help-4

[haughty jackal]

YEAH

I GET IT

So, it's (2 * 8 - 1) ways?

[graceful zinc]

yea

[haughty jackal]

Then from Case 1, number of matrices is (2 * 8 - 1)² ways, right?

[graceful zinc]

yes

think similarly for case 2

[haughty jackal]

Mm

Brb

Okay

[2(7²) - 7] ways?

[graceful zinc]

yea

[haughty jackal]

,ask 8⁴ - (15² + 2 * 49 - 7)

[rocky lotus]

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Requested by will_dows

[haughty jackal]

Yeah

Thank you

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[vale dock]

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[formal veldt]

I have a weird question. Is it possible to put smaller exponential equations that follow one big exponential equation? Kind like this:

[neat locust]

What?

You could do a piecewise function i suppose

[formal veldt]

I would like to keep it as one equation

I'm making a game and I need the enemy health to grow exponentially, but as your playing it also goes through a cycle of getting harder and easier.

[mild marsh]

you could always cycle between increasing at two different exponetial rates

[lyric sundial]

What about f(x) = x - |sinx|?

It's not exponential but the behaviour is similar

[formal veldt]

yeah

if only there was a way to make it exponential it would work

[lyric sundial]

Or e^that

$$f(x) = e^{x−\abs{\sin(x)}}$$

[rocky lotus]

Alberto Z.

[lyric sundial]

Probably this works quite well @formal veldt

[wicked zenith]

tha't s a sinusoidal on an exponential

[lyric sundial]

Yes

[formal veldt]

that does work

how could I slow it down a bit

[lyric sundial]

Instead of e you can pick a smaller number, like 2 or 1.3

[wicked zenith]

pick integral points allong your main exponential, then between pairs construct another exponential passing through both points whose derivative is flat at the first.

[lyric sundial]

He didn't want a piecewise

If I've understood correctly

[formal veldt]

yes

[knotty storm]

judging by ur name and by ur question, is this for some sort of car game?

like shifting gears n stuff

@formal veldt

[formal veldt]

Zombie game

[knotty storm]

ah ok

[lyric sundial]

Here's the graph for 4 different values of the base.

blue → e
green → 2
orange → 1.7
red → 1.3

[vale dock]

@formal veldt Has your question been resolved?

[wicked zenith]

use a sigmoid transition?

[vale dock]

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[north scarab]

@cloud coral Sorry, I had something sudden come up

[north scarab]

what were you saying?

[north scarab]

The first question I do not know how to answer

[stark wedge]

mark the angles on the diagram.

the question says angle BAC is 42° and angle ABC is 68°.

can you mark these on the diagram?

[north scarab]

@stark wedge

[stark wedge]

indeed

can you mark any other angles on here?

[north scarab]

Yes let me write it

[stark wedge]

ok those won't help you a whole lot

how about angles ACD and DCB

[north scarab]

one sec someone called me suddenly

[north scarab]

im not sure how to get the size of that angle

[stark wedge]

what do the angles in any triangle add up to?

[north scarab]

180

ok triangle ACD is 180-90-42

[stark wedge]

i think you misspoke in two places here

one place

angle ACD is 180 - 90 - 42 yes.

[north scarab]

yep

[stark wedge]

here is a slightly more polished diagram

you also know AB = 50m

[north scarab]

yep

[stark wedge]

now you'll have to do some right-triangle trigonometry in triangles ACD and BCD

would you like to try it yourself or do you want more explicit instructions

[north scarab]

ill try

solving from here

[stark wedge]

k

[north scarab]

first of all what is the

question even asking

the a) part

[stark wedge]

it's asking you to prove that w*(tan(48°) + tan(22°)) = 50

[north scarab]

so

i think I need to solve

for w?

to answer this question

[stark wedge]

not quite

in fact almost the opposite

solving this question would be a step on the way to finding the value of w

no, here's what you should do instead:

express AD and DB in terms of w

[north scarab]

Ill try n solve it rn

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@north scarab Has your question been resolved?

[vale dock]

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[merry jasper]

Im confused why on the chatgpt, it didn't use the first equation. I don't understand

[stark wedge]

!nogpt

[vale dock]

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).

[merry jasper]

Why do you use the middle piece for the left hand limit

[stark wedge]

also show the original problem please?

i think there might be some misreads going on regarding which limits exactly you might be asked to find

[merry jasper]

We don't need to graph we just find the limits

[stark wedge]

i wasn't talking about graphing

you miswrote (a)

(a) is $x \to -2^+$

[rocky lotus]

Ann

[stark wedge]

i.e. you're approaching -2 from the right

that means you're in the middle subdomain i.e. -2 ≤ x ≤ 2

thats why you use the middle piece for it

[merry jasper]

So for the left hand limit, we use the middle piece because its the closest to the as x approaches -2?

[stark wedge]

this is not left hand

this is right hand

[merry jasper]

Which equation do I use for the left hand limit

[stark wedge]

do you mean left hand limit at -2 or left hand limit at 2?

[merry jasper]

-2

[merry jasper]

ohh so the middle piece is in between -2 and positive 2

cause im struggling to understand what's the step by step to get the limits and if its continuous

[stark wedge]

that's exactly what -2 ≤ x ≤ 2 says.

[merry jasper]

My bad, I misunderstood it cause I thought its x=-2

So this is the limit for -2

[stark wedge]

no you mixed it up

completely, actually, by the looks of it

[merry jasper]

sorry, so how do I find the limit of -2

[stark wedge]

at -2

[merry jasper]

ohh so i just mixed this one up

it was already right, since my first answer was dne

since 2 and 0 are not =

instead of using the last one, i shouldve used the middle piece

[merry jasper]

cause it approaches from the right to the negative 2

is that correct

and greater than -2 approaches it from the left

so this is for the limit at positive 2?

[stark wedge]

you don't say "limit of [number]", you say "limit at [number]"

or more accurately "limit of [function] at [number]"

but yes, you worked out the limit of f(x) at 2 correctly now

[merry jasper]

alright thanks, i understood it more with the number line

[safe vector]

Where's the question

[merry jasper]

they didn't teach that at school, which is odd

[stark wedge]

@safe vector here, but we have resolved it already by the looks of it

[stark wedge]

yeah ig so but the number line is a powerful tool

[merry jasper]

thanks very much, i understand it now

[safe vector]

Oh yeah it's x tends to [number]

@merry jasper let me ask you a simple stuff

[merry jasper]

sure

[safe vector]

Why 1^∞

Indeterminent

[merry jasper]

because it has no limits?

[safe vector]

?

No

I mean why you can't write 1^∞ as 1

Let me tell you

[stark wedge]

i am sure you mean well but this is somewhat unsolicited and off-topic for OP's current question

[safe vector]

Nah I'm telling him general concept

[merry jasper]

its all good dw, i dont mind

[safe vector]

1^∞ isn't 1 because the 1 written there means that it "TENDS" to 1

It's not actually correct 1

It's tend to 1

[merry jasper]

what does it mean by like tends to 1

[safe vector]

Tends to 1 means the number is not exactly 1 but something near

Like 1.0000000000001

[stark wedge]

you're explaining it wrong.

[safe vector]

Its correct

[safe vector]

It's 100% correct

[stark wedge]

$1^{\infty}$ is shorthand for a function in the form $f(x)^{g(x)}$ where $\lim f(x) = 1$ and $\lim g(x) = \infty$.

[rocky lotus]

Ann

[stark wedge]

any talk of indeterminate forms fundamentally deals with functions not simply numbers.

[safe vector]

ABSOLUTELY DUMB

[stark wedge]

...

[safe vector]

YOU ARE WRONG HERE

[stark wedge]

HEY MAYBE TONE IT DOWN WITH THE CAPSLOCK WILL YOU.

[safe vector]

the function you meant is NOT 1

[merry jasper]

wait what's g of x

[stark wedge]

g(x) is another function

[safe vector]

F(x) is not exactly 1

Its near to 1

[stark wedge]

i specifically said the limit of f(x) is 1.

[safe vector]

Thats what I was explaining

[stark wedge]

i never said f(x) was constant 1

[merry jasper]

ohh so thats what he means by 1.00001

[stark wedge]

but your explanation of "a number not 1 but close to it" is very misleading

[safe vector]

Then how am I wrong 💀

[stark wedge]

you're misleading that's how

[merry jasper]

yeah i was kinda confused cause i thought u meant it was 0.99

[stark wedge]

see?

[merry jasper]

cause you said close to it

[safe vector]

The function is either 1+ or 1-

[safe vector]

No no

[safe vector]

Uh so my explanation was kinda dumb ig

You are new to calculus ig

[merry jasper]

not really dumb, just wrong sentence

[merry jasper]

yeah, we just started

[safe vector]

I mean i thought you would get what im talking

Explaining you with functions would mess it up for you

Since you are new

[merry jasper]

we're about to um, discuss the graphing of the functions i sent here. so we're just asked to answer for the limits

[safe vector]

Oh cool

Is the question solved?

[merry jasper]

i just got confused by the step by step of it, until ann showed me the number line

yeah

so like one more thing

[safe vector]

Alr you could close it now

[safe vector]

Sure

[merry jasper]

for example this one

[safe vector]

Piece wise functions

[merry jasper]

how would i start it, like do i find the limites at -1

of right and left hand

wait let me try and solve it and see if its correct

[safe vector]

I mean that is what piece wise functions do

In "a" you could see that we have to approach -1 from right side

So we use -x² + 1

Answer to "a" is 0

In "b" we approach -1 from left side

[merry jasper]

wait so this is correct

cause 1 approaches from the left

and x^2+1 approaches from the right

at the number line

[merry jasper]

yeah

[safe vector]

Check "c" part

That's correct

I thought you didn't solved "c" lol

All are correct

Wait

You did "a" wrong

The answer is 0

[merry jasper]

wait wym by a and c

[safe vector]

Only a part is wrong

[safe vector]

...

[merry jasper]

OH yeah its -1

[safe vector]

A

[merry jasper]

wait no its 1

[safe vector]

Plug in x = -1 in -x² +1

The answer is 0

Guess you thought -x² = 1 💀

It's -1

[merry jasper]

oh yeah i forgot the integer of x^2

its still dne right cause

both limits are not equal

[safe vector]

What are you saying, im talking about "a" partt

"c" part is already correct

[safe vector]

A in this one

[merry jasper]

Ohhhh

im sorry, i didn't notice it was given in the book

[safe vector]

Yeah 🥲

[merry jasper]

yeah i didn't put -x^2

[safe vector]

Yeah

So answer to "a" part is 0

[merry jasper]

yeah

[safe vector]

Anything else?

[merry jasper]

all good, i understand it now. I just got confused cause they only taught us how to solve one limits

thanks

[safe vector]

!close

use ".close"

[merry jasper]

.close

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[safe vector]

Question 13

Is it getting done from LH rule?

[stark wedge]

,rccw

[stark wedge]

LH is only for masochists here

[safe vector]

Nah it's possible

Can't find other way

[stark wedge]

i didn't say it was impossible, just that it is very painful and unpleasant

[safe vector]

maybe pattern observation

Let's try

You continue with your method

[hollow shuttle]

my gut feeling is that there's some closed form for that product of roots

[safe vector]

Mhm

My gut feeling really said LH

[hollow shuttle]

wait hold on

i remember this question

i think we have the same course material lmao

[safe vector]

Loll

Cengage?

[hollow shuttle]

nope

[safe vector]

Oh

[hollow shuttle]

it might be taken from there though,

i'm also in a jee course

anyway i'll try using LH because it actually seems like it might work

unless i have to do it twice

[safe vector]

Something like this

I mean we can take natural log

Idk

@stark wedge

Found a way?

[stark wedge]

your x's don't look like x's...

[safe vector]

Sry

[stark wedge]

i did it with series expansion

[safe vector]

💀

Ain't that big

That would be so long ig

[stark wedge]

$\prod_{k=2}^n \cos(kx)^{1/k} = \prod_{k=2}^n (1 - 2 \sin^2(kx/2))^{1/k} = 1 - \sum_{k=2}^n \frac{2}{k} \paren{\frac{kx}{2}}^2 + O(x^4)$

[rocky lotus]

Ann

[safe vector]

Mhm

[stark wedge]

some dust will need to settle but the idea is that you only need the x^2 term

you're gonna need to do some algebraic shuffling around

[safe vector]

I mean my general terms is correct

Wait let me try more

[stark wedge]

which i realize now i messed up in my work so i do not have anything checkable

however i can tell you for sure this will cook

[safe vector]

Ikr

Advanced mathematics 💀

Does this work?

I took natural log both sides

Now we can try differentiation

But idl

Idk*

@stark wedge found a way?

[stark wedge]

sorry i am busy now

[safe vector]

🥲

[sterile gust]

ur on the right track

[safe vector]

Nah

I messed up

Gotta try another approach

I wonder if there a result for this series

[sterile gust]

a hint would be writing $P(x) = \prod_{x=2}^n ( \cos (kx))^{\frac 1k}$

[rocky lotus]

whiteboard

[safe vector]

Why you giving hints

Just solve it and explain 😭

[sterile gust]

no im not gonna solve it.

[safe vector]

Bro why

[safe vector]

Literally this is just general term

Ann also wrote it

[sterile gust]

as i said ur on the right track.

[safe vector]

Okay 🥲

Wait what

Is it tanrx?

But then you can't observe pattern after y hops

@@hollow shuttle

<@&286206848099549185>

[nocturne sable]

Unrelated but woah we have a mutual friend

[hollow shuttle]

holy hell it's the astronomy guy

[hollow shuttle]

scratchwork

[safe vector]

🥲

[hollow shuttle]

using this

[safe vector]

Oh what let me check

[nocturne sable]

Yes!! Would you mind me friending you?

[safe vector]

Im trying wat

Wait*

[hollow shuttle]

go ahead

[safe vector]

Oh bro

Tanrx/2x

Is giving a pattern maybe

I got the same result

!!

[hollow shuttle]

awesome

[safe vector]

It was 1/2 × 1+2+3+4....+n

Now what

[hollow shuttle]

the question says that expression is equal to 10

so just solve for n

[safe vector]

Oh ye

Wait

Let solve

Le me solve

6 is the answer ig

[hollow shuttle]

yep

[safe vector]

Is it correct?

100%?

[hollow shuttle]

should be

it's pretty unlikely if we both got the same wrong answer

[safe vector]

Alr

[safe vector]

You did expansion right?

I used LH

[hollow shuttle]

yeah i used ann's expansion

[safe vector]

Betting on a pattern was brilliant by me ig 💀

Alr ty

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[woven forge]

we know from linear algebra that if a linear system contains more unknowns than equations, then the system has infinite solutions, that can be expressed as a parameterization of the free variables. but what if the system has more equations than unknowns?

[stark wedge]

it could be any of the three

it could have infinitely many solutions (if it collapses enough)

it could have a unique solution

[viscid spade]

depends if the left over equations are verified or not

[stark wedge]

or it could not have any sols at all

[stark wedge]

what if each one pays 8 euro a month to have the blue twitter checkmark

[woven forge]

if for instance we have 3 linear equations that are linearly independent in R^2 then the system if it did have a solution, should have only one soltion right?

[viscid spade]

can we have a picture of the problem ?

[woven forge]

.close

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[midnight pier]

can someone please help me analyse what the graph is trying to say especially wrt the second statement

[glass kelp]

intensity of radiation vs wavelength

this is from uhh

rayleign-jeans law

[midnight pier]

sorry i havent heard of it

[glass kelp]

vs max's distribution

[midnight pier]

what does it say

[glass kelp]

classical physics predict infintie intensity while experimental results say the intensity drops after a certain wavelength

u should look into

the uhh

"ultraviolet catastrophe"

[midnight pier]

okay, will that help me undersatnd this?

[glass kelp]

yes

i think so

[midnight pier]

okay!

i will!

thank you so much!

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[jovial edge]

What have you tried?

[somber sequoia]

just value of p?

there's no other conditions?

[lost marlin]

!xy

[vale dock]

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.

[jovial edge]

Ohh yeah, that's odd

[upper island]

What is the conditions for a/b=0 ?

[vale dock]

@twilit plume Has your question been resolved?

[twilit plume]

nopee

i think the problem is wrong 😭

[upper island]

Why ?

[twilit plume]

idk

nvm i got it guys

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[fast sentinel]

"Let S={v_1,...,v_k} be a set of vectors of the vector space V, n<k
a) we can obtain a vector basis by eliminating vectors of S
b)S is a generating system, but not a vector basis, because there's more than n vectors
c) it can be V=L(v_1,...,v_n)+L(v_n+1,...,v_k)"

I think the one that's true is b) because for S to be a vector basis of V it has to be a set of linearly independent vectors of V and if S were to be linearly independent, some vector necessarily would have to be outside of V, because there are more vectors in S than dim(V). But I'm not entirely sure. Please help

[ruby sleet]

is S said to be a generating system of V?

or is

Let S={v_1,...,v_k} be a set of vectors of the vector space V, n<k
all the assumptions we have

[fast sentinel]

No, just a set of vectors of V

[ruby sleet]

ok

so when you have more vectors than the dimension of V, you must have a generating system?

[fast sentinel]

Necessarily no, because it could be the multiples of one vector of V

[ruby sleet]

yes for example

[fast sentinel]

Like say if I have S={(1,0),(2,0),(3,0)}, S is a set of vectors of R² but you can't obtain a vector basis of R² nor it is a generating space of R² and R² is not a linear combination of anything combination of vectors of S.

So hold on

The one that's true is c. Because ot doesn't impose for it to be a basis and uses the keyword "can"

[ruby sleet]

yeah, if I understand option c) correctly

it IS possible that V = Span(v_1,...,v_n) + Span(v_(n+1),...,v_k)

though I'm wondering if it's just meant to be a "+"

or an $\oplus$

[rocky lotus]

Raphaelisius Maximus MMIII

[ruby sleet]

both would technically be possible

[fast sentinel]

It's not a direct sum

Just sum

[ruby sleet]

ok

well just take v_1,...,v_n a basis of V

and v_(n+1)...v_k whatever you want

[fast sentinel]

Sure S={(1,0),(0,1),(1,1)}

And V = R²

But like the question is about any set and any vector space so each statement has to be true for it to be correct since I found a set that doesn't work for a) and b) then c) has to be the correct one

[vale dock]

@fast sentinel Has your question been resolved?

[vale dock]

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[umbral flare]

hey can someone walk me through this

[umbral flare]

using de moivres theorem
present cos^5 x in terms of multiple angles

idk what to do like at all

[viscid spade]

use the euler's formula !

[umbral flare]

i tried using
2cosx = z + z^-1
but that kinda just circled back around

[safe fulcrum]

okay, so can you take the real part of e^(5ix)?

[umbral flare]

uhhh is there a way to do it without using the exponential forms

bc in my textbook i havent gotten to the exponential stuff yet and this exercise is right before it

[safe fulcrum]

e^(5ix) = cos(5x) + i sin(5x) by Euler's identity

[viscid spade]

maybe its an introduction to the next chapter

¯_(ツ)_/¯

[umbral flare]

idk this is supposed to be like one of the final exercises on pure de moivres

i think what sort of worked but also didnt was getting cos^5 x in terms of other powers

where i got

cos^5 x = cos5x + 10cos^3 x sin^2 x - 5 cosx sin^4 x

my textbook answer is
cos^5 x = 1/16(cos5x + 5cos3x + 10cosx)

idk... power reduction? somehow?

[safe fulcrum]

yeah you're missing a factor of 1/16 somewhere

it'd help if you could show your work on paper

then upload a photo

or if you know how to use LaTeX use that

[umbral flare]

1 sec

ive no idea what im doing but o well

[safe fulcrum]

yeah then the line where you write $\cos(5 \theta) = \cdots$ is good

then you have to replac $\sin^2 \theta = 1 - \cos^2 \theta$ I think

[rocky lotus]

south

[safe fulcrum]

hmmm let me think

ahhh right

you have to use that $z + \frac{1}{z} = 2 \cos \theta$ instead

[umbral flare]

yeah i was guessing that but idk

[rocky lotus]

south

[umbral flare]

oof

[safe fulcrum]

then you can raise both sides to the power of 5

[umbral flare]

yeah but then like

wait

ok

lemme try this

ok i think this is gonnawork

im basically expanding the

(z + z^-1)^5

[safe fulcrum]

yep

[umbral flare]

using pascals

and then ill just convert it to regular multiple angles

and take the real part

okay thats it then

thanks for helping out

[safe fulcrum]

for example, $z^3 + z^{-3} = (\cos 3 \theta + i \sin 3 \theta) + (\cos 3 \theta - i \sin 3 \theta)$

[rocky lotus]

south

[safe fulcrum]

that's the idea

no worries!

[umbral flare]

yup

tysm

bai

[safe fulcrum]

bye

[umbral flare]

.close

[vale dock]

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[coarse aspen]

Hello

[coarse aspen]

So what times does: 20x75^x get multiplied with to because 300x75^x

[glass kelp]

holy notation

[ 20 \times 75^x]

[tidal swift]

isn't this a ratio

[glass kelp]

the first one is this, right?

[coarse aspen]

Naur wait I meant uh

[rocky lotus]

k

[glass kelp]

this?

[coarse aspen]

YEAH

[tidal swift]

if this is what you meant

you know both numbers have a 75^x

[glass kelp]

divide ${300 \times 75^x}$ by ${20 \times 75^x}$

[rocky lotus]

k

[tidal swift]

what if you replace both 75^x by the word something

[glass kelp]

and u shall have ur answer

[tidal swift]

can you do it?

[coarse aspen]

Can 75^x be divided

D:

[glass kelp]

.

[tidal swift]

yes, why not

[coarse aspen]

So id say my answer is 15

[tidal swift]

there you go

[coarse aspen]

YAY

THANKSSS sorry my math teacher didn't have time to help so I just grabbed my phone

[tidal swift]

whenever you're confused and you see two equal numbers being compared

just replace those equal numbers with something else

for example

[coarse aspen]

Okayy

[tidal swift]

if you were asked for the ratio between 20(8923749802374) and 300(8923749802374)

[glass kelp]

here how i got this. let ${a}$ be ur answer. then ${a \times (20 \times 75^x) = 300 \times 75^x}$. divide both sides by ${20 \times 75^x}$ gives u ${a}$. 😄

[tidal swift]

you can just abstract that bunch of numbers as "something"

[coarse aspen]

Okay okat

[tidal swift]

then you have 20 times something vs 300 times something

[rocky lotus]

k

[tidal swift]

then it would be pretty easy from there

[coarse aspen]

THANKSS ima get back to doing math now❤️❤️

How do I end the conversation?

Close

[glass kelp]

.close

[vale dock]

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[coarse aspen]

.close

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[earnest ingot]

I think I'm having a brain fart

[silk wasp]

yooo wsgg people

help me get better at math

[smoky lily]

!occupied

[vale dock]

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

[silk wasp]

oh mb

[earnest ingot]

nvm I figured out my mistake lol

.close

[vale dock]

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[silk wasp]

its a comlicated server

[ivory valley]

Is this Algebra?

[earnest ingot]

Yes it's permutation groups

[vale dock]

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[undone dew]

Hello. The square root only has one value, right? My teacher says it has two values.

[main sundial]

there are two values

a square root can be positive or negative

thus two values

[marble furnace]

Yes only one

It can only give positive values

[marble furnace]

That's when x^2 = 9

[marble furnace]

it's different from x= √9

[main sundial]

depends on what you are solving for

[marble furnace]

Yep mod will be applied

[stark wedge]

6|x| is correct as-is, but can you show us the entire page?

i think there might be something extra that you're not showing us

[hot tide]

,rccw

[tidal swift]

i think if a square root is given to you without you having to rearrange for it, it is likely referring to the principal root (the positive answer)
but it depends on context too

[stark wedge]

hmmmm

see i was thinking there would be something at the top of the page which says "treat all letters as positive"

but i dont see anything like that

[tidal swift]

wait now that you mention it

the page starts at question 3

i don't think this is the start of the whole paper

@undone dew OP, are there any more instructions on the first page?

[undone dew]

This is the first page

[tidal swift]

so this entire practice paper is about rationalization?

[undone dew]

Yes

[tidal swift]

not sure what there is to rationalize in questions like this unless exponent rules are also used

but yes, in exercises like this, generally (and keyword: generally) they want the principal/positive root

[undone dew]

Aah, yes, It also includes the exponentiation rules that we saw in the past class.

[tidal swift]

though technically speaking, a square root does have two solutions, but only if you rearranged a square into a square root yourself

if you are given a square root directly, it is almost always the positive root unless there's some instruction saying otherwise

[undone dew]

I understand

Thanks

[tidal swift]

nps

[undone dew]

.close

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[vale dock]

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

Are we just testing for some c value that works or doesn't work here?

[stark wedge]

some f and some c.

[ebon thorn]

If I wanted my f to be x

So, f(x) = x

[stark wedge]

i mean... that won't work as a CE of the kind they're asking for.

actually wait.

there are two parts to this.

[ebon thorn]

Then we would have to ensure that the lim for this implies the existence or not?

[ebon thorn]

CE?

[stark wedge]

counterexample

(a) show that if f is differentiable then that limit exists an is equal to c
(b) show that the limit can exist without f'(c) existing

which one are you trying to do rn

[ebon thorn]

For part (a), that is just plugging and chugging into the limit formular of f(x) - f(c)/ (x - c) = f'(c)

You get the exact same equation they have

Now, that should suffice for this to be differentiable.

Now, I am on the second part of showing a counterexample.

[stark wedge]

ok

you missed some brackets there but whatever.

part (b) is about pretending you're malicious and deliberately constructing a function so that the given limit exists without f being differentiable.

im gonna make your life a bit easier and say you should concretize to c=0 and f(0)=0

that way you're tasked with defining your function in such a way that lim nf(1/n) exists but f is not differentiable at 0

perhaps f need not even really be continuous at 0

[ebon thorn]

So, f = (1/x)?

[stark wedge]

dunno what that's supposed to mean

do you mean f(x) = 1/x

[ebon thorn]

Yeah

[stark wedge]

ok can you tell me what the expression n*f(1/n) simplifies to for this f

[ebon thorn]

n*(n) = n^2

[stark wedge]

and does its limit as n->infty exist?

[ebon thorn]

No

[stark wedge]

so does that work as a counterexample?

[ebon thorn]

Well, yes, it goes to infinity

It diverges

[stark wedge]

in this problem i am 100% sure "exists" means "exists and is finite"

[ebon thorn]

It doesn't because it needs to exist, right?

And needs to be finite?

[stark wedge]

indeed, f(x) = 1/x won't cut it.

[ebon thorn]

We would use f(x) = x

[stark wedge]

back to the drawing board.

f(x)=x doesn't work because f'(0) exists.

[ebon thorn]

n * 1/n = 1

[stark wedge]

f(x)=x doesn't work because f'(0) exists.

let me repeat myself \

defining your function in such a way that lim nf(1/n) exists but f is not differentiable at 0

you want the following two things to both be true at the same time:

1. lim n*f(1/n) exists
2. f'(0) does not exist

[ebon thorn]

It says in the problem that the existence of the limit of this sequence doesn't imply the existence of f'(c)

Oh, we choose c = 0

Well, fuck. Give me 50 Intellectual Quotient points and I could figure it out

[stark wedge]

i said this earlier

i do not think this is about any intelligence points or whatever

unless your INT stat affects your reading comprehension

[ebon thorn]

Where am I supposed to pull out the high genius move to come up with a function that would give us a limit?

[turbid valve]

[in D&D I think it does ]

[ebon thorn]

They are strongly correlated

[turbid valve]

[a character in a campaign I'm in severely struggled due to their INT getting wacked down to 1, which had that effect among others]

[ebon thorn]

Dude. Fuck off. YOu are derailling

[stark wedge]

calm your tits lol

[turbid valve]

Point is, we've so far boiled this down to
"find a function f such that

• f(0) = 0
• this limit exists
• f'(0) doesn't exist"

[stark wedge]

well you only care about how your function behaves when x=1/n

so just think about a sequence

a_n

such that the limit of n a_n exists and isn't 0

and then define your function to be f(x) = a_n if x=1/n; 0 otherwise

[ebon thorn]

I am thinking of a piecewise function

How about f(x) = x^2 + x

This won't give us f'(0) = 0

[turbid valve]

But f'(0) exists there

[ebon thorn]

Oh, f'(0) shouldn't exist

[turbid valve]

We're not bothered about what f'(0) equals, just whether it exists

[stark wedge]

this is not even piecewise

[ebon thorn]

f(x) = 1/x

[ebon thorn]

I said I was thinking of it. I changed my mind

[stark wedge]

and also it is as if you aren't reading my msgs

[ruby sleet]

now neither f'(0) nor lim(....) exist and finite

[stark wedge]

or you are and then they escape you

[glass kelp]

This is definitely differentiable everywhere 🥀

[turbid valve]

This here is precisely the approach I'd take too

[ebon thorn]

Okay, let's take it back from the beginning

Too many cooks in the kitchen. Dman

[turbid valve]

...I was kinda waiting you were going to take it to the beginning...

[ebon thorn]

Welp, I have no idea

Okay, give it to me genius. Use that high intellect of yours to pull something off with your god-given intuition

[turbid valve]

Now, someone did give it to you not more than 10 mins ago:

[ebon thorn]

So, the piecewise if f(x) = a_n?

if x_n, 0 otherwise

[stark wedge]

you fumbled it

[turbid valve]

You're getting hung up on the word "piecewise"

[stark wedge]

what is "x_n" 💔

[turbid valve]

Forget about it

[ebon thorn]

Honestly, moving on.

[turbid valve]

You want a function f, such that
f(x) = a_n (i.e. the nth term in some sequence) if x = 1/n for some natural number n
and f(x) = 0 otherwise (i.e. x is not 1/n for any natural number n)

[ebon thorn]

So, how does this give us what we need?

[turbid valve]

Further, you want the sequence $(n a_n) _{n\ge1}$ to have a limit

[rocky lotus]

Waes (Wires)

[ebon thorn]

I am confused. What is even going on anymore

[turbid valve]

You're doing the SECOND part of this question, right?

[ebon thorn]

Yes

[turbid valve]

Do you understand what the question is asking you, first of all?

[ebon thorn]

It merely says to give a counterexample for why the limit from part one couldn't work.

What qualifies has not being differentiable?

It is a constant, right?

[turbid valve]

"couldn't work" in doing what?

[ebon thorn]

Be differentiable

[turbid valve]

Why "limit exists" doesn't imply "derivative at that point exists", that is correct

Okay, so

We're doing some things to simplify the work involved

Number 1:

Let's let c be 0

Then that f(c + 1/n) thing, we can just treat as f(1/n)

Number 2:

Let f(c) = f(0) be 0

Then that limit that we're assuming exists is just
lim n f(1/n)

[ebon thorn]

This f(1/n) = a_n where x = 1/n and 0 otherwise?

In order for f'(0) to not exist?

[turbid valve]

That's our goal, yeah

We're explicitly trying to find (read "build") a function f so that the limit exists, but that f'(0) doesn't

[glass kelp]

i have another idea: we can view the above limit as right derivative by considering u = 1/n

[turbid valve]

<@&286206848099549185> [sorry I'm getting too tired to continue atm, if anyone could care to take over please ]

[glass kelp]

@turbid valve does this work

[turbid valve]

[I guess it could? again, a little too tired to check this]

[ebon thorn]

Real Analysis is Real Ass

[glass kelp]

im gonna try my approach then

[ebon thorn]

Because this course is garbage

[glass kelp]

Consider ${h = 1/n}$, as ${n \to \infty}$, ${h \to 0^+}$
[ \lim_{n \to \infty} n\left(f\left(c + \frac{1}{n}\right) - f(c)\right) = \lim_{h \to 0^+} \frac{f(x+h) - f(x)}{h}]

[rocky lotus]

k

[glass kelp]

real analysis is real anal *

[glass kelp]

do u agree with this

and that the limit on the right is definition of the right derivative

[ebon thorn]

Yeah.

[glass kelp]

cool

[ebon thorn]

I am done. I am retiring for the rest of this ass of a class

[glass kelp]

can u think of a function that has a right derivative

but doesnt have a derivative there?

hint: stay positive!

[ebon thorn]

Positive lol

There is nothing positive about this

Why can they use |x| but I can't use x?

[glass kelp]

look at the graph of the absolute function

the right derivative exists

but doesnt match with the left one

[glass kelp]

the absolute function was what i was hinting at

[storm birch]

Ts is to much for my brain rn

[ebon thorn]

Huh?

[glass kelp]

u have a degree in math

[glass kelp]

.

[storm birch]

Alright u seem smart

[ebon thorn]

What is the lim of |x|?

[glass kelp]

+1

[ebon thorn]

Yeah, it works for my case as well

f(x) = x

[glass kelp]

no

derivative exists everywhere for x

[gleaming kindle]

a counterexample requires f'(c) DNE

[ebon thorn]

Same for |x|

[glass kelp]

f'(0) DNE for |x|

but f'(0) exists for x

[ebon thorn]

What is the derivative of |x|

[wraith heart]

do this

[ebon thorn]

There is a point at 0

[wraith heart]

find the slope from the left side

[ebon thorn]

-1

right side is 1

[glass kelp]

so |x| is not differentiable at 0

[ebon thorn]

x is not differentiable at 0

[wraith heart]

,tex .abs def

[rocky lotus]

riemann

[wraith heart]

$\frac{d}{dx} x = 1$

[rocky lotus]

riemann

[ebon thorn]

This didn't prove me wrong

[glass kelp]

why do u think this

[wraith heart]

yes it does if you know the definition of differentiable

[ebon thorn]

The derivative must exist at each point in the domain

It does

[glass kelp]

so it is differentiable

[wraith heart]

what is "it"

x or |x| ?

[glass kelp]

f(x) = x is differentiable

[ebon thorn]

If you knew how to read, you could figure it out with context clues

[wraith heart]

Lol

[ebon thorn]

So is f(x) = |x|

[wraith heart]

i'm not the one who thinks x is not differentiable at 0

[glass kelp]

f(x) = |x| is NOT differentiable at x = 0

[ebon thorn]

And?

[wraith heart]

did you read what you wrote

[wraith heart]

no

[glass kelp]

if u approach from the left, dy/dx is -1
if u approach from the right, dy/dx is 1

[ebon thorn]

Okay, what does it mean to differentiate?

[wraith heart]

did you actually do the work to calculate here or did you just guess

[ebon thorn]

Did you read what you wrote? I think you should go review the definition and produce examples to prove your claim

[glass kelp]

[ f'(c) = \lim_{h \to 0} \frac{f(c+h)-f(c)}{h}]

[ebon thorn]

Use your logical reasoning and figure it out

[rocky lotus]

k

[gleaming kindle]

lets chill out

[glass kelp]

ok

[wraith heart]

you're the one who's asking for help

[ebon thorn]

What is the derivative of f'(x) = x

[ebon thorn]

I never asked for your help. I don't want it lol

[glass kelp]

.

[ebon thorn]

Your bad at helping

[ebon thorn]

Good. Now do |x|

[wraith heart]

!vol

[vale dock]

Helpers are just people volunteering their time to help you. Be polite and patient.

[gleaming kindle]

@ebon thorn dont bite at helpers

[glass kelp]

thank u

[hot tide]

here

[ebon thorn]

Doesn't mean I don't have to tolerate one

[gleaming kindle]

@wraith heart its prob best to step out

[glass kelp]

and a limit doesnt exist if left and right limit arent equal

[ebon thorn]

They can go welcome themselves to another channel

Thank you.

I never want to see his face in a channel that I occupy again

[gleaming kindle]

i said dont bite

[ebon thorn]

I put the dog down if it bites me first

[glass kelp]

anyhowwwwww

[ebon thorn]

Okay, so does it exist on 0?

[glass kelp]

are the left and right limit the same?

[ebon thorn]

Take the derivative of |x|

Does it exist on 0

[glass kelp]

no

it doesnt exist at 0

[ebon thorn]

How so?

[halcyon lintel]

@ebon thorn I agree with you though. @wraith heart is terrible at helping.

[glass kelp]

the left and right limit using the lim defn of derivative arent the same

[ebon thorn]

Does the derivative of f(x) = x exist at 0?

[glass kelp]

yes

[ebon thorn]

Why does that exist on 0 but not |x|?

[wraith heart]

who is that

[ebon thorn]

Bro, get out

[glass kelp]

because the left and right limit of derivative arent equal

[gleaming kindle]

@halcyon lintel great contribution. take a day off

[wraith heart]

tell your bestie

[ebon thorn]

So, if there are two limits that aren't equal to each other, then it doesn't exist at that point?

Is that what you are saying

[glass kelp]

if the left and right limits of smth arent approaching the same thing at the same point

the limit of that smth at that point doesnt exist

since derivative is defined as a limit of this f(x+h) -f(x)/h

[ebon thorn]

I see.

Okay, so the derivative of f'(0) wouldn't exist

For |x|

[glass kelp]

yes

[ebon thorn]

Alright

[glass kelp]

can u use f(x) = |x| as the CE for the question

[ebon thorn]

I assume yes because it would give us 1

but it doesn't exist at f'(0)

[glass kelp]

can u structure a full argument for me to check plz

[glass kelp]

what does "doesn't exist at f'(0)" mean

[ebon thorn]

Can you structure the entire argument? I don't think I got everything you siad

[glass kelp]

what do u not understand

[ebon thorn]

Maybe if you laid the entire proof out, I could identify it

[glass kelp]

!noans

[vale dock]

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

[ebon thorn]

Well, how tf am I supposed ot know?

I can't know unless I see the proof

[glass kelp]

we gave u the ideas

for the proof

[gleaming kindle]

we're not obligated to give full answers. it should be smth you can figure out thru guided convo

[ebon thorn]

Well - I will just use the ss because it had it all.

[glass kelp]

alrighty then

[ebon thorn]

.close

[vale dock]

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Use .reopen if this was a mistake.

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[grim gazelle]

can someone give an example on how to solve this?🥲 Its about number systems, our prof made us do self learning, I don’t know what method to use to solve this efficiently

[cobalt crow]

like you need to go from decimal number rep to binary

no?

sorry idk how to do that, let me check youtube and i come back if i learnt it

[vale dock]

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

I am struggling with the last part of this question. How do I find the linear combination?

[keen trench]

youre just asked to find like

a,b,c such that av1+bv2+cv3 = v5

[lapis dagger]

Yeah, but, how do I do that

[keen trench]

you can setup a system of equatiosn

[lapis dagger]

Sorry I think you're gonna have to spell it out a little more

[keen trench]

so we want to solve like

a(1 0 0) + b(-1 1 0) + c(-1 2 1) = (-1 2 1)

for the left side to equal the right side

wed need that the first components are equal right

[lapis dagger]

Ah ok

[keen trench]

so you would have to have like

a - b - c = -1

then similiarly for the second component

b + 2c = 2

[lapis dagger]

Right

[keen trench]

and c = 1

for the third component

[lapis dagger]

ok that makes sense

Thank you very much

[keen trench]

👍

[lapis dagger]

.close

[vale dock]

Channel closed

Closed by @lapis dagger

Use .reopen if this was a mistake.

[vale dock]

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[rough tide]

hey, could someone check if this differentiation is correct? x and y were originally in degree form, and 2 is the multiplier for the entire function.

[vale dock]

@rough tide Has your question been resolved?

[rough tide]

<@&286206848099549185>

[minor spire]

what

is it given that x/2 is in degrees??

@rough tide

[rough tide]

yes

[rough tide]

that’s why i multiplied x/2 by pi/180 to get it in radian form

[minor spire]

ok

it is correct from what i see

is this from any book??

[rough tide]

nono

it’s my own equation

wait lemme show u why im doubtful of this

the setting is in radian

red being the equation i have, and blue the result of my differentiation by hand

and green the automaatic differentiated one

idk y they wld provide different result if it is differentiated correctly

[minor spire]

do one thing

remove the 180/pi from the second graph

[rough tide]

okok bet

[minor spire]

and lemme do it myself once the differentiation

one more thing

[rough tide]

yea

[minor spire]

had you thought first simplifying the eqn and then solving for slope??

[rough tide]

wdym

the original eq?

[minor spire]

yes

try simplifying it once to get f'x

[rough tide]

okok

in radian form or degree form

bc i hv a reason to keep the results and equations strictly in degrees

[minor spire]

why is that??

[rough tide]

bc this is for my internal assessment (dw im allowed to ask others for help to others)

where i come up w my own research question

and the question is basically about how FOV in videogames

which is always in degrees

so it has to be degrees i feel

[minor spire]

okok

umm

what about taking 2 to other side and then taking tan both sdies

leadding to tany/2 = 16/9tanx/2

try this once

x is in deg

[rough tide]

oh yea

so

basically

[minor spire]

hmm

[rough tide]

the equation was set up like that bc

i have to make y the subject

bc y represents horizontal FOV and x represents the vertical FOV

and in the game i chose u can only shift the vertical FOV

so the whole equation's purpose is to convert vFOV into hFOV

[minor spire]

oh so its a relation between the y and x axis coordinates of players fov

al;right

[rough tide]

yes

[minor spire]

lemme do it the hard way then

[rough tide]

ooo

[rough tide]

no wait like how do i put it

its relation into

how hFOV changes as vFOV changes

idk if thats what u said

but yea

[minor spire]

but on another note the result will be the same this way too since in the end your just calculating dy/dx

[rough tide]

yea thats true

[minor spire]

yea thats what i said

[rough tide]

oh ok mb

[minor spire]

how y changes wrt x

no no np

[rough tide]

yup

[minor spire]

right kay lemme try

[rough tide]

okok ty