precalculus

[obsidian monolith]

Zyzzbrah

[hasty belfry]

this is pre calc / trig functions ig

so i solved it and got sin theta > 1/2

and b/w 0 to 2pi there are 2 solutions which are pi/6 and 5pi/6

$$ (0,π/6)∪(5π/6,2π)$$

[obsidian monolith]

Zyzzbrah

[hasty belfry]

this is correct answer tho

how can i represent it in this way?

after finding out the solutions

[hybrid bane]

If it is just those 4 solutions put it in a set

Wait...

Your theta in (0, 2pi) doesn't really make sense

[hasty belfry]

okay so i solved the equations first then plotted the graph of sinx and took a line parallel to x axis with y=1/2 and marked the points intersecting it

[hasty belfry]

its principle solution (0 to 360)

[hybrid bane]

Sure but you seem to be referring to a set but talking about the interval

[hasty belfry]

oh yeah ik these

[hybrid bane]

If you are trying to say theta is an element of the interval (0, 2pi) you need to say like it is on the left

[hasty belfry]

thats what im saying the answer should be just [pi/6 and 5pi/6]

[hybrid bane]

Then just write the solution set {pi/6,5pi/6}

[hasty belfry]

yeah ikr but its not in the options

i mean ik the correct answer

[hybrid bane]

So what are the options?

[hybrid bane]

Do you have a picture of the options and or the original question

[hasty belfry]

i closed the site

but fuck it

i solved the question

all that matters

[hybrid bane]

,w solve 2sin(x)^2 -5sin(x) + 2 > 0, x between 0 and 2pi

[hasty belfry]

• 2

[obsidian monolith]

Results provided by WolframAlpha

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

Requested by Brandon7716#6524

[hybrid bane]

Yeah it isn't only 2 solutions is probably why

[hasty belfry]

yeah

anyway thx

for taking ur time out

[solar pagoda]

what do you algebraically? I am pretty sure that is the only to do it. sin(2x) = 2sin(x)cos(x) comes from sum identity for sin.

[cinder briar]

what is the range 1/x

[opal tree]

Use your asymptotic behavior

What is the horizontal asymptote of 1/x

[cinder briar]

y=0

[cinder briar]

so can you say the range is y=/0

[opal tree]

Yeah y ≠ 0

[gentle vale]

wouldn't you use the quotient rule on that?

[willow bear]

wym quotient rule

nobody asked to find the derivative did they

[gentle vale]

I've been doing derivatives all day nom stop lmao
It's just ingrained in my mind

My bad

[fringe fulcrum]

Hey guys

I'm thinking about not taking Precalculus next year and studying on my own

Does a good book cover all of this?

[hasty belfry]

sin 2x= 2 sinx cos x

[hasty belfry]

what country are you from

there are loads of good books tbh

[hasty belfry]

R - {0}

[viscid thistle]

I'm having some trouble with approaching this situation. I just not sure ir it'd be possible to find the 77th term without it being so time consuming. I also don't know why exactly n is only less or equal to 3.

[lofty sage]

use the formula from T3 to T7, after the T6 you can see that the sequence starts again from 4, 5, etc

[lofty sage]

and answering you doubt, n is >= 3 because for n = 1, T1 = 4 and for n = 2, T2 = 5

[viscid thistle]

Oops, in my text, I meant to the if not I greater or equal to 3, but I understand why now.

This sequence is basically repeating values of 4, 5, 3/2, 1/2, 1. From term 1 to 5. And those values repeat again from term 6 to 10. This mean that my 77th term would be 5, knowing that sequence repeats every 1-5 terms

[lofty sage]

yeah

its correct

[viscid thistle]

Alright got it 👍, thanks for the help @lofty sage

[loud tundra]

What's a good precalc book?

[viral bolt]

precalc for dummies

LMAOO

[viscid thistle]

like what kind book do you want, my entire text book for precalc was online btw

if u want it

[loud tundra]

I was reading some Stack posts and found titles of old school precalculus books. So idk if I want one of those or just a some other recent book. The point is that I do want to have solid foundations whilst not being overwhelmed by seeing zillions of exercises. Yeah, I want exercises, but not those kind of mechanical ones

[cinder briar]

if i have a binomial expansion of (4-x)^1/2

for what values x is the binomial expansion valid for

is it mod x < 1/4 ?

[willow bear]

no, it's |x| < 4.

[last folio]

I need help with this question under pre calculus II

[viscid thistle]

square both sides and when u get cos^2x u can substitute it for 1-sin^2x

[last folio]

okay lemme try

[uncut mulch]

squaring may generate extraneous solutions (especially since sin(x)-1 is non positive)

so you'd need to be extra careful

I'd actually recommend rearranging the equation to
sin(x)-cos(x) = 1
and apply harmonic trig identities to express the LHS in terms of a single trig function

[last folio]

okay ,thnx.

[viscid thistle]

there are also videos like from UNC on youtube teaching precalc in like 6 hours, that could be helpful. are you in America?

[loud tundra]

Yeah

[viscid thistle]

yeah that video is litterally the entire class let me see if I can find it and send it. The only problem is that its kinda short but for concepts you dont understand you can just review on youtube too

https://www.youtube.com/watch?v=eI4an8aSsgw&t=10595s

at least as a review its pretty helpful in my opinion. I just took precalc this year and this video is pretty much the exact same

[loud tundra]

Interesting. I’ll check it out. Thanks @viscid thistle

[candid cloud]

It’s pi/2+2pi(n) for any n integer

[hushed sphinx]

That gives some of the solutions, but not all.

[gloomy crater]

can someone help me with this?

[willow bear]

have you made any progress so far?

[gloomy crater]

nope, can't think of anything

[willow bear]

have you ever done things with quadratic sequences before?

[gloomy crater]

probably not

[willow bear]

maybe the words "first differences" and "second differences" may ring a bell...

there is still a way to do this if not

let $u_r = Ar^2 + Br + C$ and either set up a system of equations directly involving the first three terms (hard) or do some nonsense involving first and second differences anyway (easier mechanically but harder conceptually)

[obsidian monolith]

Ann

[gloomy crater]

oh

I think I got it

thanks

so I have to do
7=A+B+C
19=4A+2B+C
37=9A+3B+C
and find values right?

[willow bear]

that works, yes.

[gentle vale]

https://tutorial.math.lamar.edu/
I like this website for learning calculus

[remote quartz]

does anyone have a folder or something of a bunch of quick trig limits questions to practice identities etc

[frank vine]

trig practice problems?

[frigid holly]

this simplifies to x^2 - 0.5x, right?

the textbook I'm using has this coefficient 2 in the solution and I'm just wondering if I'm taking some crazy pills and have failed to understand something critical

[obsidian monolith]

The following error occured while calculating:
Error: Undefined symbol x

[frank vine]

https://photomath.net/s/VvnYQz

[frank vine]

Thar you go dude

[slate reef]

(x+2) (x^2 -0.5x) = x^3 - 0.5x^2 + 2x^2 -x
= x^3 +1.5x^2 -x
so no

[frigid holly]

true, though I can't quite understand where that 2 comes from

if (2x^2 + 3x - 2) = (x+2)(x-0.5), then why isn't
x(2x^2 + 3x - 2) = x(x+2)(x-0.5), but instead 2x(x+2)(x-0.5) ?

[frank vine]

Have to take out the two

That’s why

If you want fully and rightly factored

[slate reef]

because (2x^2 +3x -2) is not (x+2)(x-0.5)

[frank vine]

That too

[slate reef]

its 2(x+2)(x-0.5)

[frank vine]

Could also use synthetic division to divide numerator by denominator

[frigid holly]

ah

forgot that one specific part when factorizing with roots

[frank vine]

Yep

[slate reef]

yes

[frigid holly]

shows that I've been practicing mostly with 2nd degree polynomials with the a-coefficient of 1 😄

[slate reef]

yup

I don't like when the a coeff is not 1 so I always divide everything by that

so it becomes 1

[frigid holly]

makes sense. I'm guessing that in this example, it wasn't divided by 2 since that way you can use the quadratic formula with nothing but integers

[slate reef]

yeah most likely !!

[noble trench]

i have a screenshot from a class i took

gimme a sec to find it

[remote quartz]

thank you

[remote quartz]

yeah limits involving trig functions

[noble trench]

here ya go

oh wait those are derivatives

oops

i misread your question

ahhhhhh

i mean derivatves are a type of limit so they're still good practice 😉

one good limit problem i know is the following

although its kind of well known

find lim x--> 0 sin x/x

and also another one

find lim x--> (1-cos x)/x

[remote quartz]

do you know anywhere i can find like a bunch of them? are there any folders online

[remote quartz]

but i appreciate it very much thank you still lol

[frank vine]

Hmm zlib might have some books on it

[noble trench]

i found a couple online

this one has some (although probs not the most reputable lol)

https://www.cliffsnotes.com/study-guides/calculus/calculus/limits/limits-involving-trigonometric-functions

also this video has a bunch of them from orgochem tutor

https://youtu.be/HbtuSC_WOW0

u can just do them before he does the sol

[frank vine]

https://usa1lib.org/book/446983/869360

[noble trench]

and if u don't get it u can watch him go through it

[frank vine]

There’s also this https://usa1lib.org/book/640448/28a134

[frank vine]

See above

[verbal pumice]

What do I google for this to find something to help me learn to solve it? I'm not understanding how my teacher is explaining it, to the point where I don't even know what to type in to find a video showing somebody solving this type of problem

I get that it's the difference quotient, and I know it, but I don't understand how to plug in the given interval in or anything like that

[frank vine]

Search up diff quotient on YouTube I guess

[calm fern]

integration of 1/(x-1)^3/5 (x-2)^7/5 dx

help this question is making me feel like dying

[gentle vale]

isn't the average rate of change from k(5.5) and k(12) just the slope between those points?

[last wasp]

i think so

[willow bear]

it is $\frac{k(12) - k(5.5)}{12 - 5.5}$, and everything needed to know to calculate this is given in the problem

[obsidian monolith]

Ann

[unborn yacht]

I'm out of school and haven't taken a college math class yet, so I'm not even sure if this is exactly the right channel. Dot products were never brought up, so I'm absolutely stumped by them. I'm trying to figure out machine learning right now, and this is one of the steps. From my understanding and 2 days of Google searches, the dot product is just a way of multiplying two matrices?

So I'm supposed to be getting the dot product of these two matrices, but what I'm not understanding is why. According to a Khan Academy video, the dot product is a human construct and was chosen pretty arbitrarily as the standard. But how does that work? I understand how to get a dot product, but I can't find anything about what the dot product actually represents compared to elementwise multiplication

And I'm just more confused, because the dot product exists for both vectors and matrices? It represents the distance between two vectors, so how does that relate to a matrix?

[hushed sphinx]

Terminology: What your screenshots speak about are matrix products, which are not usually called "dot product".

(One can think of the dot product between vectors as a special case of matrix products, but there are some subtleties required for that which it will not make you any wiser to care about at this stage).

[unborn yacht]

Oh yeah, that's getting me some more specific results

[hushed sphinx]

So the short nice story is that a matrix can represent a linear transformation between two finite-dimensional vector spaces, each with a chosen basis (the side lengths of the matrix correspond to the dimensions of the vector spaces). If you take two linear transformations and chain them together, the matrix that represents the combined transformation is exactly the matrix product of the matrices that represent the transformation you started with.
However, the underlined terms are probably not familiar to you yet, and you'll need to learn those first before you can really appreciate why the matrix product is useful and why it has been defined the way it has. There's probably no royal road to that, other than grabbing a book (or similar source) on linear algebra and start from the beginning. The first few chapters should take you a fair way.
(And then the topic will belong better in #linear-algebra).

[unborn yacht]

Thank you for the help! I'll look into those a bit more

[smoky drift]

i need help with this please

[hushed sphinx]

First simplify the definition of the function using a well-known trig identity.

[smoky drift]

critical = d/dx

when = 0

i get these

and idk how those cancel out to 0

since if lets say 1 + -1 = 0

but idk

[hushed sphinx]

There's a sign error in your derivative.

(But you shouldn't be using derivatives if the is only __pre__calculus, should you)?

[smoky drift]

sry i was meting to want to put in ca;lc

but didn't see tag chat

so this is calc

[smoky drift]

where?

[hushed sphinx]

What is the derivative of cosine?

[smoky drift]

-sin

x

-sin(x)

[hushed sphinx]

Yes.

So how did you get +cos(x)sin(x)?

[smoky drift]

oh my god

ok im just stupid, thanks for pointing that out

ive been stuggling though this for hours

[opaque bear]

hey idk if my brain is fried from doing hw all day but- is this not the right answer??? i thought the domain of all exponential functions was (-♾,♾️) ?!!??!

[hushed sphinx]

Looks right to me.

[opaque bear]

[dark aurora]

you know what is vector ?

you know what is oméomorhfils

???

phys

[ebon barn]

If in a rational function, the curve crosses a horizontal asymptote, what would the end behavior be, would it still approach the asymptote?

<@&286206848099549185>

[tidal zenith]

@opaque bear it's (0,inf)

[hybrid pewter]

use the fact that the integrand is of the form (f(x) + f'(x))e^x for some choice of f(x)
you can derive a formula for the integral of (f(x) + f'(x))e^x by considering the derivative of f(x)e^x

dont really know any way to motivate the solution though

[calm fern]

Got the solution

[sand dune]

the function from the right will approach the asymptote, but its limit can either be positive infinity or negative infinity

the two functions $f(x) = \frac{1}{x}$ and $g(x) = \frac{1}{x^2}$ show what happens really well

[obsidian monolith]

findingsouth

[icy yarrow]

Anyone her who just finished the math 30-1 diploma?

Hereeee

[marsh solstice]

Karaly, please don't ask that in every single group

[icy yarrow]

@marsh solstice sorry just curious

[obsidian hinge]

also you probably shouldn't just give out answers :/

[zealous vigil]

what should i attach more?

[obsidian hinge]

hints

although that is a beautiful solution wow

[obsidian hinge]

something like "maybe you could try to rewrite this in terms of some sort of substitution?"

and then the sqrt is the natural choice and it's obvious from there

[zealous vigil]

okay I will take care of it

[smoky drift]

nuce

[viscid thistle]

Is it possible to repeat a function a fractional number of times? For example f(f(x)) is 2 times

[vapid plaza]

The short answer is no

Probably because you’d need a new definition of “applying f to x n times” that extends the domain of n to other real numbers…
Which is, although possibly consistent with what your f is, often uninteresting🗿

For example if f(x)=x+1 then I can define applying f n times as x+n which is consistent with our original definition…because applying f^1/2 twice will give the same result as applying f once
But for more complicated functions like f(x)= cos x, I doubt such a generalisation exists

[willow bear]

the notion of "functional square root" is a thing but it doesnt exist for all functions

[viscid thistle]

I was planning on using the W function/productlog

[void thorn]

well supposing you can, W^-1(x)=xe^x so I would maybe try the easier task of generalizing f(x)=xe^x by looking at f(f(x)), f(f(f(x))), ... to see if there might be a way through that first, then if you find something you can plug a number in, make it negative

[dawn depot]

Anyone got any thinking questions for combining functions grade 12 advanced. Not too hard but can possibly be a test question. I want to practice

[solar pagoda]

For which numbers $a$, $b$, $c$, and $d$ will the function $f$ defined by $$f(x) = \frac{ax + b}{cx + d}$$satisfy $f(f(x)) = x$ for all real $x$?

[obsidian monolith]

BenJr

[dawn depot]

Doesn’t this question have infinite solutions

[hushed sphinx]

Yes (if "infinite solusions" means "infinitely many solutions"). It's asking how to distinguish those (a,b,c,d) that do have the property from those that don't.

Note that "all real x" means that some would-be solutions need to be discarded because because there's a single x that leads to division by zero even though f(f(x))=x for all other x.

[viscid thistle]

If −7<x≤6
determine a and b for the inequality :
a<3x+8≤b

Can anyone help me out with this?

[hushed sphinx]

Try sketching a graph of 3x+8 in the interval from -7 to 6.

[little parcel]

contour integral, how?

[obsidian hinge]

that goes not in #precalculus

[viscid thistle]

🤦‍♂️ sorry mate. I guess I am exhausted, I haven’t been sleeping well lately.

[obsidian hinge]

nw

wait

no i wasnt talking to u

ur fine

[obsidian hinge]

this guy

is who i was talking to

[little parcel]

wrong channel

[viscid thistle]

What do i do here

[sage maple]

anyone good at polynomial division and can hop in a call for a couple minutes?

[willow bear]

@viscid thistle do you still need help with this?

[viscid thistle]

Yes

[willow bear]

huge delay but ok

okay so do you know how to evaluate one function at a point when you're given the function in this format (i.e. as a set of input-output pairs)?

like, if i asked you to evaluate f(-2) for example, would you be able to do it?

[viscid thistle]

Yes

[willow bear]

right

and do you know what it means when we add, subtract, multiply or divide two functions?

[viscid thistle]

Yes

[willow bear]

ok then what exactly is the issue here?

you seem to know everything you need to know for all of these exercises.

[viscid thistle]

I do not know what it wants me to do here

[willow bear]

it wants you to calculate the given expressions or conclude their value is undefined

[viscid thistle]

How

It is a point

[willow bear]

for example, (f+g)(-3) = f(-3) + g(-3).
you told me you were familiar with this. is that actually the case?

[viscid thistle]

Yes

then jhow would you do a point like(9,8)

[willow bear]

oh, my bad, you took "point" to mean "point on the xy plane"...

that's not what i meant

sorry, it's common in higher math to refer to numbers on the number line as "points" when it is convenient

[willow bear]

but coming back to this, you are able to evaluate f(-3) and g(-3) and then add them, yes?

[viscid thistle]

Yes of course

[willow bear]

ok

then proceed similarly for all the other exercises in the section.

[viscid thistle]

So how would I evaluate the point in the function

[willow bear]

??

[viscid thistle]

Can yo ushow me an example

[willow bear]

am i understanding correctly that you are asking "How do I evaluate f(-3) and g(-3)?"

[viscid thistle]

No

[willow bear]

then what are you asking

[viscid thistle]

How do you evaluate a two numbers in a fybnction

[willow bear]

...i don't understand your question.

[viscid thistle]

Can you show me an example

[willow bear]

ok sure here's exercise 51:

(f+g)(-3) = f(-3) + g(-3) = 4 + (-2) = 2.

is this what you were looking for?

[viscid thistle]

Ok let see

OH

YOu find the x value in the set and output the y value

[willow bear]

i mean... yes

[viscid thistle]

Then you add it

[willow bear]

that's how you evaluate a function that's given like this

[viscid thistle]

I did not know how to do something given like this

But now I do thank you

[viscid thistle]

How do I solve this question: suppose you have a square with side lengths a and b which is inscribed in a circle with a radius of 10 cm whats the highest area the square can have

[obsidian hinge]

figure out the largest square you can inscribe in the circle and find the area of it

[willow bear]

did you mean a rectangle

[frank vine]

200

[uncut shoal]

"How do i solve this question" "200"

[civic sierra]

"Helpers"

[frank vine]

Meant max area of square would be 200 un squared

[sterile steppe]

can anyone help me with this? I keep getting 2 * cuberoot(4), and it gets marked as wrong

[hushed sphinx]

I get that too.

[sterile steppe]

Oh ok it must be a mistake in the test

[candid ivy]

b

[last folio]

I need help with this question.

[willow bear]

??

what even is this

"Find the equation of a hyperbola given this equation of a hyperbola"

theres nothing to find???

[last folio]

sorry I sent a wrong question?

[willow bear]

ok

@last folio do you still need help with this? sorry i only saw this now - was busy irl

[last folio]

yh

[willow bear]

are you familiar with the law of sines?

it's going to be of use here

[last folio]

yh

[willow bear]

ok then are you able to apply the law of sines here?

[willow bear]

bad handwriting + bad shape for root symbol + silent introduction of the variable x

does x stand for angle B or angle C?

[last folio]

angle B

[willow bear]

ok then write B and not this strange "x" that came out of nowhere

but yes you are correct angle B is indeed 30°

[last folio]

okay

[willow bear]

are you able to continue?

[last folio]

no

[willow bear]

you know two of the angles in your triangle.

[last folio]

yh

[willow bear]

is there anything else that you can calculate from this?

ok, so what can you calculate from this?

[last folio]

angle C

45 + 30 + C = 180

[willow bear]

great then do so

[last folio]

Angle C = 105 degrees

[willow bear]

are you able to finish this exercise now?

[last folio]

yh buh I have with zero idea about it?

[willow bear]

are we going to abandon the previous problem now?

after we've walked through like 90% of it already?

[last folio]

no

[last folio]

there's more to find ?

[willow bear]

of course? there's still that third side that you haven't found.

and instead you choose to throw another, unrelated problem at me.

[last folio]

sorry bout that.

[last folio]

line A B

[willow bear]

yes, AB.

you have not yet calculated AB.

[last folio]

yh

[willow bear]

do it

calculate AB

[last folio]

how do I calculate for it.

[willow bear]

you've used the law of sines once

you are definitely able to use it again

unless you refuse to do so on principle

[last folio]

okay

line AB = 13.52

[willow bear]

sounds about right.

[last folio]

Alright.

[last folio]

I appreciate your time helping me bless up.

[steel tulip]

If $x = t^2$ then $t = \sqrt{x}$ Then you substitute that value for t into the equation involving y and t to get an equation with only y and x

[obsidian monolith]

Tiessie

[opal tree]

Basically solve for t in terms of x

Then plug that t equation into the y parameter

[bitter frigate]

If you have to use a graphing tool they give you to graph a function, and that tool only lets you graph integers, do you just have to start plugging in random numbers and hope they end up being integers?
For example let's say you have to graph f(x) = 2(x-6)^2+2, and the graphing tool they give you only goes up to 20 on each axis, do I just have to plug in each number between 1 and 5 until I find an integer <20?

[hushed sphinx]

Who are "they"? Do you not have a pencil?

[bitter frigate]

online class

which would say something like "Graph this function" and you'd have to use the tool they give you on their website to submit your answer

[celest yacht]

well this function in your example is given to you in vertex form and the vertex in this case is at (6,2) so just plug in numbers to the left and right of that?

[bitter frigate]

right and since 6 is the axis of symmetry I'd only have to worry about one side right

and then I plugged in 0 and got 74

which gave me the 1-5 for numbers to check to be <20

[celest yacht]

also in this case any integer you plug in will output an integer

[bitter frigate]

oh I am dumb

my mistake

[royal ridge]

you are in a $math$ discord server, how dumb can you be

[obsidian monolith]

Hallows

[vital ivy]

@last folio it will be (y+2)**2 = x

[warped knot]

oh

can someone help me with

[frank vine]

Here’s your answers

[frank vine]

^

[willow bear]

@frank vine don't give out answers.

[frank vine]

Alright but i included steps

[snow bronze]

You might wanna 'try' finding integral roots, for the leading coefficient is 1, check the integral factors of -192.

[warped knot]

Can someone@pls help w this

Is due vv soon

[bitter frigate]

when applying transformations to quadratic functions why does addition shift x coordinates to the left, but when applied to y coordinates they get shifted up?

specifically why do x coordinates get shifted left and not right

[sly ocean]

the unknown zero would be 3 - 3i since the polynomial has only real coefficients and according to the complex conjugate theorem: if a polynomial with real coefficients has a complex zero, the conjugate of the complex zero will also be a zero of the polynomial. after all, if a polynomial has real coefficients and complex zero, we need the conjugate of the zero to cancel out the terms with imaginary coefficients.

so the final answer would be (x + 5)²(x + 3 + 3i)(x + 3 - 3i)

[sly ocean]

if the input is added by 1 (turning x² to (x + 1)²), the graph shifts to the left since the inputs will be added by 1 before being evaluated

so if you input 1 you will get (1 + 1)² = (2)² = 4. the one becomes a 2 before being evaluated.

if you input 2 you will get (2 + 1)² = (3)² = 9. the 2 becomes a 3 before being evaluated.

basically the inputs are being "pushed" further by one

this applies to all functions and not just quadratic functions, i.e. all functions shift to the left if you add to the input before evaluating

feel free to point out any mistakes

[bitter frigate]

Oh so it's like you reach the y coordinate sooner than you would have, so the graph shifts left?

[sly ocean]

yeah

[viscid thistle]

is there any proof of that theorem

[sly ocean]

https://en.m.wikipedia.org/wiki/Complex_conjugate_root_theorem#:~:text=In mathematics%2C the complex conjugate,also a root of P

[viscid thistle]

kk thx

[sly ocean]

here's a "proof" that is more easier to understand:

[viscid thistle]

wiki one was better

[bitter frigate]

Are there any good videos explaining how this step works

[willow bear]

5-11 = -6

[bitter frigate]

oh lol

[placid cairn]

lmao

[thorn valley]

Can anyone help me understand this? I know how to do this process in reverse

But this is more intimidating and complex it seems

[willow bear]

do you know how many minutes make a degree? @thorn valley

[thorn valley]

I'm not sure. I just know that in a D M' S'' format M is x/60 and S is x/3600. So D is probably just x/1 right?

[willow bear]

ok then let's forget about angle

do you know how many minutes of time there are in an hour

[thorn valley]

60

[willow bear]

right

the relationship between hours, minutes of time and seconds of time is the exact same as between degrees, minutes of arc and seconds of arc.

1 degree = 60 arc-minutes and 1 arc-minute = 60 arc-seconds

this should give you enough info to convert 0.7° into arc-minutes.

[thorn valley]

That would just be 0.7 * 60 correct?

[willow bear]

yes

[thorn valley]

Okay I understand you this far

So what would I do for seconds? Nothing...?

yay!

I got it!

Thank you Ann!

[willow bear]

not nothing

zero

[viscid thistle]

hey can someone help with a question?

i got -2x-y^2 cos(xy^2)/ 2xy cos (xy^2)

but im not sure if its correct

[uncut mulch]

do you have a pic of your work

its hard to read your result as it doesn't feel like you have parentheses where appropriate

[prime moat]

https://www.khanacademy.org/math/precalculus would this be enough before ap calc?

[frank vine]

Probably yeah

[prime moat]

so Im trying to learn precalc since I am trying to take ap calc ab after this summer. I dont thnk i can finish this khan academy actually within a month. Do you have any good idea I can use to learn all precalc stuff I need?

[viscid thistle]

For calculus I would want to know a lot of trig, exponential properties, logarithmic properties, systems of equations, and basic algebra rules

[polar moss]

ye

[jovial cosmos]

Keeping it real, I completely skipped precalc and did fine in calc BC

I’d just review parametric equations, conics, and polar coordinates (but I think some of this is bc, don’t remember)

That’s all you really need to know from pre-calc assuming you are strong in algebra (which you should definitely brush up in). Definitely some trig as in double angle formulas, thinking in terms of rad, sum and difference of angle formulas, but I learned that in alg 2 but I know some learn that in pre-calc

[jovial cosmos]

^ good advice as well

[alpine wraith]

that’s calc? No?

[viscid thistle]

yea u kinda got confused

on where to put it

sorry

[alpine wraith]

no it’s fine I was just wondering

do you want help?

[viscid thistle]

no i got the answer the people in calc helped my thanks tho

[jaunty pendant]

starting calculus in grade 10 for gcses should i go to precalc or calc?

[rare fiber]

have you done pre calc before?

[jaunty pendant]

not sure really

the chapter im doing in differentiation 1 and 2 but if i look it up calc comes is as one of the top results

[hushed sphinx]

If you're differentiating functions, then that is surely #calculus.

[wooden field]

Anyone know what this is?

wouldn't it be 2(-x)^1/2

[solar pagoda]

No it says “Horizontally” not vertically.

[wooden field]

so -2(x)^1/2?

[solar pagoda]

No, read my previous reply. Also play around with this https://www.desmos.com/calculator and read this https://www.mathsisfun.com/sets/function-transformations.html.

[wooden field]

Thank you

[wild cypress]

what sites would you recommend for precalculus??

Is khan academy good?

[frank vine]

Yeah

[viscid thistle]

what is a function which has derivative of all orders?

[hushed sphinx]

Looking for a definition, or for examples, or for the word "smooth"?

[twilit jasper]

polynomials would be one set of examples

[placid cairn]

e^x

[solar pagoda]

That is a polynomial 😀

[willow bear]

e^x is not a polynomial

[solar pagoda]

I have to start using these emojis stuff properly.

[frank vine]

Yeah e^x exponential not polynomial

[elfin zinc]

Can someone please explain me the reason of the intersection points of a function with its inverse its found simply equalating it to X. Example, f(x)=ax²+bx+C intersection points with its inverse can be discovered equalating ax²+bx+c=x and solving it. I am seeking to know a possible mathematical explanation for that.

[uncut mulch]

functions and its inverse are symmetrical across y=x

[hushed sphinx]

That doesn't find all intersections between (the graph of) a function and its inverse, though.

For example, f(x) = (pi/4 - 1)x - arctan(x) is bijective R->R, and intersects its inverse at (-1,1) and (0,0) and (1,-1); only one of these intersections satisfy f(x)=x.

The intersections with the inverse all correspond to solutions of f(f(x))=x, though.

[elfin zinc]

Yeah but what is the logic behind equalating the function to its varible to find inverse points?

[elfin zinc]

Can you give me an example, im not sure what you mean

[hushed sphinx]

If (x,y) is a point on both the graph of f and the graph of f^-1, then one of those means y=f(x) and the other means x=f(y). Insert the former in the latter, and you get x=f(y)=f(f(x)).

This is in particular true if x=f(x), so solving that equation gives you some of the points of intersection.

[viscid thistle]

can someone give me insight on how this is wrong?

i thought it would be correct but i believe im missing something

[vapid plaza]

The denominator cannot be 0

[viscid thistle]

Oh

So will it be what I wrote + excluding - 8 and 8?

Wait not 8

Oh

-25 and 25 then

Okay that makes sense

[elfin zinc]

Thx got it

[burnt lodge]

Let f^-1 (x) be y, then swap y and x

Solve from there in the form of y=

[ornate cargo]

Hey whats up, i have a question, whats the differences between ⊆ and ⊂?

[civic sierra]

A⊆B (read: A is a subset of B) is saying that every element x in A is also an element in B

A⊂B (read: A is a proper subset of B) is saying that every element x in A is also an element in B, however there is at least 1 element in B that's not in A

[hybrid pewter]

in some places ⊂ is used for subset and ⊊ for proper subset so just keep an eye out for context

i like to use ⊆ and ⊊ to avoid ambiguity

[ember aspen]

I dont understand this quesion

[prime moat]

would it be good idea to learn and review through the calculus readyness test? I just finished algebra 2 and now thinking of skipping precalc and go into ap calc ab

[alpine wraith]

@prime moat I’d recommend you ensure your knowledge of trigonometric, logarithmic, exponential, and rational functions first

And ensure memorization of the unit circle

most of the other stuff in precalc doesn’t pertain to calc

[prime moat]

i already know full unit circle yay

[worthy coral]

what does this notation mean $f(x)=min{x,x^2}$

<@&286206848099549185>

[worthy coral]

@unborn cloak do you know this

[unborn cloak]

Yea I do

[worthy coral]

pls tell

[covert zealot]

It is the minimum between the two elements

[unborn cloak]

It's the the minimum of the function x or x² so suppose at x=1/2 x²<x so we'll take x². Continue like this

[worthy coral]

i see at [-infinity,0] and [1,infinity] we will take x

and at 0<=x<=1 we will take x^2

thanks

[unborn cloak]

Yep

[prime moat]

ive went through ecponential laws, simplifying rationals, polynomial division

how far am i for ap calc preparation

[digital lance]

Hey guys sorry to bother you studying for my exam at the moment and these logarithms equations are brutalizing me. Would appreciate some help.

[humble elbow]

Pre caluclator

[digital lance]

hm?

[humble elbow]

I need help

[viscid thistle]

hey guys. I’m a new 9th grader and I need help learning calc, dm me.

[mild swan]

Just ask questions in #calculus

[wintry tide]

Helllo

[hybrid pier]

does a reciprocal function need to have a numerator of 1?

[hushed sphinx]

What is a "reciprocal function"?

The reciprocal function is the one that maps x to 1/x for every x.

[ripe palm]

Hey, does anyone know how to do word problems like this?

<@&286206848099549185>

[obsidian monolith]

Eren Yeager

[summer basalt]

need some hints

ping me while replying

[ember aspen]

how can you simplify the logarithm in the left hand of the equation?

[daring drum]

Ty

[bitter frigate]

So graphs can intersect with asymptotes? Or only horizontal asymptotes?

And can they fully cross an asymptote if they do intersect?

And if horizontal and slant asymptotes can be crossed, what are they actually there for?

[pastel magnet]

As far as I'm aware, a graph can't intersect its own asymptotes, that's the entire point - the function approaches the asymptote, but never crosses it after a finite distance.

[uncut mulch]

As far as I'm aware, a graph can't intersect its own asymptotes, that's the entire point - the function approaches the asymptote, but never crosses it after a finite distance.
no

misconception

for example take y=sin(x)/x
that has a horizonal asymptote of y=0
but crosses the that infinitely many times

[bitter frigate]

So I'm guessing a horizontal or slant asymptote that gets crossed implies that the graph is discontinuous and there is another section that never crosses it?

Or is that not true?

[uncut mulch]

no

stuff like horizontal and slant asymtotes describe end behaviour

and by themselves don't say much about anything else

[bitter frigate]

oh interesting, I'm guessing their use will make more sense to me once I'm deeper into calc

[ripe palm]

The top photo contains the details, the bottom are the individual questions

The pencil stuff is from me, and might be incorrect

<@&286206848099549185>

[thick berry]

Does anyone know how to find the two imaginary roots

<@&286206848099549185>

[summer basalt]

you cant find imaginary roots in the real cartesian plane

[fluid yacht]

Please help me q 7 and 8 <@&286206848099549185>

[hushed sphinx]

It looks easier to start by writing an equation for f(x)-10, which seems to have three real roots on nice grid points.

[bitter frigate]

Why when somebody asks for the x intercept, you don't need to specify which axis your answer's on, but if somebody asks for the vertical asymptote, you need to specify with x= before the value?

Is the second example not implicit like the first example?

for example:

[vapid plaza]

The x-intercept is the “number” on the x-axis which the curve passes through
The asymptote is the “line” which the curve approaches

Therefore to represent that line you need to use an equation for that line, say x=7

[bitter frigate]

oh that makes sense, ty

[bitter frigate]

Is there a reason why this specifically says third row and not quotient? And is this true for normal polynomial division as well?

[burnt lodge]

Not for normal. Says synthetic for a reason

[bitter frigate]

It wouldn't end up being the same coefficients, constant, and remainder with the same signs?

[echo mortar]

hey can someone help me with a question i have?

[cobalt knot]

#❓how-to-get-help or just post it here

[echo mortar]

for sure

i dont understand how i could rewrite it

and the textbook also doesnt really help me

could anyone help me with these couple of questions?

i want to be able to understand the concept better

[uncut mulch]

try not to overthink this

a = b + c
means
b = ?

[echo mortar]

?

a=b+c

b would be log4(x-3)

if i am understanding your explanation correctly

[uncut mulch]

yes

[echo mortar]

ye

so hhow can i rewrite log4(x-3)

im sorry but idk how a = b+c plays into this

[uncut mulch]

this is currently for the first box that you left blank

[echo mortar]

yea log4(x-3) was not the answer unfortunately :(

[uncut mulch]

I know

[echo mortar]

oh what

[uncut mulch]

try not to overthink this

a = b + c
means
b = ?

[echo mortar]

when yousaid yes i thought you meant that was the answer

well i got that one wrong, but ill try to learn it still

b= log4(x-3)

[uncut mulch]

the yes was to the b representing log_4(x-3) in the equation

[echo mortar]

i c

[uncut mulch]

but what I wanted was to forget about logs for a sec

and if you were just given
a = b + c

what would you answer if you were asked for
b

[echo mortar]

b

[echo mortar]

you should have said that to begin wiht

unless you mean like

[uncut mulch]

it was implied

clearly they don't want you to its say stuff like x = x

[echo mortar]

yea i know that much

i didnt know you want to forget about the log

so would b = 4?

[uncut mulch]

not

forget about everything else for a sec

given
a = b + c
b = ?

[echo mortar]

okay forget everything else for a se

if i am just given a = b+c on a piece of paper

and you ask what b is

i would say that b is b

[uncut mulch]

forget about everything (in the question)
don't forget about basic algebra

don't forget about a=b+c

clearly they don't want you to its say stuff like x = x
I don't want you to its say stuff like x = x

[echo mortar]

i understand that

idk what you mean by basic algebra when you say that

[muted turret]

Yo why is plotting points without a calculator so hard

[echo mortar]

use desmos and get the graphing calculator

[muted turret]

idk what to do for the next steps it is so hard smh

[echo mortar]

stuff is mad helpful

[muted turret]

Does the parabola opens downwards I am not sure

Cause ax is negative

[echo mortar]

oh you probably have ot factor -x^2+2x+3

factor that, and you get your 2 x intercepts

and then try to find the vertex wiht a formula (i forgot which one) and you can plot it that way

[uncut mulch]

perhaps I might have to make it extremely clear
starting with a=b+c
if you were asked to solve for b
(as implied by asking b = ?)
what would you do?

[muted turret]

I did

[muted turret]

I found the vertex I just don’t know how to plot the four points

[echo mortar]

lmao i see what you mean, b= a-c

[uncut mulch]

to get additional points, choose some values of x, plug into the equation to get the respective y values to get your points

[muted turret]

ohhhhh

[uncut mulch]

using nice integer values will make your work easier

you can also consider symmetry

[echo mortar]

when you do b=? , i am assuming you are asking me to fill in b, instead of make it so that the equation a=b+c is in terms of b

[uncut mulch]

it was pretty clear I wanted you to isolate b I'm the equation I gave you

if I wanted you to respond with b=b
I wouldn't have even mentioned the equation a = b + c

[echo mortar]

okay sorry for being dumb ig and not being able to read it the way you read it

oh well it is what it is, ill just take the L

[muted turret]

My first parabola done without a calculator 💪💪💪

[vapid plaza]

Looks kinda not like a parabola ngl

But nice

[sinful dome]

Help pls

[fleet forge]

Can I get some help please!

[willow bear]

@fleet forge do you still need help with this

[zenith solar]

Can someone help me with this please

[willow bear]

,rccw

[zenith solar]

I’m not sure if this will help but here is my working out, I’m not sure what to do past here/if I did anything wrong

[willow bear]

your work is correct and you already have the answer

P(-1) = 5

[heady horizon]

P(x)-(4-x) is divisible by 1-x²
Since 1-x²=(1+x)(1-x)
P(x) - (4-x) is divisible by 1+x

[fleet forge]

Hi yes please!

[willow bear]

14 hour delay but ok...

do you know, in general, what an arithmetic sequence is

[fleet forge]

Sorryy

Yeah

[willow bear]

ok

can you describe in your own words what it means for a sequence to be an arithmetic sequence?

[fleet forge]

Basically a set of numbers that are in order and it’s made out of terms and is a list of numbers where the difference or distance between consecutive terms is a constant

[willow bear]

...well there is a kernel of truth in that, i'll give you that.

it's a sequence in which the difference between consecutive terms is always the same

and this difference is called, fittingly, the common difference.

[fleet forge]

Ohhh

[willow bear]

now, you have the sequence that begins -38, -47, -56, -65, ...

what is its common difference?

[fleet forge]

-9

[willow bear]

correct

now you know the first term (among others) and you know the common difference

do you understand now how to write down an expression for the general term?

[fleet forge]

Isn’t the general term tn= a + (n-1)d

So would it be tn= -38 + (n-1)-9

[willow bear]

careful there

(n-1)-9 looks like you are subtracting 9 from (n-1), which is not what the formula you quoted says

you need to multiply (n-1) by -9

so you would want -38 - 9(n-1)

[fleet forge]

Right

[willow bear]

or, if you want to be absolutely 100% adherent to how the formula is written, -38 + (n-1)(-9) or something

but that's kind of silly

[fleet forge]

Okie thank you

[fleet forge]

Just to confirm it would end of being tn= -9n -29 as the answer?

[willow bear]

if you wanna simplify it sure

that works

[fleet forge]

Thank you for the help!

[alpine wraith]

@willow bear out of curiosity what was the process of getting honorable?

[willow bear]

that is classified information.

[zenith solar]

How do you show that there is only one rational zero

[willow bear]

well you've found one

now you can factor it out and show that the quadratic that remains doesn't have any rational zeros

[zenith solar]

Thanks

[zenith solar]

Sorry for the multiple questions but I’m not sure how to approach this

[willow bear]

you can rewrite P(x) by completing the square

[zenith solar]

What do you do after that?

[willow bear]

show me what you got from rewriting the thing

then we'll talk

[willow bear]

who said c = 4?

[zenith solar]

Oh

Then I’m not sure how to factor a quartic 😅

[willow bear]

well you could modify what you have here to not be wrong

$x^4 - 4x^2 + c = x^4 - 4x^2 + 4 + (c-4) = (x^2 - 2)^2 + (c-4)$

[obsidian monolith]

Ann

[willow bear]

the equation P(x)=0 thus rewrites as (x^2 - 2)^2 = 4-c

thus if 4 - c < 0 then there are no real solutions

and if 4 - c ≥ 0 then there is at least one

[zenith solar]

Ohh that makes sense, thank you 🙏

[alpine creek]

I need some explaining on these problems

[worthy coral]

can you pls tell me where should i post questions related to set theory

[willow bear]

don't ping random ppl out of the blue like that

but #discrete-math seems to be the place you're looking for

[worn venture]

How to solve this

[viscid thistle]

First simplify it, which will give you ${1+(1/x^4)}$

[obsidian monolith]

dimi

[worn venture]

Then what

[viscid thistle]

Apply it linearly, changing $\int1/x^4dx+\int1dx$

[obsidian monolith]

dimi

[viscid thistle]

Solve it by removing the 1dx, so we can solve 1/x4 on its own: $\int\frac{1}{x^4}dx$

[obsidian monolith]

dimi

[viscid thistle]

@worn venture

Now solve $\int1dx$ which would give you x

[obsidian monolith]

dimi

[viscid thistle]

Now sub in the integrals: $x-\frac{1}{3x^3}$

[obsidian monolith]

dimi

[viscid thistle]

Done! The answer is $x-\frac{1}{3x^3}+C$

[obsidian monolith]

dimi

[viscid thistle]

I think

[worn venture]

Can you please write it on a paper

I didn't get the whole idea

If it is possible then please write it on paper

[viscid thistle]

I'm sorry, I don't have any to hand.

[viscid thistle]

When I get some I'll show you, but I'd rely on one of the other, more experienced members, who actually know how to use LaTeX

[modern valve]

hello

[modern valve]

wait what

this isn't right

[viscid thistle]

Am I doing the wrong method?

[modern valve]

square root of (1 + 1/x^4) is not 1 + 1/x^4 ?

[worn venture]

Is it possible in this case
Coz we have square root all over equation
How can we split integration just inside the square root?

[viscid thistle]

SHIT I FORGOT TO WRITE THE ROOTS

Sorry I'm still getting to grips with LaTeX

[worn venture]

That's what I am saying

[modern valve]

can you post the full problem

likely you are not supposed to evaluate the integral

[viscid thistle]

Sorry about this, I was being stupid

[modern valve]

no I mean like the question text too

[worn venture]

I just need to know how to solve this integration that's it

[modern valve]

you can't by hand

so i'm trying to figure out what the context of it is

[worn venture]

It's ok brother

[modern valve]

can you like post the full page maybe

[worn venture]

I don't have that book anymore 😑😑

[modern valve]

:|

can you state like what the full problem is

I might be able to help better

[obtuse jasper]

$\sqrt{\frac{x^4 + 1}{x^4}}$

[obsidian monolith]

MathOnNapkins

[obtuse jasper]

And maybe trig substitute / integrate by parts 🤔

[worn venture]

Find the value of L

[modern valve]

no there's no way

[modern valve]

are you allowed a calculator

[fallen pecan]

i am trying trig

[worn venture]

Yes but I have to do it on paper

[modern valve]

it doesn't work

wolframalpha gives it in terms of a hypergeometric function

[modern valve]

I don't think there is any way to evaluate this integral with the tools you have

besides just plugging it into the calculator

[fallen pecan]

[opal tree]

Why are integrals in the precalc section

[burnt lodge]

I have no idea

But they shouldn’t be there

[viscid thistle]

yes

[next cipher]

hey guys im gonna take precalc next school year and i was wondering what math i should brush up on to help me have an easier time
i feel like most of my mistakes are because i am not careful enough

[burnt lodge]

Trig

[mild swan]

Algebra I, Algebra II, Trigonometry

[obtuse jasper]

I honestly can't even remember what I specifically learned that I hadn't seen before in Precalc. Probably Trig. Otherwise it seemed like a lot of drilling preparation for Calculus

[burnt lodge]

It is preparation

Preparing for the preparation is Algebra and Geometry

[twin quest]

in terms of parametrics, when would I use the first y = formula and when would I use the second y = formula?

[twin quest]

nvm i figured the first one is in feet/sec and the other is meter/sec

[willow bear]

if you're trying to memorize both of these formulas then you're setting yourself up for overcomplication and failure

if anything the middle term should be $-\frac{1}{2}gt^2$, where $g$ is gravitational acceleration

[obsidian monolith]

Ann

[viscid thistle]

am i missing something here?

i thought it would be right but apparently not

[willow bear]

when is sqrt(x^2 - 64) = 15?

[viscid thistle]

Oh i am such an idiot

i mistook -13 and 13 with -17 and 17

that makes sense

On another note, how could someone go about finding specific functions in composite functions?

To find f(x) in something such as f(g(x))

I always used intuition to find them but it doesn't seem very efficient, is there a more reasonable approach?

[willow bear]

if all you're given is f(g(x)) then there cannot be a "more reasonable approach"

because compositional factorizations like this are not uniquely defined

[viscid thistle]

Well usually it is the other function as well

For example "Find g(x) given f(g(x)) and f(x)"

Or for f(x)

[willow bear]

i mean in that case if f happens to be invertible then g is recoverable as f^-1 composed with f o g

[viscid thistle]

Oh!

I never thought about that

Canceling the functions by taking their inverse

Yeah that does seem very reasonable thank you

[astral bobcat]

sorry i have a question
lim x -> 0, sinx/x = ?
without consulting derivatives

[willow bear]

are you allowed to use the squeeze theorem

[astral bobcat]

yes but

is it work

[willow bear]

do you mean "does it work"?

[astral bobcat]

yeah sorry

[willow bear]

prove $\cos(x) \leq \frac{\sin(x)}{x} \leq 1$ for $x$ sufficiently close to zero. that'll get you a proof that $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$.

[obsidian monolith]

Ann

[viscid thistle]

Unrelated but curious, but do you write all of that yourself for the @obsidian monolith bot to process it?

[willow bear]

yes

[viscid thistle]

ah i see, seems complicated

[viscid thistle]

i am curious now, though. How did we build up into the cos(x) leq sin(x)/x leq 1 to begin with?

I would imagine using sandwich theorem you would have something along the lines of -1/x and 1/x on both ends

[willow bear]

it isn't once you've had like 5 years of experience with it

[viscid thistle]

i guess i graphed it somewhat in my mind so it does make sense i suppose, but trying to find a way to derive them mathematically

[viscid thistle]

Fair enough

[astral bobcat]

lim x -> ∞, sinx/x

[astral bobcat]

is there any alternative solution possible rather than using sandwich theorem

[viscid thistle]

ohh i thought we were talking about x as it approaches 0

yeah makes sense

[willow bear]

well you could expand sin(x) into its taylor series whence the result would follow immediately, but that ultimately relies on derivatives so i imagine it wouldn't fly

[astral bobcat]

i understand thank you

i can't proof cosx < sinx/x < 1

$\cos(x) \leq \frac{\sin(x)}{x} \leq 1$

[obsidian monolith]

theman

[astral bobcat]

i mean this

[willow bear]

can you prove that sin(x) <= x for positive x?

and that tan(x) >= x?

i don't know if you have a formal definition of sin and cos on which you are supposed to base your proof.

@astral bobcat

[astral bobcat]

i can't

the mean value theorem?

[willow bear]

...

mean value theorem relies on derivatives so that is a no-go.

tell me what your class's definition of sin(x) is.

[astral bobcat]

the opposite side of the angle divided by the hyptenuse in a right triangle

well not exactly

[willow bear]

...

then show me the EXACT definition for god's sake

how the fuck do you expect to produce any sort of proof if you don't even have any definitions on which to base it???

[astral bobcat]

that's the deifinition i can express

i don't knoow

[willow bear]

no definition no proof

simple as that

maybe look through your notes or ask a classmate

[astral bobcat]

how about i get the unit circle

[willow bear]

ah well NOW we're talking.

[astral bobcat]

get a triangle with the field cosx.sinx/2

[willow bear]

with area cos(x)sin(x)/2.

[astral bobcat]

and a circle slice with pi.1^2.x/2pi

[vapid plaza]

You mean a sector?

[astral bobcat]

and a bigger triangle with the area tanx/2

and compare the areas

you know what i get it

[placid cairn]

circle slice

LMAO

[wise spoke]

can i ask about diffrentiation here?

[vapid plaza]

Wouldn’t that be more fit for #calculus?

[wise spoke]

How many turning points would a function $y = \frac{f(x)}{g(x)}$ have if the power of both f(x) and g(x) is two and the discriminant of f(x) < 0?

[obsidian monolith]

∫Inheritanc-e ♦

[willow bear]

do you mean the degree of f and of g are both 2

@wise spoke i believe that this info alone is insufficient to determine the number of turning points

there exist rational functions satisfying your requirements with 2, 1 and even with 0 turning points

[copper vigil]

,w d/dx (ax^2 + bx + c)/(dx^2 + ex + f)

that's a start i guess

[obsidian monolith]

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[edgy spruce]

Take it to calculus

[wise spoke]

Oh

[alpine wraith]

sigh

<@&268886789983436800>

[stuck lark]

already gotem

[alpine wraith]

gracias por te ayuda