precalculus

[whole void]

the greatest integer less than or equal to 3.87 is 3

[viscid thistle]

Like the value would be 2 for 2.9 rather than 3?

[whole void]

yes

the floor of 2.999999994 is 2

[viscid thistle]

Yeah I got it thanks

[cloud wadi]

Its all got to do with the function property like you can notice it for yourself if you compare the calculations with the function graph

It rounds down so to speak because from open 2 to closed 3 the function is a horizontal line (y=2)

[viscid thistle]

I don't really understand what u meant 😅

[cloud wadi]

The graph (the image)

If you observe it at the value you want to calculate it

It starts to make sense

From zero to 1 is zero

From 1 to 2 is 1

Its called a step function because its made up of small honrizontal lines

That go up a certain integer at a time

[viscid thistle]

Yeah but this kind of graph is also possible for other functions right?

Like other examples of step function instead of the greatest integer function

[cloud wadi]

Step function IS a greatest integer function

[lament ferry]

Do you need deep understanding of algebra to start calculus

[willow skiff]

No

You should have a reasonable understanding of algebra to not make silly mistakes / misconceptions such as that sin(x + y) is the same as sin x + sin y

If you find you don't understand a computation in calculus, you probably need to go back and revise that section of algebra

see how you go

[lament ferry]

So what do i need to undrstand before calculus

Trigonometry?

[willow skiff]

yes, and

logarithms/exponential functions, rational functions

umm what else

Function transformations in general are super important

factor/remainder theorem, long division (if you have time I recommend learning synthetic division also)

discriminants of quadratics

Ah coordinate geometry, so tangent lines, normal lines

point-slope form

[lament ferry]

What about deivatives

[hushed sphinx]

I think a good familiarity with the concept of function is the most important.
The standard functions show up in a lot of examples (and exercises), but their particular properties are not the central stumbling block.

Derivatives are calculus.

[willow skiff]

binomial expansions

if you have time also learning basic summation notation will come in handy

and prob and stats and matrix algebra are other things you could be learning at this stage, though not directly relevant for calculus

[lament ferry]

What is even calculus

[willow skiff]

the study of how functions change

so differentiation is about the slope of a function if the slope is constantly changing

Integration is the (signed) area under a curve if that area is not changing linearly

So the area of a triangle is easy

The area under a parabola, harder

[lament ferry]

Is phytagorian theorem made by calculus or just proven

[hushed sphinx]

That whole laundry list begins too look a lot like "deep understanding of (high school) algebra" ...

[willow skiff]

Depends on what you mean by 'deep'

[hushed sphinx]

The Pythagorean theorem is geometry and not particularly connected to calculus.

[willow skiff]

But if you're learning calculus you should know the Pythagorean theorem

And you should know how to apply it to the distance between two points

otherwise you'll trip up later

[lament ferry]

Why is calculus common used in geometry

[willow skiff]

Like maybe not all of those things will be immediately useful, but imagine you're studying calc 2 and you have no idea about how to write (1 + 2x)^4 in terms of a constant, x, x^2 and so on

And then your first introduction to this topic is infinite series, and Taylor series

I'm not asking them to memorise the binomial formula

It's going to be on some formula sheet anyways

[hushed sphinx]

I dunno, it sounds like you're setting the bar very high. Yes, all of these things find some kind of application in calculus exercises, but it's not like one needs to have them all super memorized before one can even start learning what calculus is about.

[willow skiff]

true

But I would expect a student to have a basic understanding of these things by the time they start calculus, I mean I don't think I'm being too unreasaonable

Of course if you're self-studying then you can learn whatever you want at whatever time

And I guess I assume that students learn calculus at roughly the same mathematics level

IDK some topics in there are much more useful than others, but the point is to not have a too limited view of mathematics at this stage

You don't need to spend hundreds of hours practicing every single topic in a lot of detail

[lament ferry]

I do not memorizing i learning with understanding, beacuse deep understanding is important

[willow skiff]

It's more about being familiar with what mathematics is out there at this level

Dweck says that facts come before skills, in other words you should know about what something is before you apply it

[lament ferry]

So, i just need to learn until im fell ready for calculus

How i understand.

[willow skiff]

You could technically start now but yeah, there might be gaps in your knowledge that prevent you from studying other concepts in calculus later on

The idea of the derivative being what the value of the slope approaches as you bring two points closer together

[lament ferry]

Ok that is helpful thanks.

I will start

[willow skiff]

https://www.youtube.com/watch?v=mwjvt0UEPQ0

Yeah start here

You could start with integration also (Riemann sums or rectangular sums is what you need to search for)

no worries

yeah honestly study how you like man, cause there's no one order you have to study maths in

like, wanting to know more about calculus can be the motivation for learning those topics before calculus

[viscid thistle]

I just watched 3blue1brown the essence of calculus and searched my way into understanding it

[willow skiff]

Ayyy good rec

[viscid thistle]

also, https://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx for practice and more deep information

[lean nest]

this is like so cool

[jovial lodge]

Quick question why can we just take out the 9 from the integral

[hallow tide]

properties assume convergence; more specifically property 5 is the 2nd fundamental theorem

[winter comet]

it's an infinite series, you can factor it out

[jovial lodge]

What are those symbols at bi) and why do I need to write them

And also for bii and biii

I don’t rly understand when to use those symbols nad when to write that x≠0 and the range and stuff

[untold sun]

For bi), It represents a condition for another function, g(x). It states that 1 is less than or equal to g(x), which is less than or equal to 3. Additionally, g(x) is an element of the real number set.

[jovial lodge]

So what does that weird e thing mean

And that weird R means it’s a real number?

[untold sun]

Apologies for the late reply, it means is an element of/belongs to
It is used in set theory

[austere hazel]

can someone explain sin (x/2) = cos (x/2)

[chrome ether]

how to solve it?

bc that equality does not hold in general

[austere hazel]

yes

[chrome ether]

ok so first we use the cofunction identity sin(x)=cos(pi/2-x)

(to see why this identity is true, either look at a right triangle or consider what happens when you shift a sine or cosine graph)

now we have cosines on both sides, what can we do to get rid of them?

[uncut mulch]

less tedious to divide by cos(x/2) and use tan

[chrome ether]

ah shit that would probably work too^

[uncut mulch]

esp if you want the general solution

[vapid plaza]

No, the set of rational numbers is written as Q. R for the real numbers is correct

[willow skiff]

the names are taken from German where there were many set theorists

no wait, Italian, Peano was Italian

quoziente for quotient

but Z for Zahlen does come from German

[alpine bolt]

exactly - it's called blackboard bold and there are a bunch of sets of numbers that have their own corresponding letter: r is real, q is rational, n is natural, z is integer, c is complex, and u is the universal set, which represents all possible numbers

but the most common are probably r and z, at least from what i've dealt with

[manic wasp]

this should help a bit

[stiff sluice]

I wanted to ask

$e^{k+4} > 1$ for $k \in \mathbb{R}$

[obsidian monolith]

938c2cc0dcc05f2b68c4287040cfcf71

[stiff sluice]

how do I find them k?

[willow skiff]

what's e^0?

[stiff sluice]

1

[willow skiff]

exactly

So $e^{k + 4} > e^0 \implies k + 4 > 0$

[obsidian monolith]

south

[willow skiff]

as e^x is one-to-one

[stiff sluice]

wow

you are smart

thank you

[willow skiff]

no worries man

[narrow briar]

Hi! One question: how im supposed to study the continuity of these two functions? Tanks

without drawing the graph

in both the limit x->1 exists and the function evaluated in x=1 doesnt exist, so, how do i know if it has an evitable or inevitable discontinuity?

its difficult😓

[thin thorn]

You just take the right and left hand limits of the function and compare it with the functional value which in this case isn't defined. So the function is discontinuous at x=1

[smoky panther]

What happened to that 2x 4x and 6x??

The explanation is written but isn’t in English and i don’t understand technical talk in my native language

[willow skiff]

Look at x = 1.0001 for example

Then 1/(1.0001 - 1) = 1/(0.0001) = 10,000

So the limit actually doesn't exist cause they both diverge to +infinity

for f(x)

For g(x), you can factor out (x - 1) from the numerator and you are left with a quadratic

So g(x) has a removable discontinuity at x = 1

[willow skiff]

,rotate

[willow skiff]

Sub $u = x^2 - 1$, then $du = 2x \ dx$

[obsidian monolith]

south

[willow skiff]

So you're left with the integral of u^(-3) du, which is (u)^(-2) / (-2) + c

The other two cases use exactly the same reasoning

[jovial lodge]

See for this question I got everything correct except I minus’d the wrong curve from the right one

How do I know what one to minus from the other

?

[paper tartan]

(4x - x^2) - (x^2 - 6) = 4x - x^2 - x^2 + 6 = 4x - 2x^2 + 6

they distributed the -1 into (x^2 - 6) and then added

[jovial lodge]

Huh

[paper tartan]

hm?

is that not what you were asking about?

[jovial lodge]

I don’t understand sorry, why did they minus one curve from the other

[winter comet]

the one that is above

[winter comet]

to find the area

[jovial lodge]

Oh

[winter comet]

they aren't finding net area, just regular area

[jovial lodge]

Just the area above the x axis?

Positive side?

[winter comet]

the area of the shaded region

the area between the two curves

[jovial lodge]

Oh

[winter comet]

no, just the positive entire area

like ignore the x axis

its all positive area

not the net area

[jovial lodge]

I’m still confused 😭

[winter comet]

ok uhm

[jovial lodge]

Why do you minus I don’t understand

I just tried to integrate that

And then sub in 3 and -1

[winter comet]

thats correct

[jovial lodge]

Is it?

[winter comet]

you're subtracting the area under one curve from the area under the other

[jovial lodge]

I just moved it all over to same side

I just set them equal idk 😂

[winter comet]

bruh

ok

so you set them equal to find the bounds, right?

does that part make sense?

[jovial lodge]

Yea

[winter comet]

alr

[jovial lodge]

I got 3,-1

[winter comet]

i'll believe it

lol

nah ill do the math actually 1 sec

[jovial lodge]

Oki

[winter comet]

yeah ok

thats right

then on the interval

which curve is above and which curve is below?

[jovial lodge]

1 sec

4x-x^2 is above

And other one below?

[winter comet]

yes

you know how integral is infinite riemann sum?

[jovial lodge]

Ah

[jovial lodge]

What

Riemann

[winter comet]

do you know like definition of integral

[jovial lodge]

Nope

I’m not good at math I just tend to learn the method and apply it

[winter comet]

it's the infinite limit of a sum of very small rectangles?

oh

alr

[jovial lodge]

I don’t actually understand

[jovial lodge]

Is it like the trapezium rule?

Or just some random thing

I don’t know.

[winter comet]

what exactly do you mean by the trapezium rule?

[jovial lodge]

To find approx area

[winter comet]

you may be talking about riemann sum, which is an estimate, or ahh

no thats just the estimate

[jovial lodge]

Yea

[winter comet]

you add up a finite number of trapezoids together

[jovial lodge]

Yea

[winter comet]

but if you add up an infinite amount of trapezoids, you get the exact area

[jovial lodge]

I have like an equation for it

[winter comet]

thats the definition of an integral, more or less

[jovial lodge]

Ah

I see

[winter comet]

so here like uh

lets just say you're adding up an infinite amount of rectangles

cuz i cant draw trapezoid lol

[winter comet]

it has an infinitely small length of dx

[jovial lodge]

Hmmm

I see

[winter comet]

and for each dx, it has a certain length that i called R

[jovial lodge]

👍

[winter comet]

for each x value, the length of R will be 4x - x^2 - x^2 + 6

it will be the distance between the curves

a.k.a top curve minus bottom curve

does that kinda make sense or

[jovial lodge]

Mmm

So will it always be top curve minus bottom

If I get a q like this again

[winter comet]

yeah, pretty much

[jovial lodge]

Oh well that helps a lot tbh

[winter comet]

sometimes it might flip and you'd have to do it with respect to the y axis

in which case it would be right minus left

but yeah

[winter comet]

im not gonna say ALWAYs cuz there "could" be exceptions, i can't think of any tho pretty much just top minus bottom

[jovial lodge]

What dis

[winter comet]

volume

[jovial lodge]

If I get it wrong I’m blaming u in test! 😠

[jovial lodge]

Oh

[winter comet]

do you know shells method or disk method?

[winter comet]

alr 😂 💀

[jovial lodge]

So that pie integral of y^2

Thingy

[winter comet]

yeah

[jovial lodge]

Ah

What are my limits

So I basically square y and integrate that

[winter comet]

it looks like they want you to stop at x = 4

[jovial lodge]

x=0 and x=4

[winter comet]

and multiply by pi yeh

[jovial lodge]

What does it mean through 2 pi

In question

[winter comet]

oh i guess

it means just a full rotation

[jovial lodge]

So does my answer change?

[winter comet]

no

[jovial lodge]

Ah

Why do they even tell me that

😦

[winter comet]

i don't know 💀

[jovial lodge]

Damn I hate maths questioners

[winter comet]

they should honestly just say "full rotation" 💀

[jovial lodge]

What does that even mean tho like why are we rotating it 💀

I’m just tryna find the volume and dip

[winter comet]

you're finding the volume of the object made when the curve made a full rotation around the x axis

[jovial lodge]

Hmm I see

Ok

Thanks

My understanding of integration is getting slightly better

[winter comet]

XD

[winter comet]

you know how theres like the trapezoidal sums?

[jovial lodge]

Yea

[winter comet]

you know how the more trapezoids you have, the more accurate the estimate is?

[jovial lodge]

Yea

The more strips

Basically

[winter comet]

yeah

so if you have an infinite number of them

thats basically the integral

its not an estimate anymore, it IS the area

[jovial lodge]

Damn

That’s deep.

I acc understand it

[winter comet]

💀

i wanna write out the definition tho gimme a sec

[jovial lodge]

What’s ornates btw

[winter comet]

no idea

[jovial lodge]

In relation to that trapzoid question

I don’t think I spelt it right

[winter comet]

uh

this is the definition

f(x_k) means like the height of each rectangle

or each trapezoid

[jovial lodge]

It was ordinates btw

[winter comet]

oh 💀

[jovial lodge]

What’s that

[winter comet]

no idea 💀

google time

[jovial lodge]

What the flip

Who’s lim

[winter comet]

limit?

wait do you know limits

[jovial lodge]

Ah

Yea

[winter comet]

oh alr

💀

its the limit going to infinity

[jovial lodge]

Ah

[winter comet]

of the sum of infinitely small rectangles

[jovial lodge]

So between 0 and infinity

[winter comet]

delta x is the width, f(x_k) is the height

[jovial lodge]

Or -infinity and infinity

Idk

[winter comet]

no?

it just means like

as the number of rectangles/trapezoids gets bigger

[jovial lodge]

Yea I see

[winter comet]

you add them all up

lol

alr

[jovial lodge]

Alright.

Thanks

I’ve done enough learning today

[winter comet]

💀

[viscid thistle]

What is the domain and range here like the input values and results? What would be the case if it's (gof)(x) instead of (fog)(x) ?

[wraith sleet]

I learned something

[smoky panther]

Alr thank you :)

[winter comet]

if it were (gof)(x), then the range of f is a subset of the domain of g

[winter comet]

the range of the inside function is a subset of the domain of the outside function

since you input the y value of the function as the x value of the new function, essentially

[viscid thistle]

Yeah but wdym by u value?

[winter comet]

u value?

[winter comet]

wdym by what do i mean by u value, i didnt say u value and idk what ur referring to 💀

[viscid thistle]

Sorry I mean y value

[winter comet]

oh

by y value i mean range

[viscid thistle]

Like in gof f(x) is y value and g(f(x)) x value ?

[winter comet]

y = f(x), g(f(x)) = g(y)
so y is the range of f(x), but its a subset of the domain of g(x)

[viscid thistle]

Got it , thanks 🙏

[smoky panther]

I cant figure out what i did wrong here can someone help?

[river drift]

your 1/2 somehow turned into a 2 in line 5

[smoky panther]

The last line?

[river drift]

the one next to the 5️⃣ i mean

[smoky panther]

was I supposed to leave the 2 in the denominator?

[river drift]

yes

[smoky panther]

Oh alr, do i always leave the number next to x where it is? cuz i think i saw some take them both

[river drift]

if you wanted to pull 2 to the numerator you'd need to give it a negative power as well

[smoky panther]

Oh ok ok, thanks

Yea i did it and it turned out right thank you :)

[chrome ether]

#calculus

[smoky panther]

Is this meant for me?

[chrome ether]

ye, integrals are calculus lol

[smoky panther]

Ah alr mb lol, where i am we literally call it all math so idk what’s considered calculus or algebra or anything

[echo sinew]

I wouldn't worry too much, actually

[stiff sluice]

help please i dont understand how to do it

[chrome ether]

a_n explodes to infinity, making that limit a 1^infty exponential indeterminate form

try taking the log of that expression and finding the limit of that first

[stiff sluice]

we need to use algebra bro

But lowkey that 4 is being very annoying

If that would be 1 this has already ended

[chrome ether]

taking the log is “using algebra” is it not?

[stiff sluice]

the limit of what first?

ok we can use log then i didnt know it was algebra

but i cannot yse log because there is no rhs of an equation

i need more hints

[chrome ether]

let L be that expression above

log L = [log of whatever that is]

now take n—>infty

when you’re done, just exponentiate back

OH wait a sec

I think this might be like one of the limit expressions for e

[stiff sluice]

you must be joking

[chrome ether]

nope

[chrome ether]

all that matters is that a_n goes to infinity as n goes to infinity

[stiff sluice]

how to get rid of the 4 so we use e limit

[chrome ether]

therefore, we can rewrite the limit as $\lim_{a_n\to\infty} \left(1+\frac{4}{a_n}\right)^{a_n}$

[obsidian monolith]

elrichardo1337

[chrome ether]

that comes out to be $e^4$ I believe

[obsidian monolith]

elrichardo1337

[stiff sluice]

Sure but 4 on top is fked

i know but i need to justify

Like there is an algebraic manipulation for this

i did a simpler exercise earlier

this is how i should use e limit but i cannot use e^x

Ohh i can double flip

[chrome ether]

ah wait

I think what you can do is

$\left(\lim_{a_n\to\infty} \left(1+\frac{4}{a_n}\right)^{a_n/4}\right)^4$

[obsidian monolith]

elrichardo1337

[chrome ether]

the stuff inside goes to e, so the overall limit is e^4?

[stiff sluice]

i can sqrt

but this limit is zero

no?

[stiff sluice]

i did double flip like in here

So i can use e

then evaluate the limit on top of e

but is 0

So answer is e^0

[narrow briar]

Isn't it assumed that if both limits tend to the same value, then the limit exists?

[winter comet]

they aren't tending towards a value, they are tending towards infinity

but both sides of the limit are tending to +infinity

so the limit is +infinity

the problem is, infinity doesn't exist

so the limit doesnt exist

and its infinity

same thing

[surreal crescent]

So close, yet so, so far

[surreal crescent]

Another way to say it is the limit is indefinite, there are so many ways to say the same thing

[warped cipher]

it is already 90%

just round it

[vernal pine]

,w $-\frac{\pi}{2} \le \frac{2\pi}{12}\left(t-0.3\right) \le \frac{\pi}{2}$

[obsidian monolith]

Results provided by WolframAlpha

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

Requested by j_jaip

[agile sparrow]

@woven remnant

[plucky osprey]

Nah thats insane, would they not round it?😭

[granite coral]

where can i learn how to do this

[sharp perch]

those are limits, search it up on youtube "introduction to limits", Im assuming that you're a beginner

[stiff sluice]

do what exactly?

I just used the e limit and algebra

[stiff sluice]

if you want tough limits, look up demidovich, or maybe other tough mir books, but beware his book has little theory

[charred epoch]

This is super easy

[flat yacht]

Heya everybody, I was just watching this video: https://youtu.be/S0_qX4VJhMQ?si=0yR95MZ6hHh-tBQB

And around 8:40 the guy from the channel says 'you can "choose" mostly x's and one dt'. What does he mean with choose? Choose for what?

I'm watching this video just to learn some calculus since me and a friend are really interested in advanced physics like particle physics and quantum physics, and our school Math curriculum isn't that far yet, since we're only in third year of high school right now.

Please do ping me if you want to give an answer :)

[vapid plaza]

@flat yacht

when multiplying/expanding expressions with multiple terms, you can think of each term after the expansion as the result of "choosing" one term from each factor

for example

(A + B)(C + D + E)

would be AC+AD+AE+BC+BD+BE

note that each of the six terms correspond to a possibility when "choosing" either A or B in the first bracket, and then either C or D or E in the second bracket

and of course this works if you have even more brackets

[flat yacht]

Ah, like that. Thank you so much! Just seems to be a struggle cos I'm not a native English speaker, terms like expressions are kinda new to me 😅

[vernal pine]

Just wondering, is R^2 the same thing as the correlation coefficient squared?

[crystal vector]

Yes, ( R^2 ) (R-squared) is indeed the square of the correlation coefficient (( r )). It represents the proportion of the variance in the dependent variable that is predictable from the independent variable(s) in a regression model.

[obsidian monolith]

Arrow ♡

[vernal pine]

Thank you very much : )

[crystal vector]

ofc

[white rapids]

Uhh is unit circle required for precalculus?

[winter comet]

wdym?

you mean as a prerequisite, or is it definitely part of precalculus

[white rapids]

I dont know

Im learning it right now for math III according to the curriculum here in california

[winter comet]

ah

[white rapids]

Im gonna assume its required

[winter comet]

i think they would cover it in precalc

[white rapids]

Cuz this stuff is confusing

[winter comet]

i dont know what you mean by required tho

[white rapids]

Oh

[winter comet]

you have to understand it by the end of precalc

[white rapids]

I mean just for necessities for it if there was trig

[white rapids]

Alright

[winter comet]

it would teach the unit circle early on in the trig part

[white rapids]

Oh alright

[golden cairn]

How do I solve for x in this, over [0,2pi]: 3sin2x+cosx=0

[hushed sphinx]

I'd start by sketching the function to get an idea of how it moves. E.g. plot its values at all multiples of pi/2.

[golden cairn]

Well I have a unit circle drawn out

[winter comet]

you'll need that later, but im pretty sure he meant the function like f(x) = 3sin(2x) + cos(x)

[golden cairn]

Well that’s not what I needed help with. I got up to here: cosx(6sinx+1)=0. And then I wrote sinx=-1/6 and cosx=0

[winter comet]

oh

[golden cairn]

I got the x values for sinx=-1/6 in my calculator

But the problem is

My answer key has different answers for cosx=0. I wrote down pi/2 and 3pi/2 but the answer key has 2pi,0

As the x values for cosx=0

[winter comet]

uhh

thats weird...

[hushed sphinx]

Uh ... cosx(6sinx+1)=0 doesn't seem to have anything to do with 3sin2x+cosx=0.

[golden cairn]

Wait

[winter comet]

sin(2x) = 2sin(x)cos(x)

[golden cairn]

Let me show my work

Yeah

That’s what I did

I used my formula sheet

[winter comet]

thats weird

[golden cairn]

So is it a mistake in the answer key?

[winter comet]

i guess so

[golden cairn]

Ok

[winter comet]

if you're sure thats all the context then yeh lol

[golden cairn]

Yeah that’s all they said

[hushed sphinx]

Ah, it looked to me like the sine was inside the argument to the cosine.

[winter comet]

oh lol

[viscid thistle]

Where's the calc channel?

[hushed sphinx]

#calculus.

[viscid thistle]

Thanks

[autumn snow]

Can someone explain this one question to me please

It’s pretty simple but I want clarification

[hushed sphinx]

The only way to find out is to ask your question and see if someone answers.

[winter comet]

no

[daring tapir]

!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/

[final pier]

Yeah I got it 😭

[golden cairn]

I need help with this question

I asked several different ai chat bots and they all get different answers which are wrong. The answer key says n=4

[charred epoch]

Oh wow I wish I had been reading the free online OpenStax textbook alongside my class.

We learn everything without it, but the textbook goes into so much detail.

[white rapids]

Can somebody help please?

[chrome ether]

!nogpt

[tender quest]

Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).

[golden cairn]

Yeah well duh. I got the wrong answer so obviously

[chrome ether]

this is just plugging into the definitions of nPr and nCr

[hushed sphinx]

If I recall correctly (but check this!) nP3 means n(n-1)(n-2) and nC4 means n(n-1)(n-2)(n-3)/24. Set these two equal to each other and divide away the common factors.

The answer cannot be 4 because 4P3 is 24 but 4C4 is only 1.

[white rapids]

@winter comet

[winter comet]

cosine graphs don't go through (0,0), for one thing, right?

i mean

not like that

they could technically i need to be more specific

uhh

i don't know how to explain it too well

[white rapids]

Uhh

[hushed sphinx]

You can plug x=0 in to find where the graph should cross the y-axis.

[winter comet]

thats true

[white rapids]

How do I find the midline

If theres like a minimum or maximum

[hushed sphinx]

Figure out what the smallest and largest possible function values are; the midline is halfway between them.

[white rapids]

What if I have an amplitude and a minimum

Do I subtract or add those values to get the midline?

Im kinda lost right now

[golden cairn]

I dont know why almost every single answer in the answer key is incorrect

This is a like a final exam review the teacher made

It’s not official so maybe that’s the reason

[winter comet]

if you have a minimum, if you add the amplitude you should get the maximum, and half way between the minimum and the maximum should be the midline

[hushed sphinx]

Twice the amplitude.

[white rapids]

Uhh

[winter comet]

oh yeah mb

[white rapids]

Alright I'll try

[winter comet]

you only have to add the amplitude to get the midline 💀

[golden cairn]

I got up to that step already but I don’t know what to do after that. It just looks like a bunch of expanding to me

[hushed sphinx]

Nono, don't expand.

[white rapids]

Uh if im trying to find midline, should x always = 0

[hushed sphinx]

Instead notice that most of the factors appear on both sides of the equation!

[golden cairn]

So first do I multiply by 24 on both sides to get rid of the denominator?

[winter comet]

the midline is a straight line, y = something for all x

[hushed sphinx]

Well yes you can do that.

[golden cairn]

Ok

[river drift]

cos(x - π/2)

[summer juniper]

how do I solve this (without l'hopitals rule)

the answer is allegedly 2/3

and I have 2 for the numerator

but i'm getting 2 for the denominator as well

i tried to rationalize by multiplying the numerator and denominator by (cube root(1+x))^2 + (cube root(1-x))^2

but now i'm stuck

[twilit oyster]

do you know about perfect cube binomials

[chrome ether]

that's not the correct multiplication to use

[hidden lagoon]

the what

[chrome ether]

that's not it either, you need difference of cubes

[twilit oyster]

Dang rlly

[chrome ether]

if you want to completely get rid of the cube roots in the numerator

[twilit oyster]

Wait that was what I was talking about

[chrome ether]

(a+b)^3?

[twilit oyster]

Ya

[chrome ether]

i'm not sure how that helps here

[twilit oyster]

Like a^3-b^3

[chrome ether]

oh yea that's difference of cubes

[hidden lagoon]

you use the formula for a difference of cubes

[twilit oyster]

Yeah do you not use that

[hidden lagoon]

(a - b)(a^2 + ab + b^2)

[chrome ether]

we just called it different things

[twilit oyster]

Lol

[viscid thistle]

👍 I was just guessing tho

[viscid thistle]

Why has the h become 1 instead of 0 in LHL?

[royal ingot]

Indeed it looks like an error

There is no reason to replace it by 1

[summer ruin]

what do the square brackets mean?

the integer part?

[daring tapir]

what is g(x)?

[summer juniper]

i'm not sure how exactly to do that

[willow skiff]

So if you let $(1 + x)^{1/3} = a, (1 - x)^{1/3} = b$, your numerator is $a - b$

[obsidian monolith]

south

[daring tapir]

(proving the formulas would be a fun exercise too)

[willow skiff]

Multiply top and bottom by $\frac{a^2 + ab + b^2}{a^2 + ab + b^2}$

[obsidian monolith]

south

[willow skiff]

then you can directly sub in once you simplify

[viscid thistle]

x-[x]

[viscid thistle]

Greatest integer function

[summer ruin]

what does that mean

[daring tapir]

there you get it why it equals one

[viscid thistle]

This is the question

[summer ruin]

so it's the integer part

[viscid thistle]

Wdym?

[summer ruin]

[c-h] = c-1 when h > 0

[viscid thistle]

👍 Thanks 🙏

[winter comet]

🫠

[chrome ether]

what do you want us to do with that 😭

[daring tapir]

cry about how weird shit can look i guess

[winter comet]

💀

"math is so complicated"
literally just didnt define it so nobody can understand because it has no meaning ._. 💀

or no singular interpretation 💀

[daring tapir]

that doesnt even look like math lmaoo, there is the gravitational constant plancks constant and idk what else floating around in that

[winter comet]

some sort of applied math lol

[daring tapir]

xD

[winter comet]

with no context yes math is very hard

even impossible

🗿

[daring tapir]

fr

[winter comet]

how many inches is in 2

[daring tapir]

4 obv

[winter comet]

nah 8

[daring tapir]

that is middle school level shit

xD

[winter comet]

☠️

[daring tapir]

oh fuck i forgot mb

[winter comet]

yeeeh

[daring tapir]

(i hope someone continues the chain and says 16 but oh well)

[sour lynx]

16

[gaunt osprey]

how is x=2 not differntiable

[river drift]

it's a jagged edge, the derivative approaching from the left is different from the derivative approaching from the right

[winter comet]

yo

how to turn r = -3cos(theta) into rectangular equation?

me blanking out XD

[river drift]

you may have an easier time if you multiply both sides by r

[chrome ether]

t 😭

oh lol

[winter comet]

r^2 = -3rcos(theta)
x = -3r(cos(theta)), x = r^2
i'm not exactly seeing how any of this information gets to a rectangular equation :l

oh wait

my bad

x = rcos(theta)

lol

-3x = r^2

._.

[river drift]

r^2 has a nice formula in rectangular form

(consider pythagoras)

[severe night]

yo based of the plus and minus how do i know what function to use

[river drift]

- means from the left, or the limit as x approaches 2 from lower values

[severe night]

so as x as comes to -2 from other neg numbers

[river drift]

well it means approaching -2 from negative numbers lower than it (think -2.0001 for example) as opposed to -2 from negative numbers higher than it (like -1.999)

[severe night]

so i would use bottom equation for - and top for +

[winter comet]

r^2 = x^2 + y^2?

[river drift]

yes

[winter comet]

i know its a circle, i dont know how to logically know its a circle

and i know its not centered at 0,0

[river drift]

well once you have
-3x = x^2 + y^2
you can complete the square to get it in standard circle form

[winter comet]

😐

so you basically just have to know its a circle

[chrome ether]

this is assumed to be well known background knowledge from alg2/geometry/wherever the heck they covered it

[winter comet]

well uh

what if its not a circle

._.

do you have to know the shape already in order to get the rectangular equation?

like there isnt a set method or anything?

[winter comet]

well I know r = sin(theta) and r = cos(theta) are circles

I don't know how I know that

I was never officially taught polar coordinates or parametric equations lol

just kinda picked up stuff :l

[chrome ether]

you try to force your expressions for r^2 or x,y into the equation

either squaring, multiplying through by r, etc.

[winter comet]

into what equation?

[chrome ether]

the eqn you’re given

that you want to convert to rectangular

[winter comet]

um

[winter comet]

in order to get what

:l

[chrome ether]

^^^^

me when reading comprehension

[golden cairn]

How do I solve this

They don’t even give the intensity of a standard earthquake or anything like that

[chrome ether]

you don’t need to know that value, it cancels out

[golden cairn]

Can you show me

[winter comet]

you force r^2 or x,y into the polar equation that you want to convert to rectangular

[chrome ether]

yeah

[winter comet]

but that doesn't tell me how to get to rectangular

._.

does it?

maybe "force" is too vague to give me an idea

lol

[chrome ether]

you then substitute $r^2=x^2+y^2$ or $x=r\cos\theta$ or $y=r\sin\theta$

[obsidian monolith]

elrichardo1337

[winter comet]

what if its not a circle :l

will it still be a circle?

oh

oh wait

oh i see

💀

hmm

let me think this through 💀

[chrome ether]

I probably shouldn’t be giving this all at once but

key idea is you can break down that log of a quotient

into a difference of logs

[golden cairn]

Yes I get it

Thank you so much

[winter comet]

so what about r = 3sin(2theta)

r^2 = 3rsin(2theta), r^2 = 6rsin(theta)cos(theta)...
x = rcos(theta)
y = rsin(theta)

[chrome ether]

double angle

[winter comet]

rcos(theta) = r^2/[6sin(theta)]?

[chrome ether]

so that gives you x^2+y^2=6y*cos(theta)

[winter comet]

:l

...

bruh

so im hearing with polar coordinates to rectangular coordinates

its basically always x^2 + y^2 = something

something = r^2, but i said something instead to indicate thats the only thing that changes

basically? ._.

[chrome ether]

I haven’t finished converting, trying to see how to finish it

[winter comet]

oh, its not converted?

[golden cairn]

Wait but what do they mean by how many times more intense was the Japan earthquake. How do I figure that out?

[chrome ether]

you need there to be only x and y for it to be rectangular

[winter comet]

oh

[chrome ether]

intensity of Japan earthquake over intensity of Vancouver earthquake

[winter comet]

by the way, i made this problem up on the spot, its like a cardioid or a limacon or something

but is the rectangular equation of that still just x^2 + y^2 = something?

🤔

no its a flower actually

mb

lol

[golden cairn]

So is 10^2.6 the answer?

Oh so is the answer (10^8.9)/(10^6.3)

?

[chrome ether]

those two are the same lol

yea

first square then multiply by r^4

@winter comet

[winter comet]

OHH

i see

[golden cairn]

Can you also tell me the answer to this? I just wanna check my answer. I got 63 times greater

[chrome ether]

10^1.8?

yea that sounds right

[golden cairn]

10^-2.2/10^-4

Same thing

Yes

[chrome ether]

yeah

[golden cairn]

Thanks I thought it was gonna be way harder

But it wasn’t I guess

[chrome ether]

npnp

[sour phoenix]

hello please who knows where i can learn improper integrals online?

[winter comet]

khan academy

[paper tartan]

seconding Khan Academy, but also consider taking a look at Paul's Online Notes
https://tutorial.math.lamar.edu/classes/calcii/improperintegrals.aspx

[daring tapir]

No one mentioned proff leonard so i will xD

[viscid thistle]

I feel it's good, but some topics aren't very in depth

[chrome ether]

it's an ok introduction

just to get off the ground with doing computations

although i agree that it either

• wasn't that good to begin with
• has fallen off quite a bit (see their peddling of whatever “AI” snake oil bullshit is on the market now)
• has been outclassed by newer better resources

or probably a bit of all three

[viscid thistle]

(Also this channel is precalculus)

[torpid ivy]

we can take the pre common factor, pre(calculus) then divide by pre and it's gonna be calculus, so they're basically the same thing

[daring tapir]

Lmfao

[golden cairn]

Can someone prove this identity

I tried so many things but it isn’t working out

[daring tapir]

write numerator as 1-sin^2(x) and then add annd subtract sin(x) in the numerator

[golden cairn]

Okay I’ll try that

Wait I just realized that’s what I did

But I did something wrong after that maybe

I’ll try it again

[silent spear]

How do I understand calculus better

[winter comet]

practice, and spending time thinking about it

what dont u understand

sometimes its rly conceptual

[paper tartan]

nothing beats practice

[viscid thistle]

perhaps this might help ya

by the way…im new here…and I was wondering if anyone could help me with the topic of interchanging of functions like log and limit ….

[daring tapir]

!nosols

[tender quest]

As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.

[viscid thistle]

ohh..ok..im really sorry…i was unaware of this fact😅

[proven night]

Hello

I tried using logarithm with base 3 and with base k, but both options didn't help me

[winter comet]

hmm

i can see the answer thru trial and error but the process...idk

thats tougher

leme think

[daring tapir]

you can write that as 3^9 = (k/3)^k

[brazen rivet]

whats q

[daring tapir]

is it given that k is an integer?

or that there is only one solution

[brazen rivet]

no solutions

[daring tapir]

k = 9 is a solution..

[brazen rivet]

yes i made a typo sorry

[brazen rivet]

i mean there will be always one, no?

cuz in the right hand we have k+9

[daring tapir]

not until we have proved that

[brazen rivet]

and left side is xlog3(x)

[daring tapir]

i mean intuitively sure

but definitely not until we have proved it

[brazen rivet]

if u divide by x

both sides

ull have 2 functions

[proven night]

I dont understand it

[brazen rivet]

one is growing up

and the right one is

the opposite

they will intersect

[daring tapir]

wdym

[brazen rivet]

divide xlog3(x) = x+9 by x

[daring tapir]

3^9 * 3^k = k^k
dividing both sides by 3^k

[brazen rivet]

ull get logx= 9/x+1

[daring tapir]

hm sure

[brazen rivet]

obviously has 1 solutuon

[daring tapir]

why

[brazen rivet]

monotone functions

left one is rising

right one is decreasing

[daring tapir]

oh- frick lol didnt realise that

[daring tapir]

though x = 9 doesnt satisfy this

but it satisfies the original equation

[brazen rivet]

its fine

[daring tapir]

we need to find the solution too not just prove its existence 😭-

[brazen rivet]

because i just proved u that there is 1 solution

u asked for it

[brazen rivet]

then u just take ur 9

and u are done cuz no more exist

[summer ruin]

k^k is not a monotone function though

[daring tapir]

but how did we get 9?

hit and trial is not a reliable method

[brazen rivet]

guessed

xd

[daring tapir]

not good enough

prove it

🙃

[brazen rivet]

why?

[daring tapir]

for positive nums it is i mean

[summer ruin]

still not monotone

[brazen rivet]

its not so hard to guess

[summer ruin]

plot a graph of x^x to see it's not monotone

[brazen rivet]

huh?

[daring tapir]

greater than 1 😭-

[brazen rivet]

of course

why should we care about this small numbers

[daring tapir]

you could say that for anything

prove everything

you do

[brazen rivet]

no, this one is really not hard

u just need to plug in 9 values

and get the result

[summer ruin]

you can divide by $3^k$ to get $\left( \frac{k}{3}\right)^k = 3^9$, take cube root to get $\left( \frac{k}{3}\right)^{\frac{k}{3}} = 3^3$, now it's easy to guess $\frac{k}{3} = 3$

[obsidian monolith]

Transparent Elemental

[daring tapir]

dont guess.

[brazen rivet]

huh?

[daring tapir]

no

[brazen rivet]

on math olympiads u sometimes need to plug into like 40 values

[daring tapir]

i know but this isnt an olympiad problem

[brazen rivet]

okay ill try to solve it

xlog3(x)=x+9
log3(x)=1+9/x
x=3^(1+9/x)
im pretty sure u cant do much about it

the problem is about guessing

and proving thats 1

[brazen rivet]

or u will constantly face logs or 3^x

[proven night]

Thank you guys a lot

But is k=9 is the only solution right?

[quartz inlet]

You could argue by inequalities, that is rewrite the equation to the “equivalent” form of

k/9*log_3(k/3)-1=0

Now consider values in which k>9 and 0<k<9 respectively

And argue that LHS must be either bigger than zero or less

For example, for k>9,
we know log_3 is a strictly increasing function,

so k > 9 => log_3(k/3) > 1

and obviously k/9 > 1.

So taking this into account we have that the LHS must be strictly bigger than 0.

[quartz inlet]

Now do the same when k is less than 9 and bigger than 0. That’ll require a bit more work however.
Hopefully this is easy enough for you to convince yourself that k=9 is the only possible answer!

[quartz inlet]

One way of motivating the solution without guessing is by “comparing” base and exponent in the form you wrote before.

That is 3^9 = (k/3)^k are equal if both

k/3 = 3 and k=9.

And then afterwards resorting to the above argument to conclude it’s unique

[quartz inlet]

I do wonder if this can directly be made to an iff statement; but not sure

[quartz inlet]

But I also realize, they just misspoke since they’re comparing too in this case, but in neater way. So it’s not really guessing

[daring tapir]

But you proved that there is a unique solution

They did not

[quartz inlet]

Well they did not claim it was unique, but I can see what you mean

[daring tapir]

Hmmm

[brazen rivet]

i did

lies

[daring tapir]

not you

transparent

[brazen rivet]

actually transparent + proof of uniqueness its a good solution

[daring tapir]

yea i have no issues with that

but the proof of uniqueness is a must

[willow skiff]

hey you got active

congrats

[daring tapir]

thank you!! (had it for the last like week or smthn but i had the not very ppl role so the colour was being overridden 🙃 )

[brazen rivet]

anyone got any other problems?

[vestal bison]

how to convert degrees to minutes and seconds form? I know how to convert otherwise (minute to degree) but i cant do it other way

for instance i got 30.43° i need to convert it to minutes

how do i do that?

[daring tapir]

1 degree = 60 minutes
0.43 degrees = 0.43*60minutes

do the same again with minute for seconds

[vestal bison]

okay so i am gonna do 30 + 0.43*60 + ?

[vestal bison]

and 0.436060?

0.43 * 60 * 60

[daring tapir]

,calc 0.43*60

[obsidian monolith]

Result:

25.8

[daring tapir]

so you get 30 degrees 25.8 minutes right?

[vestal bison]

yes

oh for the 0.8

[daring tapir]

now 1 minute = 60 seconds

[vestal bison]

0.8*60

[daring tapir]

yea

[vestal bison]

oh okay got it

thanks

[proven night]

There is quadratic equation $$ x^2 - 4ax + 4a^2 - 25 = 0 $$. It has two solutions: x1 and x2. How many values of a (integers only) that satisfy the inequality x1 < 1 < x2 are there?

[obsidian monolith]

freddie_macha1

[proven night]

I almost solved it, but

The solutions to the equation are 5 + 2a and 2a - 5 to

And i dont know which one is x1 and which one is x2

Because inequality 5 + 2a < 1 < 2a -5 doesnt make any sense because the answer is a<-2 and a>3

Whereas the answer for inequality 2a - 5 < 1 < 5 + 2a is alright, it makes sense because it's a = (-2;3)

Should i solve both inequalities and rule out the one that doesn't have solution?

[daring tapir]

2a-5 is smaller

than 2a+5

sooo x1 = 2a-5

x2= 2a+5

[proven night]

That's right, thanks. It seems like i solved the whole question but didn't notice such a detail

[daring tapir]

everyone has brainfarts

[proven night]

By the way, in terms of quadratic equations, x1 is always smaller than x2, right?

[daring tapir]

wdym?

[proven night]

I mean is it always smaller?

[daring tapir]

x1 + 10 = x2 so yea

[uncut mulch]

no, numbering the roots is arbitrary

x_1 represents one of the roots
x_2 the other

you can have x_2 represent the smaller one if you wanted to

its just usually more aesthetically pleasing to have x_1 represent the smaller one

[daring tapir]

They said x1 < x2 soo no

[daring tapir]

.

[uncut mulch]

their follow up question seemed to be asking about quadratics in general

[clever dock]

can someone help me to find the problem?

i swear I've spent like 20 minutes trying to find the problem and i can't

i feel like it's in the indicated part cause the answer should be 3ln|x+2|

but i don't seem to find where I've made the error

|| feels weird to have a 0 on the numerator from the decomposition but i checked it more than twice with the calculator lmao ||

[winter comet]

you don't have to use partial fractions because you can just factor and cancel lol

6x + 3 -> 6(2x+1)
2x^2+5x+2 -> (2x+1)(x+2)

[clever dock]

omfg

that's true

I didn't realise I could do that

but anyways, being able to factor or not, where did I make the error?

[winter comet]

uh

oh

where did the 1/2 go

lol

(x+1/2)(x+2) = x^2 + 5x/2 + 1
you need to multiply the bottom by 2, so multiply the entire thing by 1/2

its more like A/(x+1/2) + B/(2x+4)

so when B = 6, 6/(2x+4) = 3/(x+2) so yeh

[clever dock]

uh

okey

thanks hahaha

[warped wharf]

Is it even possible to solve?!

[daring tapir]

uh yea it was solved read the chats

[winter comet]

im glad its not just me who has trouble with that 😅

[daring tapir]

lmaoo

[golden cairn]

How do I prove this?

[daring tapir]

try multiply and dividing by something to get a more familiar expression

[golden cairn]

Which side do I work on?

[daring tapir]

LHS by convention but honestly either is fine

[golden cairn]

Ok

[lime parrot]

bottom is equal to sec^2(theta)

then split up the fraction into two

[daring tapir]

spoiler to the problem ||or 1/sec^2(x) = cos^2(x), then multiply that sucker with the numerator||

[white crane]

1/(1+tan^2)=cos^2 => left side=cos^2 * (1-tan^2) = cos^2 * (1- sin^2 / cos^2) = cos^2 - sin^2 = right side

[daring tapir]

!nosols

[tender quest]

As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.

[golden cairn]

It wasn’t for homework, it was just practice for my final exam

And I already figured it out

Also are both of these the same thing?

[river drift]

multiply the numerator and denominator of the left one by $\sqrt 3$

[obsidian monolith]

cloud

[golden cairn]

To rationalize it or something?

Oh I get it

It just wasn’t fully simplified

Thanks

[daring tapir]

Still

Youd learn more doing it yourself than having the solution handed out to you

[golden cairn]

No like I’m going back to old material the teacher gave me

So I already have done these questions before, I just don’t know where

I put the papers

[golden cairn]

I agree, I wasn’t asking for the solution

Somebody just gave it

I was just asking for help

[daring tapir]

Yea so my factoid call was directed to them not you

[golden cairn]

I mean I get where you’re coming from. I could have been giving homework questions even thought I wasn’t.

[winter comet]

is ln^2(x) correct notation for [ln(x)]^2? just like how sin^2(x) is correct for [sin(x)]^2?

[viscid thistle]

yes

[winter comet]

💀

what about just regular logs

thats crazy i thought you couldnt 💀💀

[viscid thistle]

you also can

[winter comet]

💀

[viscid thistle]

:))

[chrome ether]

that notation is not standard at all, i usually just write (ln x)^2 to make it totally unambiguous

[river drift]

it's not as common though, so it's good to include an explanatory note if you use that

[viscid thistle]

it is common lmao

[winter comet]

oh 💀

I just saw it in a math video but i always thought you couldnt lol

[viscid thistle]

it is standard

[obsidian monolith]

Mercury

[brazen rivet]

yes

[shadow summit]

$$(f(x))^{n}=f^{n}(x)$$
Not to be confused with
$$f^{(n)}(x)=\frac{d^{n}y}{dx^{n}}$$

[obsidian monolith]

TheLord26

[winter comet]

ln^(2)(x) may imply the second derivative of ln(x) ?

[willow skiff]

yes cause there's a bracket around the (2)

[winter comet]

ah