help-10

[ancient jacinth]

you rationalized the denominator

i see now

yes you can do that

[ruby ravine]

yip

[ancient jacinth]

thats actually easier

because the zeroes of f(x) are easier to see

[ruby ravine]

Welll, I can sort of like guess that the minima is -2, but I am not able to put it rigorously

[sinful marsh]

i guess i will just do the graph and determine the range from that

[ruby ravine]

If you write that function in even simpler format, you would write it down as

y = (t - 4)sqrt(t - 1)

[ancient jacinth]

well i dont think you are supposed to

[ruby ravine]

where t goes from (1, inf)

[ancient jacinth]

the slope of the minima is 0

[sinful marsh]

guys i dont get why x^4-5x^2+4 becomes (x^2 - 4)

[ruby ravine]

are you familiar with quadratic factoring?

[sinful marsh]

i dont think so bro

[ancient jacinth]

can you solve x^2 - 5x + 4 = 0

by factoring

[sinful marsh]

wait guys im not english im from italy

i dont understand

quadratic factoring what you mean

[ancient jacinth]

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

do you know how to do this?

[sinful marsh]

this is the same

[latent notch]

ruffini

correct?

[ancient jacinth]

ruffini?

is that italian

[latent notch]

That’s how you factor

bruh

[latent notch]

I think he is, but its a method used universally

[ancient jacinth]

what's ruffini

[sinful marsh]

i studied ruffini actually but i dont remember it lmao

[ancient jacinth]

is this a ligma prank

[ancient jacinth]

oh like that

[latent notch]

yeah

thats the ruffini method

[ancient jacinth]

hmm you can do that

[sinful marsh]

i just find the 2 solutions with delta

im noob

[ancient jacinth]

but if you look at x^4 - 5x^2 + 4
you can replace x^2 with u
and then the equation becomes u^2-5u+4

you can factor u out, and replace it with x^2

[ruby ravine]

My algebra has gotten worse, I don't know why the heck can't I move forward with such a simple equation lol

[ancient jacinth]

finding the minimum?

[ruby ravine]

y = (t-4)sqrt(t - 1)

[ruby ravine]

finding it's minima without differential

[sinful marsh]

ok i understand but i do it in a much slower way

[ancient jacinth]

ok

but in both cases you get x^4 - 5x^2 + 4 = (x^2-1)(x^2-4)

[ancient jacinth]

to answer this

go back to the original equation

what happens when you multiply the numerator and denominator by sqrt(x^2-1)

[sinful marsh]

i know that

[sinful marsh]

i think is like this x^4 - 5x^2 + 4 = (x^2-1)(x^2-4) not like this x^4-5x^2+4 becomes (x^2 - 4)

[ancient jacinth]

yes

the x^2-1 will divide

[sinful marsh]

ah ok

[ancient jacinth]

now back to the original question

to find the minimum

doing this will show us the whole range

[sinful marsh]

i dont have studied the minimum yet :/

i just started derivatives

[ancient jacinth]

hmm

[sinful marsh]

i know limits but more than that no

[ancient jacinth]

does the problem require only limits?

no calculus allowed?

[sinful marsh]

idk

[ruby ravine]

you can differentiate using limits I meannn

[sinful marsh]

i find this function online just to practice

maybe i cant do it

[ancient jacinth]

YOU CAN DO ANYTHING

[sinful marsh]

normally to find range i do the inverse and then find the domain of the inverse or i just see it from the graph

[sinful marsh]

with my knowledge

[ancient jacinth]

do you want to use derivatives or only limits?

[sinful marsh]

only limits but if is long dont worry, you already have been helpful

[ancient jacinth]

hmm

i dont know how you could do this with limits

you have to use derivatives here

[sinful marsh]

no problem man

[ruby ravine]

You can derive the derivative using limits though

[sinful marsh]

you have been a lot helpful guys

[ancient jacinth]

OHHHH

[ruby ravine]

but leave it, it will only cause more pain lol

[ancient jacinth]

wait

one sec

[ruby ravine]

lim x -> 0, (f(x + a) - f(x)) / a

if you are curious enough, evaluate this limit

after you have done so, you'll get a formula. Set that equal to 0, and the value of x you get is the value where minima's at

[ancient jacinth]

shouldnt it be lim x -> a (f(x) - f(a)) / x-a

[ruby ravine]

nope

[ancient jacinth]

this is the derivative in general

[ruby ravine]

yeah

[ancient jacinth]

actually its the limit as a approaches 0 here

[ruby ravine]

oh woops, sorry

[ancient jacinth]

and thats f'(x)

[ruby ravine]

yeah, I meant, we can derive the general formula for derivative here, and put that equal to 0

[sinful marsh]

guys i go thx for your time

i will comeback when i learn derivatives

is the next thing i will learn at school

.close

[obtuse pebble]

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[ancient jacinth]

W

[obtuse pebble]

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[manic slate]

Let d be a metric on a non-empty set X. Define ρ as follows, for x, y ∈ X:
ρ(x, y) = min {1, d(x, y)} .

[manic slate]

Prove carefully that ρ is a metric on X.

so, I have done the first two steps

so proving ρ(x,y)=0 iff x=y

and the second proving ρ(x,y)=ρ(y,x)

but in proving ρ(x,z)<= ρ(x,y)+ρ(y,z), I ran into a bit of a problem

so I used the fact that d is already a metric, thus it satisfies all the criteria

so

I found that, putting it very rough.
ρ(x,z)<= min{1,d(x,y)+d(y,z)}

reason being because we already know that the triangle inequality holds for the metric d

The next logical step would be, in my opinion to split it

as

min{1,d(x,y)+d(y,z)}

=

min{1,d(x,y)}+min{1,d(y,z)}

and that would then conclude the proof

But I don't think you're allowed to do that.

[obtuse pebble]

@manic slate Has your question been resolved?

[manic slate]

<@&286206848099549185>

[obtuse pebble]

@manic slate Has your question been resolved?

[manic slate]

<@&286206848099549185>

[obtuse pebble]

@manic slate Has your question been resolved?

[misty dagger]

check this:
p(x,y)+p(y,z)=min{d(x,y)+d(y,z),d(x,y)+1,d(y,z)+1,2}

can u show triangle inequality now?

[obtuse pebble]

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[night fossil]

Can someone fimd the An of this sequence

Its geometric

<@&286206848099549185>

I found it was geometric

But i dont know how to apply it

[supple pumice]

try separating the numerator and denominator as their own sequences

[night fossil]

Oh is it 2^n / 3^n

[supple pumice]

now simplify

[night fossil]

How to do that

[supple pumice]

i just mean to write it simpler

$\frac{2^n}{3^n} = \big(\frac{2}{3}\big)^n$

[warm shale]

please request a new nickname

[supple pumice]

do you know how to do this?

[night fossil]

Ah ok yes i know

Thank you for the help

[obtuse pebble]

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[obtuse pebble]

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[proven zealot]

https://cdn.discordapp.com/attachments/1078340536634720398/1080260578855108749/image.png

[proven zealot]

https://cdn.discordapp.com/attachments/1078340536634720398/1080260579178057788/image.png

why is c not correct

a number times a vector for example 5u is just legal I thought right?

[kind hawk]

here that dot specifically represents the dot product

not just normal scalar multiplication

but well they are being a bit strict

[proven zealot]

what is the difference?

with the dot and multiplicatioj?

[kind hawk]

in a) for example you could in theory also just define the norm of a number (which would then just end up being the absolute value)

well one is a dot product between vectors

and the other is scalar multiplication between scalar and vector

[proven zealot]

ah ye but the excercise ask why it doesn't make sense

for that one I do understand it doesn't make sense

[kind hawk]

the first gives a number as result, the second gives a vector as result

[proven zealot]

u is a vector

v times w is a number right?

[kind hawk]

yes

so it would be the dot product of a vector and a number

which doesnt make sense

[proven zealot]

ah that's why

same for d I guess

I meant b

[kind hawk]

Yes

[proven zealot]

so for b it's like

a number

• vector

ah okay I tihnk I got the hang of it now

.close

[obtuse pebble]

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[proven zealot]

do u also maybe know why this step is legal?

[civic zealot]

what is example 1.9

[proven zealot]

this but

this is the previous chapter

I think I will skip this question the proof is like 2 pages

.close

[obtuse pebble]

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[proven zealot]

btw can u maybe explain this theory

[obtuse pebble]

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[reef adder]

need help with this

[wooden cipher]

thats an equation

but

whats the question

[reef adder]

sorry

wasnt specific

i need to differintiate it

and simplify

[robust sleet]

Chainrule+product

[reef adder]

I know how to do it

however i get confused

when i need to factor

could you show me how i can factor it?

I found have it in the form f'(x)g(x)+f(x)+g'(x) already

Lmk if you can help

<@&286206848099549185>

.close

[obtuse pebble]

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[indigo coyote]

I'm supposed to solve this using law of sines

[indigo coyote]

I'm not sure how to get the height using only a side and angle

<@&286206848099549185>

[obtuse pebble]

@indigo coyote Has your question been resolved?

[obtuse pebble]

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[left jetty]

Can someone check if i did this right or not i dont have access to the answer sheet

[bold bane]

Looks good to me.

[obtuse pebble]

@left jetty Has your question been resolved?

[obtuse pebble]

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[west sigil]

Im a college freshman taking trigonometry, my highest level of math is very very basic algebra, (i failed alg1 and cheated through alg2 since i had very confusing and hard to understand teachers in high school, zoom learning didnt help either) and i have my first trigonometry exam tomorrow and trig concepts feels so alien to me, where can i start cramming knowledge and learn fast before i take my exam? im super lost

[west sigil]

this is the type of trig i need to get done

[stable rain]

ummmmmmm

thats a lot

start w khan academy

[west sigil]

this is the study guide^

[stable rain]

idk that seems to b the standard

[west sigil]

o

you think i can learn this much in less then a day?

[stable rain]

um

[west sigil]

the max i know is basic alg, simplifying square roots that kinda stuff

[stable rain]

idt im good at giving advice

TnT

i wld say its basic

maybe

but um

normally u ask specific questions u have here

do u have any specific Q u dont get

[west sigil]

bascially everything lol trig is a foreign concept to me

[sonic anchor]

you know unit circle?

[stable rain]

meeee?

uwu

[west sigil]

no

[sonic anchor]

unit circle is a circle with center in origin and radius 1

cos and sin are defined from the unit circle

[west sigil]

ah i think i seen about it in class but i didnt understand it

[sonic anchor]

every point in the circle will be $(cos(\theta),sin(\theta))$

[warm shale]

Køter

[sonic anchor]

where $\theta$ is the angle at that point

[warm shale]

Køter

[west sigil]

o

im very confused lol

[stable rain]

um

the x value u can get using tri

trig

ditto y value

[stable rain]

itll always b this

[sonic anchor]

so the x coordinate for that point is $x=cos(\theta)$ and the y coordinate $y=sin(\theta)$

[stable rain]

ay got ur own drawing

[warm shale]

Køter

[sonic anchor]

yes 💀

[west sigil]

o

you think khan academy has unit circles?

[sonic anchor]

yes they will also teach you it

you dont understand what we are saying or?

[stable rain]

khan academy has um

a lot of stuff

[west sigil]

yeah im confused, i dont understand the 6 trig functions either

[sonic anchor]

what specifically is confusing about the drawing i made

[frosty spoke]

honestly what you should probably do

if you have a few hours

is go watch all of the khan academy videos and do the exercises

[stable rain]

watch khan aca

[west sigil]

the numbers on the outside and the trig functions purpose

[frosty spoke]

but make sure to do it when you're not sleepy

you can't cram this stuff without the stuff from before

it just doesn't work very effectively

[stable rain]

sleep is impt

[sonic anchor]

the numbers on the outside is reffering to the radius of the circle

[stable rain]

sleep and eat well

[sonic anchor]

and the trig functions purpose is that when you have the point on the unit circle

the x coordinate will be cos(theta) and the y cooridnate will be sin(theta)

[west sigil]

bet ill try my hardest

[frosty spoke]

and like don't try to speedrun it

actually do it like you're trying to learn what's going on

[west sigil]

ah gocha

[frosty spoke]

skip the stuff about conic sections or whatever

[west sigil]

ight, do i learn trig on khan or should i also learn the algebra i missed out on?

[frosty spoke]

you need to start at the beginning

if that means it's algebra 1

then that's where you start

you start at somewhere where you have a good understanding of everything before

if you can't do place value or multiplication, start there

not saying that you'll manage to get everything 100%

but if there's something there, then you'll do immensely better at studying the stuff you need for the exam

so save like 4 hours before the exam, and spend the first two reviewing the exam subjects on khan academy, and then spend the next one making a "cheat sheet" paper that you write everything important you want to remember on the exam

and depending on the exam, if you're allowed to use it, then use it, if not then just toss it in the trash bin

[west sigil]

bet ill do my best, we are allowed only a ti83 or 84 and a sheet of paper thats all 😬

so bascically ima have to memorize all the trig functions 😢

[frosty spoke]

you don't really

[west sigil]

wdym

[obtuse pebble]

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[left jetty]

I dont have access to the answer sheet can someone confirm if this is right or not?

[obtuse pebble]

@left jetty Has your question been resolved?

[obtuse pebble]

@left jetty Has your question been resolved?

[night echo]

Looks correct ✅

[left jetty]

.close

[obtuse pebble]

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[timid silo]

hello

[timid silo]

can i get help with a pascal triangle question *

[royal basin]

http://dontasktoask.com/

[timid silo]

so yes?

well ill take it as a yes

“3rd term in expansion of (4m+1)^4”

[royal basin]

ok, so what's troubling you with this

[timid silo]

the 4m

wait i figured it out

nice

.close

[obtuse pebble]

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[shadow lava]

I'm really confused with the associative property

I've got brackets where they're not supposed to be, and I'm not sure why

my version looks like everything in the numerator is being multiplied, but it's actually adding and subtracting everything in the first and third terms

did I mess up this part somehow?

[obtuse pebble]

@shadow lava Has your question been resolved?

[royal basin]

yes you did

[shadow lava]

hmmm

[shadow lava]

I don't see it

and how to remove the brackets from the 1st and 3rd factors

[royal basin]

when you factor out (2x-3)^3 from

(40x+32)(2x-3)^3 - 20(20x-3)^4

you should have had:

(2x-3)^3 * [(40x+32) - 20(20x-3)]

but your (40x+32) got stuck outside

for no apparent reason, even an erroneous one

[royal basin]

i think you've got their names confused as well.

[shadow lava]

that last term..

[royal basin]

bad sign

[shadow lava]

yeah

but after I distribute

am I supposed to remove the brackets?

[royal basin]

if you keep the parentheses on, it's
...[(40x+32)-(40x-60)]

[royal basin]

now it is good, except for a stray unmatched ) at the right which should be erased

[shadow lava]

that's OK too? or should I always keep the brackets on if I distribute

it sounds like the brackets are OK on or off, as long as I distribute correctly

if there are extra brackets they will go away with + - being the only operators left in the square brackets?

I think I'm finally starting to see it, but man this was challenging for me. I just need to double check on the brackets after distribution. Good practice to keep them?

[obtuse pebble]

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[shadow lava]

.reopen

[obtuse pebble]

✅

[shadow lava]

https://tenor.com/view/phew-gif-19070342

[dark stirrup]

What is your question

[royal basin]

well, the work you've now done is correct

[shadow lava]

forgetting the rules of distribution, association, when enclosed in brackets with another distribution happening

glad to see I finally found it

[shadow lava]

thanks!

.close

[obtuse pebble]

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[true breach]

Can I get explaination of how to go from step 1 to 2

[supple granite]

First they canceled the 2 on both sides

Since 162=81×2

Seems like they also multiplied 81 on both sides

To get rid of the denominator on the RHS

[true breach]

Wait how does that work

[topaz loom]

Divide by 2 on both sides

[supple granite]

They divided both sides by 2

[true breach]

That gives 1/81 tho

O wait

Nvm

[topaz loom]

Yeah

And then multiply by 81

[true breach]

So what is rhs

[supple granite]

Right hand side

The right side of the =

[true breach]

Ok so what happened to the exponents

Idk how it went from n-1 to 1-n

[topaz loom]

I can explain that

[true breach]

Pls do

[topaz loom]

So generally

$$(1/n)^2$$ is the same as $$n^-2$$

[warm shale]

Bloxx12

[true breach]

N-2????

Isn't it n-1?

[topaz loom]

I cant use the tool sorry

[true breach]

O

[topaz loom]

But say we have (1/n)^2 for example

Any exponent works though

Then this is equal to n^-2

A negative exponent

[true breach]

Ohhhh

Ok I see what ur saying

[topaz loom]

And n-1 becomes -n+1

[supple granite]

Actually I think the RHS of the second line is wrong

It should be $3^{n-1}$

[true breach]

Maybe?

[warm shale]

jay.

[topaz loom]

No i dont think so

[true breach]

Idk I got it from brainly.in

[topaz loom]

What about the 1/3?

[supple granite]

The LHS is 3^(1-n), but the RHS is 3^(n-1)

[true breach]

Oh yeah it changed

On the 3rd line

[topaz loom]

Oh wait what

[topaz loom]

Yeah i didn't see that

[topaz loom]

Pfff what the hell lol

[true breach]

OHHH WAIT

Ok I understand the (3)^n-1 now

[topaz loom]

Thats stupid

[supple granite]

Elaborate

[true breach]

Wdym

[topaz loom]

But in the 3rd line its correct again

[true breach]

It would be 1/3 ^1-n then

[supple granite]

Yes that's why the second line is incorrect

[topaz loom]

Yep

[supple granite]

That's why this is right

[true breach]

Anyway what happened on the left side on 4th step

[supple granite]

81=3^4, right?

[true breach]

What

No like when 1-n goes into the brackets

[supple granite]

You can rewrite it as $(3^4)^2(3)^{1-n}$

[warm shale]

jay.

[supple granite]

$(a^m)(a^n)=a^{m+n}$

[warm shale]

jay.

[supple granite]

Are you familiar with this rule

[true breach]

O

Ok isee

So

How come

[true breach]

8+1-n is dragged down after 3 is canceled on both sides

[supple granite]

You can think of it as taking log_3 on both sides

[true breach]

Ok

[supple granite]

$Log_3{3^{8+1+n}}=8+1-n$

[warm shale]

jay.

[supple granite]

That way the 3 cancels and you're left with the exponents

[true breach]

Ok ty

.close

[obtuse pebble]

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[timid silo]

for the last couple of questions I plugged the x into the derivative equation and it worked fine but for these value looks like they plug into normal function. so when do I plug into function and when into derivative? (absolute minima and maxima on closed intervals)

[obtuse pebble]

@timid silo Has your question been resolved?

[obtuse pebble]

@timid silo Has your question been resolved?

[fiery birch]

They both can be used to show the presence of a critical point

@timid silo

f(x) will be rising and reaching a maxima which is also a critical point as after that f(x) changes direction or is dropping in value

Try to see why this happens , plot f(x) and then see what happens to it before and after a critical point as well as at the critical point

Also remember that the derivative is the slope of the tangent you draw to a point

[obtuse pebble]

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[rain cove]

How do you do this?

[dark stirrup]

I do not believe there is an analytical solution

length of an ellipse, right?

[obtuse pebble]

@rain cove Has your question been resolved?

[rain cove]

It doesn't have to be an Analytical solution

<@&286206848099549185>

[obtuse pebble]

@rain cove Has your question been resolved?

[tough hull]

What is interquartile range

[obtuse pebble]

@rain cove Has your question been resolved?

[obtuse pebble]

@rain cove Has your question been resolved?

[obtuse pebble]

@rain cove Has your question been resolved?

[obtuse pebble]

@rain cove Has your question been resolved?

[frosty spoke]

if it doesn't have to be analytical, any numerical integration method for a given a and b will work

[obtuse pebble]

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[sudden leaf]

so is ∫₃¹⁵ f(x) dx the same as ∫₃¹⁵ f(u) du

[lost tree]

yes the variable of integration doesnt matter

∫₃¹⁵ f(x) dx = ∫₃¹⁵ f(u) du

[obtuse pebble]

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[obtuse pebble]

@spare crypt Has your question been resolved?

[obtuse pebble]

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[trim star]

hi

[trim star]

I was having trouble solving my washer method and don't know what's wrong with the steps I did. I have included the problem and my work

[obtuse pebble]

@trim star Has your question been resolved?

[trim star]

<@&286206848099549185>

wait nvm it was right i just mistyped answer

.close

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[white knot]

Hello

[patent jetty]

JSUAT ASK

JUST

Ask

[past tide]

relevant link: https://dontasktoask.com/

[obtuse pebble]

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[proper oak]

Hey, im having a bit of trouble with this question, anybody mind helping me out

I simply don’t understand how I’d go about answering it, I’ve figured out the volume of the cake tin, but I’m a bit stuck.

[past tide]

u should first make sure the units can fit with each other in your calculations

volume over area would give you the depth, since water would fit the container

[proper oak]

Ok, i know the volume is 19,739.188cm cubed

[past tide]

what is the relationship between cm^3 and mm^3?

write it down

[proper oak]

A mm cubed is one one thousandth of a cm cubed correct?

[past tide]

hold on lemme calculate

1cm = 10mm

[proper oak]

And cubed we multiply by 10 3 times right

[past tide]

(1 cm)^3 = (10 mm)^3
1^3 cm^3 = 10^3 mm^3

yeapp ur correct

[proper oak]

So wait, should i multiply the area of the bottom by ml’s converted into cm cubed?

[past tide]

why multiply?

anyways u should convert mL to mm^3

or cm^3, whichever is more convenient

[proper oak]

That would be 200cm^3 then right since ml = cm^3

[past tide]

yes!

ok draw the shape of the water after it's poured

how do u get the length of the cylinder if you have its volume and top area?

[proper oak]

I know the diameter is 20cm, so the radius is 10cm, so the area is pi x r^2

[past tide]

yep

[proper oak]

And multiply that by the pi x diameter ( 20cm )

For volume

[past tide]

wait what

???

[proper oak]

The height would be pi x diameter

Height x area = volume

[past tide]

height = volume/area

= (volume of cylinder, which is the volume of water) / ( pi x r^2)

pi * diameter would be circumference????

[proper oak]

Oh wait yeah, and i don’t need height since I’d be using the depth of the water

As height

So how do i use the 200ml to find the height knowing that it’s filling a space with an area of 314.1592cm^2

[past tide]

.

[proper oak]

Can i convert ml’s to cm^2 or just cm^3 by the way?

Okay, i figured out, i had to turn the 200ml into 20,000cm^2, then divide that by the area of the circle, and then divide that answer by ten

To get mm’s

So thanks, i don’t need anymore help

[past tide]

yes of course both are fine. U can convert any volume unit to any volume unit, as long as the idea is the same

not just volumes, areas, lengths, energy, whatever

[proper oak]

Ok

I was a bit confused cause I haven't done stuff like that in a while, I've stopped being taught some of the basics in exchange for more formulaic stuff which makes it hard to figure out stuff i don't know the formula for

[obtuse pebble]

@proper oak Has your question been resolved?

[proper oak]

Yeah

Yes

[obtuse pebble]

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[bold siren]

hi I would like to ask how to do 7a without excel

[worn karma]

uhhh

[thick gyro]

cant you use the normal distribution set

[bold siren]

like the standard normal distribution table?

[obtuse pebble]

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[lament estuary]

I am struggling with finding this derivative using implicit differentiation. I get as far as the quotient rule but i am struggling with the algebra to solve for dy/dx.

I feel like i should have it but i cannot see the step i am missing.

[high lily]

for the initial applications of derivative rules, you can leave dy/dx as dy/dx

can you show what you've done so far?

[lament estuary]

Ill have to take a picture of my work using my phone. give me a minute please and ill post it.

[obtuse pebble]

@lament estuary Has your question been resolved?

[lament estuary]

I circled the area of algebra i was working with. I have only been working with the LHS because the RHS is pretty simple. It is just 2dy/dx.

In my pic here, the top left of my circled area is where i started and the rest is just me trying to find our where to go with my algebra.

(My algebra is wrong)

[obtuse pebble]

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[frail lake]

hello

[frail lake]

i got a question

[fierce lagoon]

Read this

Specifically the first bullet point

[frail lake]

so when you convert to polar equation

do you solve for r?

[fierce lagoon]

Typically yeah

Depends if you're parameterizing

[frail lake]

can you give me an example when you dont?

[fierce lagoon]

Well one example is where if I were to find the hypervolume of y over a cylinder

[frail lake]

here wait I think I got one

x + 4y = 2

so I changed the x and y to rcos and rsin

[fierce lagoon]

I would let
x = rcos(θ)
y = rsin(θ)

Although r and theta are both parameters

[frail lake]

my teacher has put that as the answer

[fierce lagoon]

Yeah that's fine

[frail lake]

but why about the r?

wqhat*

what*

[fierce lagoon]

I mean if we wanna graph it on the polar plane

Then having r = something makes it jice

But if you're using polars to parameterize then you don't need to

[frail lake]

so leaving it as rcos + 4(rsin) = 2 is good if you want to convert to polar equation

[fierce lagoon]

You can still isolate r though

[frail lake]

hm

[fierce lagoon]

It'll be equivalent

I guess it doesn't really matter

[frail lake]

but then there is this problem y^2 = 3x

and she did solve for 4

r*

and she got r=3cot + csc

both questions are asking to convert to polar equation

[fierce lagoon]

Then isolate r

Well actually

Did she say equation or function?

[frail lake]

equation

[fierce lagoon]

You don't need to isolate r then I think

[frail lake]

ok

[fierce lagoon]

If you're not expected a function

[frail lake]

cool thank you

.close

[obtuse pebble]

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[sour fox]

so the function g2(x):

[sour fox]

to show case that it is invertible my teacher did this:

basically, since the function is strongly increasing in the interval [-1, 4], this means it is invertible

[gleaming ridge]

you mean strictly increasing?

[sour fox]

but my question is, cant we showcase for each value for x when -1 <= x <= 4 that each input has a unique output

[sour fox]

yes

[gleaming ridge]

the same thing but an indirect route

just unwind the definition of 1-1

(assuming the function is shown to be onto)

[sour fox]

i see

ok thanks 🙂

.close

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[stable vector]

what the diff

[tardy epoch]

are you looking for the antiderivative?

[stable vector]

no like what the diffrent between this two like any example of each one to see the diffrent

[obtuse pebble]

@stable vector Has your question been resolved?

[tardy epoch]

unclear since R isn't defined

you need to show the entire context for people to help you

[stable vector]

.close

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[static beacon]

x^2+2x+2 = (x+1)^2

[severe reef]

Well, no, +1 to that

[timid silo]

Hi, I have to evaluate the oblique asymptotes for this function

[static beacon]

oops

didnt see that

[timid silo]

What I have tried after evaluating the slope is to find where the asymptotes touch the y axis

So to do this I extracted x^2 from the square root and then I took x^2 from the numerator

$\lim_{x \to +\infty} \frac{x^2(4-\frac{8}{x} + \frac{4}{x^2})}{3x\sqrt{1+\frac{2}{x} + \frac{2}{x^2}}} - \frac{4}{3}x$

[warm shale]

FriedrichDN

[timid silo]

But actually the right decision was to multiply the square on both sides and then use Taylor after extracting x^2

I don’t get why this is correct

[ancient jacinth]

think of e^x for example

the taylor series for e^x is 1+x+x^2/2+x^3/3!+x^4/4!...

consider the graph of xe^(2/x)/2

you can look at the taylor series expansion of xe^(2/x)/2

e^x is 1+x+x^2/2+x^3/3!+x^4/4!...

e^(2/x) is 1+2/x +(2/x)^2/2+(2/x)^3/3!+(2/x)^4/4!...
xe^(2/x)/2 is x(1+2/x +(2/x)^2/2+(2/x)^3/3!+(2/x)^4/4!...)/2

xe^(2/x)/2 = x/2+1 +x(2/x)^2/4+x(2/x)^3/12+x(2/x)^4/48...

as x goes to infinity, the other terms get smaller and smaller

you get left with x/2+1

which is what the oblique asymptote approaches

so using the taylor series helps

[timid silo]

But why I can’t just multiple for the square root and then extract x^2

All the times everyone comes up with an example and doesn't say why my opinion is wrong

And I end up with a checklist of things to not do instead of getting why

<@&286206848099549185>

[obtuse pebble]

@timid silo Has your question been resolved?

[timid silo]

Maybe I understand why

The numerator it’s ok

But brutally simply the denominator with 1 is not right because its like using the first order of Taylor expansion

Of the denominator

So the approximation is not good

So actually the idea is to extract what I can and then see if this lead me to a 0 caused by a bad approximation

If that’s true I have to take the next order Taylor expansion

<@&286206848099549185> am I right?

[obtuse pebble]

@timid silo Has your question been resolved?

[timid silo]

<@&286206848099549185>

[obtuse pebble]

@timid silo Has your question been resolved?

[obtuse pebble]

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[timid silo]

im very new to induction, i have a proof for Consider a chessboard of size 2n ×2n for some arbitrary positive integer n. Remove any square
from the board. Is it possible to tile the remaining squares with L-shaped triominoes could i walk someone through it

[timid silo]

im not entirely sure its correct

om charbit

my saviour

okay so the base case would be

n = 1

meaning that our board is 2 x 2

and we know that without loss of generality

we can place the L piece anywhere

because it can be oriented

and we can choose the orientation based on the missing tile

for our inductive hypothesis

Assume 2^n x 2^n is true, and show that 2^n+1 x 2^n +1 is also true

im pretty sure you need to divide your board into 4 coordinates

the idea is to apply the inductive hypothesis on the 4 quadrant

i have a drawing

actually no im wrong

i dont know : c

[unreal musk]

[I'm not actually super sure how you'd do it, but wanted to check over what you had ]

Hmmm it is a question though

I mean the subdivision thing seems like a good idea, but then if you remove just one square, while the quadrant you removed it from, you can do the induction hypothesis on

For the others, you'd still be "full", and so there's no guarantee you can tile those ones, is there?

[timid silo]

you cant pick where the missing tile will be but you can choose specific missing tiles in the other coordinate such that you know you are able to fill it in based on the inductive hypothesis?

[unreal musk]

Hmm I was thinking along those lines, you can't choose where the removed one would be sure, and of course you could remove more so that you can use the induction hypothesis on those...

Oh, what about at the "centre"?

[timid silo]

oh thats what my original drawing had

[unreal musk]

I was thinking that for the ones that don't have the missing tiles, remove a tile from the "centre" and then you'd be able to induction hypothesis to cover those, and then clearly you can cover the "removed" three with another L

[I had that typed out already haha]

Does that not work?

[timid silo]

can you say this

like for the ones that dont have missing tiles

[unreal musk]

That was my thought, and it seems to me to work, right?

[timid silo]

it should work based on the base case?

i mean i know you use the inductive hypothesis but

[frosty spoke]

oh no it's dopamine

how's the chaotic university life going

[timid silo]

dont laught at my pain

terrible

i hate my university

we are programming something to covert pre to postfix with stacks and queues and i want to kms

[frosty spoke]

LMFAOOO

they ask that in interview questions too

[timid silo]

uh wdym

i dont mind programming i just procrastinated and so now im suffering, but i only procrastinate because my courseload is terrible and my c class is not the only class im trying to pass

[frosty spoke]

like they ask the prefix to postfix notation

[timid silo]

dude i also had to like turn a queue into a dictionary to have precidence of values in the queue

i had to implement a postfix calculator

whcih wasnt that bad

non eof this is bad uh i just procrastinated and im using a late day

[frosty spoke]

sounds horrible LOL

anyway

for your problem

[timid silo]

uh yeah

right

the point is to leave the center part emtpy for any quadrant not containing the tile

?

[frosty spoke]

no

does this help

[timid silo]

thats what i said

[frosty spoke]

oh well

[timid silo]

so you have to sa

say

Assume 2^n x 2^n is true

right

[frosty spoke]

yes

assume that you can find a tiling for any board of size 2^n

[timid silo]

let me show

i dont know how to word the center thing

because technically this has four cases

based on the quadrant, what needs to be left empty is

okay i guess id have to state all of them actually

no?

[unreal musk]

Can just assume without loss of generality no?

The cases for the others, you'd argue almost identically

[timid silo]

uh i dont know how to word that the tiles we choose to be empty should be the ones facing the center if that makes any sense

this part

[unreal musk]

As for wording it depends on how exactly they would want you to state it I think

Thoughts for extra clarity would be to "coordinates" it, if you get what I mean? But then that might be a bit overkill

[unreal musk]

Part of me feels that stating something like this is "obvious", but then it might not be

"for the quadrants that do not have a missing tile, remove the tile that is closest to the centre. We can then use the induction hypothesis to deduce that the remaining tiles in those three quadrants can be tiled, and on returning those tiles, we will have an L shape that can be covered by a triomino" or something like that

[timid silo]

also unrelated but what is the purpose of the base case

might be a bad q, just want to make sure i understand fs

[frosty spoke]

basically if you don't know the base case is true

and only the inductive case

it doesn't prove that everything is true

the classic example of this is "all cars are the same colour" or variants of that

[unreal musk]

For example, "for all positive integers n, we have n > n + 1" or something

[timid silo]

i think im also confused about the inductive case

[frosty spoke]

you could number each piece as 1, 2, 3, 4

[timid silo]

like where is the inductive case derived from

like idk in cs, recursion is basically assuming your code works

is this the same thing

[frosty spoke]

and then say that piece 1 has a gap in its bottom left, piece 2 has a gap in its top right, piece 3 has a gap in its top left

[frosty spoke]

yes it is

and also say WLOG, assume that the missing corner is the top right or something

[timid silo]

is induction recursion, or is recursion induction

[frosty spoke]

hmm it's more like

[timid silo]

why?

[frosty spoke]

because you can always rotate it so the missing corner is in the top right

you can show that recursion works with induction

like given a recursive algorithm, you can show via induction that it works

[timid silo]

like rotate the individual quadrant?

[frosty spoke]

no rotate the whole thing

you can formalize this a little bit more

[timid silo]

is that even necessary?

[frosty spoke]

not really, it's pretty obvious that there's rotational symmetry

[unreal musk]

[I'd vote no]

[frosty spoke]

but if you're dealing with someone who's really hard with the logic, you can say that we'll show there exists an arrangement for the top right corner

and then by rotating everything, we now have an arrangement for the bottom left corner

[timid silo]

also how do you usually end these proofs

[frosty spoke]

the rest of the proof is left as an exercise to the grader

[timid silo]

:/ gonna pull that on my programming

[frosty spoke]

pro gaming

in all seriousness, I usually wrap up inductive proofs with something like, "since we have proven P(base case) and that for all n, P(n) implies P(n+1), we conclude that P(n) is true for all natural numbers >= base case"

but you could also end it with AAAAAAAAAAAAAAAAAAAAAAA THIS MAJOR SUCKS

[unreal musk]

"I swear down, if you say this proof is wrong, gaaah"

[frosty spoke]

you should probably say somewhere that n >= 1

actually no

it works for n = 0 too

[timid silo]

it does but its stupid to use n = 0

[frosty spoke]

looks fine

[unreal musk]

Yea I'm happy with it personally

[frosty spoke]

oh wait there's a typo at the top

the chessboard is 2^n x 2^n

not 2nx2n

[timid silo]

oh shoot yeah

latex moment

i just do it for that extra one point

[frosty spoke]

need all the points you can get before the grades go into a dumpster fire next year

and it becomes dopamine the business major

[timid silo]

ha my grades already in the dumster fire now

[timid silo]

no my parents would make my go home

my tuition too expensive for me to not study cs

[frosty spoke]

sounds like asian parents

[timid silo]

im actually white

HAHAHAHA

[frosty spoke]

you could study something else and tell your parents that you're studying math

or cs

[timid silo]

i told you many times that i like cs

i just am not good at this because ive never done it before

[timid silo]

im not some quitter

plus

math majors freak me out

[frosty spoke]

same honestly

[timid silo]

poh shen loh lovers

oh yeah se induction to prove for any n ∈ N, any set with exactly n-many distinct elements has
exactly 2^n-many distinct subsets. also confuses me

my math major friend was trying to explain but uh

i did not understand it at all

[frosty spoke]

so basically this amounts to expressing the subsets of the (n+1) set in terms of those of the n set

does that make sense so far?

[timid silo]

n + 1 is what we are trying to prove

inductive is just 2^n

[frosty spoke]

right

so consider the subsets of the (n+1) set

actually let's do a more concrete example

the subsets of the set {1, 2} are:

{}
{1}
{2}
{1, 2}

and the subsets of the set {1,2,3} are:
{}
{3}

{1}
{1, 3}

{2}
{2, 3}

{1, 2}
{1, 2, 3}

and I wrote them in that order to sorta help you figure out the pattern here

do you see how the subsets of the set {1, 2} each correspond to two of them of {1, 2, 3}?

[timid silo]

its half and half

[frosty spoke]

wdym

[timid silo]

for instance

in {1, 2, 3} 3 is in half of the subsets

[frosty spoke]

well yeah

so differently stated, you can take each of the subsets of {1, 2}, and you have two choices to make a subset of {1, 2, 3} by either including 3 or not, right?

[timid silo]

correct

[frosty spoke]

so you might say that there are twice as many subsets of {1, 2, 3} as {1, 2} right?

[timid silo]

yes

[frosty spoke]

does that help you prove the inductive case?

[timid silo]

uhh uwehuhudhuhudshufhudhfushfuh

wait so firstly the base case is just 2^1 right

[frosty spoke]

do you define the natural numbers to include 0

[timid silo]

unfortunately yes

oh shoot if orgot

yeah

0 is a natural number

thats dumb

[frosty spoke]

so then your base case has to be n=0, because you need to prove it for all natural numbers

unless you want to add in the extra bit that says it also holds for n=0 somewhere else

[timid silo]

no

[frosty spoke]

I remember all of my CS profs defined N to include 0 and all of my math profs said no

weird that way

[timid silo]

its not weird

everything in cs starts with 0

in math, 0 isnt even a real number to my knowledge

like

[frosty spoke]

especially your grades 😩

[timid silo]

its just 0

[frosty spoke]

LOL it definitely is

[timid silo]

yeah idk dude never got a 40 percent on a midterm before this year

i wish my school had grade inflation

dude at berkley you can fail all ur midterms and STILL GET A B IN THE CLASS

[frosty spoke]

surely your classes are curved right?

[timid silo]

no

absolutely not

[frosty spoke]

I remember getting a B- in a machine learning class despite totally failing the exams and messing up the final project LOL

[timid silo]

: c : c im gonna be a systems person

[frosty spoke]

yeah that's why it's berkeley

[frosty spoke]

just wait until you get into networking and all of the graph bullshit comes back

[timid silo]

im now confident i will eventually be good at this math stuff

i just need to work hard or whatever

idk

[frosty spoke]

yea you should be

eventually you get better

[timid silo]

uh okay so the base case is the empty set right

[frosty spoke]

I sucked at it compared to the math majors

yeah the base case is n=0, which corresponds to the empty set

[timid silo]

i stay away from real math majors

[frosty spoke]

I got peer pressured into going into hard math classes by my math major friends

got absolutely destroyed

I still don't even know how to solve a PDE despite having failed a class in it

[timid silo]

Assume that any set will have $2^n$ distinct elements and show that any set will also have $2^{n + 1}$ distinct elements.

[warm shale]

dopamine

[timid silo]

AY

WHAT

[frosty spoke]

the fking prof literally starts the first lecture with some Lagrangian mechanics

[timid silo]

IT TRANSLATES LATEX/

[frosty spoke]

subsets

any set with n elements will have 2^n subsets

[frosty spoke]

LOL

[timid silo]

Assume that any set with n elements will have $2^n$ distinct subsets and show that any set n + 1 elements will also have $2^{n + 1}$ distinct subsets.

[warm shale]

dopamine

[timid silo]

wait this is so fun

if i knew tis was a thing i woulda just made all my questions in latecx

latex is like vim

[frosty spoke]

gross?

and hard to use

[timid silo]

no

vim is great

[unreal musk]

You didn't know that there was $\LaTeX$ here?

[warm shale]

@unreal musk

[timid silo]

uh i knew but i didnt know it just parses discord messages into atex

[timid silo]

i get this and it makes sense but idk how this ties in with the IH

[frosty spoke]

the fact that it's {1, 2} really has no bearing

like it could have 1000 elements

well actually you have to do it in a little bit reversed order

so let's say I have a set of 1001 elements

I trim off the last element, call it THE_LAST_ELEMENT

now I have a set of 1000 elements, which we'll call B = A \ {THE_LAST_ELEMENT}

the subsets of A are precisely the disjoint union of {all subsets of B} and {all subsets of B with THE_LAST_ELEMENT added on}

you can show rather quickly that both of these have 2^1000 elements by the inductive hypothesis

also if you don't feel like working with arbitrary sets, you could insert a short remark about how we can convert these to {1, ... n} by counting, but not sure if your TA will like that

[timid silo]

we didnt learn counting yet

[frosty spoke]

--- dopamine 2023

[timid silo]

this is a weird way of putting it

[frosty spoke]

wdym

why's it weird

[timid silo]

i think i get what youre saying

{all subsets of B with THE_LAST_ELEMENT added on}

wdym here

[frosty spoke]

like added on means put into the subset

so for example, we name a subset of {1, 2, 3} like {1, 3}

this is just {1} with 3 added into it

[timid silo]

right

[frosty spoke]

or maybe I should say it this way: all subsets of A can be classed into two disjoint classes

those that contain THE_LAST_ELEMENT and those that don't

[timid silo]

disjoint meaning only UNIQUE elements

[frosty spoke]

disjoint means that they don't overlap

and if you remove all of the THE_LAST_ELEMENT from those that contain it, then they're just the subsets of B

obviously these sets don't necessarily have orders, so you can't really say something is the "last" one, but any choice of element will work

[timid silo]

i hate this

actually my brain is hurting

[frosty spoke]

"i love cs" -- dopamine 2023

"we didn't learn counting yet"

[timid silo]

this isnt programming

[frosty spoke]

hmm okay do you want a programming analogy for this

maybe that'll help you understand the logic

[timid silo]

yes

[frosty spoke]

so imagine that you have a set {a, b, c, d}

[timid silo]

oml okay

[frosty spoke]

the subsets can be represented as the binary numbers from 0 to 15:
0000
0001
0010
...
1111

[timid silo]

twos complement

[frosty spoke]

where to get from a binary number to the set, you just take each digit and use that to say whether the thing is in there

[frosty spoke]

no not twos complement, positive binary numbers

[timid silo]

hahah ik

python doesnt use twos complement

[frosty spoke]

basically you construct a subset as follows:
do I include a? (2 choices)
do I include b? (2 choices)
...

[frosty spoke]

well python doesn't overflow or underflow, but surely CPython implements integer arithmetic with some twos complement stuff for numbers in the usual range

and this makes 16 choices

[timid silo]

basically in {1, 2, 3} if we take 3, half of the subsets will include 3 and half wont

[frosty spoke]

yes

and the half that don't are precisely the subsets of {1, 2}

[timid silo]

yes

[frosty spoke]

so by equality of the halves, the total is twice the number of subsets of {1, 2}

for a set of size n+1, and an arbitrary element a picked out of it, half of the subsets will contain a, and half won't

[timid silo]

yes

yes

yes