precalculus

[storm shuttle]

does the time matter?

it does kinda ig

it doesnt?

idk

[modest bolt]

I think ending should be like "after how many seconds will they both stop at the same time?"

[versed jolt]

Those anybody have sam question ⁉️❓

[viscid thistle]

one thing we can say is, that's one ABSOLUTELY MASSIVELY GARGANTUAN skyscraper.

[lunar dawn]

Anyone care to explain why the integral of $e^ (-x^2)$ is root pi?

[obsidian monolith]

Herman Zhang

[summer ruin]

you can watch any youtube video that explains this, there are many

the short answer is you square the integral and turn it into a multivariable integral, then switch to polar coordinates and integrate it to get pi as an answer, take the root of that and that's the answer

[obsidian gulch]

im screwed

[viscid thistle]

<@&268886789983436800> there is a slur in there if you look closely

[mental stirrup]

@obsidian gulch Please don't use slurs here, thank you.

Lmao

[viscid thistle]

@mоdеrаtοrs, what's your emergency?

[dusk hedge]

Sup

😭

Again

Why derivates that look so simple end being like 10 different terms

[river drift]

1 term for each function being multiplied

[viscid thistle]

give example

[warped shuttle]

is this question wrong ?

if a quadratic with y = ax^2 + bx + a - b be tangent to y = 2ax + b
what would be the y part of the tangent point

what i did was equal both equations move them to one side because they are tangent make discriminant zero solve it you have two cases b = 0 and b = -4a and then solve for each case

if we find the x part of the tangent point for both cases and plug them into the equation we got, we will get 0 but if we put them in the original equations we will get 2a so this means 2a should be equal to 0 and thus a = 0 which means we don't have a quadratic which is in contradiction with the question's assumuptions

[simple sage]

I’m not sure I understand

Can u rephrase the question

Also dy/dx is 2ax+b, this describes the gradient at any given value of x

[simple sage]

This doesn’t necessarily imply y=2ax+b is a tangent

In fact y=2ax+b cannot be a tangent

The value of y at any tangent point with x coordinate x is simply ax^2 + bx + a-b

[warped shuttle]

If the parabola $y = a x^2 + b x + a - b$ is tangent to the line $y = 2a x + b$, what is the $y$-coordinate of the point of tangency?

[obsidian monolith]

GodOfPro

[simple sage]

Okay so my thought is

If the line is tangent it will only meet the curve once

So I would plug in y=2ax+b into the parabola equation

And set the discriminant equal to 0

Wait actually will that get u anywhere

[simple sage]

So after this just solve for x I think

[warped shuttle]

what would just make it harder

equaling them to each other and finding the x would result the same

[simple sage]

You know that bc it’s a tangent the discriminant will be 0

So the b^2-4ac in quadratic formula is 0

Therefore once u have ur coefficients in place

U can just say that x=-b/2a

Note that these a and b are dif to the one in the original question

But rather the coefficients of x^2 and x respectively

[warped shuttle]

i know that

i have done that my point is i am getting a = 0 which would make the quadratic a line

[warped shuttle]

.

[simple sage]

Ur not solving for a ur solving for x

And then for y

[fresh swan]

Can anyone help me asap

Reply if u can cuz I need it lmao

[terse pebble]

just post your question

[fresh swan]

I have an answer but my teacher has a different answer and I’m JS confused

[terse pebble]

,rotate

[terse pebble]

what’s your answer

is it what’s crossed out?

[fresh swan]

11. (-5,-4) U (0,4)

12. (-2,3)

[terse pebble]

(-2, 3) is not quite right

it needs to be both increasing and concave down

at the same time

remember, it says the rate of change of g

we don’t care about g being positive

which is what it looks like you did by ending the interval at 3

it’s increasing for (-4, 0) and concave down for (-2, 4)

take the intersection of these intervals

[fresh swan]

(-2,0)?

[terse pebble]

yes

[fresh swan]

Alright thx. Was 11 correct?

[terse pebble]

yes

[fresh swan]

Cuz my teacher had a different answer

Oh

[terse pebble]

what did they have?

did they do closed intervals or something

[fresh swan]

She had (-5,-4) U (2, 4)

[terse pebble]

no that’s definitely wrong lmao

"rate of change of g negative" means g is decreasing

g is decreasing if g(x) < g(y) whenever x > y

clearly it’s decreasing on (0, 2) as well

[fresh swan]

Yeahh alright that’s what I thought too. It had me really confused lmao. Alright but thank you for the help tho🙏🏼

[terse pebble]

you’re welcome

[wispy niche]

If f(x) = x would be the simplest function, what would the most basic integral look like?

[hidden sun]

I think f(x) = 0 would be the simplest function

[wispy niche]

Thanks cunningham ❤️ , what about the integral tho?

[hidden sun]

,w integrate 0

[obsidian monolith]

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[wispy niche]

I see now, thanks

[hidden sun]

https://tenor.com/view/so-good-wink-smile-oh-yeah-its-true-gif-24163876

[brazen cipher]

Pre-chalchulus

[dense kelp]

can someone explain what this derrivative is doing and how to use apply it for future questions?

[opal patrol]

so how does this happen?? 😓

[hidden birch]

Remember the parabola formula
[
y-k=4p(x-h)^2
]
where $(h,k)$ is the vertex and $p$ is the distance from the vertex to the focus. Also the sign determines if the parabola opens upward or downward

[obsidian monolith]

glass

[vale pivot]

yo

anyone tryna help me

[fringe cove]

Hello, could you help me? How can I write the sign of this part of the function in interval notation?

I got, negative: (1 ; 1/2) and positive: (1/2 ; 2)

I don't know if this is correct

[hidden birch]

maybe correct

[fringe cove]

;-;

[hidden birch]

like what's the context. It's obvious on the image you shown

unless the sign of a function means something else

i haven't done pre-uni math in a long time

[fringe cove]

the question asks to study the sign of the function

[hidden birch]

tbh it makes me think of increasing or decreasing functions

[hidden birch]

could the function be zero?

like no sign

iirc, we used + or minus to indicate if a function is increasing or decreasing over an interval

[jade glen]

Wouldn’t it just be 1 to 2

(1,2)

Because the line is between the two numbers

For your answer, you seem to be excluding that section instead

But i cant see the whole graph so I’m assuming it’s that odd function

[jade glen]

With what

[fringe cove]

The sign of the function is positive when it is above the X-axis and negative when it is below it.

I thought it was negative of (1, 1/2) and positive of (1/2, 2)

apparently it is correct, but there were others that I could not solve or at least understand

I was unable to determine the sign of the function in letters B and C

[jade glen]

Oh, i thought you mean domain since it wasnt specified

Also for pos; (-2,-1) U (-1,-1/2)U (1,2)

And for negative ur right

[storm shuttle]

anyone wanna help me calc the area for this>

??

[tired garden]

convert to polar form and evaluate integral

[storm shuttle]

we haven't started integrals and i only know basic ones

[tired garden]

oh

is there a reason you need this?

[regal tree]

you can see that the graph is symmetrical, so try finding the area from one of the quadrants and multiply by 4

like say in first quadrant both x and y are +ve so we can remove the absolute value

[storm shuttle]

dont need it now

[storm shuttle]

would be helpful if u sent the sol tho

[tired garden]

convert to polar:
x = r cos t
y = r sin t

r(cos^2 t / sin t) + r(sin^2 t / cos t) = 1

r = 1/(cos t cot t + sin t tan t)

notice that from t = 0 to tau/4 we get one quadrant of your shape

because we ignored the absolute value signs when we did the polar sub

now just plug into polar integral area formula

integral 1/2 r^2 dt

you get something kinda nasty so i just threw it into wolframalpha

[storm shuttle]

tau/4

[tired garden]

,w integral 1/2 * 1/(cos(t) cot(t) + sin(t) tan(t))^2 dt from t = 0 to pi/2

[obsidian monolith]

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[tired garden]

so your final area for all 4 petals is 4/6 = 2/3

[tired garden]

?

[acoustic apex]

x = tan(t) simplifies it nicely

[worldly fiber]

,help

[obsidian monolith]

A brief description and guide on how to use me was sent to your DMs!
Please use ,list to see a list of all my commands, and ,help cmd to get detailed help on a command!

[worldly fiber]

,tex f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

[obsidian monolith]

Cuber42
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[raw summit]

wtf

[worldly fiber]

?

thats the limit definition of a derivative

[raw summit]

im doing alg 2 thats why 💀

[worldly fiber]

oh ok

im taking bc calc

-# ap calc bc

[hidden birch]

-# enclose the maths in dollar signs

[winged wigeon]

What would better to find the derivative? I’m still in James Stewart chapter 1 but would the normal derivative be better or would the derivative as a function be better?

[hidden birch]

-# or just use ,mtex or ,, instead of ,tex if it's purely maths

[covert lark]

Can anyone tell what is saddle point in maxima and minima ?

[river drift]

a point where a function has zero slope but which is neither a maximum nor a minimum

,w graph x^3

[obsidian monolith]

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Requested by cloudedhorizons

[river drift]

for example here

[winged wigeon]

Eww cubic looks so weird

So zoomed in😭

[raw summit]

imma take it 9th grade 😭

[granite wolf]

can any1 link me to the general chat :/

[regal tree]

#chill

[hidden birch]

-# he was already linked there because he posted on multiple channels

[regal tree]

yea realised later lol

[verbal portal]

$$
f'(x)=\displaystyle \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}
$$

[obsidian monolith]

𝜗𝜚⋆₊˚ Usagi ₊˚⊹ ᰔ

[verbal portal]

and i mean who else uses this lol

[main pagoda]

imagine deriving a high degree polynomial with this shit

[willow skiff]

you just need to apply first principles on x^n

then for example, if you have $\lim_{x \to 0} \frac{f(x + h) + g(x + h) - (f(x) + g(x))}{h}$, what must that result in?

[obsidian monolith]

south

[willow skiff]

so adding, subtracting, and multiplying by a scalar also become easy

for the derivative (and integral in fact)

so that's why anything other than the basic cases is a complete waste of time

including all of epsilon-delta, yes

[worldly fiber]

What grade u in

I still don’t believe you’ll be taking the hardest high school math class freshman year

[raw summit]

7

[raw summit]

no linear algebra too

[jade glen]

??? What country r u from

[worldly fiber]

?

Linear alg is very easy

[tough lake]

Heli

I was studying graphing transformations from pre calc 12 and my book says to apply graph transformations in this order
1.) horizontal and vertical compressions and stretches
2.) reflections
3.) translations

And then I’m given the general transformation form:

(x/b +h, ay+k)

First of all does this general transformation follow the order above, I think yes it does

Second is the order of transformations correct

[fleet fox]

could anyone help me solve this?

[high sequoia]

Maybe separating the terms and divides both numerator and denominator of both fractions with x² can help, okay no that leaves off infinity in denominator

[fleet fox]

i think i solved it

[high sequoia]

What's ur answer

[fleet fox]

-4

ty

[cobalt cradle]

I've never felt so offended

[raw hill]

Contour integral

[river drift]

it's an integral along a closed loop (possibly either a line integral or a contour integral)

[cobalt cradle]

no I mean the i = 1 to n

[heavy pewter]

seems meaningless, it just means to show the subscript and superscript on the contour integral

[near garnet]

how cooked am i for precalc

[rustic bramble]

?

[last flax]

Hi, I was wondering what this type of equation is called. I would prefer to be led in the right direction so I can understand it through videos

[hidden birch]

not an equation. There is no equal sign in sight

[hidden birch]

variables in limits may help to know which is which in multiple integrals

or when making variable substitutions, i think

[worn orchid]

no equation, like glass said - i would search "simplifying exponents" for this. here is a video i recommend

[hidden birch]

you can call it an expression instead
-# i think

[quick jungle]

Hello, I skipped algebra 2 and I'm jumping into pre calc

Im wondering if I would need a strong foundation of algebra 2 for pre calculus.

Or if a minimal-mediocre foundation of algebra 2 would be fine

[echo sinew]

What are the topics of your algebra 2 course?

[quick jungle]

Shoot, it's on my laptop give me some minutes I can check

These were all the chapters

We didn't have a chapter 1 btw

My schools algebra 2 course covers more in depth, so I'm probably gonna self study identies and some advanced algebra 2 concepts which I do have trouble with.

So in pre calculus I'm a bit scared on what to do and stuff, and what are the usual first chapters of pre calculus

[regal tree]

you know how to solve linear equations though right?

[quick jungle]

Yes

[echo sinew]

I'd say you definitely need these on order to do precalculus

[quick jungle]

Yep, that's what they taught me

And I understood it

[hidden summit]

Would finding the average of 2 secant lines be a good mediocre "approximation" for a tangent line. in which both intersect 1 point or am I high?

[river drift]

it is possible to combine two secant line estimates for the slope of the tangent line in order to get a better estimate, although how exactly you combine them depends on how the two relate

an instance where you would actually average the two slopes would be in combining the forward difference
[ \frac{f(x + h) - f(x)}{h} ]
and the backward difference
[ \frac{f(x) - f(x - h)}{h} ]
into the central difference formula
[ \frac{\frac{f(x + h) - f(x)}{h} + \frac{f(x) - f(x - h)}{h}}{2} = \frac{f(x + h) - f(x - h)}{2} ]
which is a better approximation

[obsidian monolith]

cloud

[hidden summit]

ooooooh this might take a little for me to get but i think I can procces it.

[silent roost]

Hello guys, i’m a rookie in math and I need help in a question. And sorry if I write it wrong, my english isn’t good

[ripe heart]

What's your doubt? Is it in the proof of induction itself or arent getting any idea at the induction step?

[silent roost]

I’m begining to study induction now, so I don’t know how to start the induction at all

[ripe heart]

The idea is that you prove the statement for a starting case and then you prove that if it is true for some natural number, so it is for its successor(to falando em ingles pq nao sei se pode conversar em porutgues aqui, entrei agorinha)

it's like a domino effect

in this case, you prove it for n = 1, wich is just 2^1 = 2 > 1

Then, you suppose it's true for some arbitrary n = k (this means that k can be ANY natural number, you just have to know that the statement is true for it).

Now, you have to show that if that being true to n = k implies it being true to n = k+1

Like, if k = 1, then it'll be ttrue for 2. But then, it'll be true for 3 and so it goes

in this case you'd suppose that for some arbitrary k, 2^k > k

And then, you'll have to do some algebraic manipulation to show that 2^(k+1) > k+1 USING the fact that 2^k > k

nao sei se eu fui claro kk

[silent roost]

Certo, muito obrigado

[ripe heart]

de nada, cabecinhadegordo

[shy junco]

Im not really able to solve Precalculus ques

Can anyone give me tips of how shld i study

[uncut mulch]

start with the basics/definitions

it would be helpful to show a specific question your stuck on

[warped shuttle]

consider the solution is just one interval for $$\frac{((m^2 -1)x^2 -4mx + 4)(2x-3)}{(x-3\sqrt{x}+2)} >= 0$$
what is the value of m ?

can someone help me out with this

[obsidian monolith]

GodOfPro

[warped shuttle]

we know that the quadratic can't have a root other than these three because it will make another interval which is positive which we don't want but can't go further

[soft plover]

m²-1 < 0 and the discriminant of the quadratic factor must be equal to 0

[storm shuttle]

root x kinda throws me off cause i think there won't be a gap for any value of x for the bottom since u can just take the other sign of it but ye idk

[storm shuttle]

Can someone check this:
||2^1>1
2>1
if n=k>=1
then 2^k>=2^1
which means 2^k>1
Let's assume 2^n>n is true at n=k
then
2^k>k
2^k>1
which means 2(2^k)>k+1 => 2^k+1>k+1
since the condition holds for n=k+1 as well
2^n>n is true||

[trim salmon]

Tbh I don’t understand your way
$$
\forall n \geq 1, \quad P_n : 2^n > n
$$
First we have that $2^1 > 1$ so $P_1$ is true.
Let’s say $P_k$ is true, we'll show that $P_{k+1}$ is true too. \
We assumed that $2^k > k$, so $2 * 2^k > 2k \Leftrightarrow 2^{k+1} > 2k$ \
We know that $\forall k > 1, 2k > k+1$, so $2^{k+1} > k+1$ \
Which mean that $P_k \Rightarrow P_{k+1}$ \
So by induction, $P_n$ is true $\forall n > 1$.

[obsidian monolith]

SixScorpion

[storm shuttle]

look at his explanation

@trim salmon

[trim salmon]

I think it’s what I did

[storm shuttle]

nah u used n

i think thats a lil diff

btw this is only for regular sequences

[trim salmon]

Yes that’s the same, you assume for k and show for k+1

That’s only notations

Tbh I don’t understand your way
$$
\forall n \geq 1, \quad P_n : 2^n > n
$$
First we have that $2^1 > 1$ so $P_1$ is true.
Let’s say $P_k$ is true, we'll show that $P_{k+1}$ is true too. \
We assumed that $2^k > k$, so $2 * 2^k > 2k \Leftrightarrow 2^{k+1} > 2k$ \
We know that $\forall k > 1, 2k > k+1$, so $2^{k+1} > k+1$ \
Which mean that $P_k \Rightarrow P_{k+1}$ \
So by induction, $P_n$ is true $\forall n > 1$.

[obsidian monolith]

SixScorpion

[craggy osprey]

if i got it wrong lmk

[raw hill]

,w integrate (x^4)/sqrt(x^5+2) dx

[obsidian monolith]

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[raw hill]

You can check like this in #bots

Work looks fine as well

Also this belongs in #calculus, not #precalculus

[craggy osprey]

oh gotcha

[willow bear]

smiley face as a variable name is quite a bold choice

[somber void]

not as bold as $\mathbf{x}$

[obsidian monolith]

CyclicTree

[craggy osprey]

yes

[jagged lintel]

meh

The bot dosn't support math bold latex

well I'm curious what it does and doesn't

$\mathbb{test}\mathbb{TEST}\mathbb{N}\mathrm{test}\mathcal{test}\mathfrak{test}\mathscr{test}$

[obsidian monolith]

Stickman A

[jagged lintel]

It support maj

[hushed sphinx]

#latex-testing, please.

[jagged lintel]

mb sry 😅

[stuck fractal]

how can i self study precalc

[willow bear]

khan academy

may have better luck asking in #study-discussion btw

[stuck fractal]

oop mb

[stuck fractal]

tysm

[torpid elbow]

The organic chemist, life saver when I took calc 1

[cyan quiver]

i used a website called flippedmath.com to self study algebra 2 they also have a precalc course it can maybe help

[worldly fiber]

If f(x) = |x| why isn’t f’(0) just 0 since it’s exactly between the slopes -1 and 1

[river drift]

using the limit definition of the derivative, the limit doesn't exist

[worldly fiber]

Oh be cause lim x-> 0- ≠ lim x-> 0+

-1 ≠ 1

[uncut mulch]

yes, the slopes are different, you're not allowed to just take their average

[warped shuttle]

my teacher gave this and didn't prove it and talk further about it, what's the logic behind this and how to prove it. If

[
f(x)=\frac{\cos!\left(\frac{2\pi}{n}\right),x+\sin!\left(\frac{2\pi}{n}\right)}
{-\sin!\left(\frac{2\pi}{n}\right),x+\cos!\left(\frac{2\pi}{n}\right)}
\qquad\text{then}\qquad
\underbrace{f\circ f\circ\cdots\circ f}_{n\text{ times}}(x)=x.
]

[obsidian monolith]

GodOfPro

[hazy chasm]

What are some techniques to compute arctan limits

[willow bear]

can you explain the meaning of anndreas first?

@hazy chasm

[hazy chasm]

Uh

No meaning i guess

[willow bear]

no, no, tell me.

[hazy chasm]

Just called you that

[willow bear]

why did you call me that

[hazy chasm]

Because it sounds cool

Did it make you mad

No offense behind it

[willow bear]

i think it sounds like an attempt to mix my name with the masculine name Andreas

to which mine has no relation whatsoever

[hazy chasm]

Oh

Well its not

There you go

[willow bear]

anyway, arctan itself is continuous, therefore the limit of arctan(x) as x approaches a finite number c will equal arctan(c); additionally, the limit of arctan(x) as x approaches either +∞ or -∞ equals ±π/2 respectively

[hazy chasm]

Yeah

I know that but like

[willow bear]

if you have an example of an arctan limit on hand, please share it.

[hazy chasm]

Let me give you an example

Ill go screenshot something

See something like this

Is the a secret universal technique i can use

[willow bear]

no universal technique can possibly be given until you clearly and rigorously define a class of problems said technique must solve

[manic pelican]

hi guys im new here is the textbook provided correct because my friend is insisting it's B^2 - 4AC

[willow bear]

however, this specific problem may be recognized as the limit defn of the derivative of a particular function at a particular point

[willow bear]

this looks ripped out of context

[manic pelican]

I can provide the entire page

[willow bear]

yes, do that

[willow bear]

i do think that should be B^2 - 4AC yeah...

also "computing for the value"

[hazy chasm]

We cant derive arctan yet, its not allowed

[willow bear]

who even says that...

[willow bear]

you are not allowed to claim arctan'(x) = 1/(1+x^2) ?

[hazy chasm]

Nope cant do it because the exercices here are tied to a lesson which deriving arctan is not teached yet

[willow bear]

are you at least allowed to claim lim[x->0] arctan(x)/x = 1?

[hazy chasm]

Hence then not allowed

Yes

[willow bear]

then i believe you are fucked unless you want to mess around with formulas such as arctan(x) - arctan(y) = arctan( (x-y)/(1+xy) ) or something...

[hazy chasm]

Oh shi

Wouldnt something like

Value replacement work

[manic pelican]

I have 13 pages worth of pre-cal exam preparation, I genuinely hope this page is the only error

[hazy chasm]

pre carc

[willow bear]

you can try it; it just gives 0/0 on the limit you posted

[hazy chasm]

Okay thanks ann

[lapis shuttle]

Im in ap precalc rn and im so confused to why it's even an ap

ppl were so right

WHY IS MY CLASS ALREADY FINISHING UNIT 1 WHEN THE CLASS HAS LIKE 4 UNITS AND THE 4TH ISNT EVEN ON THE NATIONAL EXAM 💀

I'm so confuzzled rn

[winged wigeon]

Yeah that’s why it gets shit on😭

[compact bay]

whats the average age for someone doing precalculus?

[willow skiff]

16 to 17

[compact bay]

ok im 13

[pure pelican]

We are doing calculus at the age of 16/17 in india .

[regal tree]

jee student?

[pure pelican]

Kinda

[regal tree]

to be fair in jee we also do precalc at the same age right before calc

[pure pelican]

Yeah

[regal tree]

kinda how

[pure pelican]

Idk

[smoky bloom]

hello math people

[winged wigeon]

I wish I started learning advance math more early in my life😭. I am currently learning calculus as a 14 year old going into my 15 soon

[willow skiff]

why?

I mean why do you regret not studying earlier

[winged wigeon]

It was fun getting into math. I was really bored in my early age

Lowkey just staring at my wall😭

[willow skiff]

understandable

[hidden sun]

Same

And I also studied precalc at 14

[nimble wolf]

Me too, I'm also doing it rn.

[tawny flicker]

I feel so behind

[lapis shuttle]

i started studying calc when i was 12 but i wont actually have calc ab until next year (at 15)

[tender mist]

My school was too cheap in general to have courses at the same time that other schools have. I didn't start having precalc until I was about 15/16, and calc ab at 17

Not throwing shade at my school, just that it wasn't a very big school, it was pretty small, so they had a more focused approach to its classes like that, not a whole lot of variance to what people could take even if they were good enough

Mind you, at the time, I was shit at math anyway, I wasn't able to go any higher than what they gave me, but yeah, they didn't have anything resembling levels of learning really

[tender grove]

I'm in the same boat, taking calc AB rn

As a senior

[silent acorn]

The exam is super easy don’t even sweat it

Well its not easy but its curved very well

[tender grove]

Aight

[normal gale]

help

i cant believe im stuck on the first question

[willow skiff]

you know that's the same as z^12 - 1 = 0 right

oh wait I completely forgot that they overlap with z^4 - 1 = 0 and z^3 - 1 = 0, so start with the first and then take the other roots out

[tender mist]

Definitely. First course in calc is definitely not too bad

[willow skiff]

@willow bear brain not braining

[willow bear]

[cyan quiver]

is pre calc honors hard

like i learned algebra 2 in a week and a half and got an 85 on my exam should i be fine in that class

[arctic swan]

eek

[outer river]

[arctic sedge]

depends on your teacher

[cyan quiver]

ok

[hidden birch]

shouldn't the limit be zero?

sequence will have the pattern $\frac{3^n-2^{n-1}}{3^n}$ which is $1-\frac{2^{n-1}}{3^n}$

[obsidian monolith]

glass

[hidden birch]

maybe I'm wrong

,w infty-infty

,w infty/infty

nvm limit as 1 was right

it's because the denominator goes to infinity faster than the numerator so it approaches zero, ig

i wrongfully presumed infty/infty=1

[astral girder]

anyone want to teach me some coordinate geometry on call?

[willow skiff]

Assume the limit exists as L then L = 2/3 L + 1/3

[glossy heath]

calculus in #precalculus ?

[arctic swan]

i don't really know the difference sorry 😭

[arctic swan]

i didn't really undersyand where 2/3l + 1/3 came from :(

[willow skiff]

so $\lim_{n \to \infty} u_n = L \iff \lim_{n \to \infty} u_{n+1} = L$

[obsidian monolith]

south

[arctic swan]

oh wow what?? this is like a thing?

[willow skiff]

of course the challenge is now to show the limit exists, so:

1. show by induction that $\frac{1}{2} \le u_n \le 1$ for all $n$

2. show that $u_{n + 1} - u_n > 0$

[obsidian monolith]

south

[willow skiff]

yep!

so you have a monotonic sequence (u_n) and the supremum is 1, from these two facts

so the limit exists, and the supremum is attained as L = 2/3 L + 1/3 gives L = 1

you cover this material and terminology in introductory real analysis

[willow skiff]

should be <1 actually but anyways

[arctic swan]

if i get another similar style question can i also apply this?

[willow skiff]

definitely!

[arctic swan]

yayy omg thank u

[willow skiff]

so you can conclude all of this without finding the closed form

of course finding the closed form can be useful

but you don't need it for similar problems

no worries!!

[tired bloom]

its the 2nd question out of 50 this is pushing me out of my comfort zone

[fleet wren]

write it as ∫ 1/[x^2 + (1)^2]
Now using the property given in this image
you can easily get the answer

[tired bloom]

thanks for the help

[fleet wren]

np

This is the derivation incase you need it

[hazy frost]

Any lessons that can help me slowly get into precalc? I do precalc next year

[willow skiff]

KhanAcademy

[compact egret]

bro

okay

[thick blade]

quotient rule i think

[hazy frost]

Ok thx

This one?

[willow skiff]

sure, if that's what you want to cover

yeah don't assume that precalculus is all functions and algebra

it might not be depending on your school, or where you live

[hazy frost]

I won't xd

at least I have a general knowledge of it

[real prism]

f(x)=(4x+-68/6x)/0.4y

Find all possible values of x to turn f(x) = 17

[modest bolt]

what's y?

[echo sinew]

!original

[tender quest]

Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.

[mystic cloud]

can someone give me like links to learn calc again i haven't done this for a few years im trying to teach my self again

[willow bear]

khan academy perhaps?

[hollow plaza]

hello, does anybody know how to do the integral of cosec^2(x) (ik it should be -cotx) but how should I do it without knowing it's meant to be -cotx).

[echo sinew]

Make a u-substitution, such as u = x - π/2

So that you can use the fact that the integral of sec²x is tanx + C

[hollow plaza]

oh right, thx

[hollow plaza]

what about evaluating the integral of sec²x without knowing the answer is tanx?

[echo sinew]

Haven't you been taught that the derivative of tanx is sec²x?

Or have you been taught it to be 1/cos²x? Or maybe 1+tan²x?

These three are all equivalent, if you were wondering

[hollow plaza]

yeah thx i havent learnt that in school I just self learnt it (I just did my GCSE last summer) so I wasn't very sure how to work backwards without knowing the answer as tanx. Thanks

[quasi nova]

im getting this but its sayings its wrong, can somebody help

Increasing on the interval(s): (−2,1)

Decreasing on the interval(s): (−∞,−2) and (1,∞)

[raw hill]

Are you sure 1 is the cutoff

[queen grail]

are horizontal asymptotes only for end behavior or for the entire graph?
it's technically "the line that never touches y = 0 (or another number)"
the function (x+1)/(x^2) touches 0 at x = -1, however the end behavior is that it approaches zero but doesn't touch it

is it still considered a horizontal asymptote despite (-1,0)

[sick sorrel]

Guys im gna go from Alg 2 Accelerated to Calc BC

W or L move? 🔥

[willow skiff]

I said W to you in #prealg-and-algebra

[echo sinew]

"An asymptote is a line that never touches the graph" is a wrong way of conveying roughly what it means, it is a (wrong) simplification that though helps understand what they are, at least when introducing the topic
The right horizontal asymptote is just a horizontal line (say, $y = h$) for the functions satisfying $\lim_{x \to +\infty} f(x) = h \in \mathbb{R}$.
Similarly, $y = k$ is a left horizontal asymptote if $\lim_{x \to +\infty} f(x) = k \in \mathbb{R}$

[hidden birch]

an asymptote is a line that a graph approaches but never touches

-# \infty

-# also for open quotation marks, it's `` (two backticks)

[obsidian monolith]

Alberto Z.

[echo sinew]

Note that you could have:

• only right horizontal asymptote
• only left horizontal asymptote
• both left and right horiz. asymp.
• none of them
  @queen grail

[storm shuttle]

idk if its just me or is asymptote just limits till infinity lol

[queen grail]

i see

thx

[neon steppe]

What actually is a matrix? 👀

Does it represent something?
In 3Blue1Brown's explanation, a matrix is referred to as the transformation of basis vectors, is it? 👀
The columns are the individual basis vectors?
The rows would define what dimension the vectors are in? 🤔

Nvm, I don't fully understand things

Like
In that sense
A matrix is just a collection of vectors right?
Does it mean that a collection of vectors if you were to multiply with a vector would result in that individual vector to be transformed? But hmm
I don't fully process what my own doubt is, It's overall just confusing

[willow skiff]

https://web.evanchen.cc/upload/1802/r03.pdf

in case the 3b1b on matrix multiplication wasn't clear enough ^

the idea is that you just need to see where the basis vectors e1 = (1, 0) and e2 = (0, 1) get transformed to

then if you want to transform (3, 5), that's just 3 * (transformed vector of (1, 0)) + 5 * (transformed vector of (0, 1))

[neon steppe]

Hmm 👀

What happens in case of
[[3, 4] [5, 6]] . [x, y]
Here [x, y] gets transformed right? 🤔

How to actually visualize this? 👀
The result is no longer a direct transformation of [x, y] because the matrix itself has values? So is the result a scaled and transformed version of [x, y]

[neon steppe]

Oh hmm
Actually the formula
So x and y ends up as scalars which multiply the individual vectors and the sum of these is the final transformed vector oh hmm 🤔

[willow skiff]

https://www.youtube.com/watch?v=kYB8IZa5AuE

have you watched this video?

[neon steppe]

Yep 😭

[willow skiff]

you need to think of (x, y) as x * (1, 0) + y * (0, 1)

[willow skiff]

yes, so I think you're beginning to understand

[neon steppe]

Hmm 👀

[neon steppe]

Oh 👀
True 🤔

[willow skiff]

so when you have the matrix [3, 4] [5, 6]]

that tells you that the vector (1, 0) gets transformed to (3, 4)
and (0, 1) goes to (5, 6)
you just look at the columns as you said

so then knowing that (x, y) = x * (1, 0) + y * (0, 1)

[neon steppe]

Oh 👀

Aaah

[willow skiff]

(x, y) gets transformed to x * (3, 4) + y * (5, 6)

[neon steppe]

That's how it works

Thanks

[willow skiff]

= (3x + 5y, 4x + 6y)

yeah no worries!!

[neon steppe]

Hmm hmm
Understood it
For some reason my brain didn't comprehend the idea that the basis vectors could end up transforming into those sorts of values, I didn't pay as much attention to the sheer transformation and others he mentioned, so hmm

[willow skiff]

it's okay a lot of people don't get it the first time either

[neon steppe]

This isn't as related tbh~
But in the mathematics representation of a matrix, the vectors are columns of a matrix 👀
But in CS, Usually you'd take the individual arrays as a row right in a 2D array?
Would this not result in any changes in the results of the two? 🤔

[willow skiff]

the concept should still be the same though

[willow skiff]

through this process you did end up with the rows as the entries

matrix multiplication is matrix multiplication regardless of discipline

[neon steppe]

Hmm 👀
But with the rows and columns being swapped respectively, Would the results remain the same irrelevant?

[neon steppe]

Oh hmm 🤔

[willow skiff]

swapping the rows and the columns gives you the transpose matrix

[neon steppe]

Hmm

[willow skiff]

then you actually have something different, since matrix multiplication is not commutative

$(AB)^t = B^t A^t$

[neon steppe]

Hmm 🤔

[obsidian monolith]

south

[neon steppe]

Then a 2d array defined as [[1, 2] [3, 4]] , the individual vectors would be [1,2] and [3,4] oh right hmm I believe that's how it's taken as 🤔
My bad

[halcyon dragon]

Why do we take the derivative of g(x) in d/dx (fog(x))?

[willow skiff]

the derivative of f(g(x)) must depend on g(x) clearly

so to see why it depends on the 1st derivative of g(x), not g(x), or the integral of g(x) and so on

you could use the limit definition!

$\lim_{h \to 0} \frac{f(g(x + h)) - f(g(x))}{g(x + h) - g(x)} \frac{g(x + h) - g(x)}{h} = ?$

[obsidian monolith]

south

[willow skiff]

(recall the two-point form of the derivative also)

[halcyon dragon]

Thank you so much, i didn't even think about the limit definition

[sick sorrel]

im tryna ask others

[stable vessel]

This function has a minimum at x = 5/4. But what's the maximum? If a quadratic function has a minimum and I'm given an interval for x, the maximum is always at the greatest number in the interval? Here, x = 4?

[summer ruin]

no, the maximum for x^2 on [-2; 1] is not at 1, but at -2

[stable vessel]

at [-2, 1, 0, 1, 2], which one is it?

If the maximum of the function is the number with the greatest absolute value, then I assume it'd be both -2 and 2?

[summer ruin]

not necessarily since the maximum for -x^2 on [-2; 1] happens to be at neither -2, nor 1, but at 0

[stable vessel]

But its minimum would be -2?

[summer ruin]

yes

[stable vessel]

By minimum and maximum here, are we talking about the functions max() and min()?

are they related?

[summer ruin]

we are talking about the argument at which the function takes it's lowest/largest value

[stable vessel]

oh

the maximum or minimum of a quadratic function on an interval m <= x <= n, it'd be either m or n, the one whose output is the largest?

[summer ruin]

no

[stable vessel]

That's because it already has a maximum, so we are concerned about its minimum at a certain interval

[stable vessel]

like the original question, it already has a minimum, now it's to know its maximum at that interval

[summer ruin]

my example also has a minimum

you're suggesting that the maximum should be on the boundary of the interval, but I'm showing how that's not always the case

[stable vessel]

yes, because the functions goes to the infinite

[summer ruin]

the same applies to a minimum too which isn't necessarily located at the boundary, for e. g. x^2 on [-1; 1] the minimum is not on the boundary

[stable vessel]

wouldn't the minimum be either of them?

actually, minimum is 0?

cus 0 gives the smallest value

[summer ruin]

yes

[stable vessel]

so how is it not on the boundary?

[summer ruin]

because 0 is neither 1 nor -1

[stable vessel]

but it is inside the interval [-1, 1]

[summer ruin]

inside is exactly what it means to be not on the boundary

[stable vessel]

Therefore, minimum or maximum are given by the number x in the interval such that it outputs either the smallest or biggest value?

[summer ruin]

yes

[stable vessel]

but

let's now consider f(x) = x², x in [-2, 2]. It has a minimum of 0, now we wonder about the maximum. The maximum here is either of the boundaries?

[summer ruin]

yes

[stable vessel]

I see, thank you

[karmic brook]

for first principles when given f(x) and asked to find f'(x) and you use the notation in the equation f'(x) = lim h-> 0 f(x+h)- f(x)/h. when given y= something and asked to do find dy/dx (using first principles) do i use the same notation but like dy/dx = lim h-> 0 f(x+h)- f(x)/h, i know it would give the same answer but is the notation correct?

should i state that y=f(x) in my working out before substituting in the formula?

[neon steppe]

Every vector is a linear combination of the basis vectors and the coordinate scalars, Right?

You can also obtain any given vector through the linear transformation of basis vectors right?

So given a matrix, are we assuming that the given matrix is the linear transformation of its basis vectors? 👀

Any matrix can be seen as a linear transformation of basis vectors, suppose we were to multiply this transformation with a vector, It gives the transformation of that vector? 🤔
Whatever movement happens with the basis vectors happen to this vector as well? 👀
v = ai + bj
T(v) = a T(i) + b T(j)
So i and j are the basis vectors, the transformed versions are the matrix elements, ah so hmm
we obtain the transformation of the vector ends up as a linear combination of the scaled transformations of the basis vectors? 🤔
Apologies for this, I am not exactly sure what I am saying either but trying to visualize things

[neon steppe]

Basis vectors can be linearly combined to obtain any given vector in the space, You can obtain any given vector by changing the scalars 🤔

The properties of linear transformation mentioned in the Khan Academy video~
Combining the two you get what happens to the scalar 🤔

T(a+b) = T(a) + T(b)
T(ca) = c T(a)

v = ai + bj
T(v) = a T(i) + b T(j)

A transformation of a vector is the sum of the transformation of the basis vectors multiplied by the coordinate scalars

Naruhudo
That's fine 🤔
Sorry, I am repeating the same thing

I see, it makes sense now, It's easier to see why it is the way it is 🤧

Though feels like I went around in circles to reach the obvious stated by the formula

[neon steppe]

In matrix multiplication,
Is the final result always the transformation of the vectors on the right side of the operation, So we're always just visualizing the left side to be the transformation of the basis vectors which then can be used to transform the vectors on the right side of the operation 🤔

We have the rule that the number of columns of the left matrix has to meet the number of columns of the right matrix for matrix multiplication
Hmm
In this context, the reasoning for that is because to transform a 3 dimensional vector we'd need a basis vector which corresponds to the transformation in each dimension,
Each column being a basis vector, each row being a dimension

That makes more sense now as well 🤔

[obtuse island]

Hey guys do you think it’s reasonable to try to keep up with ALL types of problems at all times and the ways to solve them? Or is it normal to be able to solve the ones that you’re working on currently and actively studying in class but like problems from last year or semester need looks at notes bc you don’t know anymore?

Is this something I should be working on as I go thru precalc?

Any advise appreciated

I’m def someone who doesn’t try to keep unnecessary info until I need it but I’m open to changing my methods

[hidden birch]

it's normal to forget

and look at notes

the important thing is you understand the concept

[obtuse island]

Okay thx! :)

[viscid thistle]

integration by parts

sir

[willow skiff]

#calculus

not this channel pls

[viscid thistle]

ok ok si

[vernal quartz]

Am i missing anything here in? Can i optimize anything?

[viscid thistle]

Hello, I’m taking precalc in high school. I wanna know how I can excel in this course especially since it’s complex and I just want to take any tips from y’all

[shy vine]

lost of practice

[copper gyro]

Hi I’m currently in calc 2 but we’re reviewing precalc, how do I solve limits like this, where its a rational function with x raised to some power? Do I rewrite it in some form of ln? Kinda stuck here with what I can “legally” do

[copper gyro]

lowk I think its 0 because the denominator would grow faster than the numerator, but idk if thats a mathematically sufficient explanation. lmk what yall think

[copper gyro]

looks like (3,6) because it is going physically up but at a slower rate (its curving down), rather than some other spot like (18,30) where it is going physically up at an increasing rate

Wait new question bc what the heck can I do here

[low torrent]

I’m not in that yet

[still brook]

doesnt it include all y values??

[heavy pewter]

?

[still brook]

[5, 1)

[still brook]

does it not include all the y. alues?

[heavy pewter]

it does

[still brook]

then why cant it be written as [5, -4, 0, 1]?

help

[heavy pewter]

Because its not discrete

[still brook]

what does discrete mean

[heavy pewter]

As in only containing integers

[still brook]

?

[heavy pewter]

bro look it contains 0,5 so obviously this is insufficient

[still brook]

what

[heavy pewter]

just look up interval notation

it'll make more sense then

[still brook]

yall are supposed to help not give me resources smh

[copper gyro]

that is beautiful and you are the nerdiest nerd of all nerds

i aspire to be like you

i ended up just graphing it and visually seeing it approaches zero, i'll see what my teacher says tmrw

[quasi tartan]

i need help with my graphing pls

[royal pagoda]

Does this explanation work?

You can skip to the line just above let x_1 = 0

[daring dew]

can someone help me with math

i’m in pre calculus in 9th grade

[regal tree]

!da2a

[tender quest]

No need to ask “Can I ask…?” or “Does anyone know about…?”—it’s faster for everyone if you just ask your question! See https://dontasktoask.com/

[steel kernel]

Why the second derivative can determine if a function is a concave or convex?

[tulip hedge]

is there a resources channel or a resources webpage, I want to test my knowladge on exponents by solving problems

[echo sinew]

https://letmegooglethat.com/?q=problems+with+exponents+exercises

[tulip hedge]

tysm :>

wait-

that just mean

[fervent forge]

Can't we just differentiate it and equate to zero?

[fervent forge]

Where?

[willow skiff]

yes

[willow skiff]

you're making it harder for yourself by using both $x - a$ and $x + a$

[obsidian monolith]

south

[willow skiff]

,w -5(x+a)^2 + 40(x+a) + 1 - (-5x^2 + 40x + 1) simplify

[obsidian monolith]

Results provided by WolframAlpha

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

Requested by sooouth.

[willow skiff]

so you want -5a(-a - 2x + 8) = 0 for the sign of this to change from positive to negative

since a is allowed to be any (positive) real number

you can't set a = 0, so -a - 2x + 8 = 0

but look what happens if you take the limit of -a + 2x + 8 as a goes to 0

as you bring the two points closer and closer together

you just have -2x + 8 = 0 or x = 4 as the limiting case, so the two points are the same and that's when the function changes from increasing to decreasing

by overcomplicating it you're forcing yourself to include unnecessary assumptions and it just becomes a big mess

[royal pagoda]

Ohh right obviously thank you

[willow skiff]

No worries!

[astral apex]

because the second derivative encodes the rate of change of slopes of tangent lines to a curve

if tangent line slopes are increasing, we call it concave up; if decreasing, we call it concave down

it's best not to speak of convex in this context

convex in math means something else, and a convex shape can have pieces that are either concave up or concave down (or both)

for example a circle is convex, and the upper half is concave down, lower half is concave up

here I'm using convex to mean "the shape contains all line segments between its points"

[echo sinew]

I thought convex meant concave up

I mean, not the convex things in topology, like convex hull and so on

[astral apex]

this appears to be one of those quirks that some teachers use and others dont

[echo sinew]

It might also be something about language because for example where I live (Italy) we don't have any transnation of concave up and concave down

We use the words convex (for concave up) and concave (for concave down)

[ripe eagle]

anyone free to help me with 15 pre calc questions before my exam? just trying to study the problem solving steps

[latent flame]

can u show it

[ripe eagle]

they

they're on a few diff topics all within similar concepts etc graphs of circles,parabolas.hyperbolas,asymptotes,functions,piece-wise functions,absolute value functions,inverse function,difference quotient

[latent flame]

can i see some of the problems

[latent flame]

is the ellipse one E?

[ripe eagle]

one E?

[latent flame]

(0, -4) (0, 4)

the graph that counts as a function is figure c yeh?

cus every domain has a range matching it

[ripe eagle]

yeah

[latent flame]

oh i was right on that one too

try the number before it

is the focii 0,-4 and 0,4

oh wait im sorry

im just bad LMAO

[ripe eagle]

LOOL

ur okay

we are all in this together

thanks 4 trying

[latent flame]

LMFAOAOAOO

[proven willow]

any tips for calculus i never took it before

[shy vine]

practice

and make sure you understadn algebra

[pseudo summit]

could someone help me solve the domain for gof? The
square roots confused me😭

[modest bolt]

• Find domain of g.
• Then find range of f.
• Then see which part of that range lies inside domain of g.
• Finally, collect all x from domain of f for which f(x) falls in that overlap.

You can start by first step and tell us what domain of g you got and continue like this and if you get stuck anywhere then we can help.

||hint: for domain of √(ax+b), you have to find solutions to ax+b >= 0||

[pseudo summit]

Thank you! Someone helped me already! I will notice the steps down and try a smiler problem later!

[hidden birch]

or just inspect the new function formed by f(g(x))

[hidden birch]

what answer do you get

because your final steps is confusing to me. The answer i got is

,texsp||$-2-\sqrt5\le x\le-2+\sqrt5$||

[obsidian monolith]

glass

[hidden birch]

and graphing it on desmos confirms it

oh nvm i get your steps now

[fluid olive]

I want to start calculus. I know nothing about it. Where do i start?

[acoustic sparrow]

prerequisite: precalculus

you can do the khanacademy course for calculus I

khanacademy, professor dave

or use a book

[fluid olive]

prerequisite?

[acoustic sparrow]

yes

you should know all of precalculus(matrices and determinants are optional) before starting with calculus

[inner hawk]

https://tenor.com/view/scream-if-math-calculus-nerd-angry-gif-1020184934245494172

[inner hawk]

integral (e^root(x) dx)

😀

[willow skiff]

oh yeah, sub then IBP

[red storm]

integral e^dx

[chrome acorn]

khan acad right?

[willow skiff]

no

it's some online learning platform that schools and unis use

I know for a fact it isn't Canvas

I think it's Blackboard?

[chrome acorn]

[plush nymph]

can someone please help me with a precal assignment

I will literally pay you I need this done asap and i'm really stuggling

[heady bluff]

I need help. I don't know how to solve this backwards from the output to figure out what input(s) could've given the output of 3..

[polar marsh]

equate p² + 2p to 3, p²+ 2p = 3, p² + 2p - 3 = 0

[modest bolt]

you also think of it as p(p+2)=3
So what two numbers where one number is 2 unit larger than the other multiply to give 3?

3 has factors ±1,±3 and these are indeed 2 units apart from each other, so you can check them and guess what 1×3=3, or p=1 gives 1(1+2)=3. If you are doing it this way, I want you think and try the other solution for p.

[heady bluff]

Oh ok

[regal tree]

id much more recommend solving the quadratic that favlic made. you wouldnt always find nice, easy solutions which you could think of the way you did

[proper eagle]

https://youtu.be/beHkSchC_9s?si=wEeGQPpQNPJfGGnY

[gleaming matrix]

how are you guys learning pre-calc?

[tawny plank]

use stewart pre calc textbook

[tawdry arrow]

I like math

It is very important

[civic thicket]

why is calculus so hard 😭

[echo sinew]

Probably because you don't have a full understanding of your prerequisites

At least this is very common, I don't know if this is your case as well

[inner hawk]

🕊️

[willow skiff]

90% of all calculus struggles are caused by not fully understanding the material before calculus

[civic thicket]

no

not that

i just started it

it is extremely different from precalc and algebra

[willow skiff]

but you would agree, that you are constantly using concepts and techniques from these in calculus right?

[civic thicket]

no. its not the problems in struggling in. its understanding what it is. like i dont even understand it

[willow skiff]

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

I recommend this video series then

[civic thicket]

ty

[willow skiff]

no worries

and sorry for misunderstanding what you were asking!

[civic thicket]

nah its alr

wyd rn in math?

[willow skiff]

I'm actually not studying maths at uni anymore

the highest course which I attempted and passes was topology

[civic thicket]

ohh

[sonic loom]

is trig identities hard?

gon study soon

[trim flax]

depends what level of trig, like sin^2 + cos^2 =1 is light but inverse trig is not nice

[winged wigeon]

Derivatives inverse trig

a^2 = b^2 + c^2-2bccosA

Guys I still have it remembered right

Laws of cosines

[raw hill]

Yes

[https://en.m.wikipedia.org/wiki/Law_of_cosines]

You can also look up facts like this yourself btw

[viscid thistle]

who can speedrun calculus wi me (hw) 🙁

[celest lantern]

I think it’s also because it’s late at night but I’m so confused

Please help

[modest bolt]

All the functions are of form b^x±c

[modest bolt]

You have to add or subtract certain number 'c' from every output and see if it forms a proportional growth

for example in 1st case, adding 1 in all outputs yields 1, 3, 9, 27, 81. Notice that they are the powers of 3 as 3⁰,3¹, 3², 3³, 3⁴. So function is g(x) = 3^x - 1

Similarly try to add or subtract small numbers like 1,2,3 in 2nd and 3rd case and try to see if you notice the outputs forming powers of any number as we saw 1st one formed powers of 3

@celest lantern

[timid widget]

can anyone teach me binomial expansion using the theorem? in pre calculus. im having alot of trouble not understanding it.. like i do but only for like 5 seconds then i forgot or something always changes

[hidden birch]

pascal's triangle

[brisk crystal]

i know it's not proprely pre-calculus but i have problems in fully understanding the path of resolution of limits. I think that i understand the concept of limit but when i have to calculate it i lost the right path to follow. For example how can i know wich limit formula i have to use just by seeing what the function is?

[modest bolt]

,tex this one?\
[
(x+a)^{n} = \sum_{k=0}^{n} {n \choose k}, a^{n-k} x^{k}, \qquad n \in \mathbb{N}.
]

[celest lantern]

Thank you!

[viscid thistle]

You could also try professor Leonard on YouTube

[rigid ridge]

would appreciate any help here

I do online school and i got a really bad math teacher this year who has been horrendus

[quick jungle]

Precalc ch 1 test on Thursday

I heard that there will be quadratic word problems

But I need like a couple examples

On how they work

[wind holly]

Indicate the error

[rocky wind]

,rccw

[queen grail]

my teacher keeps saying a point can be a hole and a zero at the same time in a rational function but google says otherwise 😭

[rocky wind]

please turn it to the correct orientation, so that PC users can view them more easily

[wind holly]

For me, a point is the same as a multiplication signal(* or ×)

[uncut mulch]

do you have what they said written down exactly

or are you paraphrasing

[wind holly]

Wdym

[uncut mulch]

not u

[queen grail]

a point on the graph i meant

[queen grail]

we were solving for zeros/asymptotes/holes
she said that a point can't be a vertical asymptote and hole at the same time but 5 seconds later she said a point can be a zero and hole at the same

[uncut mulch]

did she give a specific example

[queen grail]

she showed the rational function ((x+1)(x-2)^2)/((x-2)(x+1)^2)
and said there's a hole at (2,0) and also a zero at x = 2

[uncut mulch]

ok, yeh she's wrong

[queen grail]

which doesn't make sense to me it wouldn't be touching the x axi

[queen grail]

ok

[uncut mulch]

there'd be a hole

f(2) is undefined, not 0

[queen grail]

i understand but I'm scared cuz idk if she'll let points be a zero and a hole at the same time for the test 😭

[uncut mulch]

bring it up before the test

[queen grail]

ok

thx

[eternal parcel]

can anyone help me with this

[raw hill]

Reposted: #help-45 message

[hasty heron]

To find dy/dx do i multiply by yx ?

[inner hawk]

use u/v on both terms and then transpose everything but dy/dx

[willow bear]

,rcw

[willow bear]

multiplying by xy before taking the derivative is a smart move

[inner hawk]

dy/dx = (x^2 . y - y^3) / (x^3 - y^2. x) ?

[hasty heron]

It will be x^2+y^2 = A i guess ? In the end its x/-y which is the same as y/x if im not wrong

[willow bear]

no

x^2+y^2 = Axy

[inner hawk]

how

[willow bear]

i did say to multiply BOTH SIDES by xy

oh no

i am a fool and didn't explicitly mention "both sides"

[inner hawk]

😂

[hasty heron]

Im a fool for forgetting that 😕

[hasty heron]

thanks

[viscid thistle]

im in last year of highschool and i have no idea about calculus and trignometry(inverse trig and the identities only),what respurces you guys use for learning these subjects?

its my additional subject and i just wanna pass in it

[uncut mulch]

khanacademy can get you started

and a few mathtubers

like organic chem tutor and professor leonard

[hasty heron]

as a student like you i guess just solve. diving deep into it will only give birth to new questions and curisoity .

[viscid thistle]

yep that's one of my habit of going in depth

you are right

[hasty heron]

dont like dive deep but get a superficial understanding for what it means , but it really doesnt matter in an exam

[viscid thistle]

i have kickstarted trigonometry journey from khan academy recently

thank you @uncut mulch and @hasty heron for your advice!

[left crypt]

guys can you suggest me resources where I can study and understand trignometry more better. Like the solutions and identites . I face a lot of problem in these. I fail to understand the approach . Please help if u can

[red storm]

the organic chemistry tutor

short videos

[vagrant lichen]

https://tenor.com/view/fire-writing-gif-24533171

Why does 6x sqrt of 3x become 6 sqrt of x

[viscid thistle]

The previous step is wrong
It's just 6sqrt(x)-10xsqrt(3)

[vagrant lichen]

Wait what did I do wrong

[vagrant lichen]

Is it 6 sqrt 3x

[viscid thistle]

2sqrt(9x) is supposed to be?

[vagrant lichen]

Wait I don’t know

Since its a square root

I did 3 x3

And there’s a 2 on the coefficient

So multiply by 3 x 2

[viscid thistle]

This step is right

[vagrant lichen]

Get 6 sqrt of 3x

[viscid thistle]

I'm asking what will be get by simplifying 2sqrt(9x)

[vagrant lichen]

This

I’m confused with the variable

Like what should I do when the variable isn’t a square

[viscid thistle]

Well just rewrite it as 2sqrt(9)sqrt(x)

[vagrant lichen]

oh wait

now i see the vision

it becomes 6 sqrt 3 and

sqrt of x

[viscid thistle]

What's sqrt(9) again?

[vagrant lichen]

3

[viscid thistle]

Yes then why would we have a sqrt(3) left over?

[vagrant lichen]

oh i did 3 x 3

i didn't notice that i was just a sqrt

[viscid thistle]

So what do we get in the end?

[vagrant lichen]

what happens to the x

[viscid thistle]

Nothing it stays In the square root so we have
6sqrt(x)

[vagrant lichen]

wait

so sqrt of 9 is 3

and the 3 gets multiplied by the coefficient

2?

[viscid thistle]

Yes

[shut rampart]

i’m wondering if this is what i should try to practice a lot with active recall for cal 2 (integral calculus i think it’s called)

i’m also wondering any tips for learning most of this in one semester

[vagrant lichen]

u can't

[vagrant lichen]

ur cooked

[red storm]

series and sequences as well

[silent plaza]

Bro is practicing Trigonometry in order to learn calculus😂

I mean yeah, trigonometry is one of the contributors that makes calculus hard

[silent plaza]

Learn of how to derive them bro instead of random ahh memorizations

[modest bolt]

I'll be solving calculus question easily then comes cosA cosB and I am done cause I don't remember the formula

[silent plaza]

Yeah bro that's the problem, y'all do be memorizing everything

Learn the concept and learn of how to derive them formulas

For example, deriving the power reducing formula. Suppose we have :

cos²(x)

Then we want to reduce its power, we know that :

cos²(x) + sin²(x) = 1

(Derived from the equation of the Unit Circle x²+y² = 1, and x = cos(x), y = sin(x)), then we want to isolate sin²(x), so :

cos²(x)+sin²(x) = 1
sin²(x) = 1-cos²(x)

Then we also know that the double angle formula for cosine is this following :
(Derived from compound angles formula that requires geometry for its derivation)

cos(2x) = cos²(x) - sin²(x)

We know that sin²(x) = 1-cos²(x), so substitute it in, we get :

cos(2x) = cos²(x) - (1-cos²(x))
cos(2x) = cos²(x) -1 + cos²(x)
cos(2x) = 2cos²(x) - 1
cos(2x)+1 = 2cos²(x)
(cos(2x)+1)/2 = cos²(x)

Therefore the reduced power of cos²(x) is :

(cos(2x) + 1)/2

[inner hawk]

damn 💀

[shut rampart]

how do you get stuff like $\int \frac{1}{u^2 + 1},du = \arctan(u) + C$

[obsidian monolith]

Jelly

[silent plaza]

Draw a triangle

With =

Hypotenuse = √(u² + 1)
Opposite = u
Adjacent = 1
Angle = n

Then let tan(n) = u/1

d/du (tan(n)) = d/du (u)
sec²(n)dn/du = 1
sec²(n)dn = du

Then substitute them in

∫ 1/(u²+1) du
∫ 1/(tan²(n)+1) sec²(n)dn
∫ 1/sec²(n) · sec²(n) dn
∫ 1 dn
n + c

Since tan(n) = u
We can just do some algebra so that we get

n = arctan(u)

So the answer is

arctan(u) + c

[shut rampart]

i will try to investigate this today thank you very much!

[silent plaza]

Yeah, wait i'll give you a video about it

I learned this from a youtube video honestly

[shut rampart]

yeah it seems like my teacher assumes everybody just knows but i’ve spoken with other classmates and no one seems to know lol

[silent plaza]

https://youtu.be/YAGakg39bgw?si=DVe50ENOCIbHNwr7

Here

[silent plaza]

And now you don't need to memorize shits, good luck👍

[viscid thistle]

guys how would you solve this question? Ik you need to get the formula of Sn = n/2[2a1 + (n-1)d] but how do you do ahead of that?

[inner hawk]

i got n=11

[inner hawk]

the first summation simplified is :
12 (a + 11d) = -72a/5 , take a+11d = a12 and simplify
final is a+5d = 0 --> d= -a/5

now use sum of series on 2nd summation to get a= d (1-n)/2
substitute value of d in this eqn a will cancel out on both sides and n=11

https://tenor.com/view/wake-up-gif-12100055351253481367

[storm shuttle]

try to find some kind of relation b/w frst term and common difference

[acoustic rover]

Hellooo

Is the AoPS book or Jame's Stewart's book better as a foundation for calculus

I am also taking AP Calculus btw^

[dreamy canopy]

hey guys i am having a reallyy bad time solving the question 41 can someone help pls??

[fluid sorrel]

Just apply L'hospital rule

And u will get ur answer in 2 steps

Do u know this rule?

[ripe eagle]

can not for the life of me understand this

[modest bolt]

did you find fog?

[ripe eagle]

i get stuck after i fill it all in as in i don’t know the next steps to take i might just be missing basic algebra etc

[modest bolt]

good, now simplify that

[modest bolt]

you can use brackets () [] {} to write it neatly

[ripe eagle]

that’s where i get stuck i feel like im missing out on the basics of trig and algebra, i tested into a accel pre calc course and i didn’t have the option to refuse it

any yt vids or stuff you suggest to fill that in?

i can understand the concept it’s just so hard to apply when missing parts like this

[modest bolt]

try to simplify numerator and denominator seperately

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

how would you simply this?

can you think of your first step?

You know you can write 1 as 1/1 then take LCM to get a common denominator

[ripe eagle]

i’d group like terms first apply the 2

2x+6/(3x-2) -1 i would then find a common factor like 6 to cancel them out then do 6/(-2)-1 turns to positive 6/2 =3

so x = 3

[modest bolt]

no no, you are not supposed to find value of x

How would you simply
a/b + c/d?

to get a common denominator?

[ripe eagle]

i have no idea tbh i never learned that

[obsidian monolith]

$\frac{a}{b}$ + $\frac{c}{d}$ = $\frac{ad-bc}{bd}$

[modest bolt]

do you understood how this happened?

[ripe eagle]

cross multiply denom and numerator

no?

[modest bolt]

oops i meant to add + sign there on right

[modest bolt]

yes

[modest bolt]

you have to do the same here

a = 2(x+3)
b=(3x-2)
c=1
d=1

you think you can cross multiply numerator like that?

[modest bolt]

from here

[ripe eagle]

no cause i mean anything divided by 1 is itself i mean you can but i dont see how it would move the problem forward

[modest bolt]

$\frac{2(x+3)}{(3x-2)}$ + $\frac{1}{1}$ = ?

[modest bolt]

you have to cross multiply to simply it

this is the numerator you got from fog

you have to simplify numerator and denominator of fog seperately then divide them and simplify further to obtain proper expression for fog

[ripe eagle]

do i cancel them out and leave myself with 2x+6? or group the terms then divide?

[modest bolt]

i said I mistakenly put that minis sign there

[ripe eagle]

ah

[modest bolt]

your numerator has minus

yes good

[modest bolt]

$\frac{2(x+3)}{(3x-2)}$ - $\frac{1}{1}$ = ?

[modest bolt]

It was my mistake I didn't notice, there needs to be a minus sign here like this

this is the numerator you got in your photo

now cross multiply and what you get?

[ripe eagle]

cross multiply this ? or the pic i sent

[ripe eagle]

no worries thank you for the help

[modest bolt]

here look at the numerator, it's the same thing shown in the image above

[left crypt]

hey guys sry to interrupt but whats the question asking

[fluid sorrel]

https://tenor.com/view/deep-breaths-efe-omowale-assisted-living-s3e21-inhale-exhale-gif-4389370785556430183

[ripe eagle]

trying to understand my pre calc i get lost i got into accel pre calc and i don’t know how

[fluid sorrel]

It's not calculus... Just put value of gx in place of x in fx

That's algebra

Basic

Basickkkk

I know it's calculus

But after substituting

[ripe eagle]

i understand that, i do not have good trig or algebra foundations i asked for help in where to understand those topics or etc

[fluid sorrel]

It's algebra

[modest bolt]

this is precalculus channel

[left crypt]

ya for u and me bro for ohers it isnt

[fluid sorrel]

Kryptonian my brother

[fluid sorrel]

I got eyes

That shoot laser

[left crypt]

just saw ur profile pic lost brother

[modest bolt]

well yes the question is about precalculus but idk why I need to teach him algebra to start the question

[fluid sorrel]

Ur not helping him.... Just write it n show him n save ur as well as his time

[modest bolt]

ok ok

[left crypt]

ye

[fluid sorrel]

https://tenor.com/view/beat-chest-bro-max-greenfield-new-girl-schmidt-gif-6481396707897306285

[ripe eagle]

it’s pre calc hw, i get lost in the foundation steps not really the concept

[fluid sorrel]

😭 sorry for yaa bro

[ripe eagle]

that being said do you guys have good online resources you can recommend me for those concepts?

no worries

[fluid sorrel]

English???

[ripe eagle]

where can i learn

good

[left crypt]

khan academy ig

[fluid sorrel]

https://youtube.com/@mrhtutoring?feature=shared

[viscid thistle]

thanks

[viscid thistle]

ohk thank u

[shut rampart]

hello i watched the video you linked and read what you said but i don’t understand how do you identify which is which side for the triangle.

how do you pick opposite as u instead of 1 or vice versa? hypotenuse is easy in this case bc we add but also how do we know when we do the sqrt( x^2 - a^2)

[quiet river]

How to begin this?

[astral apex]

I would regard the equation as a quadratic in x

i.e. treat everything except for x as a constant

I have no idea if that's wise

wait, I misread what we're trying to do

I think you just start by working with ðf/ðx = 0, show that it leads to the equation written next to it

and similarly for the one under that

maybe you also need to say something about why setting those partials to 0 is the right thing to do

[silent plaza]

You can pick opposite or adjacent as 1 or u, but for consistency purposes always pick dummy variable as the opposite, if the dummy variable is on the hypotenuse, that means the square root stuff goes onto the opposite side, and the constant on the adjacent.

For example, suppose you have :

√(x² - 1)

Hypotenuse = x
Opposite = √(x² - 1)
Adjacent = 1
Angle = θ

I think it's the standard triangle configuration

Here is what happens if you pick constant as the opposite :

Suppose you have this integral :

∫ 1/(1+x²) dx

Then you draw this triangle :

Hypotenuse = √(x²+1)
Opposite = 1
Adjacent = x
Angle = u

Then let,
cot(u) = x/1
d/dx [cot(u)] = d/dx [x]
-csc²(u)du/dx = 1
-csc²(u)du = dx

Then let,
csc(u) = √(x²+1)
Solve for x²+1
csc²(u) = x²+1

Plug them in :

∫ 1/(x²+1) dx
∫ (1/csc²(u)) · (-csc²(u))du
-∫ csc²(u)/csc²(u) du
-∫ 1 du
-u + c

u = arcot(x)
or
u = arccsc(√(x²+1))

Therefore the answer is :

-arccot(x) + c
or
-arccsc(√(x²+1)) + c

I think they are valid answers but not standard.

Lemme work things out with :

-arccot(x) + c
We know this identitiy :

arccot(x) + arctan(x) = π/2

Letme proof this identity,
Let,
arccot(x) = u
x = cot(u)

from the cofunction identity, we can convert the cot function, so :

x = cot(u)
x = tan(π/2 - u)
arctan(x) = π/2 - u

And we know that u = arccot(x), let substitute them in :

arccot(x) + arctan(x) = π/2
arccot(x) + (π/2 - u) = π/2
arccot(x) + π/2 -arccot(x) = π/2
π/2 = π/2

Now we know that it's true, we can solve the equation for arccot(x), so :

arccot(x) + arctan(x) = π/2
arccot(x) = π/2 - arctan(x)

Substitute them in, we get :

-arccot(x) + c
-(π/2 - arctan(x)) + c
-π/2 + arctan(x) + c
And we know that -π/2 is just a constant, then the c can just absorb that shit up and we get our good ol' standard answer

arctan(x) + c

I think the -arccot(x) + c answer is just the shifted version of the standard answer.

[sinful estuary]

hey guys i have a precalc placement test and i need to ace it, does anyone have a good practice test or/and what major concepts i should go over?

[willow skiff]

try Khan Academy's algebra 1 and 2 quizzes