help-27

[wooden veldt]

Still wrong that's now zero

[mellow panther]

You didn't have to sully me

[coarse fractal]

Sorry, just going through it again, one sec

[woven radish]

numbpy

[coarse fractal]

Brodi

[coarse fractal]

Brodi

Love you guys sm

Acc made me understand it nicely but ibr

I would have never thought of splitting the frac like that

You reckon with more experience it’ll just come to me?

[devout snow]

@coarse fractal Has your question been resolved?

[mellow panther]

It should

[devout snow]

Channel closed

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[coarse fractal]

❤️

[devout snow]

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[ashen tendon]

Hey so I was wondering on how I would get started on this? Because I have the standard form but am a little confused on what to do next

[scarlet sequoia]

,rccw

[ashen tendon]

Huh

[wooden zodiac]

Hello

[ashen tendon]

Hi!

[wooden zodiac]

Di u know what slopeninteret formats?

Do u know what slope and intercept form is?*

[ashen tendon]

You mean. Y=mx+b?

[main gull]

Your autocorrect is buggy

[wooden zodiac]

Yes

[ashen tendon]

Ye

I do

[wooden zodiac]

Did u try to put it into rhat form?

From standard form to that

[ashen tendon]

In the top right, I tried, but 100 ÷48 is 2.088888 so that felt wrong

[main gull]

Fractions and decimals do exist

[ashen tendon]

Ikk but like

Idk just felt off for me

[wooden zodiac]

So u solved for y?

Ohh

I see it niw

Now

U did 1 small mistake

How did u bring 100x to the other side?

[ashen tendon]

Well I did what my teacher told me to do

And

Basically first step was to get rid of the 100x on one side and move it to the other

[wooden zodiac]

100+x=0

[wooden zodiac]

Solve for x here

[ashen tendon]

-100

[wooden zodiac]

Yes

You subtracted left side by 100x, but added right side by 100x

Should be y=(4000-100x)/48

[ashen tendon]

Huhhhhh

That's like a totally different format though

Cause for what I learned it should be 48y = -100x + 4000

The divide out the 48 form the y so then it's just y

And then do -100x and 4000 ÷ 48

[wooden zodiac]

Wdym by this

[ashen tendon]

Well this

The thingie in the middle

[main gull]

It might be best to keep it in simplifies fraction form

[ashen tendon]

Ok so like how do I do that

[main gull]

Simplify 100/48

[ashen tendon]

25/12?

[main gull]

Yes

[ashen tendon]

Kk so then do I also just simplify 4000/48 or

[main gull]

Yes

[ashen tendon]

Sorry meant 48 lol

Kk

Kk so 250/3?

[main gull]

Yes

[ashen tendon]

Kk so then

Total amount of square feet they have is 4000

[wooden zodiac]

Should be y=-25x/12+250/3 btw

[ashen tendon]

now I gotta figure out how much square feet for the for the 2 different types of offices

[ashen tendon]

Tyy

Nvm got that

I think

Dang it

How do I do the table ahhhh

Only 5hing that comes out for the first one is 83.33333333

Healppppp

Pls

[wooden zodiac]

How did u do the table? For x=0?

[ashen tendon]

Oh that was just there 0.o

[wooden zodiac]

Huh

[ashen tendon]

Ya

It's part of a worksheet I got

And it was already there

Just filled out

[wooden zodiac]

Ahh

[ashen tendon]

Ikr

Well it would mean that each cubicle for the y should be 48 right?

Out of 4000 Sq ft

[devout snow]

@ashen tendon Has your question been resolved?

[ashen tendon]

Idk

Sorta

[devout snow]

@ashen tendon Has your question been resolved?

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[bitter patrol]

Write this quadratic function in vertex form. Give the coordinate of the vertex and the equation of the axis of symmetry. Then describe the transformations from f(x)=x^2 to g.

[bitter patrol]

$g(x)=-x^2+2$

[woven radish]

Hehehehaw

[bitter patrol]

I know how to find the vertex form

im struggling with "Then describe the transformations from f(x)=x^2 to g."

[vivid bridge]

this is g(x)=x^2

this is g(x)=-x^2

up side down

this is g(x)=-x^2+2

[bitter patrol]

oh yeeee

[vivid bridge]

up for 2 block

[bitter patrol]

x^2 is a parabola

and +2 determines the y axiss

ahh i got it

thx

.close

[devout snow]

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[vivid bridge]

nvm😇

[devout snow]

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[quasi yoke]

hi!

[lusty sapphire]

Question?

[quasi yoke]

I’m not sure why (v) is pi/2

[arctic field]

whats I_2n

[lusty sapphire]

Taylor series of tan?

[quasi yoke]

hi!

oh sorry one minute let me take a picture of the entire thing

for the first 3 parts

[devout snow]

@quasi yoke Has your question been resolved?

[arctic field]

,w sum r=1 to infinity of (-1)^(r+1)/(2r-1)

[woven radish]

Results provided by WolframAlpha

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

Requested by 夢雪#2250

[arctic field]

its pi/4

[devout snow]

@quasi yoke Has your question been resolved?

[quasi yoke]

oh okay! ty!

[devout snow]

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[paper wren]

$2sin^2 θ-sinθ-1=0$

[paper wren]

$Solve for 0°<=θ<=360°$

[woven radish]

Zyme><SOL

[paper wren]

$2sin^2 θ-sinθ-1=0$

[woven radish]

Zyme><SOL

[upper schooner]

Any ideas

[paper wren]

Not any good ones lol

I can't cancel out the sin's

Not unless I'm missing something

[restive river]

Take sin theta = x

[paper wren]

Is it supposed to be some sort of brackets thing

[upper schooner]

What do you mean? You might be on the right track 🛤️

[paper wren]

So turning it into two brackets

Only got it to be 2x^2+x-1 that way

$(x+1)(2x-1)$

[woven radish]

Zyme><SOL

[upper schooner]

,w expand (x+1)(2x - 1)

[woven radish]

Results provided by WolframAlpha

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Requested by chartbit#3171

[paper wren]

$=2x^2+x-1$

[woven radish]

Zyme><SOL

[paper wren]

Yeah

[upper schooner]

Yep what's this equal to?

[paper wren]

To this

Wait

No nvm I don't think we can rearrange

Can we?

[upper schooner]

Alright another question, what was this originally equal to?

[paper wren]

(X+1)(2x-1)

[upper schooner]

(hint: don't say anything in terms of sin)

[upper schooner]

Not what I was looking for

[upper schooner]

[paper wren]

I don't think the two are equal

Wait

Yeah nvm

It isn't equal to the initial eq.

[upper schooner]

Well they are: remember that $x = \sin(\theta)$, so substituting that into anywhere gets us the same thing

[woven radish]

chartbit

[upper schooner]

But the answer I'm looking for is super simple

$2x^2+x-1 = 0$ was what I was looking for

[woven radish]

chartbit

[paper wren]

That's this

Oh are you saying =0 wasn't in there

Bruh

[upper schooner]

That's it 🥳

[paper wren]

I never pay attention to that

[upper schooner]

Haha sometimes it's tough when it's the simple stuff as well

But yep, solve that quadratic for x please

[paper wren]

😭

Quadratic formula

Thank you, I didn't know you could use that

[upper schooner]

[paper wren]

$\frac-b+-\sqrt{b^2-4ac}{2a}$

[upper schooner]

(well you could use the quadratic formula, but there's no need to in this case if you already factorised)

\sqrt{}

Put the stuff inside, like $\sqrt{b^{2} - 4ac}$

[woven radish]

chartbit

[upper schooner]

Frac is like $\frac{top}{bottom}$

[woven radish]

chartbit

[upper schooner]

$x_{\pm} = \frac{-b \pm \sqrt{b^{2} - 4ac} }{2a}$

[woven radish]

chartbit

[upper schooner]

[paper wren]

Bruh nvm

So you'd sub it into here?

And solve?

[upper schooner]

You can from there to find what x is

[paper wren]

Alright, I'll do that. Thanks for the help

[upper schooner]

(or just remember that $(x+1)(2x-1)=0$ and solve for $x$ there)

[woven radish]

chartbit

[paper wren]

X=-1, 2x=1

lmao

It's a hit or miss whether that cat emoji is a yes or no because it's so small

[upper schooner]

Remember to get the other one in terms of x="something"

[restive river]

The eqn is
$ (x-1) (2x+1)

The eqn is
(x-1) (2x+1)

[upper schooner]

It's my

A cat saying yes haha

[paper wren]

I keep forgetting its as stupid as reversing the sign on the inside numbers

Anyways, thx

.close

[devout snow]

Channel closed

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[upper schooner]

Don't forget to solve for $\sin(\theta)$ btw!

[woven radish]

chartbit

[paper wren]

By inversing it?

[devout snow]

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[azure granite]

area of AEF = 5 cm² HCG = 7 cm²

it asks the area of ABCD

E H F and G are the mid points of the lines they are on

[devout snow]

@azure granite Has your question been resolved?

[devout snow]

@azure granite Has your question been resolved?

[devout snow]

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[sage socket]

For #12, I expanded the lhs with the hint that I was given

This is (4)

I honestly don't know what to do

I keep asking myself what I want to prove but I'm struggling on this

Like, would I just cancel stuff out or

[devout snow]

@sage socket Has your question been resolved?

[sage socket]

<@&286206848099549185>

[devout snow]

@sage socket Has your question been resolved?

[devout snow]

@sage socket Has your question been resolved?

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[restive river]

Hello! Another limit to be evaluated with neither L'Hopital or Taylor series.
[
\lim_{x \to 0} \frac{3^x + 5^x -2}{2x}
]

[woven radish]

♡LexQa♡

[restive river]

I tried rewriting it several times but I am not getting anywhere

[supple knot]

Split it as a sum. Then observe it's the sum of two derivatives evaluated at 0

3^x - 1 + 5^x - 1

[restive river]

Hmm alright let me give it a shot

[arctic field]

lhopital in disguise

[restive river]

Actually

[pseudo basin]

no

[restive river]

there is a hint under this question

where it says

[
\lim_{x \to 0} \frac{e^x -1}{x} = 1
]

[woven radish]

♡LexQa♡

[arctic field]

yeah thats derivative of e^x at 0

[restive river]

yeah i guess there is no escaping the derivative

[arctic field]

there is

maybe

well

you have to use another standard limit

[restive river]

hmmm

turning the cogs in my brain

uh

[dim perch]

do you happen to have the answer?

[restive river]

no

but like

easy to find

[dim perch]

if i use the hint you give, i think i get a natural log

LOL

idk thou

[arctic field]

evaluate using this first

[restive river]

yeah i think so too buttt

alright sure

so

\begin{align*}
\lim_{x \to 0} \frac{3^x +5^x - 2}{2x} & \to \lim_{x \to 0} \qty(\frac{3^x -1}{x} + \frac{5^x -1}{x}) \
& \lim_{x \to 0} \frac{e^{\ln(3^x)} -1}{x} + \frac{e^{\ln(5^x)}-1}{x}
\end{align*}

[arctic field]

x

all that work for an x

[restive river]

wtf i dont get why it is crying

[arctic field]

[dim perch]

[arctic field]

where is the error

you have an extra }

on the 5^x

i think

in the last line

[woven radish]

♡LexQa♡

[restive river]

yeah

phew

anyways

[restive river]

uhhh

well

[dim perch]

rnt u forgetting the 2x

?

[restive river]

\begin{align*}
\lim_{x \to 0} \frac{3^x +5^x - 2}{2x} & \to \frac{1}{2} \lim_{x \to 0} \qty(\frac{3^x -1}{x} + \frac{5^x -1}{x}) \
& \to \frac{1}{2}\lim_{x \to 0} \frac{e^{\ln(3^x)} -1}{x} + \frac{e^{\ln(5^x)}-1}{x} \
& \to \frac{1}{2} \lim_{x \to 0} \frac{e^{x\ln(3)} -1}{x} + \frac{e^{x\ln(5)}-1}{x} \
& \to \frac{1}{2}\lim_{u \to 0} \frac{e^{(u)} -1}{\frac{u}{\ln(3)}} + \lim_{t \to 0} \frac{e^{(t)} -1}{\frac{t}{\ln(5)}} \
& \to \frac{1}{2} \lim_{u \to 0} \frac{\ln(3)(e^{u} -1)}{u} + \lim_{t \to 0} \frac{\ln(5)(e^t -1)}{t} \
& = \frac{\ln(3) + \ln(5)}{2}
\end{align*}

[arctic field]

you split the 2x

in the denominator

[restive river]

wait am i tripping

[dim perch]

latex is having an aneurysm

yeah the denom in line 1

[dim perch]

yerrr

[restive river]

oh right

then just

[arctic field]

halve everything

just remove the 2 in the first limit

add it back later

easier

[restive river]

yeah okay

[dim perch]

careful on the fourth line btw

idt u need the extra ln on the e

you'll end up w ln^2(3) and ln^2(5)

[restive river]

uhh

oh wait

i think i got it

its like

[arctic field]

your substitution is a bit messy

[dim perch]

should just be

$$\ln(3)\lim_{x \to 0}\frac{e^{\ln(3)x} - 1}{\ln(3)x} + \ln(5)\lim_{x\to 0}\frac{e^{\ln(5)x}-1}{\ln(5)x}$$

[woven radish]

blanket

[dim perch]

think that looks right

[restive river]

right?

[dim perch]

looks abt right

[restive river]

wait i missed brackets there

[arctic field]

yeah

lots of brackets missed

[restive river]

nooo

[arctic field]

[woven radish]

♡LexQa♡

[restive river]

there u go

[dim perch]

TeXit having a field day

[restive river]

i wish i could write faster with altex

latex

pain

alright thank u a lot u two!

[arctic field]

okay good

i feel like

[dim perch]

out of curiosity does latex support code blocks?

TeXiT*

[arctic field]

your sub couldve been cleaner

[restive river]

code locks?

blocks

[arctic field]

because all you're doing is like

[restive river]

cleaner?

[restive river]

wait isnt that extra work

[dim perch]

not really, ur just rewriting

its your substitution but not a substitution

[arctic field]

like you see that the exponent and the denom are the same

[dim perch]

you just multiply by a natural log on top n bottom

[arctic field]

they approach 0 at the same rate

and so its going to be the same limit

[restive river]

oh i see

yeah that makes sense

[arctic field]

you can

also prove the limit

[arctic field]

well

"prove"

not really "prove" but

[restive river]

limit substitutions are harder than integrals istg

[restive river]

uhh

like

[arctic field]

and now you get

[restive river]

ohhhhhhhhhhhhh

yeah i see that

okay

[restive river]

feels good to finally understand your weird substitution methods

[arctic field]

[restive river]

actually snow

yesterday i asked another limit

it was so fucking weird

i wonder how you would do it, i figured it out but lemme show

[arctic field]

[dim perch]

@arctic field would u happen to know if texit can recognize code blocks

[woven radish]

blanket

[restive river]

$\lim_{x \to 0} \frac{\sin(x) - x}{x^3}$

[woven radish]

♡LexQa♡

[restive river]

no l'hopital or taylor

[arctic field]

texit can do anything you can do in a standalone environment

so pretty much anything

[dim perch]

errr so would it recognize

if i wrote in a code block?

[arctic field]

on

you mean

[dim perch]

can i write latex in here and it would be recognized

discord

[arctic field]

\[ \parens {a + b}^2 = a^2 + b^2 \]

[dim perch]

oh woah

that is useful to know

[restive river]

wtf

[arctic field]

only preamble reviewers will understand

[arctic field]

this is kinda sus

[restive river]

its answer is interesting

[arctic field]

well its just like

1/6

[restive river]

yeah

[arctic field]

what can you use

[restive river]

but how do u show that

[restive river]

anything u want algebraically tbh

just no l'h or taylor

[arctic field]

[restive river]

it is solveable

u have to do

a smart substitution

[uncut crow]

that's incorrect ms snow unless 2ab is 0 in the number system a and b are taken from

[restive river]

[arctic field]

the characteristics of my fields are always 2

[uncut crow]

oh phew

[restive river]

the idea is to get a linear equation in the limit after u do some substitution, i am actually kind of disappointed i didnt see it faster

[arctic field]

[restive river]

i got the whole server riled up on this limit yesterday it was kind of funny

[tulip swan]

This limit is trivial.

[restive river]

oh shut up slurp

[wicked turtle]

what's the rationale for avoiding the taylor series for sin?

[restive river]

smh

[tulip swan]

Okay!

[restive river]

NO DONT

bruhh

[restive river]

to make your life harder, basically

[dim perch]

we don't like Taylor

[wicked turtle]

i suspect that any argument that avoids the taylor series is gonna be circular

[dim perch]

they're rude

[uncut crow]

who needs taylor

[wicked turtle]

like if it involves taking the derivative of sin, how do you do that without a definition for sin

[uncut crow]

just define sin as a series

[dim perch]

who needs sin

[uncut crow]

me

[restive river]

it does not

[wicked turtle]

that's how any self-respecting person defines it 😀

[restive river]

you do not need to take derivatives at all

[dim perch]

you don't need sin

nobody needs sin.

[wicked turtle]

what was the substitution you used?

[restive river]

should i just say it?

[uncut crow]

finally someone who gets me 🥺

[tulip swan]

What's the purpose of computing limits without lhopital though

[restive river]

its ||x = 2t||

[wicked turtle]

walter rudin gets you too!

[uncut crow]

walter rudin is why i get myself tbh

[restive river]

mainly like

to think intuitively about how to solve it

but like

you just stare at it for an hour lmao

[uncut crow]

what's the purpose of staring at it for an hour?

[wicked turtle]

avoiding l'hospital does tend to give you practice at finding clever manipulations which can be useful elsewhere

[restive river]

build intuition, tbf i think it develops your skills in picking substitutions for things that actually need it (integrals or whatever)

[uncut crow]

you cn use mathematica for those

[restive river]

lmao

[wicked turtle]

MSE already has a tag for limits-without-hospital, maybe it needs a limits-without-wolfram as well 😆

[restive river]

lmaoo wolfram is bad with limits for some reason

it thought $\lim_{x \to \infty} e^{-x}\qty(1+ \frac{1}{1+\cot(x)})$ went to 0 for some reason yesterday

[woven radish]

♡LexQa♡

[restive river]

anyways i assume snow died so like

.close

[devout snow]

Channel closed

Closed by @reef sable

Use .reopen if this was a mistake.

[uncut crow]

she's crying in her grave rn

[restive river]

lmaoo

[arctic field]

wait

[restive river]

should i reopen

[arctic field]

.reopen

[devout snow]

✅

[restive river]

oh

okay then

we are back in the game

[arctic field]

where was the limit

[restive river]

[arctic field]

https://cdn.discordapp.com/attachments/903486480075354182/1060845173338296360/1022141568737607750.png

[restive river]

yes

x = 2t solves it

[arctic field]

thats what i did

[restive river]

oh okay good

was curious if u had another method

[uncut crow]

||series||

[restive river]

no

smh

[uncut crow]

yes

[restive river]

oop

hacky preamble breaking

oh u finally switch to align

[restive river]

but yeah thats about it

[arctic field]

the second limit can be done with half angle

[restive river]

there is another one after it thats like

[
\lim_{x \to 0} \frac{x-\sin(x)}{x\sin(x)}
]

[woven radish]

♡LexQa♡

[restive river]

i managed to solve this using this

but i have not a single clue how to do it independently

yeah thats what i did

[arctic field]

thats exactly the same shit

[restive river]

yeah

just

[arctic field]

you can convert the sinx to an x

[restive river]

it goes to 0 i think

[restive river]

sinx = x?

[arctic field]

its the same limit multiplied by x

[arctic field]

assuming limits exist

[restive river]

makes sense makes sense

alright enough with random limits ty snow

.close

[devout snow]

Channel closed

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[devout snow]

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[alpine drift]

i need help to complete the square here

[alpine drift]

i got a negative square root 💀

[solemn wedge]

Are you expected to find complex roots?

[neon aspen]

💀

[alpine drift]

um idk my teacher just said complete the square

[solemn wedge]

well, this quadratic clearly has complex roots

given that the determinant is negative

[alpine drift]

here is wat my teacher said abt it

so confused :(

[warped relic]

$(a\pm b)^2 = a^2 \pm 2ab + b^2$

[woven radish]

VulcanOne

[solemn wedge]

Perhaps this helps?

[alpine drift]

there's no (a+b)^2 i've seen here ?

[alpine drift]

a bit but still not sure on some of it

[warped relic]

$-4x^2 + 3x + 5 \
-(2x)^2 + 3x + 5 \
-( , (2x)^2 - 3x) + 5 \
-( , (2x)^2 - 3x) + 5 +\left(\frac{-(-3x)}{2(2x)}\right)^2 - \left(\frac{-(-3x)}{2(2x)} \right)^2 \
-( , (2x)^2 - 3x) + 5 +\frac{3^2}{4^2} - \frac{3^2}{4^2} \
-\left( , (2x)^2 - 3x +\frac{3^2}{4^2}\right) + \frac{9}{16} + 5 \
-\left(2x -\frac{3}{4}\right)^2 + \frac{89}{16}$

[woven radish]

VulcanOne

[warped relic]

Nope wait

Something went wrong

Wait wait

Okay it's correct

[warped relic]

This is the complete process of completing the square for your specific problem

[alpine drift]

y is the sign now a -

[warped relic]

I factored a negative sign out of +3x

So it can be -(-3x)

So I can move it inside the brackets with the -negative sign

[torn field]

pls help

[warped relic]

!help

[devout snow]

Please read #❓how-to-get-help

[warped relic]

Please open your own channel

[devout snow]

@alpine drift Has your question been resolved?

[devout snow]

Channel closed

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[devout snow]

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[heady orbit]

https://gyazo.com/7c38a0378aa8d351113b5a6cdc2921ea

[warped relic]

@heady orbit Ask all your questions here

Use only 1 channel

[upper schooner]

From the circle they've given, you know its centre and radius, so then can form an equation of the form $(x-a)^{2} + (y - b)^{2} = r^{2}$

[woven radish]

chartbit

[upper schooner]

Rearrange that and see what you get, with the appropriate values for a, b and r

[heady orbit]

i honestly have no idea what any of that means. thats all the info they give and its just between answer 1-4 if you could specify that

[upper schooner]

Are you familiar with the equation of a circle?

[heady orbit]

im in my junior year of hs i havn't been present for a math class in two years

last shit i remember was adding letters to math

im fucked bruh

[upper schooner]

Well ouch I guess...

[upper schooner]

Well, can you at least tell me what the centre of the circle here is, and what its radius is?

[heady orbit]

center is the dot

raidius is halfway to the middle of a circle

or all the way through it

one of those ikkkk

[upper schooner]

Yes, what's it's coordinates?

[upper schooner]

This is probably what you're thinking of - can you give me its value from the circle in your picture?

[novel dome]

peanut i pray for your future

[devout snow]

@heady orbit Has your question been resolved?

[heady orbit]

-2, 1

[heady orbit]

uh would it be 3 or -3?

[novel dome]

a radius is a line

[upper schooner]

Good track, but remember distances are positive

[heady orbit]

yk

i neva thought people on discord would make me feel dumb 🤣

okay so center is 1-, 2 and the radius is 3

so wha?

[upper schooner]

Relatable

[upper schooner]

Well (-2, 1) and 3, but yep, from that, we can find an equation of a circle given its centre and radius

[devout snow]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

[upper schooner]

.reopen

[devout snow]

✅

[upper schooner]

A circle with a centre $(a, b)$ and radius $r$ can be given the equation $(x - a)^{2} + (y - b)^{2} = r^2$

[woven radish]

chartbit

[upper schooner]

To intuitively see why, all the points on the circle $(x,y)$ must have distance $r$ to the centre $(a,b)$

[woven radish]

chartbit

[novel dome]

wait peanut have you learnt about (a+b)²?

[upper schooner]

Oh yeah good question actually!

[devout snow]

@heady orbit Has your question been resolved?

[devout snow]

Channel closed

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[devout snow]

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• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
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[tacit wedge]

,rotate

[tacit wedge]

how do i do this

when i use substitution i always get 180=180 bc i cant figure what to substitute

(ignore my work)

[upper schooner]

Your work is a bit messy

But you know

Triangle ABC is isosceles with AB = AC and [angle] BAC = 40 [degrees]

Work out this angle here

Then you know that triangle BEC is isosceles too, as...

...CE = BC

That allows you to get angle BCE, but then you also know angle BCA [isos triangles]

[devout snow]

@tacit wedge Has your question been resolved?

[devout snow]

Channel closed

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[devout snow]

Available help channel!

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Remember:
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• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
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• Be polite and have a nice day!

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[peak walrus]

When we are calculating concavity, what do we do if y''≠0?

For example:
y=x+1/x
y''=2/x^3

[warped relic]

You find the infliction points and see where y'' is negative and where y'' is positive in between those infliction points

And you find the infliction points when y'' = 0

[peak walrus]

But y" cannot equal zero

[warped relic]

In this example?

[peak walrus]

Yes

[warped relic]

$y'' = \frac{2}{x^3}$

[woven radish]

VulcanOne

[winter patrol]

indicates no inflection points

[warped relic]

That means it has 1 behavior

Either concave up or concave down

[winter patrol]

not necessarily

[warped relic]

Hmm

Oh yeah right

[peak walrus]

It has 2 behaviors though, because it's got an asymptote

[warped relic]

At discontinuities

Yep got it

So you check your limits at the points where y'' can have an error

As well as infliction points

Like at x=0

Because the domain of y'' is all real numbers except for 0

[peak walrus]

So basically at 0+ it's infinity positive and at 0- it's infinity negative

[warped relic]

You check the limit from the left and from the right

Yep

[peak walrus]

And therefore it'd be concave down left of 0 and right of zero is concave up?

[warped relic]

Yeah

[restive river]

a point is a possible inflection point if the second derivative is either 0 or undefined

[peak walrus]

So I have to check all the local extreme points?

[restive river]

however, you would have to see that the second derivative changes signs as you cross the x value

[peak walrus]

Do I use limits at all then?

[warped relic]

It's helpful but I think you can manage without it

[peak walrus]

Okay

Also would it be fitting to post 4 functions with solutions asking for someone in this channel to check it for me?

I'm making a study sheet for my class and don't want to give incorrect information

[warped relic]

Hmm

Okay sure

[warped relic]

Are these all the problems?

[peak walrus]

Missing the last one because I haven't put in the concavity yet

[warped relic]

Okay lemme rotate the first image

https://cdn.discordapp.com/attachments/903486480075354182/1061628370959487086/rn_image_picker_lib_temp_fb0e1b9a-764c-4815-9073-2cff64ed4172.jpg

,rotate

[warped relic]

https://cdn.discordapp.com/attachments/903486480075354182/1061628429038014484/rn_image_picker_lib_temp_e092097a-a301-49f2-a3f7-1f275192ced8.jpg

https://cdn.discordapp.com/attachments/903486480075354182/1061628479159943229/rn_image_picker_lib_temp_968a059c-549c-4424-b9ed-5e0aac000073.jpg

[peak walrus]

Haha sorry mb

[warped relic]

No worries

[peak walrus]

Here's the final question

[warped relic]

Correct

[warped relic]

Correct

[warped relic]

Correct but the asymptotes need a bit of work

[peak walrus]

Hmm, as in writing it all out you mean?

[warped relic]

No like defining the asymptotes as lines

Like the asymptote at x=0

You should write "Vertical Asymptote at x=0"

Also the asymptotes ate x approaching infinity and negative infinity

[peak walrus]

Ah, I just take that as givens when it's in the domain, but I'll write it out

[warped relic]

You can argue that at both infinities, the behavior of the function behaves more and more like x

Because as x increases, 1/x tends to 0

[peak walrus]

Yeah

[warped relic]

So your function has an asymptote at y=x

The line y=x is an asymptote

[peak walrus]

So a diagonal line?

[warped relic]

Yep

You can have diagonal lines as asymptotes

[peak walrus]

Got it

Wouldn't there always be an asymptote then?

[warped relic]

Wdym?

[peak walrus]

Also for #1 and 2 they'd have to include an asymptote

[warped relic]

Problem 1 did include an asymptote

At y=0

[peak walrus]

In addition to that wouldn't there be one that's ever increasing

[warped relic]

Wdym? Can you specify a problem where you faced that doubt?

[peak walrus]

Hmm nevermind

[warped relic]

No this concept is pretty important

https://math.stackexchange.com/questions/850745/can-asymptotes-be-curved

[peak walrus]

Oooh I understand

I just never used oblique asymptotes

[warped relic]

Did the teacher talk about oblique asymptotes?

[peak walrus]

No

[warped relic]

Then I guess that's extra info for you then. Keep it as you wrote it so that you stay in line with your teacher

[peak walrus]

It's fine either way. We were told for 20pts on the final will be using 1 of these 4 functions, I doubt he'll take off points from writing extra :p

[warped relic]

Well sometimes the teacher can get upset that you're writing more things than what they taught

Idk

I am pretty used to that with my professors

[peak walrus]

I doubt it. Personally I already took calculus last year in another university. I'm just retaking it for the free 100 to help my average. So he's pretty used to me adding things already 😂

Also we have a small class, so he won't mind checking it

[warped relic]

Oh lmao

Alright then if it's that way then sure

You can write that f(x) has an oblique asymptote y=x

[peak walrus]

Thanks :)

[warped relic]

Okay now for the last question

https://cdn.discordapp.com/attachments/903486480075354182/1061630941564522506/20230108_150206.jpg

There is no y-intercept for this function

Other than that, it's all correct

[peak walrus]

Thanks a lot!

.close

[devout snow]

Channel closed

Closed by @peak walrus

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[devout snow]

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• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
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[ocean hinge]

Show that the triangle with as sides the x-axis, the y-axis and a tangent line to the graph of the
function f(x) = 1/x has constant surface area (i.e., the surface area of the triangle does not depend
on the choice of tangent line).

[ocean hinge]

i have just taken f'(x) = -1/x^2

and i have no clue what to do next

[brave lily]

what do you think?

[ocean hinge]

maybe the three sides of the triangle are the x axis, y axis and the tangent line?

[pseudo basin]

indeed

it might do you good to introduce a variable for the point of contact

then express all relevant equations and coordinates in terms of that

[ocean hinge]

okay, but how do i apply these ?

[pseudo basin]

well

let c be the x-coordinate of the point of contact

we can assume that c is positive because symmetry

write down the equation of the tangent line to y=1/x at (c, 1/c)

[ocean hinge]

point of contact?

.close

[devout snow]

Channel closed

Closed by @ocean hinge

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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.
• After 15 minutes, feel free to ping <@&286206848099549185>.
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[restive river]

uhh help I have like a group task tomorrow and the topic is solving for c

[restive river]

though I don't know how to do it

[hazy quail]

Question

[restive river]

I mean I dont have a question though I need to learn it so I can participate in my group assignment

solving quadtratic equations through factoring, finding the pst, and quadtratic formula

Thats the topics

[earnest trellis]

pst means?

[restive river]

perfect square trinomial

like x squared and 36

[hazy quail]

I have never heard of that in my life

[restive river]

u think u can help me?

I rlly dont understand anything of it

[hazy quail]

Oh ok it’s just that

[winter patrol]

if you're starting from scratch, try doing the free quadratics course on khan

[restive river]

hmm that's a binomial

but by perfect square trinomial

im assuming you mean

[earnest trellis]

you should watch some videos how to do it

[hazy quail]

He just means a binomial squared

[restive river]

hmm alr

[restive river]

I tried to

Didnt understand anythig

[winter patrol]

link to vid, timestamp which part you don't understand

[hazy quail]

^

[restive river]

are you expected to self-learn quadratics for school?

[restive river]

What I mean by I watched a video of it was like

Before like I had a different topic I tried watching a video on it and didnt understand anything

[restive river]

what does that mean

I dont understand anytigng

can u guys give me a video to watch?

[winter patrol]

you said you've watched vids

what were you watching

[restive river]

I remember watching a video on like khan university way before but didnt understand anything

[restive river]

it was a different topic

not this one

[winter patrol]

well try watching a vid from khan on this, the topic you're interested in

[earnest trellis]

https://youtu.be/JvfOwEWqL6I

[winter patrol]

organic chem tutor is also very good

[restive river]

these 3 topics

[restive river]

okay ill watch if

can u guys still teach me though even if I do watch it?

I need to be sure I do understand ot

[earnest trellis]

sure if you don't understand anything then just ask

[winter patrol]

watch, timestamp stuff you don't understand

[restive river]

okay

How did he go from x=-3+- (3) and so on

To 3+-9+40

I legit dont understand whats happening in this video bro

[earnest trellis]

that is quadratic formula

do you know quadratic formula

[winter patrol]

also the square root isn't invisible

[restive river]

No

[winter patrol]

the quadratic formula is in the top right of that screenshot

[restive river]

I MEAN I UNDERSTAND THIS

cuz if u add -2 and 5 its 3 and -2 times positive 5 is 10

So I guess I slightly understand it though idk

[earnest trellis]

-10

[woven radish]

ℝamonov

[restive river]

Oh yea it has - infront of it

[winter patrol]

which is pretty much outlined in the screenshot you posted

[restive river]

u only change b ac and the other b and a right?

[winter patrol]

identifying the values of a,b,c,
sub those values into the formula

u only change b ac and the other b and a right?
whut?

i don't understand your question

[restive river]

The b and b squared ac and a is the only things u should change there?

[winter patrol]

whut

[winter patrol]

step 1 identify the values of a,b,c

[restive river]

these things highlited in red would just be the only rhing u change

[winter patrol]

step 2 substitute those values of a,b,c into the formula

well yeh.

[restive river]

can u give an example

[winter patrol]

which is pretty much outlined in the screenshot you posted

they identified
a=1,
b=3
c=-10
and then simplify substituted those values into the formula

and proceeded to simplify

[restive river]

oh I see now

how do we simplify them

[winter patrol]

basic addition/multiplication etc

[restive river]

im gonna be honest

I don't understand the video

[winter patrol]

which part

[restive river]

everything

[winter patrol]

i mean you implied that you understood basic factoring

and what we just went through that was supposedly part of the vid

everything
don't overexaggerate

[restive river]

Yeah like the equation u factor it to a b and c

[winter patrol]

whut

wdym factor it to a b and c

[restive river]

this equation

u factored x or 1 to a

[winter patrol]

wdym factor it to a b and c

[restive river]

this

[winter patrol]

that isn't factoring

[restive river]

oh

then what is

[winter patrol]

i've typed it above

$a$ represents the coefficient of $x^2$\
$b$ represents the coefficient of $x$ and \
$c$ the constant term

[woven radish]

ℝamonov

[restive river]

oh

[winter patrol]

pls read

i don't like having to copypasta repeat myself

[restive river]

what is the coefficient of x squared

[winter patrol]

x^2 = 1x^2

all this is explicitly in the screenshot you posted

its literally ALL in the screenshot

[restive river]

I am slow in the head

[winter patrol]

its all there

[restive river]

ah I see it now

what after that

[winter patrol]

pls read
i don't like having to copypasta repeat myself
step 2 substitute those values of a,b,c into the formula

which again is all in the screenshot you posted

[restive river]

this formula?

[winter patrol]

yes

[restive river]

how do we substitute it?

[winter patrol]

replace the variable with whatever value they represent
use additional ( ) around your value to make sure you don't mess up the order of operations if you're new to this

sometimes they're needed, sometimes they aren't, better to be safe.

[restive river]

what

I have no clue what to do

[winter patrol]

replace every a in the formula with the value of a, which was identified to be 1, to be extra safe, replace a with (1)

[restive river]

Oh I see

[winter patrol]

similarly replace every b in the formula with the value of b which is 3, and to be safe, use (3)

and c with (-10)

[restive river]

what if its ac?

[winter patrol]

ac is the product of a and c

it isn't a big issue if you use () as i mentioned

which pretty much gives what they've written in the screenshot you've posted

[radiant drift]

[restive river]

how about b squared?

I mean it kinda counts as B so

[winter patrol]

b^2 is b^2

focus on the initial substitution first

worry about simplification after

if anything just humor me and literally replace all
a with (1)
b with (3)
c with (-10)

keeping those () there

do nothing else other than that

and show me what you have

[restive river]

like thos?

[winter patrol]

can you rewrite it separately without those scribbles?

you're also missing that $\pm$ sign

[woven radish]

ℝamonov

[winter patrol]

you haven't replaced that b or a under the root either

surely you've done substitution before right?

you must have if you've done stuff with straight lines, (which comes before quadratics)

e.g. you must've done stuff like
find the y-coord of the point on
y = 3x +4
where x=2
right?

[restive river]

okay

this better?

im actually gunna fail bro

I dont think we have enough time for me to learn all of this

@winter patrol

u there

[restive river]

I did I forgot

[restive river]

uh no

I dont think so

[winter patrol]

have you done anything with straight lines before?

[restive river]

What does that suppose to mean

[winter patrol]

don't see how there can be any issue interpreting that question

[restive river]

like vertical or horizontal line?

[winter patrol]

vertical, horizontal, slanted

[restive river]

is straight line something u do in math or somethig u can just do with a ruler?

I mean I don't think I do but I was forced to in art back then

[winter patrol]

like surely you've done stuff with

,w graph 2x+5

[woven radish]

Results provided by WolframAlpha

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

Requested by ramonov#0921

[winter patrol]

stuff like that before right?

[restive river]

oh

I mean yeah I 5ink we did something like this

or something similar to this

[winter patrol]

make a new fraction, now every time you see
a, write (1) instead
b, write (3) instead
c, write (-10) instead

keep everything else as is

i should see NO a,b,c in the result

[restive river]

Okay

that cool

@

?

[winter patrol]

no

[restive river]

what did I do wring

[winter patrol]

you don't have () around the 1

you didn't follow my instructions

[winter patrol]

there aren't () around the 1 for the a that was under the root

[restive river]

im on an ipad its kinda more hard to do

[winter patrol]

not really, if you followed the instructions from the start

[restive river]

sorry I don't rlly understand how the topic works

[winter patrol]

i didn't say replace a with 1

[restive river]

wait huh

[winter patrol]

i said

now every time you see
a, write (1) instead

[restive river]

oh

[winter patrol]

those () weren't just for kicks

[restive river]

okay

[winter patrol]

they're there to ensure that you don't make any algebraic screw ups

[restive river]

I uh tried

[winter patrol]

fk it close enough

[restive river]

its just so much harder to do it on ipad

[restive river]

im so sorry bro

[winter patrol]

you could just rewrite it

[restive river]

I MEAN ITS CLOSE ENOUGH

[winter patrol]

$x = \frac{-(3) \pm\sqrt{(3)^2 - 4(1)(-10)}}{2(1)}$

[woven radish]

ℝamonov

[winter patrol]

now simplify this

[restive river]

Uh

What

[winter patrol]

wdym what

the substitution is done, the rest is just simplification

[restive river]

Bro I know how to simplify fractions like in 5th grade like 2/4 = 1/@

[restive river]

So -10 = -5?

What

[winter patrol]

no

-10 is definitely NOT -5

[restive river]

Si yh

so uh

how do I simplify it

[winter patrol]

don't overthink this,

this current step is basic multiplication and subtraction

[restive river]

Oh so what if I do -(3)+-(3)^2-(1)(-10)(1)

does that work

[winter patrol]

no

[restive river]

Actually no it does not

[winter patrol]

don't overthink this,
this current step is basic multiplication and subtraction

[restive river]

Make (3) squared? (3)(3)?

[winter patrol]

and can you simplify that further?

[restive river]

no

Its 3 squared not 3 cubed

Did I get that right?

[winter patrol]

do you know what 3 squared means?

[restive river]

3^2?

[winter patrol]

are you implying you know what 3 cubed is but don't know what 3 squared is?

[restive river]

Why is this so scary

Uh

[winter patrol]

you're overthinking

[restive river]

I actuallt do not know

[winter patrol]

can you give a simplified numerical value for
3^2?

[restive river]

OH I GET IT NOW

CANCEL OUT

[winter patrol]

no

[restive river]

subtract 1-1 and 3-3

[winter patrol]

no

[restive river]

Okay

[winter patrol]

do you know the definition of squaring?

[restive river]

Square root?

[winter patrol]

no

[restive river]

Uh

[winter patrol]

squaring and square root are not the same

[restive river]

No?

[winter patrol]

can you give a simplified numerical value for
3^2?

[restive river]

6?

[winter patrol]

no

[restive river]

9?

[winter patrol]

yes

[restive river]

okay

[winter patrol]

to square something is just to multiply something by itself

[restive river]

ph yeah

[winter patrol]

you even initially implied you knew that
3^2 = (3)(3) which is 3*3

which is 9

[restive river]

Oh so I was right

[winter patrol]

well (3)^2 = (3)(3)
nfi why you said you didn't know how to simplify taht further

[restive river]

CUZ I DIDNT KNOW

[winter patrol]

anyway focussing only on the expression under the root atm
$$(3)^2 - 4(1)(-10) = 9 - 4(1)(-10)$$
can you try simplifying that part further

[woven radish]

ℝamonov