really appreciate it

[tawdry flame]

Np

[blissful field]

have a good day

[tawdry flame]

I’m just killing time since i cant sleep, Gn

Do .close when you’re done

[blissful field]

.close

[calm coral]

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[blissful field]

hey! ive been asking questions here for like two years

[blissful field]

how dare you /jk

[tawdry flame]

Yh

[calm coral]

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[plucky walrus]

Angle CAB > angle DAB.
Prove that minor arc(DB)/DB < minor arc(BC)/BC

[plucky walrus]

If I prove this, I can prove that limit x->0 sin(x)/x = 1

No trignometry allowed.

[velvet osprey]

....

the fuck do you MEAN no trig allowed.

feels like you are bullshitting

[plucky walrus]

No.

[velvet osprey]

how are you gonna get any relationship between arc lengths and chord lengths if you do this stubbornness thing where you forbid yourself from using any trig

plus if your goal's to prove lim sin(x)/x = 1

that's a trig function right there

[plucky walrus]

It's just an inequality.

[velvet osprey]

the sin(x)

you're trying to do trig without doing trig

[mellow harbor]

I mean $\overline{CB}$ is longer than $\overline{DB}$ by the fact that $\overline{AD}$ intersects with $\overline{BC}$ inside the circle.

[potent lotus]

@mellow harbor

[mellow harbor]

Which then implies that $\angle CAB > \angle DAB$

[potent lotus]

@mellow harbor

[velvet osprey]

that one's given then

[plucky walrus]

Yes.

[mellow harbor]

Oh you want the comparision for arcs, mb

Can you use the arclength formula?

[plucky walrus]

theta R
How will one find CB and DB in terms of theta and R?

[mellow harbor]

Yes that one

[velvet osprey]

nah he's gonna scream it's not allowed

knowing him

How will one find CB and DB in terms of theta and R?
which angle is theta again?

also i am sorry to disappoint but that WILL involve trig.

[plucky walrus]

A general angle.

[mellow harbor]

$\because r_1 = r_2, \theta_1 > \theta_2 \implies r\cdot\theta_1 > r\cdot\theta_2$

[velvet osprey]

no, which angle in your diagram are you calling by the name theta?

[potent lotus]

@mellow harbor

[plucky walrus]

theta = angle DAB and theta + beta = angle CAB

[mellow harbor]

Coolios then, does this work? @plucky walrus

[velvet osprey]

ok right theta = angle DAB

in that case $DB = 2r \sin(\theta/2)$

[plucky walrus]

Yes.

[potent lotus]

Ann

[plucky walrus]

This proves that DB < CB

No.

[velvet osprey]

fym no

[plucky walrus]

arc(DB) < arc(CB)

[velvet osprey]

what exactly are you NOing at

[mellow harbor]

This is the formula of arc length

this isn't chord length formula

[plucky walrus]

This.

Yes.

Let's assume that I can prove that DB < CB.

[dense lagoon]

just read this log lol why is anyone even asking to prove this without trig

[velvet osprey]

arcDB / DB < arcBC / BC is equivalent to arcDB / arcBC < DB / BC

if y'all are gonna compare angles to angles and lengths to lengths

you'll have proved that arcDB / arcBC and DB / BC are both less than 1

which tells us nothing of their relationship to each other

[velvet osprey]

it's OP, he is known to introduce stupid constraints into problems of his own authorship

[mellow harbor]

Why is everyone just ignoring the arc-length formula 😭

[dense lagoon]

points for being adventurous ig

[plucky walrus]

Not me.

[mellow harbor]

Then what other query do you have?

[plucky walrus]

What have you proved?

[plucky walrus]

What is this?

[velvet osprey]

your REAL goal is to prove the limit of sin(x)/x as x->0 is 1, yes?

@plucky walrus

[plucky walrus]

No.

[velvet osprey]

bruh ok

[plucky walrus]

I should not have told that.

[mellow harbor]

Would you agree that $\overset{\frown}{CB} - \overline{CB} > \overset{\frown}{DB} - \overline{DB}$?

[dense lagoon]

how about this i just thought of something

[dense lagoon]

as central angle subtended increases, the rate at which the 'arc length' increases is faster than the rate at which the 'chord length' increases

so we can say if a bigger central angle is subtended, the ratio of arc/length is bigger

[plucky walrus]

I don't think so.

[velvet osprey]

how does that shake out

also i think you meant CB not CD?

[potent lotus]

@mellow harbor

[mellow harbor]

Yea, sorry mb lol

[velvet osprey]

but nonetheless i do not see how one could derive this

[dense lagoon]

couldnt this be explained by that msg i sent

[plucky walrus]

Nah

[mellow harbor]

Yea sorry i just read this, I was too busy looking up how to get the arc symbol lol

[plucky walrus]

What you said isn't any proof

[dense lagoon]

its a thought

which might be true idk

any thoughts?

cuz im led to believe that an increase in radius of curvature would lead to an increase in arc length. just seeing how much 'curvier' an arc is

[mellow harbor]

I mean if you really really wanna go that route, I won't help you do this but you can construct a rohumbus for each triangle you have (because they're already isocelas) and you can use half of the area of that rohumbus to get the area of the triangle from wher you can use heron's formula to reverse calculate the third side and soothe your heart

[dense lagoon]

can we just consider the arc/chord ratio as an increasing linear function

[mellow harbor]

Or you can use the determinant formula and make an inverse matrix

[mellow harbor]

I don't think he wants to

[dense lagoon]

for increasing angels

[dense lagoon]

that just makes it hard doesnt it

[mellow harbor]

Also, only true till you reach $\pi$ radians

[potent lotus]

@mellow harbor

[dense lagoon]

yea

i mean clearly he has some plan in his mind

[mellow harbor]

What do you think of this @plucky walrus

[mellow harbor]

Maybe he wants a non-definition dependent proof of $\lim_{\theta \to 0}\frac{\sin \theta}{\theta} = 1$

[potent lotus]

@mellow harbor

[mellow harbor]

Since @velvet osprey mentioned it

[dense lagoon]

ic

[dense lagoon]

i see this working nicely

will be hectic though

[mellow harbor]

that's why I said I am out but you can do this if you want to lol

[dense lagoon]

honestly ill give it a go when im free

[velvet osprey]

he specifically denied it being his goal

[dense lagoon]

where do i put my result when im done with it btw

ill prolly just dm him

[mellow harbor]

Whatever he wants to do

he can do it

[calm coral]

@plucky walrus Has your question been resolved?

[plucky walrus]

oh

I will see this.

Thanks y'all

.close

[calm coral]

Channel closed

Closed by @plucky walrus

Use .reopen if this was a mistake.

[mellow harbor]

I'll send you a desmos graph because I feel bad for you

Where should I send it @plucky walrus

[plucky walrus]

graph?

[mellow harbor]

Yes

interactive garph

[plucky walrus]

ok

Why do you feel bad for me?

[mellow harbor]

I said I won't help you

so I felt that was rude

sorry weird phrasing lol

[plucky walrus]

ok, no problem

[calm coral]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
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• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[proper ocean]

HELP????

[proper ocean]

WDYM ITS

[nimble harbor]

,rotate

[proper ocean]

Y+ 3X =11

[distant fog]

Yes

What you want my man

[proper ocean]

HOW DID

[distant fog]

How did what

[proper ocean]

THIS

QUESTION B

[distant fog]

Yeah what to do with question b?

[remote mural]

what do you even have to do ?

[proper ocean]

FIND AD

LIKE HOW

Y = ⅓X + 1 FOR AB

IGNORE THE PENCIK

D (0,y)

[distant fog]

That looks 0,4 to me

[proper ocean]

gradient of AD is -3

[proper ocean]

FORGET THAT

I WAS WRITING

,rotate

[potent lotus]

Couldn't find an attached image in the last 10 messages.

May 3, 2025, 3:38 PM

[proper ocean]

,rotate

[proper ocean]

OKAY SO

WHAT

[distant fog]

Soo hmm

What have you tried so far

To find equation of AD

[proper ocean]

IDK BRO

I KNOW THAT

[distant fog]

,rotate

[distant fog]

,rotate

[proper ocean]

X =0
y = ½x + 1

For D

[distant fog]

Well dang

I dont know man

All i can say is that you have nice hand writting

Better than my

[proper ocean]

WHY ARE YOU HERE THEN

BSGDIDISJE

Dear prime my apologies

[distant fog]

Brain storming

[distant fog]

🙏

[proper ocean]

Aren't u in college

[distant fog]

What made you thought i am in college

[proper ocean]

Nvm

<@&286206848099549185>

[distant fog]

I am in 9th dang

[proper ocean]

🙏🙏🙏

[proper ocean]

Grade 9?

[distant fog]

Yeah, grade 9.

[proper ocean]

Same gng🙏

I already finished my syllabus

[distant fog]

🙏🙏

[proper ocean]

Im doing grade 10 add maths

🙏🙏

[distant fog]

My syllabus just started

[proper ocean]

FUCK COORDINATE GEOMETRY

BRO GRADE 9 MATHS VS GRADE 10 ADD MATHS IS WAY HARDEE

[distant fog]

I thought thats easiest chapter smh

[proper ocean]

OMH

[proper ocean]

I FUCKING HATE IT

[distant fog]

Are you by chance from southeast asia

[proper ocean]

No

[distant fog]

Ok i guess

[distant fog]

You should ask this to chatgpt

[proper ocean]

me rn

Someone help

[proper ocean]

Okay

Im gonna

Tweak

[unreal isle]

!nogpt

[calm coral]

Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).

[unreal isle]

alright

hello

first off i need you to calm down

[proper ocean]

HELP

SO

Okie

[proper ocean]

Yes

[unreal isle]

.

thank you

now

[proper ocean]

Tech vs daq

Tech f4

[distant fog]

Dang

[unreal isle]

,rccw

[potent lotus]

Couldn't find an attached image in the last 10 messages.

May 3, 2025, 3:46 PM

[proper ocean]

This

[distant fog]

Then i guess use gemini

[proper ocean]

,rotate

[unreal isle]

he's in 10th grade

[proper ocean]

Use deepseek

[distant fog]

I was joking 🙏

[proper ocean]

Im in grade 9th🙏

[unreal isle]

anyways ignore him OP

alr so q16?

[proper ocean]

Yes

[unreal isle]

have you done any work?

i think i saw some up there

would love if you could reshare

[proper ocean]

,rotate