help-28

[spiral vigil]

,tex .log rules

[glossy valve]

hayley

[spiral vigil]

in particular the power rule and the change of base rule will be very helpful here

[lucid sierra]

yeah

i think ive got right so far but idk what next step

[spiral vigil]

get everything into base 2 or 3

[lucid sierra]

yup

this is my work uhh

[spiral vigil]

yeah

[lucid sierra]

so the divisor

idk what its called uh

becomes a - a + 2 right

which is 2

[spiral vigil]

yes

[lucid sierra]

and the part above idk

hello

[spiral vigil]

hello you said wait no

which made me think you had something

[lucid sierra]

ohh xd

the divisor

3 log 2

i though its 3 log 3

[lucid sierra]

this

it was 1 - 1 + 2 not a - a + 2

[spiral vigil]

yes the divisor is 2

[lucid sierra]

i need help about the uhh

[spiral vigil]

use the power rule up top

[lucid sierra]

dividend

wait

idk next step

does that makes ²log 4

and where do the superscripts goes

or exponent idk what its called

[spiral vigil]

i don't think that 3/2 makes sense there

usually they're subscripts so I'm having a hard time following

[lucid sierra]

ohh

let me write it differently then

its just my teacher idk

[spiral vigil]

at any rate, remember that √x = x^1/2

[lucid sierra]

Ohh i see

[spiral vigil]

and 4 is 2²

[lucid sierra]

yep

do i turn it root (3)^2 or turn it to 3^(1/2)

[spiral vigil]

i mean more broadly like
(√3)² = 3...

since that's what square root means

[lucid sierra]

NOO its so weird

idk how to ask it

letme write it again then

log2 (root (3))^2 + log(root (3) (2)^2
or
log2 (3) + log(3^(1/2)) (2)^2

sorry idk how to use texlit

[spiral vigil]

those are equal

[lucid sierra]

yeah but which one is better

[spiral vigil]

um the bottom one seems better

[lucid sierra]

kk

[spiral vigil]

now change the bases so everything is in your favorite base

whether that's 2 or 3

[lucid sierra]

uhh

[spiral vigil]

it does not matter which one

[lucid sierra]

idk how to do that

[spiral vigil]

use the change of base rule

,tex .log rules

[glossy valve]

hayley

[lucid sierra]

i mean

there multiplications

[spiral vigil]

ok

and?

[lucid sierra]

let me try

[spiral vigil]

once everything is in the same base you can stop writing the base it's really useful

[lucid sierra]

oh tehres' also the exponent

idk whre to put them

[spiral vigil]

wdym

[lucid sierra]

log3^3 (3)^2 for example

that makes log3 (3)^2/log3 (3)^3 ?

[spiral vigil]

$\log_{(3³)} 3²$ will be $\f{\log 3²}{\log 3³}$ yes

[glossy valve]

hayley

[lucid sierra]

what abt the base

[spiral vigil]

doesn't matter which base you use there

but sure they're base 3

[lucid sierra]

kk

ohh a sec

the logarithm multiplication rule

i still dont understand this part sry

can i use this for something idkk

[spiral vigil]

i guess you could

but literally just convert everything into log3

and then a lot of stuff cancels

[lucid sierra]

correct so far?

[spiral vigil]

it's unclear to me where you're starting from on that first equation

[lucid sierra]

im turning them into log 3

[spiral vigil]

yeah but like

you've posted a lot of images

what was the step right before that?

[spiral vigil]

oh. then no, that's entirely wrong.

draw a big ass fraction bar and put 2 below it

and nothing else

then focus on the nimerator

you'll have subfractions

it'll look like this...

[lucid sierra]

this kinda takes too long ig

let me do it then

uhh like this?

[spiral vigil]

the more you do this the faster you get, but you must do it correctly

[spiral vigil]

yeah like that exactly

now use power rule

in like four places

[lucid sierra]

should i just make it 2log3 (3) or just 2

[spiral vigil]

if you're comfortable doing that in one step you can do that

[lucid sierra]

do what

[spiral vigil]

both of those are equal

so either one works, 2 seems simpler

[lucid sierra]

i miss something?

oh its 1 not 3

and what the rest

[spiral vigil]

again, √3 is 3^½

[lucid sierra]

@spiral vigil

[spiral vigil]

ok yes good

now do you see something you can cancel

[lucid sierra]

log3 (2)

[spiral vigil]

yeah

[lucid sierra]

@spiral vigil

[spiral vigil]

sure seems fine

[lucid sierra]

and i dont know if its actually correct or no

and i havent even study exponent equation

it took me a dam hour

tysm

[spiral vigil]

and the next one will take you significantly less time

[lucid sierra]

uhh its currently 1am and i have exam at 7

i have some more subss to study

and next one is geometric and arithmetic sequence ugh which i need a lot of formula to remember

is this correct or wrong and or i can simplify it?

well

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[spiral vigil]

looks right, idk if it can be meaningfully simplified

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[limpid bane]

,rotatw

[limpid bane]

,rotate

[limpid bane]

What do I do from here?

<@&286206848099549185>

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[limpid bane]

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[surreal spindle]

What kind of function would generate a curve like this? I would've tried an exponential but it's almost like a reverse exponential, I'm confused
I'm not sure if I'm in the right place to ask so please point it out if I'm not, thanks <33

[quaint prawn]

,w plot y=-log(x)

Not quite

[velvet sedge]

looks like a power law

if its something with like word counts or ranking, it would be expected

[surreal spindle]

oh it does

[velvet sedge]

what are you graphing?

yeah definitely looks Pareto Distribution-y/Power Law

[surreal spindle]

amount of identifiers in relation to the character count

[velvet sedge]

like x^(a) to be clear, with a clearly being some negative

[surreal spindle]

Okayy I see

[velvet sedge]

this is in the context of coding I assume? Are there other ways you can phrase this? what dataset are you looking at?

[surreal spindle]

yeah it is in the context of coding

[velvet sedge]

well idenfitiers and characters are like words of a language

because identifiers are just...variable names right?

and words of a language and their usage follow zipf's law/power law

[surreal spindle]

it's just
y: unique identifier count divided by character count
x: character count

[velvet sedge]

yeah, word lengths

[surreal spindle]

Right i see

[velvet sedge]

https://en.wikipedia.org/wiki/Zipf's_law

https://en.wikipedia.org/wiki/Brevity_law

this one

[surreal spindle]

Not quite

[velvet sedge]

yeah

[surreal spindle]

it's the amount of unique words in relation to the size of the entire thing

[velvet sedge]

OH

https://en.wikipedia.org/wiki/Heaps'_law

[surreal spindle]

oh damn right on the money I think

[velvet sedge]

maybe? or maybe I'm misunderstanding how you're ranking/assigning

https://en.wikipedia.org/wiki/Rank–size_distribution

there's a lot of these, but they all tend to follow power laws

[surreal spindle]

I think this is exactly it

[velvet sedge]

yea

well, glad to have helped :)

[surreal spindle]

thank youuu

i'm gonna go check it against the data

Yep

those two look pretty similar

Thanks a lott!!!

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@void arch Has your question been resolved?

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[warped sapphire]

would this be correct

based off of my notes it looks correct but it marked it as incorrect for some reason

[shrewd hamlet]

Maybe it wants neg exponent

@warped sapphire

[warped sapphire]

nvm it was some stupid error I fixed it

thx

there was like a space in the answer box

[shrewd hamlet]

Bruh webassign garbo

[warped sapphire]

tru

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[marsh tusk]

Find the number of times the digit $9$ appears in the list of all integers from $1$ to $500$. (The number $ 99 $, for example, is counted twice, because $9$ appears two times in it.)

[glossy valve]

龖

[weak iron]

in 1-89 the digit ‘9’ appears 9 times
in 90-99 the digit ‘9’ appears 10 times
this can be extended for each multiple of 100 (up to 800)

[marsh tusk]

so 19*5?

[left bone]

their work isn't right

you should double check it before using it

[weak iron]

i would say so

[left bone]

what?

[weak iron]

?

[left bone]

what would you say so?

[marsh tusk]

theres only 7 in the first group

i think

[weak iron]

the number 9 appears 19*5 times in all integers between 1-500

[left bone]

this is both answer giving and wrong

[marsh tusk]

whats th ansewr then

[left bone]

I'm not giving you the answer

you can see what aysob did, the idea is right, then check it and apply the idea to get the right answer

[marsh tusk]

bruh i dont see what he did wrong though

[left bone]

did you count both groups?

it's possible I'm the one making a mistake as well, in which case you should do what you know is right

[full forum]

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

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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.

[marsh tusk]

2 groups?

do you mean 1-89 and 89-99

[left bone]

1-89 and 90-99

yes

[marsh tusk]

yeah i got 9 values for first

and 10 valjues for second

so dont you jst plus the two

so 19 values for 100 digits

and 19x5

[left bone]

{90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 99}

I count 11 values for the second

[marsh tusk]

u wrote 99 twice

[left bone]

I sure did

that was very intentional

[marsh tusk]

doesnt the question say distinct though

OH

nvm

i loked at th wrong question

oh

so its 100

thanks

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[left creek]

"Sketch the solid by the given definition"

[left creek]

I can't think of a way I can math this out to make it make sense

I have half a cylinder right now

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[torn jolt]

Hi, im just learning sine/cosine graphing and my question is how can I get the starting point (reference point) from simply the equation. I know you can use (-h,k) but i was told at school by a friend that you cant always use h,k

This is the equation im trying to figure out right now, would the starting point be (-pi/6,-2) or am i wrong?

[rocky vale]

yeah that's right

if I understand what you mean by reference point

like the point on the graph that would normally be at the origin?

[torn jolt]

Yeah

[rocky vale]

yeah

[torn jolt]

is there any difference with cosine?

[rocky vale]

No, but the cosine graph wouldn't go through the reference point, right?

just to make sure I understand you

[torn jolt]

uhhhh cuz cosine is like min or max only right

[rocky vale]

yeah the normal cosine graph doesn't go through the origin, it starts at a maximum

[torn jolt]

im getting reference point questions though for cosine

y = 3cos(x - pi/2) + 1

[rocky vale]

does the question actually use the term reference point?

I just want to make sure we're talking about the same thing, becasue I don't think that's a standard term

[rocky vale]

,rotate

[rocky vale]

ok yeah it does seem to mean what I thought

so yes you have the right idea

I think what your friend was referring to, is if you have a coefficient in front of the variable, then you need to factor it out to find h

for example if you had cos(2x-pi/2)+1, you would factor out the 2 and make it cos(2(x-pi/4))+1

[torn jolt]

he was addressing this problem

ahhhh

nah we aint that far yet 😭

i got confused because i thought its always h,k when its in that form

these questions are from a team quiz we did a day ago and i dont remember exactly what we put since i just copied

[rocky vale]

wait, do you mean (h,k) as opposed to (-h,k)?

[torn jolt]

sorry -h,k

[rocky vale]

it is always the opposite of what appears in the function for h

[torn jolt]

Yeah

[rocky vale]

the only caveat is what I mentioned here

[torn jolt]

hm

wait so what would the reference point be on cosine equations then

[rocky vale]

you tell me

[torn jolt]

😭 😭

[rocky vale]

no, you had the exact right idea from the beginning

just tell me what you think

exactly like the sine one

[torn jolt]

1sec let me graph it

wait im so confused

the "reference point" for the cosine graph doesnt exist unless i didnt graph right

,rotate

[rocky vale]

,rotate

[rocky vale]

lol

[torn jolt]

😂

[rocky vale]

graph looks good 👍

If I'm understanding what you mean by reference point, it's (pi/2, 1)

and yes, that's not actually on the curve for cosine

[torn jolt]

exactly

so what does that mean

wouldnt that be a wrong answer then

[rocky vale]

everything there is definitely right

[torn jolt]

but (pi/2, 1) doesnt exist

[rocky vale]

hm

you might need to ask your teacher to clarify what "reference point" means. I've never seen that term used like that and I can't find it online

[torn jolt]

yeah thats why i came here 😭

starting point is exchanged with reference point

previous page asked "What is the starting point (reference point)"

[rocky vale]

I'll ask around but I don't think it's common terminology

[torn jolt]

i just found some notes "reference point (h,k) - the new origin"

[rocky vale]

ah

okay, that's basically what I thought

then we shouldn't expect the graph of a cosine function to go through the reference point

the normal cosine graph doesn't pass through the origin, after all

[torn jolt]

Ah

[rocky vale]

and yes, it's (h,k), not (-h,k), but that's only because the - sign is in the general form of the equation: cos(x-h)+k

[torn jolt]

wait would it be that since sin graphs start at this "new origin" since they start at mid prehaps this reference point thing is just an artifical 0,0 or something

[rocky vale]

yes

artificial (0,0) is a good way to think of it

sin passes through the origin

cos does not

[torn jolt]

yeah cuz like u do x-h=0 and u move the h over which is x=h so u inverted the sign in other words

okay welp idk what that applies towards but thanks for ur help

🙏

[rocky vale]

sure thing 👍 no problem

[torn jolt]

.close

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[torn jolt]

,close

ahhhhhhh

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[solemn lance]

Alina wants to make keepsake boxes for her two best friends. She doesn't have a lot of money, so she wants to make each box described so that it holds as much as possible with a limited amount of material.
For Jen, Alina wants to make a box with a square base whose sides and base are made of wood and whose top is made of metal. The wood she wants to use costs 5 cents per square inch, while the material for the metal top costs 12 cents per square inch. What is the largest possible box (in terms of volume measured in cubic inches) that Alina can make for Jen if she only has $30.00 to spend on materials? (Round your answer to three decimal places.)

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[dense fable]

#1(27)->#2(35)=29.6%
#2(35)->#3(48)=37.1% +7.5%
#2(7.5%)>#3(6.15%) -18% Delta
#3(48)->#4(63)=31.25% +6.15%
Scaling increases #4(6.15%)>#5(22.75%) +269.9% Delta
#4(63)->#5(97)=54% +22.75%
#5(97)->#6(~?)=
#6(~?)->#7(219)=

#5(97)->#7(219)=125.8%

I'm trying to find what #6 equals while factoring in the % scaling from the other numbers

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[spark vapor]

I am confused about how to compute low rank approximation

[fast peak]

compute svd and then just ignore a few of the singular values

[spark vapor]

im going to leave this open i have more questions

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[languid lion]

guys i need help with a geometry task it says: Imagine your math book is a rectangle. This rectangle stands on a corner and is held by the opposite corner. Your book rotates around the imaginary line (s. Fig.), which encloses a right angle with the table.

A) Draw an exact oblique image on white paper on a scale of 1:2. Enter all values and guidelines that have been measured or calculated (with a solution).

B) Calculate the volume and surface content of the resulting body.

For A I kind of know what to do but i’m stuck on B. The Book is 20x26,5cm and the rotation axis is 33.2cm. Now i don’t know how i’m supposed to calculate the volume and surface content 🥹

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[languid lion]

nope

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[stiff musk]

is this a test?

[thorny spire]

.close

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[stiff musk]

<@&268886789983436800> (not sure if this is actionable, but user deleted second screenshot that indicated this was a test)

[reef sparrow]

i'll just time out for a few hours to make sure the test deadline passes

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[ancient sequoia]

Hello, I'm a chemistry student and i haven't officially taken MVC yet but i understand the basics. a variation of this statement with different variables is used for my lecture notes, and i am not understanding how this statement holds true. would it be possible for anyone to explain this verbally or write a step by step process to prove this? even a link to a page describing whatever law or rule this follows would be fine, i just don't know how to start.

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[lethal temple]

hiii

[lethal temple]

could you help me prove that my teacher's wrong?

well, that is the assignment

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[lethal temple]

pls help

[vague moss]

Do you see (5-9/2)²=(4-9/2)²?

[lethal temple]

yes

[vague moss]

What do the numbers within each bracket evaluate to?

[lethal temple]

like, when you solve it what do you get?

[vague moss]

Yes

[lethal temple]

it's 25-45+(81/4)=16-36+(81/4)

[vague moss]

No no

Just tell me what the numbers inside the brackets evaluate to

Let the square remain

[lethal temple]

so (1/2)^2=(-1/2)^2?

[vague moss]

Yes

So when you take the square root

And cancel the exponent

[lethal temple]

it's 1/2=-1/2

[vague moss]

Do u see the problem?

[lethal temple]

not actually

[vague moss]

Ok can u gimme 5 mins

[lethal temple]

k

[lethal temple]

so technically i can't demonstrate them they're wrong?

[vague moss]

No, they're wrong because when we take a square root, we have to account for both solutions

[lethal temple]

ohhhhh

true true

[vague moss]

Sqrt(4), algebraically, gives 2 and -2

[lethal temple]

thx, really appreciate it

[vague moss]

The reason there is a mistake here is that we forget about the +/-

Aight

[lethal temple]

yes yes

thx

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[gritty robin]

"We can label the height and the slant height of the small triangle as H and L respectively. On the diagram below, h is divided in the ratio 2 : 1 (because "the strut is one third of the way up the ladder");"

[gritty robin]

I don't get it

how do you find that the ration is 2:1

[elfin stream]

similarity i suppose

[gritty robin]

but how

it tells that it is similar

but we don't know the length of the base of the other triangle

[elfin stream]

do u mind giving the full question

[gritty robin]

ok

Ohh shoot

Hehe 😅

Mb

[elfin stream]

lol its given

[gritty robin]

I guess I missed it

This is why you should read the question 2 times

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[stiff swan]

Can someone explain why this is true by the Markov Property: $P(X_3 = 3 | X_1 = 1) = P( X_2 = 3 | X_0 = 1)$.
I dont really see how the Markov property : future depends only on present works here

[glossy valve]

LeGM

[gritty rose]

Not enough context

[stiff swan]

given a transistion probability matrix for example and I am asked to find P(X_3 = 3 | X_1 = 1)

I want to know why this is true $P(X_3 = 3 | X_1 = 1) = P( X_2 = 3 | X_0 = 1)$

[glossy valve]

LeGM

[stiff swan]

is this just a time homogeneity property?

[gritty rose]

Yea that matrix is very important to the question

Notice the lack of indices less than t on the right side

Only the difference between t and t+n matters

Apply that to your problem

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[stiff swan]

ok I got the answer 1/15 but what's the point of this $P(X_3 = 3 | X_1 = 1) = P( X_2 = 3 | X_0 = 1)$ equation

[glossy valve]

LeGM

[stiff swan]

is it just to clearly show how many n-steps are required?

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@stiff swan Has your question been resolved?

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[olive horizon]

I dont know if its easy or not guys

But i couldnt make it

[summer echo]

what's the question?

[olive horizon]

it wants x lenght

[summer echo]

are you supposed to use trigonometry btw?

[olive horizon]

No u shouldnt for now

[summer echo]

Maybe you should draw a diagram yourself, this one seems inaccurate

not sure how to continue without trig though

[olive horizon]

No, this question is correct, it's just very difficult to see, at least for you and me. And you shouldn't do it with trigonometry, because according to the topic order in the book, I haven't seen that topic yet.

Please anyone?

<@&286206848099549185>

[digital wedge]

find x right?

is angle ADB a right angle?

[olive horizon]

Yep its 90 degree

Btw if anyone wonders this is a geometry question for YKS (Turkiye university exam)

[summer echo]

I tried for a bit but the only way I can think of is finding the value of sin(15) with geometric arguments

You can look up how the exact value of sin(15) is derived

Here's a more accurate version of the drawing by the way:

[olive horizon]

Thank u for ur attention but i wonder is there no way to solve this question other than trigo?

[warm flicker]

We could use assumptions

But it's geometry so

[olive horizon]

Oky guys go ahead use trigo

i dont know how

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@olive horizon Has your question been resolved?

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@olive horizon Has your question been resolved?

[charred canopy]

@olive horizon u here?

ig i solved it

forgot to write it, u get BH by doing pythagoras in BDH triangle

ig it's like this

[digital wedge]

we need to find x

from what i understand, you have proved that dbc=dcb=15degrees

but that could be proved by taking the angle sum property for triangle adb and dbc=dcb because they are base angle of isosceles

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[halcyon prism]

i have no clue as to how to do this question

all i was able to find so far is that AEC=84

clearer version

<@&286206848099549185>

[digital wedge]

.reopen

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[digital wedge]

Gordon Freeman, you can use pythagoras in the end for your answer. the value of tan 15 is 2- root 3

in terms of x?

IMz

not much info cuz of the question

aint being full

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[outer dust]

so i find r'(t) = <2t,5,3t^2>

ah wait i need to find the unit tangent vector right?

the unit tangent vector looking realllll weird here

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@outer dust Has your question been resolved?

[red blade]

hi keaton!!

[outer dust]

hi

[slate violet]

you don't need to actually

converting to parametric form, the line has direction vector (-2, 5, 3)

so you just need to dot r'(t) with (-2, 5, 3) and set it equal to 0

the value of t you get will give you the point

[outer dust]

i tried that but i just got 9t^2-4t+25=0?

[slate violet]

yeah

[outer dust]

that cant be factored, oh do i need to use the quadratic formula?

[slate violet]

that's weird

has no real roots

[outer dust]

huh

[slate violet]

oh wait the normal plane of the curve

ahhhh so when (t^2, 5t, t^3) is some multiple of (-2, 5, 3)

cause the normal plane of the curve is perpendicular to the direction vector of the curve

so the curve and the direction vector are parallel

[outer dust]

yea im not sure though how to proceed, its looking for a point but i cant stop thinking about cross product

which i know is wrong

[slate violet]

so just t^2/-2 = 5t/t = t^3/3

which gives t = 0

[outer dust]

the answer is (1,-5,-1)

wait what did you do to get t=0

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[paper kraken]

for this problem you are trying to determine if the shapes are congruent. If they are what transformations are needed. My friends told me how the figures are not congruent as point N is odd whille all the other points are even. Can somewone explain this further please

[white geyser]

i think what they mean by even and odd is that triangle ZXY's vertices are all on coordinate points, while triangle LMN has the point N that lands somewhere in between coordinates.

[paper kraken]

Meaning the y value of point N is odd right?

[white geyser]

i wouldn't call it odd, we could say its y value is not an integer.

[paper kraken]

The cordinates of P N are (8,-5)

[white geyser]

ah i wasn't reading the whole graph mb, yeah then what they said about even/odd is true.

the triangles are not congruent and thus cannot be mapped onto eachother.

[paper kraken]

What do the odds and evens represent

[white geyser]

just points on the coordinate plane

you could also just find an angle or side that is not similar to the other triangle and prove they are not congruent that way

[paper kraken]

True

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[violet minnow]

I thought i was right

[worn matrix]

yea u are right.

[violet minnow]

hmm it didn't work tho

[worn matrix]

wdym what was the question asking

for?

to find the value of x?

[violet minnow]

ah i see

just the solution set not the intersection

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[violet minnow]

thank you

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[graceful peak]

how do i approach this problem

i tried to simplify it

which gives me [abc + a + c]/[bc + 1] = 3.14

no idea how to go from that (or if its even right)

[hot herald]

consider that since b,c are positive integers
b + 1/c > 1,
what will be the range of 1/(b+1/c)?

[graceful peak]

it would still be greater than 1 right

[hot herald]

wdym by "it"

are you saying that
$$\frac{1}{\text{something greater than 1}} > 1$$
?

[glossy valve]

ℝαμΩℕωⅤ

[graceful peak]

OHHH

okay so

sry i thought since a>0 then 1/a>0 it also applied to a>1 😅 im dum

since 1/(b+1/c) is less than 1 and a is a positive integer

a = 3

and 1/(b+1/c) = 0.14 ?

b + 1/c = 7 and 1/7

so b = 7 and c = 7 ?

so a + b + c = 17

[hot herald]

yeh

[graceful peak]

thank you!!

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[boreal folio]

Im trying to solve this problem and keep getting lost, here is my work:

[devout valley]

A valid basis, but not unique - different ways to do it, luckily you know that each option is linearly independent by not being scalar multiples of each other, so you could test whether each vector is a valid element of the plane

[boreal folio]

so if the answer was a, i should be able to plug in x_1 = 1, x_2 = 2, x_3 = -1 and get 0?

hypothetically lol

[devout valley]

Yep, that's the idea - do you get zero when you do that? similar for the other one (they're also nice to repeat the options a bit!)

[boreal folio]

ah

well tysm that makes it so much easier

was there anything wrong with my approach?

[devout valley]

Not really - as mentioned, your basis is valid, so if they asked you for one, and you gave that, it would be fine

[boreal folio]

ah ok, just multiple vectors

again tysm

[devout valley]

Another way you could do it is to see if you can get any of the other vectors from your basis ones

For example, $\pmqty{1 \ 0 \ 3} + \pmqty{0 \ 1 \ -2}$ is a pretty nice choice \catthink

[glossy valve]

@devout valley

[boreal folio]

just to make sure I understand, d would be the answer because you can reverse plug them back in?

[devout valley]

Yep it's d

[boreal folio]

Ok bet thank you 😄 ive been at this for so long lmao that explanation really helped

I kept thinking i did something wrong and not finding my mistake, didnt know there were multiple answes

answers*

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[devout valley]

Also note that this gets you the $\pmqty{1 \ 1 \1}$, you can get that $\pmqty{1 \ 2 \ -1} = \pmqty{1 \ 0 \ 3} + 2 \pmqty{0 \ 1 \ -2}$, so from your basis you can create theirs

[glossy valve]

@devout valley

[boreal folio]

.reopen

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✅

[devout valley]

sorry was just trying to add that in

[boreal folio]

wait so if I just factored mine a little I could get that too?

ah

well thats good to know lol

[devout valley]

But yep, (uncountably) infinitely many in fact(!)

[boreal folio]

to build on that they have a couple more equations with the same style of questions

can I do the same thing here? like reverse plug them back in?

[devout valley]

Basically for those ones, you want to figure out if the sets are linearly independent (for a set of two vectors, that's if they're not scalar multiples of each other) and they satisfy the equation

Note the importance of the linear independence comment!

(one you should immediately be able to rule out as a basis from just looking at it...)

[boreal folio]

so both the vectors cant be a multiple of other ones? Like for ii they arnt liniarly independednt?

sorry I came from diff eq lol this is my first lesson on linalg

[devout valley]

Yep, (ii) is linearly dependent, in fact any set with the zero vector in it is linearly dependent!

[boreal folio]

hmm ok thats neat

so for these, as long as they are not linearly independent and they satisfy the equation they work, no more rules?

and I can do what i was doing before to find it if ans choices are not given?

[devout valley]

For this one, pretty much (might change a bit for other questions though!)

[devout valley]

Yep, you can!

[boreal folio]

ah ok perfect, so for this one the last two work

second one isnt liniarly independent

and first one jsut doesnt satisfy the equation

[devout valley]

nice nice!

[boreal folio]

ok bet

again tysm lol you just explained something to me which 30 minutes in the book did not

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[torn jolt]

How do I simplify $\frac{1}{\frac{x+2}{3x-6}}$

[glossy valve]

Matt

[lime ether]

kcf

[torn jolt]

?

[lime ether]

keep change flip

when u take the reciprocal of any fraction

it just makes the denominator the numerator

1/(x+2/3x-6)

[torn jolt]

ohh

[lime ether]

is the same thing as 1 * (3x-6/x+2)

then factor the numerator

and u will see it simplifies nicely

ur final answer is

just

[torn jolt]

so $\frac{3x-6}{x+2}*1 = \frac{1}{\frac{x+2}{3x-6}}$

[glossy valve]

Matt

[lime ether]

yes

then factor

the numerator

for simplest form

and ur done

[torn jolt]

$\frac{3(x-2)}{x+2}$

[glossy valve]

Matt

[lime ether]

👍🏻

[torn jolt]

i see

[lime ether]

yup

[torn jolt]

ty!

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[lime ether]

no problem

[hot herald]

factorisation not needed

[torn jolt]

why

[hot herald]

doesn't make it objectively simpler

[lime ether]

nah u always factor

atleast i do

[hot herald]

its situational

[lime ether]

u don’t have to ig

but

i would

[hot herald]

here there's no significant difference

[lime ether]

true

[hot herald]

so not much point

[lime ether]

🤷🏼‍♂️

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[plush egret]

I need some help understanding this conclusion.

[glossy valve]

jan Niku

We have some function $\rho(t) = \phi(t) + i \psi (t)$ is analytic. Assume we are moving along some contour $C$ where $\psi$ is constant, and that $\phi$ has a maximum at some $t_0$ on the interior of $C$. 

Now, what Strogatz says is that let $\hat T$ be the unit tangent vector at $t_0$. Then the directional derivative $\dv{\psi}{s} = 0$ at $t_0$ since $\psi$ is constant on $C$ (here $s$ parameterizes $C$). But, $\phi$ satisfies $\dv{\phi}{s}$ at $t=t_0$ because $\phi$ has an interior maximum there. so $\dv s (\phi + i \psi) = 0$ at $t=t_0$, and $\dv s \rho$ in direction $\hat T$. But $\rho$ is analytic, so its derivative is the same in all directions. Then $\rho ' (t_0) = 0$, and $t_0$ is a saddle point.

[plush egret]

I'm not sure that i understand why t_0 is a saddle point

the fact that both the directional derivatives are 0 along C at some point would mean that both phi and psi should increase if we moved orthogonally off of it there, right?

I get that this means that the level curves have crossed, I just dont understand what that means for the global curvature there

why cant we just have a maximum or a minimum?

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@plush egret Has your question been resolved?

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@plush egret Has your question been resolved?

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@plush egret Has your question been resolved?

[simple totem]

@plush egret by interior of C, do you mean a point not on the endpoint? or do you mean that C is a closed loop and you're talking about the area bounded by C

i think you were talking about the former. let me see if i can explain

[plush egret]

C is just some counter along which we have $\psi$ is constant

[glossy valve]

jan Niku

[plush egret]

I just dont think i undestand some kind of critical relationship between the directional derivative, the level curves, the gradients, the function, and the CR equations

[simple totem]

my thinking is that by CR you immediately have that p'(t) = 0 implies a saddle point
phi_x = psi_y, and phi_y = -psi_x
if you were to have a maximum, you would need phi_y = psi_x as well

[plush egret]

i guess i dont believe p'=0

i mean, i dont really get why we have that

strogatz just constructs it

out of p=phi + ipsi

but i am not convinced thats uhh

i am not convinced

like its a dot product ... does that give us that we can just assemble it this way?

[simple totem]

p'(t_0) = 0 by construction

phi' = 0 on C, and psi' = 0 by virtue of being a maximum

or

i have it backwards

but yeah

[plush egret]

i guess okay maybe by stretch of me saying i understand and believe

direction derivative along C of p is 0

but why does that give p'(t_0) = 0?

[plush egret]

i guess critically at t_0 in the direction of C

[simple totem]

what does p'(t_0) mean to you

[plush egret]

im not really sure

i have in my head either a minimum or a maximum of the function locally

and there it makes sense

[simple totem]

i mean rigorously

[plush egret]

i dont know

[simple totem]

i understand it as p' = phi' + i*psi'
where (-)' = d-/dt

or ds

whatever the parametrization variable is

[plush egret]

but it should depend on direction

right?

[simple totem]

the direction is implicitly given by the fact that we're on a contour

so like

[plush egret]

i guess we get interactions between phi and psi maybe cancel some of that dependence out

[simple totem]

(\dv{\rho}{t} = \pdv{\rho}{x}\cdot\dv{x}{t} + \pdv{\rho}{y}\cdot\dv{y}{t})

[glossy valve]

maximo

[simple totem]

where dx/dt and dy/dt are coming from the countour C

[plush egret]

bleh i might not know enough multivar to understand this

[simple totem]

well like

[plush egret]

i think strogatz is trying to lean into multivar calculus

to be handwavey about complex stuff

[simple totem]

the countour C can be parametrized via (r(t) = \langle x(t), y(t)\rangle)

[glossy valve]

maximo

[simple totem]

or generally

x(t) + iy(t)

[plush egret]

sure

[simple totem]

we then have the tangent vector (\langle x'(t), y'(t)\rangle)

[glossy valve]

maximo

[simple totem]

the point being

when you talk about dp/dt

the derivation wrt t

[plush egret]

oh

[simple totem]

only makes sense while you're on C

[plush egret]

then this almost looks like grad r dot tangent

[simple totem]

yeah

[plush egret]

but each component function has its own gradient

should i not be putting too much neccesity on understanding his like

construction of dp/ds?

youre talking about it like i can just jump to dp/ds at t_0 along C directly from just understanding how its constructed there

the point, i mean, not dp/ds

does that make sense

[simple totem]

what i am saying is that dp/ds only makes sense along C

if you forget about C

and forget it's parametrization wrt s

then dp/ds loses meaning

what is ds?

[plush egret]

i understand that, i mean, then you don't have a tangent vector so what are you even dotting it with

[simple totem]

yeah exactly

[plush egret]

but strogatz moves right from dp/ds to p'

so idk if understanding how he builds up from the other directional derivatives is critical here

[simple totem]

i think it's back to the original question now

what does p' mean?

[plush egret]

should it be clear why mechanically we can do that

[plush egret]

idk anymore i used to think of it as a linearization

but i dont think thats right either

i mean, i can give the definition

but i dont understand what it means on an intuitive level

[simple totem]

maybe im jumping ahead of myself
have you shown that analytic implies smooth?

[plush egret]

i heard ryc say this the other day

i dont really get what it means

doesnt analytic just mean its once differentiable?

in whatever blah domain

so this is saying if you can differentiate once, you get infinitely many as a result?

but i hadnt heard it until ryc said it

[simple totem]

differentiable -> smooth on C is another result yeah. im actually not sure about the relationship

what i was going to say is that you know p is analytic

and so smooth

[plush egret]

okay

[simple totem]

so what you should expect is that the derivative at a point behaves well, regardless of where you approach the point from

[plush egret]

this is just because you can break it apart right

i mean or maybe uhh

you arent restrained in your choice of what distance means in the limit defn of the derivative

wrt direction

or i guess you mean more intuitively otherwise itd indicate some kind of not niceness

a jump or something somewhere

but that wouldnt make sense

because its smooth

[simple totem]

yeah what i think of is that our complex derivative exists, and so the limit converges "regardless of path"

so this contour is not particularly special when it comes to the derivative, that is, p' is going to be 0 regardless of which contour we approach x(t_0) + iy(t_0) by.
C just has some outstanding properties that let us compute the derivative

namely that psi is constant and phi has a maximum at t_0

[plush egret]

so its like

if we find some path through the complex plane

and some tangent unit vector along that path gives dp/ds

then if p is analytic, it must be that p'(that point) = 0, too

i guess it didnt make sense to translate from like

derivative is notion of as you approach a point

and to me directional derivative is like

[simple totem]

if dp/ds = 0 here, then yes i think we are saying the same thing

[plush egret]

as you move away from it

so i think it doesnt seem like they should connect in that way

okay so

so why does p'(t_0) = 0 mean t_0 is a saddle point

should that be clear

you said something before, sorry

[simple totem]

CR

[plush egret]

this

[simple totem]

phi_x = psi_y, and phi_y = -psi_x

[plush egret]

i don't really get why this would mean saddle 😳

is it intepretable through gradients/level curves

[simple totem]

let me see

maybe it's not as obvious as i thought
i was thinking that if phi_x = psi_y and phi_y = psi_x, that would amount to a max/min

so changing the sign in one of the directions would result in a saddle instead

[plush egret]

i guess i dont even get why that would mean max 😅

[simple totem]

let me see if i can justify it somehow

[plush egret]

i was more wondering like

[simple totem]

the problem is that this is in C^2 or R^4

so it's tough to visualize

[plush egret]

i just imagine, two level curves

each one is at level 0 or however you would say that

and theyre orthogonal

and the gradients are pointing some direction orthogonal to the level curves

so really, at t_0, you have gradients pointing along each level curve, like an overlayed capital L ontop of an X

yea i guess i dont know

@simple totem I appreciate your help, my sleeping pills are kicking in

maybe itd be better if i slept on it then toyed with this and see what happens

i can come back with something better than idk

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[simple totem]

yeah i was trying to think of something but nothing comes to mind

maybe i should look at the definition of a saddle point in C 😂

good night jan

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[simple totem]

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[verbal adder]

can someone explain what formula theyre using

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<@&286206848099549185>

[inland beacon]

@verbal adder its the exponetial decay model f(t)=A*e^kt

[torn jolt]

Prove that (abc)^2>4∆/√3, where a, b and c are the sides of a triangle and ∆ is its area. Give me a hint

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[light steppe]

tiny question about cauchy sequences and the right intuition

[light steppe]

so the way i understand it cauchy sequences are those where the elements get arbitrarily close together

however, in my lecture notes it says that it is not sufficient to require that the distance between successive elements converges towards 0 and i have a hard time understanding why

[ocean lark]

Because cauchy says something stronger

not just about the difference between consecutive elements

But about the differences between every element

if that makes sense

I remember a question on a quiz or something from my analysis class that provided some intuition as to why exactly this is different

but its been a while so i cant remember

[light steppe]

but if i think about it visually, the elements of a sequence whose consecutive elements get closer and closer, wouldn't that mean that if i go along infinitely long in the sequence all elements will converge towards 1 point

[light steppe]

in the lecture script we got it just says that these notions are not equivalent but not why 😕

[ivory cairn]

cauchy says for any value n and m greater than some N, we have |a_n - a_m| is less than a given epsilon.
What you have is that after some N, |a_n - a_(n+1)| is less than a given epsilon

take something like a_n = log(n)
as n increases log(n) and log(n+1) get closer. So at some point |log(n) - log(n+1)| will be less than a given epsilon.
however, log(n) and log(1000n) will always be far apart, so log(n) is not a cauchy sequence.

[light steppe]

okay thank you both, i don't totally get the intuition but i see how the statements differ

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[misty fractal]

Hello guys, could someone help me with this?

[misty fractal]

I guess i'd have to use simon's favorite factoring trick here

Not sure how though maybe i'm dumb

x + y = 1/4 xy

[minor crater]

yeah

notice that x+y is an integer

thus 1/4 xy is also an integer

so xy must have a multiple of 4

[misty fractal]

hmm okay i see

4x + 4y = xy

4x + 4y - xy = 0

x(4-y) + 4y = 0

x(4-y) -4(4-y) = 0

(4-y)(x-4) = 16

then list out all the combinations?

[minor crater]

not sure what you did here

[misty fractal]

i rewrote it so that i can factor?

i subtracted by 16

so i added by 16 on the other side to compensate for it

[minor crater]

oh you wrote = 0

so x(4-y)-4(4-y) = 16

[misty fractal]

oh whoops typo

yes since i'm subtracting 16 from both sides i add it back to both sides so rhs should've been 16 not 0

[torn jolt]

||dont want to disturb you guys,
i just have one idea to solve it,
according to the question we need to consider a line y=x and all the points above this line and on this line would be the points where x<y
so you might try sketching the graph or taking intersections maybe
might help ya||

[misty fractal]

is a hyperbola right

not a line

[torn jolt]

all points are asked ? 💀

or maybe a new function which satifies the condition given ?

[misty fractal]

i'm not sure what you mean 😭

[misty fractal]

it should be -16 right

or am i dumb 😭

(x-4)(4-y) = -16

(x-4)(y-4) = 16

not trying to be pedantic, just making sure

[torn jolt]

never mind me
although i dont think it would work

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[kindred raven]

derivatives

$e^{5tanx} + 5^{2x}$

[glossy valve]

odokawa

[kindred raven]

why I can't to use the sum formula

[torn jolt]

are you trying to compute [
\m[\bigg]{\dv x}{e^{5\m\tan x} + 5^{2x}}
]

[kindred raven]

$f(x) \pm g(x) = f'(x) \pm g'(x)$

yes

that

[glossy valve]

odokawa

[kindred raven]

or i dont know

the problem is

just that

[torn jolt]

yes that is valid (not the way you wrote it)

[torn jolt]

Use chain rule

[kindred raven]

chain rule for this?

[torn jolt]

Ofc...

How will u derivative e^5tanx

[kindred raven]

5sec^2

[torn jolt]

What 💀

[kindred raven]

or am I wrong

[torn jolt]

Yes

Derivative e^x is e^x* x'

[kindred raven]

i have mental issues and skill issues if you could help me to understand better it would be great

[torn jolt]

That is the chain rule

[kindred raven]

$e^x = e^x (x')$

[glossy valve]

odokawa

[torn jolt]

You are smart my dude, just missing out on a stupid theorem

Yes

That

[kindred raven]

$e^{5tanx} (5sec^2e^{5tanx})$

[glossy valve]

odokawa

[kindred raven]

this for the left side

[torn jolt]

Why would e^x appear again

[kindred raven]

agh

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[clear turret]

This has to be in cos

I don’t understand the period

[clear turret]

so far I have this 5 cos ( x + pi/3 ) -2

<@&286206848099549185>

<@&286206848099549185>

[woven eagle]

Do you have problem finding period ??

[clear turret]

yes

[woven eagle]

There's a property stating thar

If fundamental period of f(x) is T then fundamental period of f(ax ± b) is T/abs(a)

Just use that

[clear turret]

can u simplify what u said cause i did not understand a single word

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[coarse tundra]

Hello