help-38

[bitter quest]

.solved

.closed

[viral shore]

.close omegalul

[bitter quest]

.close

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[reef plaza]

https://cdn.discordapp.com/attachments/951973195035078707/1024713559726370826/318024849333288960.png

[reef plaza]

how would i come across to solve this

[cunning cliff]

what have you tried

[reef plaza]

divided by -2x and than sin^-1(0) = x^2

so x is 0

idk how to get other values tho

[limpid sun]

double angle identity?

[cunning cliff]

no

you should be using the zero law, not dividing by -2x (-2x might be 0 and you can't divide by 0)

if 2 numbers multiply to make 0, then either of them must be 0

[reef plaza]

Yea but if i dont do that idk what to do

im a bit lost

[cunning cliff]

ab = 0 => a = 0 or b = 0

[limpid sun]

so your options are x=0 or sin(x^2) = 0

obviously x = 0 is a sol'n

[reef plaza]

Yea

that is one

[limpid sun]

so the hard part is finding x such that sin(x^2) = 0

[reef plaza]

Yea

[sharp arrow]

well sin(x^2) = sin(0)

so we can use

general solution

of

sin

[limpid sun]

so when is sin(Φ) = 0?

[sharp arrow]

oh

[reef plaza]

YEP never heard of it

[limpid sun]

when Φ = 2nπ for some integral n, right?

[sharp arrow]

just periodicity

[cunning cliff]

🤔

[reef plaza]

Yea

it repeats every 2pi

[limpid sun]

so just set x² = 2nπ and solve for x

[cunning cliff]

uh

[reef plaza]

but yea

im looking at the graph

[cunning cliff]

draw the graph of sin

[vernal briar]

Actually every pi*n

[limpid sun]

yeah, right strike the 2

[vernal briar]

n being an integer

[reef plaza]

and i dont like it it looks scary

[left oriole]

how scary can it be to take a square root

[reef plaza]

mhm

[limpid sun]

so you end up with all x such that $x = \pm \sqrt{n \pi}$ where $n \in \mathbb{Z}, n\ge 0$

[left oriole]

(just make sure you don't take square roots of negative numbers)

[sharp arrow]

\pm

[reef plaza]

YEA

[left oriole]

$n \in \mathbb N$ assuming we're looking for real solutions

[solid kiln]

Bungo

rome of oxtrot

[limpid sun]

thanks for the adjustments

and note also that the other solution is subsumed in this

[reef plaza]

so is N is any natural numberd above 0 ?

or equal too

[limpid sun]

depends on where you live

[reef plaza]

??? England

[sharp arrow]

lol

[left oriole]

varies by author

[limpid sun]

from what i've seen in the US most authors include 0 i nthe natural numbers, while in Europe most authors exclude it

[left oriole]

even if it doesn't include 0, you already identified that as a solution

[reef plaza]

oh my prof was talking about this yesterday

[limpid sun]

but there are exceptions on both side of the pond

[left oriole]

i like $\mathbb Z^{\geq 0}$ for unambiguity

[solid kiln]

Bungo

[reef plaza]

thanks guys

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[olive carbon]

Does anyone here knows how to use GeoGebra for Parabola?

[olive carbon]

Like the app or website

[drowsy dirge]

For Parabola, as in, do what with the Parabola?

[olive carbon]

the app

I want to put focus, endpoints and directrix but idk how

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[vital saddle]

The daily production cost (in lakh ₹) of manufacturing an electric device is p(x) = 7400-60x+15x^2 , where x is the number of electric devices produced per day and the daily transportation cost (in lakh ₹) of x number of electric devices is given by the slope of the function p(x) at point x.
A)How many electric devices should be produced per day to yield minimum production cost?
B) if the transportation cost of the electronic devices on a particular day is 30(in lakh₹),then find the number of transported electric devices.

I don't understand the question '-'

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[vital saddle]

<@&286206848099549185>

[past comet]

You are given a function over cost of production

You are asked to find amount to product such that the cost is as low as possible

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[sage plover]

I am having trouble understanding how the first two questions are solved using unit vector

[sage plover]

I am stuck in the part on how exactly the problem is solved using tan

Usually you put the vector into the unit vector formula and solve like in #25.

But the ones using tan have different phrasing and i do not understand how to solve then.

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[sage plover]

No

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<@&286206848099549185>

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[leaden prism]

Upload a picture for better results

People aren’t gonna wanna download something

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[covert path]

,rotate

[covert path]

how do i approach this

its review and i missed this class

[slender shard]

n is a 5-digit number. That means it has a domain from 10000 to 99999.

Pick any number n in that domain and divide it by 100, what will the quotient and remainder be?

[covert path]

the last two digits?

thats the remainder right

anf the front three is the quotient?

[slender shard]

Yes.

12345/100 = 123 with a remainder of 45.

[covert path]

do i just brute force all the possibilities then

[slender shard]

It would be useful to brute force a few solutions that are divisible by 11, then try to generalize a solution based on your results.

[covert path]

is there another method?

[slender shard]

Of course, and you are trying to figure that out.

[covert path]

.close

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[tepid shoal]

got a question

[tepid shoal]

thanks

[trim lichen]

you should begin by posting your question

[lament jewel]

thanks

[long basin]

thanks

[tepid shoal]

Yeah my fault

Anyways so on these 3 idk how I’m getting them wrong

There are other questions exactly like this but in CSC instead of SEC

There must be some rule for SEC that makes it different than csc

So if anyone knows plz help

<@&286206848099549185>

[stable bear]

hm

Does sec pi exist?

[tepid shoal]

uh idk im just learning this stuff and my teacher is like sick or something so ive got no assistance in the last 2 weeks

[stable bear]

X = 1 is an asymptote

Is that saying you were wrong?

[tepid shoal]

yes it was

thats what im confused with i thought i was right too

[stable bear]

no wait it isn’t

It’s not an asymptote

[tepid shoal]

ok

ok how

[stable bear]

I forgot the bottom

[tepid shoal]

i dont think i did when i was solving it..

so how do you solve it?

[stable bear]

Secant is the peaks and crescents of cosine right?

At each peak and crescent a parabola looking graph occurs right?

[tepid shoal]

yes

[stable bear]

You plug and check here basically

[tepid shoal]

wait so is that the same for cosecant to sine?

i really dont know much on this topic

[stable bear]

Plug in 1/2 and see what you get

[tepid shoal]

its very new. we just started this chapter and like i said my teacher is sick

K

[stable bear]

does sec of pi/2 exist?

[tepid shoal]

yea

[stable bear]

does it?

You sure?

[tepid shoal]

i mean i think it should be 1/sqrt3/2pi

wait does it not?

im so lost

[stable bear]

Put sec(pi/2 in a calculator

Ignore how to find it algebraicallg

[tepid shoal]

ight. we arent allowed to use calcs tho

like for my actual class

is sec on a calculator sin-1

if it is then it doesnt

[stable bear]

sec on a calculator isn’t sin-1

[tepid shoal]

then idk bruh

[stable bear]

It’s 1/cos you have to type it that way

1/cos(pi/2)

Bottom line, it doesn’t exist

[tepid shoal]

alright

[stable bear]

And if it doesn’t exist, the point can’t either

The asymptote is at pi/2

I need sleep

[tepid shoal]

ok

gn

so that should translate for all 3 of those problems right]

close.

.close

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[inland cedar]

Can someone analyze this ?

[inland cedar]

Give me answers please

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[inland cedar]

<@&286206848099549185>

<@&286206848099549185>

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[maiden monolith]

can someone help w these simple trig problems

[maiden monolith]

idk why im so confused

[normal kernel]

What is the problem asking for in number one?

[maiden monolith]

find the measure of theta

the second is asking to solve the triange

triangle

so probably find all values or smtn

[normal kernel]

For number one, set up an equation: cos(theta)=12/13

in order to solve for theta, you need a calculator to find the cos-1

so plug in to your calculator cos-1=12/13

[maiden monolith]

im trying to prep for a test, so are these possible w no calculator?

wait so when I do 12/13 what is my result

it's like 0.92

does that number mean anything or do i just plug in if i have a calc

[normal kernel]

Do you know if the teacher will allow calculators?

I havent learned any way to solve arc functions without calculator

[maiden monolith]

yeah its a no calc test so they probably won't have problems like that on the test

one more question

[normal kernel]

so to find the measure of that angle, you go to your calculator and press the cos-1 button. and then it you put in cos-1 (12/13)

and whatever value it gives you is your angle

[maiden monolith]

and to convert to degrees do i just divide by 180

[normal kernel]

to convert radians to degrees?

[maiden monolith]

yeah

or i mean multiply by 180

[normal kernel]

to convert a radian to degrees just substitute 180 instead of the pi. For example if you want to convert 3pi/2 radians into degrees, you substitute 180 in for the pi....so it would be 2(180)/2....and then simplify that fraction

that would be your angle in degrees

[maiden monolith]

ohhh i see gotcha ok tyty i get it

one more question

its about graphing

how would u approach graphing this

like i dont understand how to get started

i know about even roots and all that stuff

[normal kernel]

First I would determine the end behaver of the function by looking at the leading term. If the degree of the leading term is an even number, the graph will look like a parabola

if its odd, the graph will look like a trig graph

Then find the zeros of the function. The x intercepts.

for that function you would have to use synthetic division to know where is crosses the x axis

honestly tho...

to make life easier

to graph that, just find points

so plug in a random number for x in that equation and whatever value you get from solving it is your y value

plot that point and then continue doing the same thing until you get a graph

[maiden monolith]

oh i see so the leading term w the be x^4

so it's even

[normal kernel]

yeah

[maiden monolith]

and it ends positive

[normal kernel]

yeah

[maiden monolith]

wait so say I did the division what would i divide by>

?

because we learned about division and stuff so i'm assuming she wants me to

do that

[normal kernel]

have you covered synthetic division yet?

[maiden monolith]

stuff like

this

yeah

[normal kernel]

ah yeah that kind of division would work

if you do that division you will be able to find the x intercepts of the function

[maiden monolith]

oh ok so w the function i showed above

what would i divide by?

or how do i find what to divide by?

[normal kernel]

its honestly trial and error....usually you divide by factors of the last term. so you could divide that equation above by (x-2)

if you get no remainder for that division, then 2 is a x intercept

if you get a remainder, then you have to choose a different number and divide by a different linear factor

[maiden monolith]

oh wait what, so will the factor always follow a (x - someting) format

or do i need to think about smtn in front of the x too

[normal kernel]

yes, you will always divide by (x - something)...that is your linear factor

[maiden monolith]

wait ok say say i get a problem and they give me one intercept like (3,0) or smtn then the factor would be x - 3 right

and i can divide to find the rest?

[normal kernel]

yes exactly!

if they give you a negative number like (-3,0) then make sure you divide by (x+3).....it would be like this: (x-(-3)) and that would turn into (x+3)

[maiden monolith]

ohhhh i see ok gotcha, I have a feeling they'll give me 1 x-int if it's on the test

so that's probably what it'll be

also

back to cos and sin say I get smtn like cos(7x/4)

how do u use the unit circle to help w that

or sin (180)

no calculator

[normal kernel]

ummm honestly just memorizing the unit circle is the best way to help

sin of 180 is 0

[maiden monolith]

oh yeah I think ur right

ok last question how would u solve a problem like "if an eight foot stop sign casts a 10 foot shadow, what is the angle of the elevation to the top of the sign"

like I get it makes a triangle

u only have 2 digits

and the hypotnous is like the squareroot of 164 or smtn

do u use the law of cosine or whatever?

[normal kernel]

no you use arc functions again to find the angle

your given the opposite side length

and the adjacent length

so you would plus into your calculator tan-1 (8/10)

and then whatever it gives you would be the angle

[maiden monolith]

ok so if i'm given 2 side lengths should use my calculator and find the inverse right?

the onoy situations whre I don't is if im looking for an angle

and I can do soh cah toa?

rigth

[normal kernel]

if your looking for an angle and you are given side lengths, you need to use either tan-1, cos-1, or sin-1. depending of what sides it gives you. if it gives you the opposite and the hypotenuse, and it asks for the angle then you do sin-1 (opp/hyp)

[maiden monolith]

oh I see but if it's a right triangle and im given one angle aside from the 90

i can figure out the other

and use the law of cosine to find sides?

is that how it works?

wait no i just read abt it i get it now

so law of sine when given 2 angles one side

anyways I appreicate all the help sm

i needed to prep for a test and this helped tons

thank u

[normal kernel]

yeah np

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[proven bobcat]

i know i subtract a+4 to get the whole function = to 0

but where do i go from there?

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[austere sierra]

you can then multiply both sides by (a-9)^2

after you’ve taken denominator common

[austere sierra]

this makes sure the inequality doesn’t change

[proven bobcat]

.close

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[tawdry lantern]

When math books explain formulas with variables such as (a + b)(a-b) = a^2-b^2. Should I read the right side actually as (a)^2-(b)^2 ?

[lament jewel]

um

if u wanna touch on how to actually read it

mayb write in latex first

so

$(a + b)(a-b) = a^2-b^2$

[sonic echo]

whats the difference?

[sharp arrow]

$(a + b)(a-b) = (a)^2-(b)^2$

[lament jewel]

$(a + b)(a-b) = (a)^2-(b)^2$

[sharp arrow]

ha

[lament jewel]

i mean its the same

ur contrast it horrible

unreadable

[valid moss]

$x^2 = (x)^2$

[solid kiln]

illuminator3 (#eric4honorable)

[lament jewel]

yea ig

makes no diff

[sonic echo]

unless x itself has an exponent, i suppose

[tawdry lantern]

i made a mistake because i interpreted it wrong so that is why I asked.
I wrote 16y^2 as 4y^2

[valid moss]

no?

[sonic echo]

(x)^2 may be safer?

[valid moss]

what?

[sharp arrow]

safer

[lament jewel]

$16y^2 = (4y)^2$

[lament jewel]

like the ^2 doesnt apply to the number

unless u put it in ()

[sonic echo]

its the same, might be easier to see how exponent laws work though

[tawdry lantern]

Yup that is why i asked should you put braces around the variables in textbook formulas.

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[vagrant delta]

yo

[vagrant delta]

help a brother out

[wraith hinge]

$(- 1, 1), (5, - 6)

BRUH

[lament jewel]

hi

[vagrant delta]

go get your own channel bruv

[wraith hinge]

Sorry

[vagrant delta]

all good

<@&286206848099549185>

[split chasm]

!15m

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Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.

[vagrant delta]

what do i do with fractions

@split chasm

@lament jewel

@valid moss

[lament jewel]

LOL

THIS IS NEXT LEVEL

no the amount of pings are too much, im not helping

[vagrant delta]

alr bro mb

[split chasm]

you've been here long enough to know you shouldn't be doing these things

[vagrant delta]

there are no rules against pinging individual people

bruv is there

mb if there is

ill wait patiently

[sonic echo]

what do you usually do to simplify fractions?

[vagrant delta]

what do i do?

idk

give me the starting steps

im confused

[sonic echo]

3/2 = x

[vagrant delta]

the bases different

[sonic echo]

what do you do to find x

[vagrant delta]

isnt x=3/2

[sonic echo]

okay lets not take that example, lets take y=5/8

[vagrant delta]

alr

[sonic echo]

how do you get rid of the fraction?

[vagrant delta]

isnt y 5/8?

8y = 5?

[sonic echo]

there we go

[vagrant delta]

ayeee

how does that apply here?

[sonic echo]

same laws here, multiplication is your friend

except you have to multiply exponents now

[vagrant delta]

do i log both sides?

[sonic echo]

how do you multiply exponents?

[vagrant delta]

uh

multiply with the base?

OH

you raise it to a power

[sonic echo]

yep

which power would be suitable here?

[vagrant delta]

3

so i cube root?

because 3/2 is 3*1/2

bro am i dumb

[sonic echo]

no no

[vagrant delta]

im kinda lost

[sonic echo]

not quite yet

[sonic echo]

are you sure? will it get rid of the fraction?

[vagrant delta]

so i cant do (x1/2)3

oh i multiply 2

is that it

[sonic echo]

raise to the power 2, yes

[vagrant delta]

i got x3?

[sonic echo]

x^3 = ?

[vagrant delta]

16

64

mb

[sonic echo]

there we go

[vagrant delta]

OHH SHT

[sonic echo]

then x=?

[vagrant delta]

cube root

thx bruv

much love

[sonic echo]

[vagrant delta]

.close

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[twilit mauve]

hi
I'm having trouble understanding this part of the steps to a question I'm doing

[twilit mauve]

this is the whole question

the one I have to find is i'm guessing 3d =1(mod 20)

[slender shard]

What are you having difficulty understanding?

[twilit mauve]

idk how 1 (mod40) is 81

isn't it 1?

[slender shard]

1 (mod 40) is like saying 40n + 1. It could be 41, 81, 121, etc.

[twilit mauve]

well the one i'm trying to find is 3d =1(mod20)

[slender shard]

20n + 1 = 3d

[twilit mauve]

so it can be any number

like 21

41

[slender shard]

You're looking for a value of n that results in an integer d.

[twilit mauve]

sorry... how do i do that

[slender shard]

41/3 does not return an integer.

[twilit mauve]

oh so 7

?

[slender shard]

Yes.

[twilit mauve]

so 7 is d

[slender shard]

Yes.

[twilit mauve]

ohhhh

so like if y=cd (mod m)

that means

y=c7(mod 33)

but if i don'T know c how can i get the answer

[slender shard]

$y = C^{d} (mod m)$

[solid kiln]

Kookiemon

[slender shard]

Use the encoded message for the values of C.

[twilit mauve]

okay

wait but then it will be

y=0008 7 (mod 33)

[slender shard]

Use one character at a time

[twilit mauve]

oh

y=0?

[slender shard]

,wolf 0^7 (mod 33)

[solid kiln]

Results provided by WolframAlpha

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

Requested by Kookiemon#3404

[twilit mauve]

so 0

okay

and i'm guessing i have to do that 3 more times to get the answer

[slender shard]

That depends on if 00 and 08 are two letters or four letters.

They way it is written in the problem would indicate they are just two letters.

[twilit mauve]

yeah

okay

[slender shard]

So you would just decode 8 next.

[twilit mauve]

okay

how do i make e=3 into p and q

because m=33 was p=3 and q=11

[slender shard]

And it says theta is (p-1)(q-1).

e=3 is used to determine d.

[twilit mauve]

so theta will stay the same weather its m or e

oh wait

huh

[slender shard]

No, m is used to determine the prime factors p and q.

Then (p-1)(q-1) is how you find theta.

From there, you use e in the formula

$de = 1 (mod\ theta)$

[solid kiln]

Kookiemon

[slender shard]

And that will allow you to solve for d.

[twilit mauve]

so the answer will still be 7

7 3 =1 mod 20

[slender shard]

Yes, you use the d you found to decrypt the message.

$y = 00^7 (mod\ 33)\
y = 08^7 (mod\ 33)$

[solid kiln]

Kookiemon

[twilit mauve]

so if the first number is 0 and the second number is 3 does that mean the answer will be ac?

wait how did it change to 8 to the power of 7?

[slender shard]

The encrypted message is 00 08.

[twilit mauve]

oh oh woops okay i get it ow

i got it

thank you so much for the help

[slender shard]

yw

[twilit mauve]

@close

i'm i supposed to do that

sorry i'm new

.close

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[wraith hinge]

Evaluate the limit of 𝑓(𝑥) numerically or state that the limit does not exist.

𝑓(𝑥)=(4cos(𝜃)−4)/5𝜃
(Give your answer as a whole or exact number. Use the symbol ∞ for infinity. Enter DNE if the limit does not exist.

lim𝜃→0𝑓(𝑥)=

bruh i am stuck on this for a lot of time

[wraith hinge]

<@&286206848099549185>

pls

welp

i have tried approximating it from both sides(x=0.001, x=0.0001 and x=-0.001, x=-0.0001) but i cant find it

[marble wharf]

factor out 4/5

then look at the limit again and see that it is the limit of a certain derivative

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[cerulean hearth]

An idea?

[cerulean hearth]

When I'm trying to transform the cos(3x) in a cos(2x) thx to the cos(a+b) formula, i have an expression who looks more complicated

[sharp arrow]

,tex \trigangles

[agile halo]

ew what are these colours

[sharp arrow]

lets change it then

[agile halo]

my eyes

white on black

or black on white

[cerulean hearth]

In what it helps me x)?

[onyx lark]

I think u can make a cubic equation out of this

what is the question?

anyways this might help u

I'm guessing u have to find the general solution for x, there u go u can move on from here

ping if u need more help

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[oak topaz]

State a single vector equal to:
b) 𝑃𝑊⃑⃑⃑ − 𝑉𝑃⃑⃑⃑ c) (𝑅𝑄⃑⃑⃑ + 𝑅𝑆⃑⃑⃑ ) + 𝑉U

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[oak topaz]

If |𝑎⃗| = 3 and |𝑏⃑⃗|= 2, and the angle between these two vectors is 60, determine (3𝑎⃗ + 2𝑏⃑⃗) ∙ (4𝑎⃗ − 3𝑏⃑⃗)

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[lone mauve]

So I have been told that a monomial is not a polynomial

I do see how x^4 would be a polynomial because you could add 0x^3 + 0x^2 + 0x^1 + 0 to make it multi term

[split chasm]

depends on the monomial

[lone mauve]

I would just like to make sure on the other two examples

Pi is a constant?

[split chasm]

pi is a constant yes,
and would be classified as a polynomial

[lone mauve]

So it is not a rational function?

[split chasm]

it is

[lone mauve]

So this is not the correct definition of a rational function?

Oooo wait

I forgot

[split chasm]

polynomial/(another polynomial; not 0) is correct

[lone mauve]

Constant is a polynomial too

Because a constant can be multiplied by x^0

but 0 is a special case. F(x) = 0 is not a polynomial function but a zero function iirc?

[split chasm]

0 is a constant

which is a polynomial

[lone mauve]

Zero degree

[split chasm]

which makes it a rational function

[lone mauve]

Special case no?

[lone mauve]

I mean just this on its own no division

[split chasm]

if you can express it as polynomial/(another polynomial; not 0)
its a rational function

0 can be expressed as 0/1
or whatever
which satisfies the definition

[lone mauve]

We can call this a zero function according to Professor Leonard

[split chasm]

its also a polynomial

you can call it the zero polynomial if you want

[lone mauve]

Oh.. he stated it’s a weird one.. not the same as constant like 9

[lone mauve]

Alright thanks.. I will make note of this

[lone mauve]

Thank you so all three examples above are rational functions got it

So that would mean this last one is a rational function too then? By definition as per above?

[split chasm]

no, division by 0 trumps that

[lone mauve]

Lol.. but we already stated 0 is a polynomial

[split chasm]

technically it would be poly/(non 0 polynomial)

[lone mauve]

Ooo OK, I will add that in definition

[split chasm]

but no division by 0 is pretty much standard that people don't bother saying it

[lone mauve]

It’s undefined right? The last example

[split chasm]

yeh

[lone mauve]

Calculator will explode

Black hole emerges

All that stuff

https://tenor.com/view/うわぁー-助けて-怖い-恐怖-ブラックホール-gif-15781987

.close

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[lyric nymph]

Hi

[lyric nymph]

I want to check if what I did is correct for proving theorem 1.9

First let Y be an infinite set

So Y is in N or N is in Y(N is the set of natural numbers)

If Y is in N then Y is countable

Let a_n be the greatest element in Y

[reef agate]

Uh, you want to be careful with "is in" there, what you said isn't true

[lyric nymph]

I mean that it's included in it

[reef agate]

Right, that's not true

[lyric nymph]

What's wrong with that

[reef agate]

For example, the set of real numbers (0,1)

Neither is N included in it nor is it included in N

[lyric nymph]

Oh yea you are right

So it's not that easy to prove after all

Then how should I work this

Can you lend me a hand in this

[reef agate]

Let's see, I would probably do something like: given S infinite, try to construct a function f(n) which is an injection from N to S

Then the image of N under f is the subset you want

[lyric nymph]

But what if S is included in N

[reef agate]

No problem

There are injections from N to proper subsets of N, like f(n) = 2n

That's injective, and it maps into just the even naturals

[lyric nymph]

This won't work in this case beacuse there will be elements of N that won't have images in S

[reef agate]

That's okay, S can be really really big, it just has to be at least as big as N

You don't have to hit all of S to prove that S has an infinite subset

A countable subset, rather

You just need to hit countably infinite many things

[lyric nymph]

If S is infinite then it directly have an infinite subset

But the countable part is the deal

[reef agate]

Yes

Do you see how providing the injection I mentioned above shows that S has a countable subset?

[lyric nymph]

I don't know if I really got that or no

[random citrus]

you're creating a countable subset because you're listing the elements. f(k) is the k'th element

[lyric nymph]

Ok what if S is the set of real numbers in (0;1)

Then this set is uncountable

[reef agate]

Yup

Can you think of a countable subset of the reals (0,1) ?

[lyric nymph]

Doesn't this contradict the theorem bc this is an infinite set and doesn't have a countable infinite subset ?

[random citrus]

it does

[reef agate]

Why do you think it doesn't have a countable infinite subset?

Oh

Do you think countable sets have to be made up of natural numbers?

They don't -- there has to be a bijection between them and the set of natural numbers.

[lyric nymph]

Yea you are right bc if there is a bijection between the set and natural numbers then there cardinalities are equal

[lyric nymph]

So I can use this as the function that maps from N→S ?

[random citrus]

that's how rsb defined it

[lyric nymph]

He just asked me to find an injection between N and S he didn't say anything about this function

Ok but after defining this function I'll have to show that it is a bijection ?

[reef agate]

Yes, a bijection between N and ... ?

[lyric nymph]

S

[reef agate]

Not quite

[lyric nymph]

And an infinite subset of S

[reef agate]

Right

[lyric nymph]

Which can be S\least element of S

[reef agate]

S might not have a least element

These are just sets, they might not even have a sense of ordering

[lyric nymph]

S is non empty so it should have a least element

[random citrus]

what function?

[reef agate]

What's the least element of (0,1)?

[lyric nymph]

0

[reef agate]

0 isn't in that set.

[lyric nymph]

Oh yea you are right

[reef agate]

(0,1) means: all x that are strictly greater than 0 and strictly less than 1

[lyric nymph]

Yea ik just made a stupid mistake

This has no least element ..

Bc you cant reach a least element

You can take an element closer and closer to 0 as much as you wish

[reef agate]

Let's start by trying to find a countable subset of (0,1) though, that seems like a good start

[lyric nymph]

Ok

[reef agate]

how about this one: {(1/2), (1/3), (1/4), (1/5),...}

[lyric nymph]

Ok

[reef agate]

Do you agree that that's countable?

[lyric nymph]

This should be countable bc the (n+1)th element is 1/n

Isn't that right

[reef agate]

Correct

[lyric nymph]

And n is in N

[random citrus]

no

[reef agate]

Actually I think the nth elemen t is 1/(n+1) but close enough

[lyric nymph]

Yea you are right I made this as a mistake but ik the concept

[reef agate]

Yeah, I think you know the concept

Coming up with this for an arbitrary set S is a little trickier

The proof goes something like this: since S is infinite, it must have some element, call that s. So we let f(1) = s. Since S is infinite, S{s} must also be infinite, so it must have some element, call that t. Let f(2) = t. Since S is infinite, S{s,t} must also be infinite, so it must have some element, call that u. Let f(3) = u.

Etc.

To get the exact formal definitions and so on depends on the conventions of the book you're reading, but a construction like that is broadly how I would prove this theorem

[lyric nymph]

oh so you are doing something similar to using least element principle but instead here doesnt depend on a least element

[reef agate]

Just need there to be any element at all

[lyric nymph]

ok but after defining this function i have to prove that it is bijective am i right ?

[reef agate]

Yeah, but its injective by definition

[lyric nymph]

yea

[reef agate]

And every injective function is a bijection with its image

[random citrus]

if it's bijective then N and S have the same cardinality which is not necessarily true

[lyric nymph]

wdym by that

[reef agate]

No

[random citrus]

if they had the same cardinality the problem would be trivial

[reef agate]

Because it's a bijection with the set {s,u,t....}

not with the set S

It's a bijection with some subset of S

[lyric nymph]

yea and since it is a bijection with a subset of S then this is what proves it

[reef agate]

Right

[lyric nymph]

every injection is a bijection with its image but there may be elements that dont have images

so shouldnt i prove that its a surjection too

??

[reef agate]

"elements that don't have images"?

[lyric nymph]

oh wait {s,u,t...} are images of elements in N

[reef agate]

Right

X = {s,u,t...}

$X \subseteq S$ and $f$ is a bijection between $\mathbb N$ and $X$ is what we need

[solid kiln]

RSB

[lyric nymph]

what i meant is that if it is an injection that doesnt mean that all images in the codomain have elements that map them in N

[reef agate]

huh?

That's word salad to me, sorry. Can you put it a different way?

[lyric nymph]

i mean that elements in S should have 1 element in N at most

so there may be elements in S that doesnt have an element in N

[reef agate]

Right, that's $X\subseteq S$

[solid kiln]

RSB

[reef agate]

OK is it clear that $f$ is a bijection between $\mathbb N$ and $X$ ?

[solid kiln]

RSB

[lyric nymph]

but then the subset X has all the elements images of elements in N

[reef agate]

Right...

So it's a countably infinite subset of s

we're done!

[lyric nymph]

and we need X we dont need S anymore

yea

bc f is an injection between N and X and since all elements in X are images of elements in N then it is bijective which settles it

but how do i define the function f

[reef agate]

We defined it right here

[lyric nymph]

i mean how to define it symbolically

[reef agate]

?

You want an equation?

There isn't one

[lyric nymph]

ohh

[reef agate]

S could be anhything, the set of reals, of cats, of hairs on my head, of words, of rubiks cubes positions...

Gotta run now, though. Hope that helped!

[lyric nymph]

so its ok to say let f be a function where f(1)=s where s is in S and so on ?

[reef agate]

Yeah I think that's fine

[lyric nymph]

yea that helped alot tysm

have a very nice day

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[strong vault]

yo can someone do my whole algebra test

[strong vault]

did u know that 2+1=4

[onyx lark]

well uh #rules

[strong vault]

oh mb some indian scamme rkid took mu phoen

[onyx lark]

?

[strong vault]

some kid took my phone

and started typing randoms stuff ins ervers

[onyx lark]

ok

well then ig u can close this

.close

[strong vault]

.close

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[shrewd anvil]

can someone help me with 3a

,rotate

[shrewd anvil]

man

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[shrewd anvil]

3a

[stark bison]

Just find f'(1)

[shrewd anvil]

im confused

[stark bison]

Differentiate 3x^2 + 4x and plug in x = 1

[shrewd anvil]

7?

[stark bison]

No, first differentiate and then put 1

You know what differentiation is, right?

[shrewd anvil]

no

i haven’t started calclus yet btw

[vernal briar]

Then why are you trying to solve these questions, without knowing how to do the math?

[shrewd anvil]

i mean that’s kinda why i asked cuz i don’t know where to start

[lament jewel]

learn differentiation

https://www.youtube.com/watch?v=eiI_m6FVANU

[shrewd anvil]

my teacher never taught us that yet so i’m assuming there’s another method

[burnt mulch]

Prolly approximate using the slope of a secant whose endpoints are near the desired value

The computation is standard (I assume) so I'll leave that to you

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[midnight jackal]

I'm trying to show this isnt a vector space

but something is confusing me

if I have the constant function f(x) = 1

axion 6 states (c+d)x =cx +dx needs to hold

but how do I handle c+d? the way I would normally or as stated by V = F(R)

c and d are constants but just like f(x) we can have constant functions

[wraith hinge]

is F(R) functions from reals to reals?

[midnight jackal]

yea

[wraith hinge]

V is an R vector space, so c,d are reals

and x is a function

(assuming I correctly guessed what axiom you are referring to, might help if you post the full list)

[midnight jackal]

yes

exactly

im just confused with how to resolve c+d in this situation

[wraith hinge]

c+d is a real

so you use the second line of the definition

(notation is a bit trippy here)

[midnight jackal]

Ok so there is a clear distinction between c = 1 and f(x) = 1?

[wraith hinge]

on the left of the equals in the definition, yes. On the right, no

So (c+d)*f is the function (c+d)+f

on the left, (c+d) is still formally a real number

but on the right, its something you add to f to make a function (so you can treat it as a function)

[midnight jackal]

I see

Ok I think I understand

[wraith hinge]

(it might probably help to use slightly different notation for each side)

[midnight jackal]

ty

@wraith hinge one more question for this situation

does axiom for additive inverse hold

since x + 1/x = 1
and h(x) = 1 is a zero vector for F(R)

or well actually I guess this doesnt work for f(x) = 0?

[wraith hinge]

yeah f(x)=0 doesnt have an additive inverse

what you've said sounds right

[midnight jackal]

Ok thanks

does cd(x) = cd+x

and c(dx) = c(d+x) = c+(d+x)?
or (cd+c+x)?

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[gloomy fern]

hello

[gloomy fern]

can someone help me w derivatives

what does instantaneous rate of chang emean

[chilly mauve]

just the same as f prime

[vernal briar]

The slope at a specific point, you could say

And yeah, that is just the derivative as ZO37 said

[gloomy fern]

How can I find it

[gloomy fern]

Do I j have to find the derivative

[vernal briar]

I mean, yeah that is what they are asking for

Using the limit definition.

[gloomy fern]

so instataneous rate of change

is derivative as well

[vernal briar]

Yeah

You should go back and remember what the meaning of the derivative is.

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[hexed lynx]

This isn’t making sense to me

[lament jewel]

the Q?

[hexed lynx]

Hold on it’s not uploading

[lament jewel]

lol

[hexed lynx]

Smh

[lament jewel]

[hexed lynx]

Find the equation of the normal to y=x^3 -4x + 2 at x=2

That

@lament jewel

[fresh pecan]

what is the normal to

[lament jewel]

hv u learnt derivatives

[fresh pecan]

tangent line?

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[languid ocean]

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

[solid kiln]

wowowin123
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[languid ocean]

this

lol

as x approaches 1 to the right

how to i find this kind of limit

am i allowed to factor the polynomial inside the sin func?

[quaint gazelle]

you are allowed

[languid ocean]

it would be sin(x-1)*sin(x+4)/(2x+1)(x−1)?

[quaint gazelle]

oh not this way

it would be

sin((x-1)(x+4))

[languid ocean]

sin((x-1)(x+4))/(2x+1)(x-1)? Am I allowed to use the trix identities sin(x)/x = 1 on this?

[quaint gazelle]

nope

its a good intuition though

and with more advanced math this idea is essentially good

but for now i suggest l'hospital

[languid ocean]

is the answer would be similar if i use that to the conventional way?

[quaint gazelle]

wdym?

[languid ocean]

lhospital wasn't part of my discussion on school yet

so i might think if i approach a problem

using that

[quaint gazelle]

yeah its hard

but maybe you could force it to appear

like what if you multiply on top and on the bottom by (x+4)?

then you would get something of the form

sin(y)/y

do you see what i mean?

[languid ocean]

yeah i see where you are going

i might study lhospital to make sure of my answers

thanks for this!

[quaint gazelle]

just use this hint i gave you and post the answer back and ping me

i'll tell you if its the right answer

[languid ocean]

ok thank you

.close

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[lone mauve]

How can you have two equal signs? is this legal?

Is this another way of saying make any term equal zero and due to the zero multiplicity rule, the entire output will equal zero?

[gloomy basalt]

yeah that's pretty normal it just means that you have two equations
f(x) = 7x²(x-2)(x+1/2)⁴(x+3)⁵
And
7x²(x-2)(x+1/2)⁴(x+3)⁵ = 0

and yes to that second part

[lone mauve]

So underneath it should say 7x²(x-2)(x+1/2)⁴(x+3)⁵ = 0 but maybe the instructor got lazy?

[gloomy basalt]

It "shouldn't" do anything because both notations work

[lone mauve]

OK, it's just weird to see equals and then another equals, I will have to get used to that...

[gloomy basalt]

In fact, you can have any number of equals signs
2/2 = 4/4 = 6/6 = 8/8 = 10/10

[lone mauve]

so I can read this as: "put in a certain input to make the entire function equal zero as the output?

and there are a few ways to tackle it (4 that I can spot), to make any term equal zero

and due to the zero multiplicity rule, all of it equals zero

having a zero term is like pacman it will eat them all up

[gloomy basalt]

I prefer to think of zero like a bomb, because eating seems like you eat one thing at a time, while zero blows them all up and leaves nothing

But yeah

[lone mauve]

lol

unless it's 0^0

[gloomy basalt]

abcde = 0 means a or b or c or d or e = 0

[gloomy basalt]

That's like blowing up zero times, which means you leave only 1 thing

[lone mauve]

it's like a magic trick is what 0^0 is lol

something from nothing

[gloomy basalt]

lol

[lone mauve]

i will call this the big bang operation

and regular zero as bang operations

[gloomy basalt]

So yes to that, f(x) is defined like ( ... ), and we want to know when is f(x) is equal to 0 so in other words f(x) = 0 => ( ... ) = 0

[lone mauve]

Got it 🙂 Thanks!

https://tenor.com/view/okay-ok-see-what-i-did-there-jim-carrey-thumbs-up-gif-15801589

.close

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[upbeat plover]

Hi I am having trouble proving this lemma from real analysis

[upbeat plover]

below is my work so far, I am a bit confused on how to show $r < s^2$ will get me $r(\frac{1+s}{r+s})^2 > 1$ since we know that $r(\frac{1+s}{r+s})^2 < s^2(\frac{1+s}{r+s})^2$

[solid kiln]

OhGumbus

[upbeat plover]

I think I may have already proved it, but I am unsure if just saying that since s^2 is a positive number bounded by one and r is also a positive bounded by 1, that the claim holds.

<@&286206848099549185>

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[dense breach]

I'd start probably backwards

Try to turn the target inequality into something without fractions

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[rancid wadi]

hi, this is not exactly a math question, but do anyone know a book where I can practice nonlinear system of equations like this?

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[jade flint]

idrk where to start tbh

[sharp arrow]

try to remember how 1/sinx graph looks like

[jade flint]

ummm

ik what a sin graph looks like...

how do u sketch 1/sinx

[sharp arrow]

hmm okay well then nvm

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[floral solstice]

the bottom statement

[floral solstice]

if we do

3 ≡ 5 (mod 2)

b should be 1

when using a = b + kn

wait nah

wont b be a decimal

[lusty pike]

5 is not a number in mod 2. If you have a, you're solving for b by dividing by n, and the remainder is b.

[sudden obsidian]

3 = 5 + k * 2, implies k = -1, or
5 = 3 + k * 2, implies k = 1

both -1 and 1 are valid residues here

[floral solstice]

so its not saying a ≡ b (mod n) can be written as a = b +kn

[sudden obsidian]

why not?

[floral solstice]

because if we use 3 ≡ 5 (mod 2)

b changes

[sudden obsidian]

3 = 5 + (-1) * 2
it still works

k is an integer

[floral solstice]

so how do we find k

surely we rearange a = b + kn

[lusty pike]

Am I fundamentally misinterpreting 5mod2

[marble wharf]

a-b = kn. so k=(a-b)/n. is that your question?

they shouldn't have called b the remainder in the image

a remainder is usually implied to be less than n

[floral solstice]

yeah thats what i was confused with

[floral solstice]

idk we talking about (mod 2)

like

1 in mod 2

[lusty pike]

It should be 1

[floral solstice]

is the same as

5 in mod 2

im not solving 5 mod 2

i think i understand now anyways