help-19

[quasi sparrow]

https://www.google.com/

[pulsar tiger]

I looked there. I couldn't find any

Hence, why I am asking for specifics

[quasi sparrow]

google harder

[pulsar tiger]

I did.

Just saying that won't change anything

[quasi sparrow]

even harder

[pulsar tiger]

I did

[quasi sparrow]

https://tenor.com/view/doubt-press-x-la-noire-meme-x-button-gif-19259237

[pulsar tiger]

I did. Cope

[tidal matrix]

unfortunately, neither does anyone else. you can try to formalize what Denascite said (take delta = min(delta1, delta2) ), and you can ask for help along the way. if not, try either copy-pasting this question into google, or searching "bartle & sherbert chapter X exercise solutions" - this is a pretty well-known book so there are quite a few solutions

[odd edge]

@pulsar tiger Has your question been resolved?

[pulsar tiger]

Okay, fine.

How is this:

We will establish the continuity of h at b. Since f is continuous at b, given
ε>0 there exists δ1 >0 such that if b−δ1 <x<b, then |f(x)−f(b)|<ε. Similarly,
there exists δ2 >0 such that if b<x<b+δ2, then |g(x)−g(b)|<ε. Let
δ := inf{δ1, δ2} so that |h(x)−h(b)|<ε for |x−b|<δ, whence h is continuous
at b.

[woeful vine]

are you sure here? Supposing δ₁ > δ₂ if you take b-δ₁ < x < b+δ₁ then you have no control over numbers between b+δ₂ and b+δ₁ from the given info

[meager juniper]

you're right

I'm a dummy

[quasi sparrow]

Lmao did you plug your question into chatgpt

[pulsar tiger]

I found the solutions text book

These are partial solutions though

[quasi sparrow]

Great you've done this

[pulsar tiger]

Yeah, but they are incomplete.

Partial solutions. It is a bit underwhelming

[quasi sparrow]

Keep using

[odd edge]

@pulsar tiger Has your question been resolved?

[pulsar tiger]

I've been looking for so long. I need more than just partial solutions.

[quasi sparrow]

What's missing

Try to write whatever you think is missing yourself and put it here

People will help when you actually demonstrate effort and ask a more specific question

[quasi sparrow]

Also doubt

[hollow ginkgo]

What is partial about the solution you just wrote out to you?

[pulsar tiger]

I guess nothing is missing. I am stilltrying to understand why we take the lower bound interval of b-delta1 and the upper of b +delta 2 for f and g, respectively.

[woeful vine]

<@&268886789983436800>

[hollow ginkgo]

<@&268886789983436800>

[hollow ginkgo]

They are unwrapping the definition of the limit from the left and right.

[low locust]

the definition of continuity requires that you go the same distance to the left and right

you cant go delta1 to the left and delta2 to the right

so you just take the minimum of those two, cause that much you can go in both directions

[hollow ginkgo]

One of the sided limits gives you one delta, the other sided limits gives you the other delta. Pick something smaller than both and you have a delta that works on either side.

[pulsar tiger]

Oh, I see. That makes sense

[hollow ginkgo]

Another option is to try and rephrase using sequential characterizations of limits and continuity but that seems like overkill

[pulsar tiger]

My professor sucks at teaching. I will be ecstatic once this class is over

[hollow ginkgo]

Analysis is kinda hard the first time you encounter it.

[pulsar tiger]

It is too hard for me

[amber schooner]

bro aren’t you a mod?

[pulsar tiger]

I don't have the aptitude and background for this

[hollow ginkgo]

Not currently

[amber schooner]

so you don’t have the power to ban them?

what does reserve mod do then

[hollow ginkgo]

As you get used to it it gets easier. A lot of the sort of standard tricks become familiar and intuitive after you see them a lot.

[amber schooner]

why is it all his fault

how does blaming other people for your own shortcomings help?

[hollow ginkgo]

I mean, they could be bad. It's complicated because analysis is challenging and that causes frustration in people.

[amber schooner]

sure but you don’t need the greatest teacher in the world to learn first year analysis

and having the attitude of "oh it’s just because my teacher sucks, that’s why i can’t learn" is unproductive

[pulsar tiger]

I am more than halfway done with this deplorable course.

Nothing has clicked

[hollow ginkgo]

Idk there's no way to actually tell if their prof is good or not so it's best to just not worry about it.

[pulsar tiger]

Because I feel it is justified. My life is in his hands to graduate in August

[amber schooner]

no it’s in yours

[hollow ginkgo]

A lot of it is shit you just get used to rather than some big clicking moment.

[pulsar tiger]

His trash pedagogy approach isn't good

[amber schooner]

if you study enough and properly then you can learn and improve

[amber schooner]

there are online lectures

go to mit ocw real analysis

go to bright side of math

there are countless resources at your disposal

so there’s really no excuse

[pulsar tiger]

Dude. It isn't. I can't continue schooling if I fail. I will literally be a university dropout with one credit left.

[amber schooner]

ok and you’re in control of how you perform by the effort you put in?

[hollow ginkgo]

I don't see how pseudo-chastising them over a situation we aren't part of is helpful.

[pulsar tiger]

I haven't gotten used to it.

[hollow ginkgo]

Keep trying

[amber schooner]

have you done this?

have you watched online lectures

[pulsar tiger]

I invested my day to day life in this class. It will be his fault.

[hollow ginkgo]

Throwing around blame and fault isn't helpful on your end either

[amber schooner]

you mentioned previously that everyone else in your class had like 80+ averages

[hollow ginkgo]

We can't do anything about your situation with your prof ya know

[amber schooner]

how is it his fault then if his teaching seems to be successful with other students

you’re attending the same lectures

[quasi sparrow]

seems like amount of time blaming and complaining >> amount of time reading actual real analysis

[pulsar tiger]

No. I've only been using the book, other online resources, and a few discord servers to solidify the content

[hollow ginkgo]

How are you typically studying?

[pulsar tiger]

Because it should accommodate those who are geniuses and those who are below average.

[amber schooner]

i suggest watching the bright side of math real analysis playlist, mit ocw real analysis playlist, and the michael penn real analysis playlist

they’re all great

[hollow ginkgo]

They are good resources

Doesn't peyam have some math vids on analysis?

[amber schooner]

i doubt that every last kid besides you is just some sort of prodigy

[pulsar tiger]

Re-reading the notes, going to OH, reading through the book, writing down definitions, forming a study guide, doing more practice problems.

[amber schooner]

real analysis is a hard course sure but if others can succeed in it in the same class then you should be able to succeed as well

[amber schooner]

yes

[hollow ginkgo]

Do things make sense when you do these things?

Like maybe not your first time doing a problem

[amber schooner]

wrath of math is good too

i think that’s his name

[pulsar tiger]

No

[hollow ginkgo]

But when you review it looking back at the solution does it make sense?

[pulsar tiger]

It takes a long time redoing all of the above to understand.

[pulsar tiger]

After many iterations, sure.

[hollow ginkgo]

Okay, but if you do eventually reach a point where you understand things that is good.

[pulsar tiger]

Give me new problem I haven't seen before, it will take me just as long.

[amber schooner]

when does your course end?

like when’s the final

[hollow ginkgo]

Without more context that is a normal part of learning.

Like, external factors may make you want to be faster

But when you are fresh to a topic trying to learn all the tricks and ideas it's kinda just a slow uphill battle in which you beat your head against the wall until it clicks.

[pulsar tiger]

2nd week if August, I believe.

2nd week if August, I believe.

[amber schooner]

ok you have a whole month then

[tidal matrix]

especially in an introductory analysis course! the formal definition of the limit is, from my observation, difficult to grasp for students

[amber schooner]

if you do sufficiently well on the final i assume you can pass

you probably still have homeworks

what does your course go up to?

in terms of topics

[amber schooner]

probably because people who teach it just throw quantifiers at the students instead of using words and translating what it actually means

[tidal matrix]

i thought you were on the prof's side

[amber schooner]

i am

if the rest of the class is doing exceptional (80s/90s) but you’re getting 30s then clearly the professor isn’t the problem

[tidal matrix]

i think it just takes students a while to internalize the arbitrary inequalities and what they mean visually, i.e. there are a lot of mistakes in the absolute value inequalities

[amber schooner]

|x - a| could easily be replaced with d(x, a) and would save so many students hours of their time trying to understand

which is why i think the general metric space definitions should be introduced earlier or at least the notation

after i learned metric spaces the initial limit stuff became obvious

translate everything into distance and open balls/sets

[tidal matrix]

sure, calculus is easier if you learn measure theory first, but you cannot start with measure theory and go to calculus

[amber schooner]

i never mentioned measure theory

[tidal matrix]

d(x,a)

[amber schooner]

metric spaces are covered in intro analysis courses

[amber schooner]

this isn’t measure theory

[amber schooner]

essentially first 7 or so chapters of rudin

other than spivak like all of the analysis books cover topology/metric spaces

[tidal matrix]

they do, but usually introduced in the context of |x-a|, then generalized later

[amber schooner]

bartle might not or is that the one where they do it as the last chapter

i think it is actually

yea drop that book then lmao

horrible

[tidal matrix]

disagree, but this is a discussion for not here

[odd edge]

@pulsar tiger Has your question been resolved?

[odd edge]

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[errant plank]

aren't inverse functions and functions meant to undo each other? if so how come the answer to this problem isn't (3pi)/4

[narrow crypt]

What is the

Range of arcsinx

[mystic saffron]

Cuz it isnt in the range

[errant plank]

oh

of

[-2pi,2pi]

[mystic saffron]

Bruh

[errant plank]

right?

[mystic saffron]

Its pi/2

Not 2pi

[errant plank]

oh

wait

[calm raft]

it goes -1 to 1

[errant plank]

this actually makes so much sense now

lol

[mystic saffron]

So u have to change the angle withing this range

[errant plank]

(posting this here to help me)

so

it would just be pi/4

[narrow crypt]

👍

[mystic saffron]

Ye

[errant plank]

ok thank you so much you two

.close

[odd edge]

Channel closed

Closed by @errant plank

Use .reopen if this was a mistake.

[errant plank]

.reopen

[odd edge]

✅

[errant plank]

nevermind I don't think I fully understand

for this one

theta would = (2pi)/3

[narrow crypt]

Range of arccosine

Is 0 to pi

[errant plank]

its the same range as the last problem right?

[narrow crypt]

No

[errant plank]

wait why?

[narrow crypt]

Cuz the range of cosine is fully in the top semicircle

[errant plank]

hmm?

I can't say I follow what is the "top semicircle"

[errant plank]

wait

I got mixed up I was looking back at the wrong problem

[errant plank]

and that one had arcos

so

has to be in quad 1 or two

[narrow crypt]

Yup

[errant plank]

and its the inverse angle of 2pi/3

[narrow crypt]

For arccos

[narrow crypt]

Yup cuz cos(2pi/3) = cos(4pi/3)

When u do this

It’s appropriate to convert ur angle to match the range of these arc functions

[errant plank]

wait

but (2pi)/3

is in quad 2?

so shouldnt it be good?

[narrow crypt]

It should

[errant plank]

oh

but not in domain

[narrow crypt]

?

[errant plank]

so pi/3

[narrow crypt]

2pi/3 is in range of arccos

[errant plank]

ok

[narrow crypt]

,w arccos(cos(4pi/3))

[clever fjord]

Results provided by WolframAlpha

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

Requested by nuclearinteraction

[narrow crypt]

2pi/3 is correct

[narrow crypt]

Just get the angle into the appropriate quadrants here

[errant plank]

I swear I entered that in before

but apparently I didnt

ok I got it

[narrow crypt]

👍

[errant plank]

.close

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[terse mauve]

can i have some help on 1.2.1 a and b?

[terse mauve]

im not sure i fully understand the significance of the proof they gave

it was just showing that p and q must have a common factor which is for whatever reason an absurdity and makes it irrational

didnt follow that logic wel

[final ether]

Exactly

[terse mauve]

why does that imply that tho?

[steep mantle]

have you done this with proving sqrt(2) is irrational?

[terse mauve]

i read their proof yeah

but my uncertainty with their logic carried here

[steep mantle]

ah didnt see this

its either they did proof by contradiction, or infinite descent but im going to assume they did proof by contradiction here

[terse mauve]

yea they did

[ivory grove]

they're doing it with proof by contradiction; they assumed that it wasn't irrational, which meant there are some numbers p and q such that p/q is a fraction, and it's in reduced form. and then they proved that this assumption doesn't work because p and q actually do have a common factor

if they assume that

which leads to a contradiction

[final ether]

it is reasoning by the absurd we want to show P so we suppose -P and we show that -P not true so necessrly true

[terse mauve]

but why does the fact that they must have a common factor then imply it is irrational

[ivory grove]

and hence the assumption isn't true

[ivory grove]

because they assumed that p/q is in reduced form

[terse mauve]

but we forged the assumption, why was the common factor part ever important

[ivory grove]

and did some work with it, and then it turned out not to be in reduced form

which is a contradiction

[steep mantle]

basically, at the start they assumed it could be written as a fraction p/q, since any fraction can be simplified to a fraction that cannot be simplified further, thus they assumed p and q share no common factors, then they proved that p and q must share a common factor, however that contradicts with p and q not sharing a common factor, thus that raises a contradiction thus the original assumption is incorrect

[terse mauve]

but why do we need to assert it is fully reduced

[ivory grove]

think about it like this: if we assume something like, birds can't fly. Then, we did some work with it and realized that, if that were the case, then we would be able to catch birds much easier than now. But catching birds is still hard. So, birds can fly

[short topaz]

else they could hav a common factor

[ivory grove]

-# idk weird example

[steep mantle]

if you assumed i have an object in my hands, and with that assumption you got that i dont have an object in my hands, then your original assumption was impossible/flawed

[ivory grove]

it'll be much easier for us, bc why not? a rational number can always be expressed as p/q with gcd(p,q)=1

[terse mauve]

yeah sure i understand its flawed, but i dont aee the reason to ever impose its reduced

[steep mantle]

i mean you can get away with not letting it be fully reduced, but its more complicated, its called "proof by infinite descent"

[final ether]

So supposing -P is supposing that sqrt(3) is not a irrational so he is a rationnal and the définition of x is rational is there exist p,q which p,q have no common factor such as x=p/q and we want to show that’s not possible so the only way to find a contradiction is working on our supposition here it’s p and q have a common factor se we must show they don’t have

[ivory grove]

I think ur question is, if we don't assume that it's reduced, then the proof doesn't work, right?

[terse mauve]

yea

i dont see why we need ever mention it

it doesnt seem to be relevant

[ivory grove]

do you agree that a rational number can always be expressed as p/q with gcd(p,q)=1?

the writer of the proof thought of that, and did some work with it, hoping to find a contradiction. and then they did

[ivory grove]

it is relevant, bc it's about the definition of a rational number

[terse mauve]

yes it can be

ohhh

wait

so if it cant ever, then it cant be rational

and since we showed for all p and q it never is its never rational

[ivory grove]

precisely

[terse mauve]

okay also on a side note im trying to practice my pure symbolic proof writing, could you show me how to do that example?

id like to practice it on the exercises

[ivory grove]

that's not a question for me sry I can't do pure symbolic proofs (well I did, but I forgot )

[terse mauve]

😔

alr thanks

[ivory grove]

np

[terse mauve]

!close

what is it again

[ivory grove]

it's .close

[terse mauve]

been a minute 😭

.close

[odd edge]

Channel closed

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[steep mantle]

it is ".close"

damn my internet sucks

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[terse mauve]

b

[terse mauve]

@ivory grove i dont see how this breaks for root 4?

[ivory grove]

well, it can be written as p/q in reduced form

and when we assume that, we don't actually get a fraction not in reduced form

[terse mauve]

wdym

[ivory grove]

what happened with sqrt 3 and sqrt 2 was that, once we did some work while assuming p/q is reduced, we actually got that p/q is not reduced

so basically, we assumed that the sky is red and did some work and realized the sky is purple

[terse mauve]

yea

[ivory grove]

-# wild example but ykwim

here, that doesn't work with sqrt 4

bc when we assume that it's reduced, it is reduced, and we don't get that the sky is purple

[terse mauve]

uhh

but i can show p and q and divisible by 2

meanig its not reduced

[willow swift]

What can I help you?

[steep mantle]

show it then

[steep mantle]

you are assuming that p can be written as 4r, thats an assumption as you only got that p is even, which implies p=2r, not 4r, also you should write r is an integer, not r is rational

[terse mauve]

oh mb yea

okay but then what else

[steep mantle]

also for sqrt3 being irrational for some reason you qrote the proof for sqrt2?

[terse mauve]

ye ignore that

just look at bottom

[steep mantle]

should be fine, but fixing your mistake would get you why it doesent work

[broken schooner]

what question are we having here?

[terse mauve]

okay i see

q^2=r^2 always holds

@steep mantle is this all i need to do?

[steep mantle]

it looks fine i think

[terse mauve]

alr thx

preciate you🙏

.close

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[sonic pewter]

So my text book work solutions does this step in one but i don't quite understand how they did it

[sonic pewter]

This is what I’ve done

[broken schooner]

um it is identity

sec^2=1+tan^2

and sin^2=1-cos^2

those are 2 identities

tell me if you want proof

[green elm]

just take sin^2 + cos^2 = 1 and divide it by cos^2 to get the first identity

[sonic pewter]

ohhhhh

i think i understand

wait sorry how do i get the second identity

[green elm]

take sin^2 + cos^2 = 1 and subtract cos^2 from both sides

[sonic pewter]

thank you :)

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[frosty sundial]

i set u= x and dv=tan^-1

[amber schooner]

ibp

[broken schooner]

cowking right method wrong way

[amber schooner]

tan^-1 differentiates to something nice

differentiate that and integrate 1

[broken schooner]

what you do is set u=tan^-1x and dv=dx

[frosty sundial]

ohhh

that makes more sense

oh

and bc dx is already in differential form

it can be set =dv

[amber veldt]

im concerned that you attempted this though

[frosty sundial]

how do i differentiate tan^-1 (x)

[amber veldt]

it suggests you are very confused

[amber schooner]

you should remember this before starting integration

[broken schooner]

um do you want straight answer or a process?

[frosty sundial]

no i just didnt realize that x was part of arctan

[wooden python]

its derivative is 1/(x^2+1), which is frankly sth you should know for integration

[frosty sundial]

i dont remember the ivnerse trig derivatives

[wooden python]

shame then you gotta

[broken schooner]

I mean you can derive it yourself

[amber schooner]

you could always figure this out on your own if you forget just by using implicit differentiation

[amber veldt]

so derive them

1/dy/dx=dx/dy

[frosty sundial]

ok alr

[broken schooner]

to derive y=tan^-1x

you have tany=x

sec^2y*y'=1

y'=1/sec^2y

y'=1/(tan^2y+1)

y'=1/(x^2+1)

there you have it, derivitave of arctan

[amber schooner]

derivitave

[frosty sundial]

whyd you replace tan with x

[broken schooner]

tany=x

tan^2y=x^2

[frosty sundial]

oh yea

ok so i have my u, du, dv, and v

then its

uv * integral of vdu

right?

[amber schooner]

nope

[frosty sundial]

integral of udv?

[amber schooner]

that’s what you have originally

not *

[frosty sundial]

oh shoot

it would be vdu

[broken schooner]

integral of udv=uv-integral of vdu

[frosty sundial]

it would be - not *

alr gimme a sec

how can i integrate x/x^2 +1

i keep getting x*ln(x^2+1)

but thats not part of any answer choice

[amber schooner]

wut

[broken schooner]

omg

integrate x/(x^2+1)

[amber schooner]

u = x^2 + 1, du = 2x dx

[frosty sundial]

is it that bad 😭

[amber schooner]

yea

[amber schooner]

x is not a constant multiple and just because something is in the denominator doesn’t mean the integral is just ln(denominator)

[broken schooner]

why did you have to integrate x/(x^2+1)?

[narrow crypt]

[frosty sundial]

vdu

[narrow crypt]

Another integral from ibp

[frosty sundial]

v=x bc dv=dx

[amber schooner]

yep

[frosty sundial]

bc we set u=atan(x)

[amber schooner]

you’re correct

no need to explain

but you integrated wrong

[frosty sundial]

i dont get it

[broken schooner]

okokok

[amber schooner]

what’s confusing

[frosty sundial]

whats the integral of x/x^2+1

[broken schooner]

tell me

what is v

[frosty sundial]

is x

[narrow crypt]

U sub

[frosty sundial]

and du=1/x^2+1

[amber schooner]

i’m guessing your thought process was "oh so just ln(denominator) but then we have to multiply by the numerator because of constant multiple"

[narrow crypt]

Knief gave u the sub choice already

[frosty sundial]

OHHH

I SEE IT NOW

wait

holdup

let me cook

ok wait

i have an answer

ladies and gentle man

xtan^-1 (x)- (ln(x^2 +1))/2

+C

BOOM

[broken schooner]

I was about to yell

XD

[quasi sparrow]

,w int arctan(x)

[frosty sundial]

mic drop

[clever fjord]

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Requested by rie.mann

[quasi sparrow]

Looks right

[frosty sundial]

bro

are you serious right now

wait

[quasi sparrow]

You should also differentiate your answer to check

[frosty sundial]

whys it using log instead of ln

[broken schooner]

some place does it

[frosty sundial]

alr

[broken schooner]

both means log base e

you are good

[frosty sundial]

bro

my next question

is the same thing

but instead of tan^-1 (x)

its 2x

💀

[broken schooner]

same procedure

[frosty sundial]

,w int arctan(2x)

[clever fjord]

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Requested by cowking9831

[frosty sundial]

got a 100 on my workshet

[broken schooner]

sounds nice

[frosty sundial]

thanks @broken schooner and @amber schoonerand @quasi sparrow thank you for the end clutch

.close

[odd edge]

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Use .reopen if this was a mistake.

[frosty sundial]

hell nawr bro

[amber schooner]

is this khan academy

[frosty sundial]

didnt teach my trig sub which made me sad

so i taught myself that

[amber schooner]

are you using a book as well?

you should pick up a book

[quasi sparrow]

Also didn't teach how to use leibniz rule for differentiation under the integral

[amber schooner]

lol

[frosty sundial]

after calc 2 i plan on doing ODE, PDE, and Applied Physics

[frosty sundial]

yea

[frosty sundial]

il look that up

[amber schooner]

khan academy isn’t sufficient for learning these things

[frosty sundial]

i got a really good one too

[amber schooner]

youll need linear algebra

[frosty sundial]

apparently the book was used by feynman to teach himself calc

[amber schooner]

don’t tell me

[frosty sundial]

which is what ima do

[amber schooner]

was it calculus for the practical man

[frosty sundial]

yee

[amber schooner]

🤣

yea i used it

garbage

[frosty sundial]

i have a book on dat

[frosty sundial]

lmao

is that bad

[amber schooner]

i found it from math sorcerer when i was still naive

it’s like my first chat

riemann helped me with some bullshit problem

yea it’s bad

[quasi sparrow]

That explains why he kept reinventing the wheel

[amber schooner]

the solutions in the back were wrong multiple times

it’s a book from like 1930

what do you expect

[frosty sundial]

im using 3rd edition or sum so idk tho

ima jsut read through it

if i need other books il get them

[amber schooner]

he didn’t introduce anything even remotely related to a limit until the very end where he made a brief comment like "in recent years people adopted this idea of a limit"

likes it’s that ancient

doesn’t define it at all

he used basically just applied math the entire time

like physics problems to teach calculus

[frosty sundial]

oh yea

[amber schooner]

or chemistry problems

[frosty sundial]

but i took ap phys 1 so irdc

[amber schooner]

use a better book

[frosty sundial]

and i also know limits so

[frosty sundial]

ima read through it and then use other books

[quasi sparrow]

Cs get degrees amirite

[frosty sundial]

cuz theres always SUM knowledge to gain

[amber schooner]

please don’t waste your time brother

he defines derivatives using velocity

[frosty sundial]

gang i have like 8 other books on my desk on the same topic

im in no rush

[frosty sundial]

sal khan did something similar with negative integrals

[amber schooner]

lemme find it one second

[quasi sparrow]

.

[frosty sundial]

plus after reading both that book im gonna do a thorough self study of any topics i missed through online books, curriculm, etc

[tribal ingot]

[amber schooner]

fuck i cant find it

[amber schooner]

yikes

why do people fanboy over feynman

😭

i’ll never get it

[frosty sundial]

leave no page unturned and no topics uncovered

[amber schooner]

it’s a terrible book dawg

[frosty sundial]

i just heard it was a good book

[amber schooner]

i did every exercise

[frosty sundial]

makes sense

[frosty sundial]

lmaoo

im only goona read half the book or so anyway

alr gn gang

.close

!close

close ts

.close

[quasi sparrow]

Some people have to learn the hard way

[amber schooner]

some people like to think they’re learning more than they like to learn

feynman bro!

too much pop physics on youtube

[odd edge]

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[shrewd trellis]

would my answer be correct, i said minimum turning point but in the answer it said local minimum

[wooden python]

yes it's ok

[shrewd trellis]

alr thanks

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[mighty epoch]

Hey there
I have an upcoming speed math comp and lots of questions are like these:

I understand and can solve them but I have trouble solving it in the time given. If anyone is able to provide me with a faster solution that would be much appreciated. 🙂

If a=1 and b=4 and c=9 and so on, what is the value of M?

[modern ingot]

what number is m in the alphabet?

[odd edge]

@mighty epoch Has your question been resolved?

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[analog mantle]

first year math student in linear algebra

textbook says that every linear transformation between finite dimensional inner product vector spaces that preserves inner products is also bijective

i.e. if V,W are finite dimensional inner product vector spaces and T:V -> W is linear, and

[analog mantle]

$\langle T\alpha\vert T\beta\rangle=\langle \alpha\vert\beta\rangle,,\forall\alpha,\beta\in V$

[clever fjord]

K

[analog mantle]

where lhs inner product is in W and rhs is in V

then T should be isomorphic

[wooden python]

hmmmmmmmmmmmm that sounds a bit sus. what if W has way higher dimension than V

can you post the whole thing from the textbook

[analog mantle]

yeah that is what I was thinking

[wooden python]

just in case you missed some detail

[analog mantle]

these are actually lecture notes and not the textbook itself so maybe professor made a long typo or something

[wooden python]

textbook says that every linear transformation between finite dimensional inner product vector spaces that preserves inner products is also bijective

where does it say that

[analog mantle]

in the footnote, "operators which preserve inner products are guaranteed to be isomorphisms"

[wooden python]

mmm

i mean this is patently false as stated, idk what else to say

[analog mantle]

oh is it maybe that "linear operator" is only V->V whereas "linear transformation" is V->W

[wooden python]

i guess so

[analog mantle]

ok makes sene I guess 👍

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[elfin zodiac]

It doesnt say every

[odd edge]

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[pulsar tiger]

Clarification question:

Is f [a,b] -> R uniformally continuous on closedi ntervals but not on open intervals?

[sinful grove]

Hm, could you clarify your question? If f is continuous it’s uniformly continuous the way you’ve defined f

[pulsar tiger]

If we had an f: (a,b) -> R, then would it be guarenteed to not be uniformally continuous?

[sinful grove]

All subsets thereof (of [a,b]) it’s uniformly continuous on

[low locust]

no

[sinful grove]

You can’t guarantee

It depends

[pulsar tiger]

I guess 1/x from 0 coming from the right would go to infinity slowly, but on the left side, it would go quickly?

[sinful grove]

Yeah if (a,b) is say (0,1) then 1/x would be such an example, it’s more to do with how it behaves near 0

[low locust]

1/x goes to +-infinity "at the same speed", no matter from the left or from the right

[sinful grove]

But if (a,b) is arbitrary then we can’t guarantee 1/x to not be uniformly continuous if that answers ur question (just avoid 0 a closure away)

[pulsar tiger]

It must have a satisying f for the Lipschitz Condition to be uniformally continuous, right?

[sinful grove]

Not necessarily, but sufficiently so

[pulsar tiger]

If sufficiently, what else?

[sinful grove]

There are functions which are not Lipschitz but uniformly continuous

So f being Lipschitz is not a necessary condition

[pulsar tiger]

I had to take the derivative of sqrt(x) to see that now.

Not sure how to do it without evaluating a derivative.

[ebon harness]

since domain of f is compact, continuity implies uniformly continuity i think you can proceed with that

[copper quarry]

no way cyrenux in a help channel

[sinful grove]

How to determine if a fucntion is uniformly continuous?

[pulsar tiger]

Yeah

[sinful grove]

There are multiple ways, that you can combine aswell. Like for sqrt you could just cut up the domain in overlapping bits, one where you know its Lipschitz and one where the set is compact. Then it’s uniformly continuous on both bits; and since they overlap then the sqrt is uniformly continuous on all of its domain

Like by [0,69] and (3, infty)

For example

One I like that is an iff (which helps you to see if it’s either uniformly continuous or not) is that if say f is defined o a bounded set, then it is uniformly continuous iff there’s a continuous extension (of f) on the closure of the bounded set

Take like before f: (0,1) -> R, with f(x) = 1/x. we can directly see that it’s not uniformly continuous by the above theorem

[odd edge]

@pulsar tiger Has your question been resolved?

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[wicked cloud]

Whats happening here? I searched for help with my dad but he didn't know, i searched for websites that may have this and they have different answers from each other

I even attempted it but ended up with multiple answers
(2nd image for more clear image)

[ivory grove]

Which part are you having trouble on? part (a), part (b), …?

[wicked cloud]

part a

[ivory grove]

give me a sec to read this

[wicked cloud]

oki

[ivory grove]

try to use the fact that triangle BCD is isosceles

can you find angle BCD?

(or DBC, they're the same thing)

[wicked cloud]

I can try

[mystic saffron]

Hi

[wicked cloud]

Hi

[ivory grove]

[brittle plinth]

if you're having a problem try dropping a perpendicular from D to BC

[wicked cloud]

180-70-38 = 72
12/sin 70 × sin 72 = 12.1451cm?

[ivory grove]

wait!

[wicked cloud]

Oh wait angle

[ivory grove]

38º is not part of BCD

[wicked cloud]

[wicked cloud]

Oh yeah when you remove the line

Now im just staring at 12/sin70

[ivory grove]

but BCD is isosceles

doesn't that mean you know the other two angles?

[wicked cloud]

Angle C is 40° isnt it

[ivory grove]

which C?

[wicked cloud]

BCD

[ivory grove]

there are 3 angle Cs

[wicked cloud]

THERES THREE?

[ivory grove]

mmm, no, it's not 40º

[rich snow]

which grade question is this lmao

[ivory grove]

please don't chat here pls

[rich snow]

mb gng

[ivory grove]

let me grab a picture to show u wait a sec

[ivory grove]

ur all good

[wicked cloud]

Oki

[rich snow]

i was trying to solve it depending on which grade so that the user may understand it in their level

[wicked cloud]

10th grade(?)

We use a different naming system here

[ivory grove]

ACB, ACD, and BCD are all angle Cs so it's important that you refer to them by three letters

[wicked cloud]

Ohhh

[ivory grove]

ok anyway, you know that BCD is isosceles.

[wicked cloud]

Ye

[ivory grove]

BD=CD, right?

[wicked cloud]

Yes

[ivory grove]

so

[brittle plinth]

focus on ∆BCD only

don't let the other elements make you go 😵‍💫

[wicked cloud]

[ivory grove]

can u find the other angles?

[wicked cloud]

55 ?

[ivory grove]

mhm

now u can use the law of sines

and don't forget that BD=CD

that'll make things easier

[wicked cloud]

Wait this one is related to the 70° in the question

And then when it splits into two cause its an isosceles its 55°

[ivory grove]

yeah, this one is ∆BCD

[wicked cloud]

[wicked cloud]

10.4607?

[ivory grove]

what is this the value of? sin C? a side length? etc.?

[wicked cloud]

The side length of C

BCD

[ivory grove]

of CD I'm assuming you mean

[wicked cloud]

Ye

[ivory grove]

ok lemme calculate

yeah that seems right

[wicked cloud]

Yipeee

[ivory grove]

do u have any more questions?

[wicked cloud]

Why did you ask me about this? Is it so i can imagine it easier with this rather than the question?

[ivory grove]

yeah

We only need ∆BCD

[wicked cloud]

I didnt think BCD was 55° considering I was just taught you can just 180-70-70

[ivory grove]

so I took it out and drew it like that

[ivory grove]

oh, the problem is that 70º is the apex

not the base angles

there is double of 55º, not two 70º s

[wicked cloud]

THATS WHAT IT WAS ???

[ivory grove]

mhm!

[wicked cloud]

Wait so

[ivory grove]

I was a bit @.@ at the problem too, until I focused on ∆BCD

[wicked cloud]

I put the B angle a 70° too

[ivory grove]

,rccw

[ivory grove]

yeah

and that is also the same for BD

[wicked cloud]

Ok so

BCD is 55° because its an inclined plane

We got 55° from 70°

[ivory grove]

no, not because it's an inclined plane

[wicked cloud]

[ivory grove]

because it's an isosceles triangle

DB=DC

^^ this is given in the problem

[ivory grove]

yeah, this is true

[wicked cloud]

Oh i see it now

Ohhhhh

[ivory grove]

yeah

[wicked cloud]

🧍‍♀️

Yeah that makes everything easier actually

[ivory grove]

wait no

that was not the triangle we worked with

give me a sec

[wicked cloud]

[ivory grove]

this one

the base

@wicked cloud

[wicked cloud]

Oh my poor internet hold on

[ivory grove]

[wicked cloud]

So the DBC is also 70° or is it just BDC

[ivory grove]

hold on, lemme send another picture

(the green marks just means that those two lines are equal)

@wicked cloud

[wicked cloud]

oh yeah BD = CD

Oh

[ivory grove]

mhm

so it's CBD=BCD

(angles)

[wicked cloud]

Ohhhhhhh

[ivory grove]

not CBD=BCD

[ivory grove]

[wicked cloud]

Yeah give me a sec to comprehend it

Oh because its an isoscles

Isoceles

[ivory grove]

yeah!

[wicked cloud]

Same edges

That took me a while

Tyyy

[ivory grove]

[ivory grove]

no problem! if you have more questions, feel free to ask here

welcome to the mathcord btw

!done

[odd edge]

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

[wicked cloud]

Ty

.close

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[tawdry valley]

Clearly I’m very close here. I decided to input my first answer (2 + root 3) and it was correct
However, for the bottom part of the matrices, I obtain that k = 2 - root 3, but the mark scheme states otherwise. Am I messing up simple algebra here?

[tawdry valley]

mark scheme

[civic storm]

ah first of all

It would be easier to notice that A is a rotation matrix

then it becomes a simple geometry question rather than an algebra heavy question

[civic storm]

the bottom row of your matrix should also lead you to $k = \frac{1}{2-\sqrt{3}} = 2+\sqrt{3}$

[clever fjord]

sed

[odd edge]

@tawdry valley Has your question been resolved?

[odd edge]

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[carmine marsh]

Could someone explain question d (f(2)) to me? I thought it was DNE.

[nimble blaze]

why do you think its dne

[carmine marsh]

because theres 2 points on x=2

[wooden python]

only one is filled

[carmine marsh]

and one of them is a hole

[wooden python]

you ignore the hole and only go for the one that's a filled circle

[mystic nova]

Yeah that means it doesn't define any value of y

[wet wolf]

That means does not exist

[carmine marsh]

Ok, thank you. If thats the case, then why is c DNE?

[nimble blaze]

i know...

[nimble blaze]

that concerns limits,
the one sided limits don't match

[mystic nova]

$\lim_{x \to x_0}f(x)=L$ only defined when $\lim_{x \to x_0^+}f(x)= $\lim_{x \to x_0^-}f(x)=L$

[clever fjord]

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

[merry finch]

(this is a great question, it helps clarify why the limit symbol is important)

[carmine marsh]

what if you had 2 points that were filled in (not holes) as x apprached 2? Would it still be DNE?

[carmine marsh]

Thank you 🙂

[nimble blaze]

there won't be two points filled in (at the same x value)

[carmine marsh]

oh ok

[nimble blaze]

as then you'd no longer have a function

[merry finch]

why would two points filled in make it not a function?

-# question directed at OP here

[frigid isle]

[wet wolf]

Oh. Misread 😂

[carmine marsh]

.close

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[covert root]

lim as x->0 (sin(pi-2x)/cos3x-1)
lhopital gives -2, symbolab says it diverges

[weary pelican]

when are you allowed to use l'hopital

[covert root]

0/0

[weary pelican]

wait

cos(3x-1) or cos(3x) - 1

[covert root]

cos(3x) -1

[weary pelican]

so

[covert root]

or inf/inf

[weary pelican]

$\lim_{x\to 0}\frac{\sin(\pi-2x)}{\cos(3x)-1}$

[clever fjord]

Raphaelisius Maximus MMIII

[weary pelican]

is that exactly your limit

[covert root]

good

yes

[weary pelican]

ok

[covert root]

i really dont need solving im after the logic

[weary pelican]

we can rewrite as $\lim_{x\to 0}\frac{\sin(2x)}{\cos(3x)-1}$

wait

[clever fjord]

Raphaelisius Maximus MMIII

[weary pelican]

better

[shut nymph]

lol

[weary pelican]

and then yes l'hopital is allowed

but this should diverge anyway after one l'hopital

[covert root]

for this sin(alpha-beta) formula right?

[weary pelican]

what happens after you l'hopital-ed?

[weary pelican]

well sin(pi-x) = sin(x) is an easy formula

[sand horizon]

its more about geometric reflexion of sin

[covert root]

sure if you know it otherwise i used sinacosb - cosasinb

[weary pelican]

ok, a bit overkill but why not

[covert root]

with lhopital i end up with -2

[lyric marlin]

and that is wrong

[weary pelican]

but like, show me the fraction you get

after l'hopital

bc it's wrong indeed

[covert root]

im not well accustomed to the shortcuts

[covert root]

how do you instantly see that?

[covert root]

i lhopitaled from the base form of this

[weary pelican]

ok, but show what you get right after l'hopital

[lyric marlin]

well I did the calculation myself and I am not getting -2, in agreement with a graph of the function

[covert root]

sec ill screenshot

[weary pelican]

I bet ||you forgot that a constant's derivative is 0||

[sand horizon]

using equivalent i got diverging also

[covert root]

camera died, (cos(pi-2x)*(-2))/(-sin(3x)**3-1)

[amber schooner]

lol

[weary pelican]

aaaaand we have a winner

[shut nymph]

crazy work there

[amber schooner]

💀

[shut nymph]

deriving 1 is hard sometimes yk

[weary pelican]

so

derivative of 1 is not 1

1 is a constant

[covert root]

oh fuck

[weary pelican]

its derivative is 0

so

[covert root]

ok lemme resolve

[weary pelican]

something/0

[tidal matrix]

i keep using epsilon-delta, i wish there was a "constant rule" or something :/

[weary pelican]

mmh?

By the definition of derivative

[shut nymph]

(isajoke)

[weary pelican]

(1(x+h) - 1(x))/h = (1-1)/h = 0

eh my joke detector didn't pick up

[covert root]

btw are you guys like teachers irl or what

otherwise resolved, its clear now

[amber schooner]

mostly no

[weary pelican]

you can see lots of people here are undergrads

[amber schooner]

yep

or high school

[weary pelican]

I am an undergrad doing post-grad math

so not a lot of teachers

[covert root]

pretty cool then, thanks and sorry to bother

[reef tartan]

Bold of you to assume people are employed here 💀

[covert root]

hey its a discord for math

a contradiction on its own

.close

[odd edge]

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

[wispy remnant]

Good evening gentlemen

[wispy remnant]

Need just a quick hint for this

[vernal yacht]

integral of 1/x is lnx

Is this enough?

[wispy remnant]

So we're doing parts?

Well I got that menacing x⁶+1 so I do not get that hint unfortunately

[manic sleet]

but we have that extra x, in fact that helps a lot :) In particular, notice:
[x(x^6+1)=x^7\qty(\frac{1}{x^6}+1)]

[clever fjord]

PajamaMamaLlama

[wispy remnant]

Uno momento

Like this, right?

Ah the minus shouldn't be there

[manic sleet]

[manic sleet]

minus is right

the only crime I see here is no +C

but doesnt matter we have definite integral anwyays

[wispy remnant]

Don't worry comrade, the integral had 2 and 1 in the problem

[wispy remnant]

Ohhh right!

Okay, it's late at night

[manic sleet]

no worries, happens

now that just turns into a plug n chug

[wispy remnant]

Should be b but

Ah wait

It's basically 2/(65/64)

[shut nymph]

the 2 multiplies the 64, not the 65

[manic sleet]

and you forgot your minus

bringing that into the log

[shut nymph]

You have (65/64)/2 = 65/128

[wispy remnant]

I ate it! But I brought it back

Excuse my legs

Anyways

I got it!!

[shut nymph]

for free??

[manic sleet]

dropping feet pics is crazy work

[wispy remnant]

🤣 🔥

In this economy!!

[shut nymph]

lool

[wispy remnant]

So the trick here is to look at any fraction derivative one power above what we got in the starting function?

[manic sleet]

yep, or one less if we're not dealing with negative exponents

especially being mutliplied

because that makes some very cancellations most of the time

[wispy remnant]

And people use damn AI when this is 10x better

Thanks y'all. I feel educated

[shut nymph]

AI is ass at math

[wispy remnant]

I meeean, it kinda got better ngl

[shut nymph]

still not reliable

it gives good Looking answers

[manic sleet]

yeahhhh it's getting better, but until it can start thinking critically and reliably humans dominate 🙏

[shut nymph]

key word looking

[wispy remnant]

Well, until then, we'll become disgustingly educated in the most hated subject ever. Peace, lads!

And have a nice one

[shut nymph]

Most *Loved subject ever

:)

[manic sleet]

integral calculus my beloved

[wispy remnant]

Ha. I wish, at least here 🙂

.close

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• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
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[supple cypress]

When I submit the answer shown, it says "So close! Can you try again checking for rounding errors? You can redo the calculations and only round at the last step." I am not sure how to round this properly.

[ivory grove]

!show

[odd edge]

Show your work, and if possible, explain where you are stuck.

[ivory grove]

Welcome to the mathcord btw

Still there @supple cypress ?

[odd edge]

@supple cypress Has your question been resolved?

[supple cypress]

Oh whoops

Im here sorry

!close

[ivory grove]

!done

[odd edge]

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

[supple cypress]

.close

[odd edge]

Channel closed

Closed by @supple cypress

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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!

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[mystic saffron]

Hi can someone show me the steps on how to solve this problem? Also the answer key says its B but everywhere I search they say the answer is A so I'm wondering if its just an error in the answer key or something else.

[ivory grove]

!show

[odd edge]

Show your work, and if possible, explain where you are stuck.

[mystic saffron]

I don't know how to solve it at all

[ivory grove]

If you haven't tried anything, maybe it's best to start with drawing a diagram

[indigo dirge]

what is the shape of the base of the large square?

[mystic saffron]

Uh

[ivory grove]

Have you drawn a diagram yet?

It'll be easier to answer that

[indigo dirge]

sorry box lmao

[ivory grove]

once you draw a diagram

[vivid wyvern]

Uh no not that way

[ivory grove]

it (kind of) works

[vivid wyvern]

O
I would ay use area here since u dont know the number of squares

[vivid wyvern]

(How so?)

[ivory grove]

So, you have to find out how many of the 3x3x3 boxes you can squeeze into the larger 9x9x9 box

can you do that?

[indigo dirge]

what would be the volume of each small box?

[ivory grove]

@vivid wyvern @indigo dirge don't worry, I got this under control

[indigo dirge]

alr

[ivory grove]

I would rather not have the "too many cooks in the kitchen" effect

[mystic saffron]

27

[ivory grove]

How about the large box?

[mystic saffron]

729

[ivory grove]

Yes. So do you know how many of the small boxes you can fit into the large box?

You know how much "available space" there is, and you know how much each small takes up

[mystic saffron]

Oh wait so I divide 729 by 27

[ivory grove]

mhm! exactly

[mystic saffron]

Thank you 🙂

[ivory grove]

[mystic saffron]

How do I close this channel

😭

[ivory grove]

.close

[mystic saffron]

Yy

.close

[odd edge]

Channel closed

Closed by @timid drift

Use .reopen if this was a mistake.

[odd edge]

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>.
• Type the command .close to free the channel when you're done.
• 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.

[silk crescent]

can anyone help explain to me how you’d find the amount of solutions for a trig equation (below is an example of a question asking for it)

[fluid tundra]

a fairly reliable way of doing problems like these is drawing a diagram of cos(x/2) and y = 1/4 and counting the number of times they intersect on the interval

[ivory grove]

You can also look at the unit circle

[silk crescent]

what about the 1/2 in the cos1/2x?

how does the 1/2 affect the amount of solutions

[vivid wyvern]

It's clear that u will find positive values and u can hv that in 2 quadrant

[vivid wyvern]

No i dont think so

It doesnt affect

[silk crescent]

thing is my answer key told me 1 solution is the answer

idk why and how

[fluid tundra]

graphically the 1/2 factor corresponds with a horizontal stretch

[fluid tundra]

this is incorrect

the 1/2 factor certainly does affect the number of solutions

[vivid wyvern]

How

[fluid tundra]

this is not your help channel

[silk crescent]

does the 1/2 basically split the amount of solutions in half?

like normally its two solutions so 2/2=1

[fluid tundra]

heuristically, yes
but this alone is not a sufficient explanation if you are doing this problem in practice

but that is a good intuitive way of thinking about it

[silk crescent]

so if it was cos2x instead, then theoretically there should be 4 solutions?

[vernal yacht]

pretty straightforward approach

[strange aspen]

makes period shorter means extra solutions

[silk crescent]

then

since a full cycle is 2pi

would it basically make it so that the range is 0<1/2x<pi

therefore i would only look at the first triangle in the cycle?

aka like

the only triangle within that range

[lone bison]

ok so for this, it would be helpful to change the variables in order to reframe it
say cos(2x) = 1/4 on x = [0, 2pi]. you can create a new temporary variable z = 2x to change the question to cos(z) = 1/4. but then you also have to change the new interval to be in terms of z not x. so you have to consider z on [0, 4pi] instead of [0, 2pi]. so yes you would technically get double the extra solutions