help-49

[proper swift]

how about decrease

[slate ferry]

how about you answer that

[proper swift]

(- infinity, + infinity)

wait

o

[slate ferry]

||(I see. I just suspected that the graph might be concave up)||

[proper swift]

no

[slate ferry]

hm?

[proper swift]

help

[slate ferry]

what is your thought process?

[proper swift]

i suck at math is what im thinking

[slate ferry]

where do you think the graph decreases

[proper swift]

everywhere

[slate ferry]

actually good point

that is true

but...

[proper swift]

but what

[slate ferry]

this is correct

altho more famously written as R

the REAL numbers

I'm not sure which form your fill in the blank thing would accept

yeah so its always decreasing on R and never increases

you already figured the asym thing

what about the domain

[proper swift]

• inf , pos inf

idk what that dot is i put neg inf first

[slate ferry]

actually that was a pretty silly thing to ask

[slate ferry]

but yeah sure

[proper swift]

ok cool

[slate ferry]

y intercept

from the name, find where the graph intercepts with the y-axis

[proper swift]

this is where we're at

[slate ferry]

or "touches" it

[proper swift]

is the inc/dec the x values?

[slate ferry]

the what?

[proper swift]

Inc or decrease

is it the x value

[slate ferry]

yeah

that's how we write intervals

[proper swift]

its always x value?

[slate ferry]

yes...

[proper swift]

ok nice

[slate ferry]

I think?

I mean- I'm study functions and I've never used y-repreesntation for intervals

[proper swift]

ok

[slate ferry]

so yeah, always x values

espiecally for functions like

y=3

[proper swift]

here we are

now what

[slate ferry]

actually yeah, the domain corresponds to x values

the y-values thing is called the range

in fact, its asking for it!

[slate ferry]

we already discussed the pos and neg part

pos is the part above the x-axis

and the neg is the part below

[proper swift]

so how do i write that

[slate ferry]

https://youtu.be/Ww7xtG2S7IM?si=MAhin-5oc9exE4fL

intervals. they're a big deal 🙂

you sure learned how to do them but maybe forgot

so I'd def recommand watching the video.

[proper swift]

thats not what my teach has down on other questions

[slate ferry]

can you show me

the said "other questions"

[slate ferry]

these are intervals.

exactly whats in the video

do watch it 😔.

they will teach you how to represent hte "pos" and "neg"

and everything else really.

[slate ferry]

the -inf , inf is also an interval.

[proper swift]

but it doesnt have the u

what is the u for

[slate ferry]

Union you mean

yeah that's other stuff you def learned about

its when you want to combine two intervals

[proper swift]

do we want to do that here

[slate ferry]

no

but you will most def need it in the future

its still worth understnading.

[proper swift]

ill get to that when i need it

so that vid shows the greater than and less than signs, forgot what those are called, and looks easy, but how am i supposed to find that out from our question here

[slate ferry]

what? the union part is not the main concept taught

[slate ferry]

your teacher used intervals

[proper swift]

i already learned that last semester

[slate ferry]

yes but you dont remember :((

[proper swift]

didnt understand that much

but whatever

dont need it right now

[slate ferry]

you see how your teacher represented the range here

(1 , inf)

thats an interval

[proper swift]

ok

[slate ferry]

sigh

I can explain intervals myself

because they are what you need here

will you promise to stay focused?

[proper swift]

ill try

[slate ferry]

alright

so this is a not-to-scale numberline

its basically all the numbers you can think of (real numbers)

you know it!

it extends to infinity to the right and negative infinity to the left

are you with me so far

[proper swift]

so far yes

[slate ferry]

amazing

so

we'll call this number line the x axis (it represents x values i guess.)

[proper swift]

ok

[slate ferry]

alright! if I wanted to circle where x=1

where would I put the circle

[proper swift]

1

[slate ferry]

so like- here

[proper swift]

eys

yes

[slate ferry]

if I wanted to circle x=-2

this would be it

[proper swift]

yes

[slate ferry]

alright

now what if i wanted to have x equal to many many things at once

for example

if I had a brother and I dont remember how old he is

but I siad

"my brother is in between 3 and 5 years old"

you can interpert that my brother is

3

3,5

4,5

one of those

5

[proper swift]

ok

[slate ferry]

aha

give me a bit.

alright back

[slate ferry]

so

if i wanted to represent my brother's age in terms of x

I'll have to say

my brother is in ebtween x=3 and x=5

[proper swift]

ok

[slate ferry]

so he's in-between these two points

but now you see

[proper swift]

yes

[slate ferry]

its kinda hard to put a point on EVERY x-value my brother can be

like

my brother can be

3 and a half years old

3.75

3.599

4.512

like a ton of ages in between 3 and 5

in fact, there is an infinite amount of numbers between 3 and 5

yeah there is- a lot.

[slate ferry]

we CANNOT represent them the way I did here

I dont have enough points or time for the amount of x-values i want to put here

so mathematicans created this NEW way of representation.

called intervals!

intervals are a way we represent values that are just too much to write down

[proper swift]

nice

[slate ferry]

so yeah

graphically

this is how we denote an interval;

with the brown filling denoting that

EVERY value between 3 and 5

can be my brother's age

can be my x values

and

on paper

the way we denote them...

differs for one country to another

I denote them with [ ] but you seem to be denoting them with ( )

im not sure how these work...

but yeah let me teach you how to read and type an interval

so

at last

[proper swift]

what do you mean can be

isint it always x vaules

[slate ferry]

I wanna say, every number between 3 and 5 could be my brother's age

[slate ferry]

it is always x values. I'm just miscommunicating

[slate ferry]

the way we'd write this on paper is...

(3 , 5)

with the 3 on one side and the 5 on the other

[proper swift]

i understand the basics of intervals

[slate ferry]

basically the** boundaries** of my x values

perfect

now

if i perhaps

wanted to say

I want my x values to be

in between 1 and 20

how would you write that downh

[proper swift]

(1,20)

[slate ferry]

good now

[proper swift]

https://tenor.com/view/congrats-noice-noice-job-nice-job-good-job-gif-16436948

[slate ferry]

in ebtween

-2 and 4

[proper swift]

-2, 4

with the brackets

[slate ferry]

yes

notice soemthing very important here

the SMALLER number is ALWAYS on the left.

[proper swift]

ok

[slate ferry]

( 4, -2) doesnt make sense

we ALWAYS start with teh smaller number when it comes to intervals

alright

now lets make it a bit trickier

what If I wanted to say....

I want my x values to be bigger than 4.

think about it

how could we represent htis

[proper swift]

(4, infinity)

[slate ferry]

GOOD

[proper swift]

https://tenor.com/view/zerobaseone-zb1-ricky-happy-pretty-gif-1424023175268138638

[slate ferry]

but now here coems the part i dont fucking now

[slate ferry]

what do you guys do when you DONT want to include the 4 for example

... you dont understand dont you

okay what If I said

a human being cant be 0 years old

and lets say they only live to 60 years old

if i wanted to write the interval for these x-values

ME

i would do this

]0 , 60]

the ] denotes that "my x value CANNOT be 0"

if i wanted to include 0 as a part of my x values

I would [0 , 60]

[proper swift]

what

[slate ferry]

but what do you with (0 , 60)

....

[proper swift]

give me 10 mins

[neat coral]

how muuch have you done

ima put this here so i dont gotta keep going up

[slate ferry]

a good chunk. Now I'm trying to explain how intervals work

I dont understand how people ues ( ) instead of [ ] for intervals.

[civic lynx]

in NA, we use [a,b] to mean [a,b], and (a,b) to mean ]a, b[

and ofc infinity is never closed

[proper swift]

guess whos back

ok lets go

hello

@slate ferry

[slate ferry]

do you know the difference between [ ] and ( )

and when each is used

[proper swift]

bracket is when the number is included

[slate ferry]

and ( ) when numbers are excluded

alright so

I- thought ( ) was how we say included

[proper swift]

bruh

[slate ferry]

yeah so

its fine

we cna just rephrase my previous example to be "my brother's age is inbetween 3 and 5"

which would still be (3 , 5)

in any case

[slate ferry]

yeah so

if x can equal to 2 and can be bigger than it

how would you write that

[proper swift]

x is greater than or equal to 2

[slate ferry]

TRUE

how would you put that in interval form

[proper swift]

x > line under it 2

[slate ferry]

😭 yes

good

perfect

amazing

so

we can go back to your original question.

[proper swift]

nice

[slate ferry]

for where it is postiive

you said

"where the graph is above the x axis"

[proper swift]

yes

[slate ferry]

so ALL of these values

and whats before -3

notice, pink area is where the graph goes below the x axis

we dont want that yet

we want the x values that gets the graph to be above the x axis.

which are the valeus i put in black + the values that are before -3 that we cannot see

are you with me?

[proper swift]

no

i am saying that positive values are where the graph line is above x

[slate ferry]

yes

[proper swift]

axis

[slate ferry]

true

but

we need to repreesnt that

in itnerval form

write it in the form of ( smth , smth )

or include stuff if we need to.

[proper swift]

(-inf, 0}

0]

"?

[slate ferry]

hmm

not quite

it sure does start with - infinity

but does it end with 0?

the graph is still abnove the x axis even when x=1

and 2

and 2.1

and 2.2

so 0 is def not the end of it.

[proper swift]

(-inf, 2.585]

[slate ferry]

PERFECT

exactly that

[proper swift]

bang

[slate ferry]

notice at x=3, the graph is-

wauit a minute

[slate ferry]

I dont think we include 2.585 tho

[proper swift]

idk when to use that

[slate ferry]

at the exact point of 2.585

the graph-

I might eb mistaken...

but yeah

in all

decaying

increasing

negative

positive

all these stuff

we only use ( )

[proper swift]

ok nice

[slate ferry]

only these stuff tho.

ranges and domains require [ ] sometimes!

[slate ferry]

anyways yeah you did amazing

[proper swift]

lets go

[slate ferry]

so now

for the nmegative part

[proper swift]

(2.585, inf)

[slate ferry]

and there you go

[proper swift]

LETS GO

https://tenor.com/view/clapping-clap-applause-standing-ovation-gif-15648815

[slate ferry]

alright what's next

we already did domain, pos, neg, inmc, dec

asym

right?

[proper swift]

this is where we are

[slate ferry]

great

next on the list is range.

[proper swift]

range

[slate ferry]

so you know in intervals, we always use the x-values

well that was al lie 🙁

[proper swift]

i know range is y values

[slate ferry]

the range is the only execption of which we represent it with y values

oh-

amazing.

[slate ferry]

ok

read from up to down.

let me make myself clearer

give me a bit

[proper swift]

(3, -inf)

[slate ferry]

LMAO

correct

you did skip me quite the yapping.

great job

[proper swift]

lets go

[slate ferry]

y-intercept, can you do that yourself?

[proper swift]

here we are

y int

is next

[slate ferry]

yes

now idk if you're supposed to approximat- actually, where is the y-intercept on the graph

[proper swift]

just from eyeballing it i would say it is (0, 2.5)

[slate ferry]

yeah

so-

idk if eyeballing is inteneded

if it is then sure

[proper swift]

i dont think it is

[slate ferry]

if its not, we need to solve for it algebriaclly

yes

[proper swift]

how

[slate ferry]

thats actually why we're given the function itself!

[proper swift]

do we plug something in

[slate ferry]

yes

[proper swift]

what

[slate ferry]

you see, y-intercepts are just where the graph touches the y axis. do you know the equation for the y-axis?

[proper swift]

theres an equation for the y axis?

[slate ferry]

yeah

a defining equation which

[proper swift]

so i dont

[slate ferry]

represnets soemthing that is applicant on all points of the y axis

ill give examples

( 0 , 3)

( 0 ,-1)

(0 , -9485)

(0 , 28)

these are all points on the y axis

notice a pattern?

[proper swift]

x is always 0

[slate ferry]

EXACTLY

sooo

the equation of the y axis is just

x=0

yeah.

[proper swift]

so plug 0 in

[slate ferry]

you plug in x=0

yes

for the x.

LMAO.

[slate ferry]

||nice guess.||

[proper swift]

ok nince

[slate ferry]

yeah

so whats the point

[proper swift]

here we are

[slate ferry]

do you know what f(1) means)

[proper swift]

plug in 1 for x

[slate ferry]

yes indeed

you can do that

or you can use the graph

[proper swift]

let me calculate it

[slate ferry]

you can use the graph.

[proper swift]

or yea i can use the graph

[slate ferry]

f(1) means the value of y when x=1

[proper swift]

so 2

[slate ferry]

look at x=1 and-

yes.

[proper swift]

next one is 2

here we are

[slate ferry]

perfect

now- what the fuck is end behavior 💀

[proper swift]

when x is approaching - inf what is y doing

[slate ferry]

can you show me your teacher's example

[slate ferry]

oh

ok

[proper swift]

and when x in approachig inf what is y doing

[slate ferry]

do that

[proper swift]

seems easy

[slate ferry]

yes.

[proper swift]

is y going to 3

first one

[neat coral]

oh I highkey lost focus and got lost where are we in the problem

[slate ferry]

yes

[proper swift]

end behavior

[slate ferry]

its okaty we're done

very almost.

[neat coral]

ohk alr

[proper swift]

was this yes to my question

[neat coral]

end behavior with interval notation right?

[slate ferry]

yes

[slate ferry]

I'm not sure if you use interval notation for it

[proper swift]

example from different questio

[slate ferry]

yeah

you dont need intervals

and indeed

[proper swift]

so first one is 3 and second one is - inf

[neat coral]

depends on the teacher

[slate ferry]

for x appraoches -inf the value y approaches is 3

[slate ferry]

andddd you're done!

I didnt even do that much you did amazing pudgy.

[neat coral]

for -infnity try to see what happens as it the graph moves left and for +infnity try to see what happens as the graph moves right (I know you're done but for future reference, yk)

[proper swift]

https://tenor.com/view/clap-gif-25211694

[neat coral]

https://tenor.com/view/cinema-absolute-cinema-peak-cinema-salmon-jjk-gif-6981231960394212546

[slate ferry]

any more questions?

[proper swift]

thats just one part of the quiz

[slate ferry]

well- solve each one and send them here

let people check your work

as for me...

I shall study

great luck

[proper swift]

thank you very much

[proper swift]

will send grade as soon as i take the quiz tommor

w

[slate ferry]

you're welcome

[slate ferry]

oh- you dont need to!

keep your grade to yourself and fill yourself with pride

[proper swift]

what is there to be prideful of

[neat coral]

if you need any extra help you can bother me

[slate ferry]

this is where math gets fucking annoying with derivatives. you're almost there ❤️ (its not hard, its just an interesting process)

[neat coral]

Im writing 3 essays anyway so Id be up for it 😭

anything but working on my essays 🤣

[slate ferry]

alright, goodbye

[proper swift]

why did teach write range is 1 , inf

[neat coral]

alright so look at the graph

[proper swift]

i would say it is inf to 1

or car said the lowest number always goes first

[neat coral]

is inf>1?

[neat coral]

yes

[proper swift]

so teach is right?

[neat coral]

yea

lowest always first

infnity is not less than 1 so it goes after got it?

I'll have to go offline for a bit but any questions you can just dm me and il try to get to them later @proper swift

[proper swift]

ok tahnk yu

how to know when the graph is neither

anybody else

.close

[midnight plank]

Channel closed

Closed by @proper swift

Use .reopen if this was a mistake.

[proper swift]

.end

[slate ferry]

neither?

[proper swift]

yes

[slate ferry]

isnt growing anmd isnt decarying

[proper swift]

well yes

[slate ferry]

is when

first

reopen the channel

[proper swift]

.open

[slate ferry]

type .reopen

[proper swift]

.reopen

[midnight plank]

✅ Original question: #help-49 message

[twilit jetty]

didnt I just tell you what the difference was

[slate ferry]

Its my fault, I didnt see that you told him.

[proper swift]

its better here

[twilit jetty]

pudgy this is why youd dont open multiple channels

[slate ferry]

I told him to reopen you can go on tho.

[proper swift]

its my first time

[twilit jetty]

go read what I said then go back here

[proper swift]

here, f constantly heads downward instead of upward, so it wouldnt look like "growth", that could be why

you can tell it goes downward because its -(4)^x instead of +(4^x)
the - would mean youre subtracting larger and larger numbers as x gets larger

this doesnt make sense

[twilit jetty]

what about the picture

[twilit jetty]

let me ask you about context clues

what do you think this is a graph of

[proper swift]

decay

[twilit jetty]

not what I meant

[proper swift]

ok nice

[twilit jetty]

what function is this graphing

[proper swift]

i dont know what that means

[twilit jetty]

you know how you need a function to graph something?

[proper swift]

yes

[twilit jetty]

so what function did I use to get this graph?

and remember, this is what I said directly before that

[proper swift]

the one in the picture

[twilit jetty]

thats good

now heres the big problem with "growth" or "decay"

mathematically we call things getting faster and steeper "growth"

while we call things that settle or more lenient "decay"

so in this graph of 5 functions, all the ones going up or going down Id call "growth"

because its growing, two of them are growing up, two of them are growing down

this is different than what your teacher seems to be doing

more complicated I think

[proper swift]

it might just be that when a is negative it is neither

[twilit jetty]

yep

but it might not be

in this case it should be safe, |b| > 1 and a < 0 should be "neither"

[proper swift]

what

[twilit jetty]

you said "when a is negative", so youre using f(x) = ab^x in general

if |b| > 1 and a < 0, then that should mean "neither" to your teacher

[slate ferry]

I thought

neither

meant

for when f'(x) = 0

@twilit jetty

the slope of the tangent when its horizontal

[twilit jetty]

car, as I said earlier, the teacher is using a different, more complicated definition of growth/decay/neither

so we have to go with whatever the teacher is cooking

[twilit jetty]

also, this is not true

sin(x) does not grow or decay as x -> ∞

sin(x)'s derivative though is cos(x), not 0

[slate ferry]

I meant for whatever values makes f'(x) = 0 aka when the slope of the tangent line is 0 at some point, at that specific point, the function doesnt increase or decrease 😟

[twilit jetty]

this is also not true, growth is not the same thing as increasing

[slate ferry]

oh...

OH

[twilit jetty]

this would mean sin(x) is "growing" when it goes up and "decaying" when it goes down

[slate ferry]

oh noo 😭

I was- helping this person earlier....

I interperted decaying as decreasing and increasing as growing

[twilit jetty]

growth/decay as math words are usually taught along exponentials, if they dont teach negative a then its easy to think thats what the words mean

[slate ferry]

the function I described to him was an exponential one

give me a minute

oh... wait the growth part is out of the- okay nevermind

sorry.

[midnight plank]

@proper swift Has your question been resolved?

[midnight plank]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

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

[blissful trench]

heyo i need to make a proof of product rule and ill show some process I did

[blissful trench]

⁨```latex
\begin{align*}
p'(x) &= \lim_{h \to 0}\frac{p(x+h)-p(x)}{h}\
&=\lim_{h \to 0}\frac{f(x+h)g(x+h)-f(x)g(x)}{h}\
\end{align*}

[grand pond]

# real car fan

[blissful trench]

what I can do after this?

[fossil knot]

I think something to try is expand out f'g + fg' with the limit definition

then see how you can maybe transform what you have into that

[blissful trench]

no idea

yes

[slow tiger]

do what he said

[blissful trench]

can I just write $-f(x)g(x+h)+f(x)g(x+h)$?

[grand pond]

# real car fan

[fossil knot]

Yes that's a good idea

[slow tiger]

go on

[fossil knot]

you can since it's just +0

but it helps you change it into what you want

[blissful trench]

alright i think I got it

ill continue the rest :)

thanks

.close

[midnight plank]

Channel closed

Closed by @blissful trench

Use .reopen if this was a mistake.

[midnight plank]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

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

[fickle sierra]

Hello, could someone check if this proof looks good please?
⁨```latex
\begin{Theorem}
Assume $a,b \in \bR$. If $a \in \bQ$ and $ab \notin \bQ$, then $b \notin \bQ$.
\end{Theorem}

\begin{proof}
Assume for a contradiction that $a \in \bQ$, $ab \notin \bQ$, and $b \in \bQ$.
Since $a$ and $b$ are rationals, we can express both as
$a = \frac{m}{n}$ and $b = \frac{p}{a}$
where $p,q,m$ and $n$ are integers.
Then, $$ab = \frac{m}{n} \cdot \frac{p}{q} = \frac{mp}{nq}$$
Since $mn$ and $pq$ are integers, $\frac{mp}{nq}$ is rational.
This is a contradiction, since we assumed that $ab$ is irrational.
Therefore, if $a \in \bQ$ and $b \notin \bQ$, then $b \notin \bQ$.
\end{proof}

[grand pond]

Mor Bras

[jaunty ivy]

looks fine aside from the typo in the last line

also b=p/q

u wrote b=p/a

we need mp and nq to be integers, not mn and pq

and nq nonzero

[midnight plank]

@fickle sierra Has your question been resolved?

[fickle sierra]

Here's the the fix
⁨```latex
\begin{proof}
Assume for a contradiction that $a \in \bQ$, $ab \notin \bQ$, and $b \in \bQ$.
Since $a$ and $b$ are rationals, we can express both as
$a = \frac{m}{n}$ and $b = \frac{p}{q}$
where $p$ and $m$ are integers, and $n$ and $q$ non zero integers.
Then, $$ab = \frac{m}{n} \cdot \frac{p}{q} = \frac{mp}{nq}$$
Since $mp$ are integers and $nq$ non zero integers, $\frac{mp}{nq}$ is rational.
This is a contradiction, since we assumed that $ab$ is irrational.
Therefore, if $a \in \bQ$ and $ab \notin \bQ$, then $b \notin \bQ$.
\end{proof}

[grand pond]

Mor Bras

[fickle sierra]

Thanks for you comment!

.close

[midnight plank]

Channel closed

Closed by @fickle sierra

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[midnight plank]

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[halcyon forge]

Help for number 1 for slop and secant line equation

[midnight plank]

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[halcyon forge]

Ok

[cerulean oyster]

,rccw

[cerulean oyster]

Well, to start with, m should be a coefficient, which doesnt seem to be the case here.

[slow thorn]

.. find and name the points through the secant properly first

[midnight plank]

@halcyon forge Has your question been resolved?

[midnight plank]

Channel closed

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Remember:
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[twilit field]

Let the projection map be $\phi$. Then $\phi(p(x))= p(x) +((f(x)))$

[grand pond]

wai

[twilit field]

I suppose I now have to determine what $((f(x))$ looks like

[grand pond]

wai

[twilit field]

oops, I have a RA quiz today, guess have to study for that first

.close