precalculus

[wintry coral]

It cleans up decently but I'm having trouble with limits

Also I'm having some issues with the asymptotes at x=0,1/2 and 1

[midnight quiver]

What do y’all think is the best way to learn set builder notation?

[worn wraith]

i have no idea

i just learned with excercises

[midnight quiver]

Do you think a memory game would help?

[raw spear]

memory game? Like an actual game? Games are horrible for memory, they will only fill up the mind with ways to be better at the game.

[tepid fable]

i can get the first two equations by solving the basic data given ( got c=3) but for the third , i dont know how to get it , i applied y'=0 for x=-2 and it gave me the right answer and the solution shows the same but i dont get the reason behind it

can someone please help

[tepid fable]

i got it

i didnt see that it said "touches " the x axis which should make it a local minima or maxima , hence y' should be equal to zero at x=-2 ( if im getting it wrong please correct me )

[viscid thistle]

Why does it say the server is down

[silk rock]

april 1st

[opal tree]

No that's by coincidence

The server was down

But it's stuck as is

The channel renaming is the actual joke

The server being wonky isn't

[silk rock]

totally

[crude plover]

1^2 = 1, 1^3 = 1, 1 = 1, 1^2 = 1^3, log_1(1^2) = log_1(1^3), 2(log_1(1)) = 3(log_1(1)), 2(1) = 3(1), therefore 2 = 3

[tawdry idol]

,tex 3 + 4

[obsidian monolith]

kvomi

[tawdry idol]

,tex 1^2 - 4x sqrt-10

[obsidian monolith]

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

[viscid thistle]

wtf is precals

[tawdry idol]

1^2 = 1, 1^3 = 1, 1 = 1, 1^2 = 1^3, log_1(1^2) = log_1(1^3), 2(log_1(1)) = 3(log_1(1)), 2(1) = 3(1), therefore 2 = 3

,tex 1^2 = 1, 1^3 = 1, 1 = 1, 1^2 = 1^3, log_1(1^2) = log_1(1^3), 2(log_1(1)) = 3(log_1(1)), 2(1) = 3(1), therefore 2 = 3

[obsidian monolith]

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

[tawdry idol]

lolll

[wintry coral]

?

The solution I sent is wrong btw

I just happened to get the right answer

I have the actual solution if anyone cares

[night spoke]

wtf is precalc

[deft compass]

wtf is precalc

[viscid thistle]

bro channel name

[weary saddle]

wtf is precalc

[viscid thistle]

wtf is precalc

[hexed wraith]

wtf is precalc

[near willow]

wtf is precalc

[shrewd flint]

but like actually, precalc's content for my highschool was just a review of algebra2+trig+inverse functions

[opal tree]

wtf is precalc ong

[dull tendon]

wtf is precalc

[vast matrix]

wtf is precalc

[silk rock]

wtf is precalc

[bitter bolt]

precalc is wtf

[white bear]

wtf is precalc frfr

[foggy flicker]

channe just changed

[viscid thistle]

XD

okay

that was sick

[hollow orchid]

Honestly I’m about to take it next year

What do I expect

is it just algebra 2 just even more stupid

[hexed wraith]

Smth like that

You'll feel even more stupid

[weary saddle]

tis april fools

[viscid thistle]

😂

[floral ore]

wtf is precalc

[strong swallow]

budget calc

[viscid thistle]

Yea
Tf is precalc??

*clac mb

[nova bison]

wtf is precalc

[viscid thistle]

wtf is precalc

[jovial plinth]

Wtf is precalc

[solar tiger]

wtf is precalc~

is it like before calculator or some shit?

[dense cove]

precalc wtf

[solar warren]

Dafuq is precalc

[hexed wraith]

Alexa, what is precalc?

[hoary ferry]

,w precalc

[obsidian monolith]

Wolfram Alpha doesn't understand your query!
Perhaps try rephrasing your question?
Click here to refine your query online

Requested by Chalknoise#9545

[viscid thistle]

wtf is precalc

[viscid thistle]

yeah man, i only know real anaylsis.. now wtf is thsi precalc?

[floral ore]

You guessed it.
According to the Wikipedia the precalc era refers to the time before calculators were invented and used. During that period people used to perform mathematical operations by hand.
This era lasted from the birth of mathematics to the birth of the calculator. Now, we're in the calc era.

[solar tiger]

DAYUM

[turbid sparrow]

hi

[hollow orchid]

well shit

am i gonna be forced to not use my calculator

[topaz coyote]

clacerp si ftw

[tepid sorrel]

how do i improve my algebra

there are some topics like

[hushed sphinx]

Practice

[tepid sorrel]

sequence and series

binomial theorm

[tepid sorrel]

any

book suggestions

i need some good materials

[hushed sphinx]

Sorry, I don't have that.

[tepid sorrel]

itsalr

[viscid thistle]

are you preparing for JEE?

If so, for algebra, I'd recommend Adas gupta.

[full condor]

In mathematics education, precalculus is a course, or a set of courses, that includes algebra and trigonometry at a level which is designed to prepare students for the study of calculus, thus the name precalculus. Schools often distinguish between algebra and trigonometry as two separate parts of the coursework.

[floral ore]

I'm sorry but that definition is factually incorrect.

What source do you have to back your claims?

As I've mentioned above, precalc refers to the era before the invention of calculators, hence the name precalc.

[solar tiger]

go practice

[solar tiger]

i think its wikipedia

[viscid thistle]

It stands for precalculus, preparing you for calculus, according to Bing (Cited 2 sources - definitions.net and cuemath.com)

Hi guys. I'm not sure where to ask that type of questions. Why for a that approaches 0, log_a(x) = a^x?

Ping me If you know

[hexed wraith]

There are help rooms for reason

[white carbon]

That question easy but I’m too lazy

[ionic kettle]

Delta math!

[undone garden]

i hate partial fraction decomposition

[tepid cargo]

Apply Euler’s identity
Polar to rectangular conversion

Then do the multiplication and division of complex numbers

[inner spoke]

im new but wtf is precalc

[tepid cargo]

Studies which prepares u for calculus

Algebra and trig knowledge which are necessary for it

[bronze summit]

What happened to the pre-alebra chat?

I cannot see it

[cursive cairn]

#prealg-and-algebra

[floral ore]

wtf is precalc- wait

[tame belfry]

its precalc

[north remnant]

Could somebody help me find what I did wrong?

[vapid plaza]

the differentiation part is wrong; the derivative of (x^2 sinx) is not (x^2 cosx)

[north remnant]

Isn’t the coefficient always left alone if multiplying to a function?

[summer ruin]

x^2 is not a coefficient

[north remnant]

Ok so the derivative of x^2 is 2X and the derivative of SinX is CosX. Then use U’V +UV’?

Then that’ll give the entire derivative of the function which I can then use to find the equation for the line tangent?

[summer ruin]

yes

[north remnant]

Ok thank you 😊

[slim steppe]

Use product rule for that

[silver otter]

blyat

[slim steppe]

No cursing

[indigo halo]

Question: is integrals precalc or calc

[hushed sphinx]

There's no way they could be PRE-calculus.

[autumn wadi]

[tight vortex]

Is there a trick to this one I don’t know

d is the correct answer btw not c

[hushed sphinx]

Have you heard about multinomial coefficients?

[tight vortex]

nope

[hushed sphinx]

Okay, that makes it a bit more involved to explain.
Basically, we're asking ourselves: when we multiply out (x-y+2z)(x-y+2z)···(x-y+2z), how many times do we get a term that simplifies to x^3(-y)^4(2z)^3?

We get such a term for each way there is to choose three of the ten factors to supply the x's, four of the remaining seven factors to supply the (-y)s, and the remaining three factors to supply the 2z's.

[tight vortex]

I think that makes sense but why is the y negative

[hushed sphinx]

Because the thing you raise to the 10th power is (x-y+2z) rather than (x+y+2z).

(I fixed some typos, btw, hope it is correct now).

[tight vortex]

oh ok

[hushed sphinx]

So the number of ways to do that is (10 choose 3) times (7 choose 4).

Which turns out to be 4200.

[tight vortex]

Yep

[hushed sphinx]

We need to multiply by the constants we can pull out from x^3(-y)^4(2z)^3, giving 4200·1^3·(-1)^4·2^3.

(The only constant that makes a net contribution is the 2 in front of z. The negation of y disappears when we raise it to an even power).

A shorter way notation for the combinatorial factor $\binom{10}{3}\binom{7}{4}\binom{3}{3}$ is $\binom{10}{3,4,3}$, which is called a \emph{multinomial coefficient}.

[obsidian monolith]

Troposphere

[hushed sphinx]

If we substitute in the formula for binomial coefficients and simplify a bit, we get
$$ \binom{n}{a,b,c} = \frac{n!}{a!b!c!} $$
but beware that the notation is only defined when $a+b+c=n$.

[obsidian monolith]

Troposphere

[tight vortex]

which makes the answer

[tight vortex]

can you explain this part again please

Oh nevermind I got it

Thanks a lot

[stuck lark]

@robust bear this isnt the place for memes

[restive seal]

what to expect after pre cactus ?

[slim steppe]

cactus

[stable pond]

cactus 1

[restive seal]

cactus D:

[old isle]

can some1 help w this q? the question is to find k

[gritty wren]

k is potassium ?

[old isle]

No

k is a variable

[hollow badge]

lmao

[vapid plaza]

wow omg u so funny

[gritty wren]

yes I know

i failed math 3 years ago xx

/jay

[vapid plaza]

k not even in capital letters so it wouldn’t be true

:/

[gritty wren]

im good at math at year 8 level 😹

[teal jetty]

you may make it like (1^2+2^3+3^3+...+2024^3)-2(2^2+4^3+....+2024^3)

[old isle]

could you elaborate on what effect would that have? I'm really struggling with the question.

[teal jetty]

you have a formula for sum of cubes

it says 1^3+2^3+3^3+....+ n^3 = ((n(n+1)/2))^2

[teal jetty]

so in this the first part you get easily by this formula

and for the second part ie the substraction part

you can take 2^3 common from all terms so that you get 2^4(1^3+2^3+3^3+....+1012^3)

[teal jetty]

now by this formula things can be simplified easily

did you get it??

[old isle]

how did you get to this part?

[old isle]

I get this part and how to solve it

[teal jetty]

you see a-b

can be written as

a+b -b-b

= a+b -2b

similarly i just added and subtracted all the negative terms

[teal jetty]

and got this

you got it now??

[old isle]

im still a bit confused. From this equation, did you factor something out or rewrite it with a different method?

[teal jetty]

rewrite it

doing some useful manipulations

like if you add and subtract 2^3 simultaneoulsy

it won't affect the result

did you get it now??

[old isle]

OHHH okkayy I got it

thanksss!!

[teal jetty]

sure na??

[old isle]

yupsss

[teal jetty]

nice

[old isle]

so I got until here but idk how to proceed further to find k

its a non calculator question

the final eq i got is written down

[solid elm]

I failed my logarithmic test. How do I relearn logs?

[mortal urchin]

So you want 1012^2 * k so maybe try factoring it out first.
The remaining bit, do you notice anything about esquire numbers, and looking at their difference?

[teal jetty]

you can simplify by taking terms common

[viscid thistle]

first thing that should come to your head is khan academy

just search for it there, the dude will probably teach it to you better than most of your teachers

( at least what its like from my experience )

[viscid thistle]

Do harder problems. Search for a problem set online, and practice.

You'll find out what you don't understand by doing exercise

[dreamy prairie]

what is precalculus

[slim steppe]

It's the foundation for calculus

[stable pond]

or o-chem tutor on youtube

I’m doing conics now. I use his vids

[safe oak]

Current state of my quadratics program I’m making on Desmos

Looks bad because it’s on mobile, I’ll also probably make a mobile version

When finished it will show the expression in factored form, it should show roots in radical form if they are irrational, it will show the expression in vertex form, it will show the axis of symmetry and more

[viscid thistle]

i dont get the deal with derivatives

they're defined as the instantaneous rate of change of a function

but i don't know how to really visualise that or really do anything with that information

[dull tendon]

It's just the slope of the tangent to the function at a point

instantaneous just means keep zooming in (means keep approaching a particular point)

[willow bear]

there's a commonly cited visual in that if your function is the position of an object in time, then its derivative is velocity

[visual hill]

Can someone help me evaluate this step by step

[cerulean bronze]

Is that pi or x?

[visual hill]

Pi

7pi/4

[shadow sequoia]

dont cos^-1 and cos just cancel out because theyre inverse

[visual hill]

Im not sure

Thats y im asking lol

I just know ok that 7pi/4 is sqrt2/2, sqrt2/2 according to the unit cricle

So is the ref angle pi/4

[viscid thistle]

personally i think ochem tutor doesnt teach super well, he just throws a lot of problems at you and shows you how to solve them, whereas khan academy has explanations, and then questions. both are good depending on what you are studying for

[stable pond]

fair

I like khan academy too

[slim steppe]

Inverse cos of 45 is pi/4

[viscid thistle]

1st cos(7pi/4) can be written as cos(2pi - pi/4)which is equal to cos(-pi/4) and cos(-pi/4)=cos(pi/4) as (-pi/4) is in the 4th quadrant in the 4th quadrant cos is positive. So now cos(pi/4)=1/root under 2 and cos inverse(1/root 2) = pi/4

[slim steppe]

meant to say root 2 over 2

[dark niche]

f^-1(g(x))

it is just

1/(f(g(x))

or not

🤔

-1 should be arcfunction

arrcos(cos(x))

[old isle]

thats jus equal to x

[dark niche]

or it is curve

[dark niche]

or 1/cos(cos(x))?

[viscid thistle]

Anyone know about using the discriminant and putting a point “inside” of the line that the line passes through (not tangent to), and then putting some point P (x_1, y_1) in the line to, that could be tangent to another given function you have?

Then using the discriminate to either see if the line that passes through the point and tangent to another point exist or doesn’t exist

It’s not your typical discriminant / tangent problem

[old isle]

no, its a straight line

[dark niche]

ok now we have three(!) options

[old isle]

none of them are correct

[dark niche]

using -1 is wrong

[hardy rivet]

Help 😭

I’m genuinely so lost doing word problems as you can see

[dark niche]

cos^(-1)
arccos()

[near cove]

This problem seems to be the only one I’m having issues with, I tried getting a new one but it just gives me the same numbers with sin or cos

[summer ruin]

do you see how this is a quadratic equation?

[near cove]

Yes

I tried factoring it but -1+-isqrt(7)/4 is what stumps me

[summer ruin]

have you made the substitution of the new variable in place of cosine before solving the quadratic equation?

[near cove]

Like just putting X instead of cosin

[summer ruin]

yes

[near cove]

So writing it like 2x^2-1-1?

[summer ruin]

2x^2-x-1, yes

[near cove]

Ooh actually this one is different

This one has a negative 1 in the middle

[hushed sphinx]

The discriminant is indeed -7. No it isn't, I'm a moron.

[summer ruin]

you can just apply the quadratic formula

this one has real roots though the discriminant is positive

[near cove]

So it would be like 1+-isqrt7/4

Same thing tho basically

[summer ruin]

you must've made an algebra mistake along the way because the roots are real numbers and not complex

[hushed sphinx]

You seem to have missed that c is -1, not 1, when you computed the discriminant.

[near cove]

Ah yeah

I see now

1+-sqrt9/4

So the solution is -1/2 and 1?

[summer ruin]

it's not

think about again what equation you were solving

it wasn't the original equation

[near cove]

Sorry lemme rephrase, is that the correct answer to factor this?

Just want to be more specific so I’m clear

[summer ruin]

yes

[near cove]

Oof I’m overcomplicating it
(2cos(w)+1)(cos(w)-1)
2cos(w)^2 - 2cos(w) + cos(w) -1
2cos(w)^2 - cos(w) - 1

Legit was just +1 -1

Meaning -1/2 and 1 for the answers

[summer ruin]

you figured out that x was equal to -1/2 or 1

but x was equal to cos(w)

solving cos(w) = x for w seems much easier now

[near cove]

So cos(1) is 0

[summer ruin]

you're plugging in w = 1 when that's not what w is equal to

x = 1

[hushed sphinx]

What you should be concluding now is: "cos(w)=1 or cos(w)=-½".

[near cove]

Yeah that’s what I concluded

[hushed sphinx]

But the question was what w is, not what cos(w) is, so you're not done.

[near cove]

No I know

[hushed sphinx]

This was a somewhat confusing statement, in that case :-)

[near cove]

Yeah my bad

So cos(w)=1 is 0, wouldnt be pi because it would need to be -1 right?

[hushed sphinx]

"cos(w)=1 is 0" does not make sense.

[near cove]

Well I’m looking for exact value answers

at 0, cos is 1 at 1,0

Right?

[hushed sphinx]

It doesn't make sense to say that the equation cos(w)=1 "is" any number, 0 or otherwise.

You could say "cos(w)=1 holds when w=0".

[near cove]

Okay

cos(w)=1. To find the answer I am looking for I figure out where the cos(w) would equal 1, which is at 0 degrees on the unit circle.

Then, to find the exact values of cos(w) = -1/2, I figure that these are also on the unit triangle as 60 degrees angles. Because its negative, I use 180 to figure out that the angles are 120 and 240, which when multiplied by pi and divided by 180 simplifies to 2pi/3 and 4pi/3

[hushed sphinx]

Right, except you're still taking about "values of cos(w)=-1/2". An equation is not a number. It does not have a value, exact or not.

[near cove]

My solutions are what need to be exact values here

Which they are

[proven siren]

I think they just skipped some words and they mean values of w which satisfy that equation

@near covein the future you might want to explicitly say that

because that is what you are asking for

[summer crag]

Is there a general way to solve inequalities involving cubic polynomials , in order to break down the cubic in quadratic or linear would require atleast one root , how can we find any "one" root of the cubic polynomial, hit and trial is not always applicable and cardano's equation is difficult to remember , factorization is also not always applicable, is there any other more general way which is always applicable ?

[summer ruin]

search the divisors of the constant term for possible roots

otherwise no, there isn't a general way

but you will never be given an inequality with a cubic polynomial which you cannot factor, in cases when that does happen I'd look at whether the inequality is always satisfied for that polynomial for all x

[summer crag]

What about expressing the cubic polynomial in (ax+-b)³ +- c form , this way we can graph it and then solve the inequality , I guess it can't be done always , right ?

[summer ruin]

that's effectively completing the cube and that's essentially what the cardano does

[summer crag]

that means it's "doable" ? , quadratic formula is also derived using completing the square and as you stated that cardano formula aslo effeftively does that , but how can we do it exactly ?

[summer ruin]

for a general cubic that's as doable as deriving the cardano's formula yourself

sometimes it's just obvious that the given expression is something that was cubed previously in which case the problem was setup that way intentionally

[summer crag]

a cubic polynomial "inequated with 0" will have atleast one real real root , so how can it satisfy the inequality ∀ x ∈ ℝ

[summer crag]

Alright , I will check out the "completing the cube method" for solving cubic polynomials.

[summer ruin]

I guess that's not what I meant, what I meant is that the solution of the inequality is irrelevant in the broader context of the problem

[summer crag]

oh , sorry I misinterpreted your statement , it makes sense that the solution of inequality is irrelevant in the broader context in such case.

[summer ruin]

I don't have an example on my hands right now, but there are examples of logarithmic inequalities with cubic terms in them, which you would want to figure out sign of to find the domain of the problem before tackling the primary inequalities

but it might just so happen that both sides of the inequality are undefined for some points and hence there's no need to worry about bringing in extra solutions and hence solving the difficult cubic

[summer crag]

okay , so it depends on the context of the situation , (whether the cubic inequality is with other expressions such as log , trig or hyperbolic trig ) , thus solving explicitily for cubic inequality is irrelevant to the actual inequality's solution and firstly we need to check the domain of the expression and subsequently the range, then try to solve the inequality , right ?

[summer ruin]

so like this one, before you proceed you need to make sure that the inner parts of the logarithms are positive, but you will end up with a cubic inequality in the log on the right side which you will not be able to solve

but you can make a few clever moves to still solve the problem without solving the cubic inequality

[summer ruin]

it depends

[summer crag]

yeah , I guess we can't generalise especially when it comes to inequalities,

[summer crag]

let me try this scary looking inequality

[summer crag]

It's all coming down to solving 8x³-5x+1 and this cubic is difficult to solve.

[summer ruin]

the hint would be to transform the left hand side using properties of log and stare at the obtained inequality

[summer crag]

When I transformed LHS using log properties , I got
log₇ (2x(1-x)) ≤ log₇(8x³-5x+1 / x) ; x ≠ {0,1} how should I proceed , can I use logᵤ(x) ≤ logᵤ(y) and compare x and y , x ≤ y (wherever x and y are defined)

[summer ruin]

yeah the log is a monotonic function

I guess you meant 2x^2 / (1-x)

[summer crag]

I'm getting log₇(2x(1-x)) in LHS of inequality , how you got log₇(2x² / (1-x))

[summer ruin]

$2\log_7(x\sqrt{2}) = \log_7((x\sqrt{2})^2) = \log_7(2x^2)$ given that we already know that $x > 0 \$ and $\log (x) - \log (y) = \log (\frac{x}{y})$ provided $x > 0, y > 0$

[obsidian monolith]

Transparent_Elemental

[summer crag]

LHS , log₇(2x²) - log (x/(1-x)) = log [2x² (1-x) / x] since x≠0 , we can cancel x , then we get log₇[2x(1-x)]

[summer ruin]

ah right I forgot there was an x multiple

[summer crag]

Now , we have log₇(2x(1-x)) ≤ log₇((8x³-5x+1)/x) how should I proceed from here

[summer ruin]

the log is a monotonic function, if it's value at one input is bigger than at the other then the inputs are related by the same inequality

[summer crag]

so , I need to solve 2x(1-x) ≤ (8x³-5x+1 / x) for x ?

[summer ruin]

for now you need to justify why the right hand side is positive

[summer crag]

I'm sorry , but I don't get it , so you are saying I need to justify why the right hand side viz. (log₇((8x³-5x+1)/x) is greater than 0 , what do you mean by justify ?

[summer ruin]

the right hand side of this inequality

otherwise the removal of logs was not a legitimate thing to do

what I mean by justify is that you need to prove that the right hand side is positive

[summer crag]

okay , but how will proving right hand side positive will help us solve the inequality , i.e. how can it give value of x for which the original inequality is true

[summer ruin]

it won't just now, but it will justify the removal of logs

$\log (x) \leq \log (y)$ implies $x \leq y$ only if $x > 0, y > 0$

[obsidian monolith]

Transparent_Elemental

[summer crag]

So , 2x(1-x) > 0 ∀ x ∈ (0,1) and
when I try to do the
(8x³-5x+1 / x) > 0 , again it all comes down to finding one root of the cubic polynomial 8x³-5x+1

which we wanted to avoid , and you said there is a way to solve without finding the root

[summer ruin]

rewriting 2x(1-x) as 2x^2 / ( x/(1-x) ) would help

[summer crag]

2x²/(x/(1-x)) > 0 ∀ x ∈ (0,1)

I got the same result.

[summer ruin]

that implies something about the cubic

ah I see what you mean, I didn't notice you say 2x(1-x) > 0 earlier

[summer crag]

Can you help with that, I'm unable to find what does that imply about the cubic

[summer ruin]

$0 < 2x(1-x) \leq 8x^2 + \frac{1}{x} - 5$

[obsidian monolith]

Transparent_Elemental

[summer ruin]

now the cubic inequality 2x^2 * (1-x) <= 8x^3 - 5x + 1 is much easier than 8x^3 - 5x + 1 >= 0

[summer crag]

I'm unable to find any root of this cubic inequality as well. Factorization is also not possible.

[summer ruin]

$2x^2 - 2x^3 \leq 8x^3 - 5x + 1 \ 10x^3 - 2x^2 - 5x + 1 \geq 0 \ 2x^2(5x-1) - (5x-1) \geq 0$

[obsidian monolith]

Transparent_Elemental

[summer crag]

x ∈ [-1/√2 , 0.2] ∪ [1/√2 , ∞)

[summer ruin]

well that's only an answer for this inequality, but not for the original one

original logs imposed restrictions on the domain of x

[summer crag]

yeah , so x ∈ (0,∞) ∩ (0,1) ∩ ( [-1/√2,0.2] ∪ [1/√2,∞) ) - {0,1}
implies x ∈ (0,0.2]

[summer ruin]

no there's another segment which x belongs to

[summer crag]

I haven't missed any upto my calculations , the original log restrictions are being taken care of , the restrictions due to the cubic are also taken care of.

[summer ruin]

the intersection of (0,1) and [1/sqrt(2), infinity) is [1/sqrt(2), 1)

[summer crag]

ohh, I'm sorry I just went over it quickly, and missed that part during the intersection , So the intersection of (0,1) and (1/_/2) will give us x ∈ [1√2,1) which is the final solution.

[summer ruin]

if you also include (0, 0.2], yes

[summer crag]

yes, I missed it again , So finally x ∈ (0,0.2] ∪ [1/√2,1)

[lyric cloak]

guys i need help in this task from pre-calculus by james stewart: Show that proj(v) u and u-proj(v) u are orthogonal.

[autumn wadi]

Could u be more specific? What is proj(v)?

[lyric cloak]

it's the projection of u onto v

[willow bear]

ain't that part of the defn of proj

[lyric cloak]

i didn't understand what you mean

[willow bear]

as far as i am concerned, the fact that proj_v(u) and u - proj_v(u) are perpendicular is literally part of the definition of proj_v(u).

unless, of course, you are supposed to take the formula as the primary definition of this thing, and to pay zero attention to the geometry of it all.

[lyric cloak]

ok, thanks

[proven siren]

!show

[tender quest]

Show your work, and if possible, explain where you are stuck.

[plush umbra]

Why is pmi wild

[willow bear]

what is "pmi"?

[summer crag]

Principal of mathematical induction ?!

[red prism]

Why do we get an extra dy/dx while differentiating implicit functions

[summer crag]

What do you mean by "extra" dy/dx ?

[red prism]

For instance if i do
d/dx x²=2x
But d/dx y²=2y*dy/dx

That dy/dx

Someone was telling me about the dx we get while doing the former if you could light that too

[summer crag]

d/dx (y²) means differentiation of y² w.r.t x , thus we use the chain rule to differentiate it , d(y²)/dy × dy/dx , therefore we get "dy/dx" here.

[red prism]

So do we consider the y as an x

[summer crag]

no , not like that , you see d/dx is a operator, and dy/dx can be thought of as a "ratio" (it is the differential coefficient of y w.r.t x) , so when we do d(y²)/dy × dy/dx , the dy term "sort" of gets cancelled , but at the same time we can use the d/dy term as a operator.

[summer ruin]

if y = y(x) is a function of x, then derivative of y^2 is 2y*y' by the chain rule

and that's it

[summer crag]

Yeah , but ig d(f(y))/dy can be thought of as a operator on a f(y) function.

How can we "justify" the use of chain rule of differentiation, otherwise ?

[summer ruin]

just use the definition of derivative, that's how chain rule is proved

[summer crag]

First principle of derivatives ?

[summer ruin]

I have no clue what that is

[summer crag]

f'(x) = lim h->o ( f(x+h) - f(x) )/h

That's the definition of derivatives as far as I know.

[red prism]

Oh okay i get it so the sole use of dy/dx's appearance is for comfort of calculation as we can differentiate y² to 2y

[summer crag]

yep, so that we can differentiate y² w.r.t y because y² is "not directly" differentiable w.r.t x, so we have to use the chain rule of differentiation.

[red prism]

Okay man thankyou so very much i was just stuck at that thing the whole day

[summer crag]

yep, have fun solving those derivatives 🙂

[red prism]

Sure!

[floral ore]

I'm having a little trouble with a specific functional equation

Say, the equation is simplified to [f(x) -1]×[f(1/x) -1] = 1

I understand that the two terms must be reciprocal to each other

But I don't understand why we assume the first term to be x^n

Like, I understand why we take n as the exponent, but I don't get why we take x

Like, I do get it but at the same time I do not

It's an uncomfortable feeling

[summer ruin]

you just try some function and see whether it's a solution or not

same functional equation can have tons of solutions involving various functions e. g. polynomials, trigonometric, exponentials, logarithmic etc

and even as series

[floral ore]

So I try to write f(x) in terms of x?

[summer ruin]

what do you mean "in terms of x"

f(x) is a function of x, it's already written in terms of x by definition of a function

we just don't know what that expression is

[tiny tiger]

yall is this still precalculus

[summer ruin]

yes

[tiny tiger]

.

damn i didnt know derivatives and limits were included

[floral ore]

Yeah, so to find that expression out I try to write f(x) such that it gives an expression of x

I meant equations asking us to find out f(x)

[summer ruin]

well that's one way

but you have to understand what it is that you're trying to do to make functional equation true

simply plugging in random expression for a function isn't going to work most of the time

[floral ore]

I don't understand, what random expression are we plugging in while solving it this way?

[summer ruin]

nothing

I'm not suggesting doing that, in fact quite the opposite

[floral ore]

Oh alright

Also while we're onto functions, since functions basically give a single output for a single input, can relations give multiple outputs for a single input?

[summer ruin]

relationship isn't a function

so it's not even clear what you mean by "output of a relatinoship"

[floral ore]

Our teacher explained it that way, but yeah I meant mapping

[summer ruin]

mapping is just a name for a function

[floral ore]

oh

[summer crag]

Can someone explain what "eccentricity " means with regard to conic sections / curves formed by the intersection of a plane and a double napped right circular cone ? , I looked up a lot but it's just formulas and ways to calculate it for different curves , but I don't get a intuitive idea of what it is, 3Blue1Brown says it's "squisification" or something like that?!!

[pulsar stream]

anyone know how to solve this question?

[summer ruin]

a circle has eccentricity 0, an ellipse has eccentricity greater than 0, but less than 1

[red prism]

For my understanding i see it as
How much of a curve is uncurved to put it simply
That what are its needs to be a proper circle .
The assumption needed here is all the curves are circles at the start until put on some condition like you'll see below
Now let's start with a parabola. You know it has two refrences one being the directrix and the other focus. Parabola is a locust of the points equidistant to those two references.Now the eccentricity of parabola is 1, and if you sacrifice this up what you get is a circle. So a parabola is uncurved by 1 or its eccentricity is 1

*This is my personal understanding

[red prism]

Can someone help.me.on this

[solar olive]

this is precalculus

[summer crag]

Hint : take log of the expression y(x) = (2x+3)^(x-5) , then differentiate both sides w.r.t x.

[red prism]

Oh yes man got that thankyou

[crystal raft]

Does anyone here know how to solve this? I've been trying to understand this but I can't 😭

[deft harness]

that's easy no?

[crystal raft]

No💀

[deft harness]

you just compare it with the standard conic equation

[crystal raft]

Ohhh

[deft harness]

since i cant write in latex

[crystal raft]

I'll try to answer it hang on

Is this correct? I'm not sure tho...

Excuse my writing I'm using it as a scratch

[deft harness]

Hmm

a's term is 0

as well as h

is it a parabola?

[crystal raft]

Yes

My prof said use two circle and parabola

I don't know the 4🥲

[deft harness]

is 4th an ellipse?

[crystal raft]

They only told us to use 2 methods which is circle and parabola

[deft harness]

sorry idk then

🤷‍♂️

[hushed sphinx]

Hmm, eccentricity is weird. Suppose we have a nondegenerate conic in $\bR^2$, and rotate and translate it such that it is symmetric across the $y$-axis and passes through $(0,0)$. It must then have an equation of the form $$ax^2 + by^2 + cy = 0.$$ Since the conic was nondegenerate, we can assume $a=1$ without loss of generality, so we have $$x^2 + by^2 + cy = 0.$$ Now the coefficient $c$ is just a scaling factor: if we dilate the entire conic around $(0,0)$, $c$ will change in response while $b$ stays the same. So the overall shape up to similarity is controlled by $b$ alone, so the eccentricity is a function of $b$. But if we actually plot this function it turns out to have ugly singularities at $b=0$ (parabola with $e=1$) and $b=1$ (circle with $e=0$), and there's no nice expression for it that works in more than one of the intervals $(\infty,0)$, $(0,1)$, and $(1,\infty)$.

[obsidian monolith]

Troposphere

[hushed sphinx]

On the other hand, Wikiedia also explains:

Any conic section can be defined as the locus of points whose distances to a point (the focus) and a line (the directrix) are in a constant ratio. That ratio is called the eccentricity, commonly denoted as e.
and that seems to suggest that the eccentricity ought to be a nice smoothly varying function of the shape of the curve at least in a neighborhood of a parabola (in which case the directrix and focus are well-defined).

[autumn wadi]

[hushed sphinx]

Hmm, perhaps I botched the plotting of e as a function of b.

Yeah, I did. It should be e = sqrt(1-b) for all b<1. So things are nice and smooth around the parabola case.

And e = sqrt(1-1/b) for b > 1.

[grand lily]

X and y intercepts?

[proven siren]

!show

[tender quest]

Show your work, and if possible, explain where you are stuck.

[gray maple]

how does the help channel work

[proven siren]

@gray maple#❓how-to-get-help

[bitter compass]

quick question how did my teacher end up with (x+4)?

shouldn't it be (x+2)

or did I miss sum

[autumn wadi]

It seems like Yr teacher made a mistake

[floral ore]

Does a basic understanding of derivatives come under precalc?

since we do have to use it in Physics but haven't really studied limits yet

[autumn wadi]

It comes under calculus

[floral ore]

Damn

[shell bobcat]

I might need some help, so my school at the end of 11th grade does an entrance exam for Pre-Calc (for 12th grade) to see whether your skills and whether your able to do Pre-Calc and if you do good you'll be able to get into Pre-Calc, how would someone get prepared for that Exam?

[dire aspen]

You guys do precalc in 12th grade??

[shell bobcat]

Government gets us to do that

[dire aspen]

That’s super late

[shell bobcat]

Algebra 1 in 9th
Geometry 10th
Algebra 2 11th
Pre-Calc/Foundational Math in 12th

[dire aspen]

[shell bobcat]

Yeah IK its how it is here

But any answers to my question?

Cause I wanna do Engineering

[dire aspen]

just study

[short knot]

What is the difference between pre calculus and calculus

[proven siren]

Not really

That’s basically average

[proven siren]

Precalc is intro to limits, complex numbers, parametric

[dire aspen]

Precalc is the math to prepare you for calc so like functions, trig,logs

[proven siren]

That’s not precalc

That’s trig..

[dire aspen]

Isn’t trig in precalc

[proven siren]

No trig is it’s own class

Same thing as alg 2

[dire aspen]

Depends what country I guess

[proven siren]

In the us it’s average

[dire aspen]

Idk I’m not american so I’m not familiar with the names of the courses

I thought most schools would start calc at 11th grade

[terse ridge]

maybe I’m stupid but I’m in 12th grade and doing pre-calc

[dire aspen]

What country

[terse ridge]

Usa

[dire aspen]

Well Kefir says that’s normal so

[terse ridge]

Okay

[dire aspen]

for australia at least calc starts at 11th

[daring basalt]

Hi.

Im trying to find a polynomial that raised to n is equal to

i^(n+1) + 2i^(n) + 2i^(n-1) + 2i^(n-2) + ... i^(n-n)

Any idea on the process to find such polynomial?

I would prefer if the polynomial is not sent here, I just need some push in the right direction, I want to learn.

Thanks.

[willow bear]

??

is i your variable?

[daring basalt]

Yes

[willow bear]

so you want to find a polynomial whose n'th power is what you wrote, a polynomial of degree n+1?

with all coefficients 2 except the leading and constant, which are 1

[daring basalt]

Is there a way to just write the variable one time?

and raise that expression to match the polynomial

ex:

write (x+1)^2

to represent x^2 + 2x + 1

[willow bear]

you still have not confirmed your goal to me

and i am somewhat confused bc your goal as i understand it is impossible

[daring basalt]

I tried to find the formula of compound interest by myself.

At the end, I got this expression

x(i^(n+1) + 2i^(n) + 2i^(n-1) + ... i^(n-n))

being i the interest in percentage

Im trying to find a math expression in which I only need to write i 1 time, and raise that expression by a number that gives the polynomial.

n is the time in which the percentage is applied.

if is 20% by year, then
i = 0.2
n = number of years

Is there a way to find that expression?

[willow bear]

i think you screwed up at some point.

also obligatory http://xyproblem.info/

also did you by any chance start with x * (1+i)^n ???

[daring basalt]

No

[willow bear]

you started with your principal x and you multiplied it by (1+i) n times, yes?

[daring basalt]

I started with x(1+i) for n = 1

Then added to that x(1+1) * i

which would be the interest over the current quantity

x(1+i)+xi(1+i)+xi^2(1+i) ...

If I had n = 6 then the ladder would end with a monomial being multiplied by i^(n-1)

I tried to go backwards from there

Maybe I did something wrong seeing my notes now. What should I do at the step I just explained?

[willow bear]

you should probably backtrack to almost the beginning

and understand that applying your interest rate once means multiplication by (1+i) no matter what quantity it is applied to

[daring basalt]

Oh

Thank you

I see it now

[near cove]

How do I continue down the path of figuring out the sin half identity here

I have sqrt((6/7)/2)) rn

But idk how to continue. do I multiply 6/7 by 1/2 inside the sqrt or do I split it up so that its sqrt 6/7 over the sqrt 2, then multiply by sqrt 2 over sqrt 2?

[obsidian monolith]

hannibal

hannibal

[near cove]

Hm?

I can use the double-angle identity to solve for this?

I was using this

The one on the top

[hazy hollow]

Its been a while since I did systems of equations and graphing them etc. can someone remind me how to write the equation of this.

[jaunty raven]

set them equal to each other?

then simplify?

$x^{3}- \frac{1}{2}x^{2}-4x-1 = 0$

[obsidian monolith]

roshan

[jaunty raven]

that should be it

roots of this eq are also the intersections of the two separate expressions

[dull tendon]

i was wondering

can we integrate dx^2?

i don't mean double integration

integrate only once, but instead of w.r.t dx, do it w.r.t (dx^2)

for example this equation-
ds = u dt + 1/2 (a) (dt^2)

[night flame]

Can someone tell me if I am doing this problem properly?

[uncut mulch]

,rotate

[uncut mulch]

@night flameyou're missing an n on the right side of the equation
nor should be writing a hanging expression/sign like that in the last line

[night flame]

Thx. I finished it now.

[safe oak]

I have made a discovery

I was messing around with the gamma function

And as it turns out

This is sine

[autumn wadi]

What

[safe oak]

You put Pi/x on the second one

Instead of x/Pi

[autumn wadi]

Oops mb

Wholly crap it actually is

HOW

[bitter compass]

is this correct?

because as x approaches 1 f(x) lands on 2?

[slim steppe]

No

Approach x=1 from the left and right side

You’ll see what you’re approaching two different values

So, it doesn’t have a defined limit but a defined value at x=1, yes

[short saddle]

https://math.stackexchange.com/a/13558

i don't understand how they derived the recurrence relation

[ocean scroll]

Question, could I define "Mapping" as a relationship between two elements? Like, the mapping itself, generally.

I understand it's a function, but, I was meet with something like this:

You have a vector.

(x,y)

and get a new vector by applying an operation

(x_1 , y_2, z_3)

Especially talking about the idea of "rotation" when it comes to complex number multiplying and so on and so forth.

and im meet with this question, there are functions that are not displayed like 2x, 2x^2, etc.?

[celest yacht]

Definition 3.31 (Functions). Let $X,Y$ be sets and let $P(x,y)$ be a property pertaining to an object $x \in X$ and an object $y \in Y$, such that for every $x \in X$, there is exactly one $y \in Y$ for which $P(x,y)$ is true (i.e inputs are only mapped to exactly one output). Then we define the function $f \colon X \to Y$ defined by $P$ on the domain $X$ and the codomain $Y$ to be the object which, given any input $x \in X$, assigns an output $f(x) \in Y$, defined to be the unique object $f(x) \in Y$ for which $P(x,f(x))$ is true. Thus, for any $x \in X$ and $y \in Y$, $$y = f(x) \iff P(x,y) \text{is true}$$.

[obsidian monolith]

josemom2

[ocean scroll]

Ah, so really it's just... For every X there is an input Y.

[celest yacht]

no

read it

[ocean scroll]

i did

and, it seems... uhm

So, there is no short definition essensially?

If I want to be correct.

[celest yacht]

"a thing that sends every element of one set to another by some property such that each input has exactly one output"

[ocean scroll]

So it is a relationship, but maybe it's not correctly defined as a relationship.

It's more correctly defined as: An element gets operated on, leads to an other element

[celest yacht]

sure

[ocean scroll]

Yeah, what I ment in terms of relationship is that.

There is an element that leads to an other element

but, that's not right? Is it too vague?

[celest yacht]

if it works for you that's ok

but this is the explicit definition

[ocean scroll]

Eh, I feel like it would be fine "An element gets operated on, leads to an other element"

yeah

[ocean scroll]

In this case. The function 2x, is where you take an element, x in domain A.

Multiplying it by 2. Which gives you an element, y in codomain B.

[celest yacht]

sure

[ocean scroll]

And the important thing is, "Multiplying it by 2", is an operation.

Yeah.

[ocean scroll]

Could this be generelised to multiplication of 1xn matrices?

[celest yacht]

?

[ocean scroll]

like, multiply a matrice by an other matrice, to get an other matrice.

Can be set up to be a function yeah?

[celest yacht]

that can be viewed as a composition of functions

[ocean scroll]

which, can be viewed as one giant function of itself?

[celest yacht]

that's what a composition is

[ocean scroll]

right, a function that contains smaller functions

sorry, other functions

[celest yacht]

Definition 3.3.13 (Composition). Let $f \colon X \to Y$ and $g \colon Y \to Z$ be two functions, such that the codomain of $f$ is the same set as the domain of $g$. We then define the composition of $g \circ f \colon X \to Z$ the functions $g$ and $f$ to be the function defined explicitly by the formula $$(g \circ f) \coloneqq g(f(x)).$$ If the codomain of $f$ does not match the domain of $g$, we leave the composition $g \circ f$ undefined.

[obsidian monolith]

josemom2

[ocean scroll]

AYE

[celest yacht]

<@&268886789983436800>

[ocean scroll]

<@&268886789983436800>

[celest yacht]

gross

[ocean scroll]

yep

[stuck lark]

3 nsfw spammers in a row, par for the course

[ocean scroll]

this will be fun to understand

lets see

[celest yacht]

it's not bad

it just means f sends stuff from X to Y

then g sends stuff from y to Z

so the composition of these functions overall is a function from X to Z

[ocean scroll]

i can see the relationship

So, the set of all outputs in f, is basically the domain of g.

g(f(2)

f(2) = an output, which is an element of y

[celest yacht]

an element of y

[ocean scroll]

oh right, big Y

sorry

[celest yacht]

here's some random functions

[ocean scroll]

neat

[celest yacht]

or:

[ocean scroll]

this is implying, the definition, that f(x) is actually our input, since it acts as a domain in the function of g.

[celest yacht]

indeed

which is why the explicit function g(f(x)) is defined as it is

[ocean scroll]

right

[ocean scroll]

i can see now

[ocean scroll]

But basically, the interesting thing is.

For example:

f(g(t(x)))

You need to break the problem down right if you want to define it.

g(t(x)) can be defined first. Then, you can define for f.

[celest yacht]

sure

[ocean scroll]

It's a stepping up process

[celest yacht]

this is nothing more than the fact that the composition of functions is associative

standard result which many texts ask you to verify

[ocean scroll]

i see

[celest yacht]

you can either do $(g \circ f) \circ h$ or $g \circ (f \circ h)$

[obsidian monolith]

josemom2

[celest yacht]

or simply leave it written as $g \circ f \circ h$

[obsidian monolith]

josemom2

[ocean scroll]

nice

well, im going to have fun with this in school, cause im planning to write a OK definition of G(t(f(x))) etc

see what i can learn

[floral ore]

Could somebody explain to me a better way to think about why graph of modulus function changes if say, in |x - 1|, the value of x is greater than or lesser than 1?

Like, when I'm presented with variables I can tell that it'll change its behavior because we write x<1 as -(x-1)

But when I'm actually plotting the graph and taking x as different numbers, I can't apply the logic

[summer ruin]

the definition of absolute value

|t| = t for t >= 0, |t| = -t for t < 0

[floral ore]

I understand that, I also get why the behavior of the function would change, but when it comes to substituting t for numbers my mind just goes blank

Maybe it's because this is a new topic, maybe it's because I haven't fully grasped what variables are

[void thorn]

yeah it can be a bit confusing to wrap your mind around at first, we're replacing everywhere you see 't' with 'x-1'

maybe seeing side by side helps,
|t| = t for t >= 0, |t| = -t for t < 0
|x-1| = x-1 for x-1 >= 0, |x-1| = -(x-1) for x-1 < 0

also notice you can put parenthesis around x-1 to make (x-1), in particular we had to do it so that -t would become -(x-1) and put the negative everywhere appropriately

[floral ore]

yeah, my main problem is mostly when I try to plot the graph

Like say, I put x=1 in |x-1|

the answer should be 0

So a value greater than 1 or lesser than 1 should affect how the function behaves

But it's way easier to see it in variables than it is in numbers

Like I can understand why |x-1| might show different behavior in x>1 and x<1, but when it comes to plotting the graph I can't apply this knowledge properly

It might be because this stuff is new to me but I don't know

[summer ruin]

for x = 5, |5-1| = |4| = 4
for x = 0, |0-1| = |-1| = 1

i don't really see the problem

[floral ore]

Yeah it was just my brain coming to terms with this

I think I understand it more now

[tropic sinew]

Can anyone teach me differentiation and integration from the scratch

[cursive cairn]

try Khan Academy: https://www.khanacademy.org/math/calculus-all-old

[tropic sinew]

Thanks

[red prism]

Is Khan academy sufficient from basics to advanced?

[autumn wadi]

Depends on what u refer to by advanced

[red prism]

Point

[ocean scroll]

Question, is this correct?

I'm defining T(G(f(x)))

Wait, this technically counts as a help channel

oh whatever, doesn't matter 💀

[modern valve]

If x is an element of X, f : X→Y, G : Y→Z, and T : Z→I, then yes, T(G(f(x))) is an element of I

[severe olive]

looks right to me

[modern valve]

And we call (T \circ G \circ f) a composite function, from (X) to (I)

[obsidian monolith]

boolean_satisfiERIC

[ocean scroll]

Oh, interesting, but, the definition i provided was not correct?

or lacked

thereoff

[modern valve]

No I'm just restating what you said

[ocean scroll]

oh

So it's the same thing

[modern valve]

Yes

[ocean scroll]

just, more elaborate

[ocean scroll]

so thats your condensed definition

oh wait

wrong

there

ah, i didn't know you could type it that way, cool

[modern valve]

yes the stuff before the "then" is a definition and the stuff after is a conclusion you can make

[stone ridge]

Trying to solve the missing angle but in degrees. In the end, I keep getting 0.72772…. And I have to use only 3 decimal places. Yet when I put in 0.727. It’s wrong. I also round it and I get it incorrect. What could I be doing wrong???

[autumn wadi]

U should be getting 46.7 degrees.

$\arcsin{18(\frac{\sin{76°}}{24})}$

[obsidian monolith]

Kiameimon | Welt Rene

[grave forge]

Hint: 24 18 30 is a Pythagorean triple

Nah wait that wont help here

Sry

Im dumb

[still dove]

Can anyone quickly explain cross products to me. I missed my math class

[mild swan]

You can probably google this, but it's essentially a process which combines 2 vectors in 3D space and gives you a 3rd, which is perpendicular to the original two, and has a magnitude which is dependent on the magnitude of the first two in a certain way

[viscid thistle]

Chatgpt

[hushed sphinx]

#changelog message

[viscid thistle]

They never saw me

[viscid thistle]

I love how it's only stated in the changelog and not the actual rules

Unless I'm blind

[viscid thistle]

Yea but where is that?

[viscid thistle]

Dude I can't find it anywhere

Ugh dw I'm too tired for this have a good night 💀

[autumn wadi]

I'm referring to the fact that the original rules already imply that chatgpt is nto allowed

[viscid thistle]

It doesn't doe-

[autumn wadi]

.

please r e a d

[viscid thistle]

No no I did but when looking in #❓how-to-get-help and #rules there isn't much on that anyway

And the implied rules don't really imply well

[autumn wadi]

in any case this is not the place to discuss server rules

[viscid thistle]

||I'm just kinda salty because chatgpt was the only reason I scored 29/30 on my test. It taught me everything from start to finish 💀||

[viscid thistle]

I agree

[last magnet]

using ||chatgpt ||to study is crazy 😭

[elder marten]

Hey guys. I was working through this question for practice, got it wrong, and I'm trying to understand it so I'll be able to successfully tackle similar problems in the future. Looking at the markscheme, I'm a bit confused. Can anyone tell me why the curve touches the circle when r is maximum?

[proven siren]

If r were any smaller it would be intersecting with the curve

[elder marten]

Why?

Oh wait

Actually

I get it now

It's because the function shows the values of r for which they intersect. Therefore, the maximum value of r is when each "hump" of the graph has one intersection, while values below that the circle is small enough to have multiple intersection points

Thank you!

[proven siren]

no problem

[viscid thistle]

Which is the diference between range and codomain?

[hushed sphinx]

Codomain is where the function is allowed to pick its outputs from. Range is the outputs it actually picks.

[viscid thistle]

Okay, thank you

[hushed sphinx]

(Beware that some people use "range" as a synonym of "codomain" and call the set of actually picked outputs "image" instead).

[viscid thistle]

It's so much easier doe when you're falling behind because it breaks down the steps for you and doesn't assume that you already know something.

[bleak dune]

Range is all the possible outputs you can get from a function. The range is also called image of the function. For example, the range or output image of a function f(x) = x^2 is [0,infinity).

there are not negative numbers in the range of f because x^2 is always positive and never achieve negative values

[proven siren]

They asked for the difference not the definition

Also someone already responded with a good answer

[willow bear]

#chill is that way 👈

[floral ore]

I know what co domain means, but i don't really understand - can I find the co domain of f(x) = x² for example?

[summer ruin]

yes, the codomain of this function is the set of all real numbers

but it's range is non-negative real numbers

[hushed sphinx]

You can't see just from the expression what the codomain is.

I could define f: R -> [0,infty) or f: R -> C by f(x) = x² just as well.

It's a datum you need to give in addition to explaining to evaluate the function (though there are default guesses that will often match what people have in mind).

[autumn wadi]

please no advertising here

<@&268886789983436800> person has been spamming this message in multiple channels

[viscid thistle]

im a sophomore and i feel like precalc is kinda hard for me

[slim steppe]

If something is hard for me, I isolate myself from everyone so I can just focus in on the work and dive deep into it and unravel its mechanics

Works extremely well for me, and that may be because I have a good visual imagination, may or may not work as effectively for you

[cursive fiber]

you just gotta get through it

[torn lark]

"How many times is the digit 0 written when listing all numbers from 1 to 3333".

My doubt is: wont we be double counting numbers like 1000, 2000 etc.

I've attached the solution.

[willow bear]

we're SUPPOSED to double-count numbers such as 1200 and triple-count numbers such as 1000. these have two and three zeros each respectively!

[torn lark]

damn, ok I understood. Thank you very much.

[quick crest]

anyone know how I'm supposed to figure out IAL pure mathematics 4 integration?

I've got 0 problems with differentiation but as soon as I get an integration question my brain just shuts down

[safe oak]

My brain just shuts down in general

[summer ruin]

it's continuation of your school math program

[safe oak]

There’s a lot of polynomials, quadratics is a big part of it, a lot to do with exponents

There’s a lot of applications of higher level mathematics in computer science so taking precalc is a good idea

And also if you are familiar with functions in computer science, functions will make a ton of sense in mathematics

[stable pond]

my pre calc class had trigonometry as well

We are doing sequences and series, but we are nearing the end of the course

in the fall im gonna take calc 1

I’m kind of excited and terrified at the same time

[twin peak]

Can someone help me with these questions?

[old isle]

for 3a) use product rule

for 3b) let your derivative equal to 0 and solve for x

[floral ore]

okay so I'm at a bit of a loss here

we were taught before that log written without specifying the base usually means the base is 10

now our teacher is saying that log written without specifying the base means the base is e??

I thought we took ln for base e

[old isle]

you are correct

ln(x) implies log base e of x
log(x) implies log base 10 of x

your teacher's notation is wrong there ig

[floral ore]

I asked my teacher about this like thrice

He simply said that when we were taught in smaller classes we were taught log without specified base means the base is 10 because it's "easy"

[old isle]

💀

LMAO

[floral ore]

But now whenever log without specified base is written we have to assume that the base is e

Like what???

What's the point of ln then?!

[old isle]

ln looks cooler either way

[uncut mulch]

the intended base of log without an explicit base written depends on context

[floral ore]

yeah I searched it up, but why not just write log a different way instead?

Why are you making life harder on students if you're too lazy to write ln, or lg

[uncut mulch]

stems from history i suppose

[summer ruin]

the deeper in math you dive the more you realize the base is irrelevant and e is the nicest outside of very specific contexts like information theory or data structures (but even in data structures you make big O bounds and logs of different bases differ from each other on a constant multiple and thus it's irrelevant too)

[old isle]

the base is relevant when simplifying stuff sometimes

[bitter frigate]

why is (-oo, -1) concave up?

wouldn't positive to negative be concave down?

oh would that be for the first derivative

[frail sandal]

idk what that domain restriction is for cause idk precalc

but do you think you could teach me the entirety of precalculus from discord before I take the class

[twin peak]

Hi, let dydx =0, then is 1+2sin2x=0,then solve for x and get x is either 15 or 165. After that, substitute into y? But the answer i obtained very weird 😅😅

[ocean plank]

is this the way

[slim steppe]

That’s right

[solar olive]

does anyone know an identity like this but with tan (what matters is that the highest power is 3)

[proven siren]

Maybe double angle plus angle addition identity?

oh wait that’s just an identity in itself

It is always true for any theta

Oh wait

Just search up tan3x identity

[viscid thistle]

This may be a dumb question, but how do you find the principle angle

Of a radian

Welp

No one can help 😦

[lusty sail]

Pi (Radians) = 180 degrees. Use the ratio.

[uncut mulch]

as
0 < 1 rad < pi/2 rad
the principle angle of 1 rad is 1 rad

[willow bear]

fade attempted to solicit help from me earlier by pinging me in the middle of a convo, and iirc the angle she wanted was not 1 radian. instead, "a radian" is supposed to mean "an angle whose value is given in radians"

[uncut mulch]

did that end up getting resolved?

[willow bear]

not to my knowledge.

[torn lark]

1. Does the " - n " mean that we are removing 'n' outcomes- one from each x_i (since x_i is assumed to include 0, and we don't want 0).

2. Why is " r - n" supposed to be equal to " n - 1 "? Is this similar to "17 choose 6" being equal to "17 choose 11"?

Equation 1 is x_1 + x_2 + x_3 + ....... + x_n = r. Thanks.

[ocean lily]

The purpose of -1 from each of the positive integer solutions y_i is to transform it into x_i, which is now a 'non-negative' integer (in other words, can take the value 0). With this, the problem is now equivalent to distributing r-n identical ones into n distinct bins (or commonly called the bars and stars method). The -n can be seen as removing the constraint that each y_i must be at least 1 (or have at least a one in each distinct bin)

for your second question, correct it's basically nCr = nC(n-r)

[bitter frigate]

is writing dx/dt as x′(t) valid notation?

[summer ruin]

everyone would question what is meant by that

[indigo halo]

Shouldnt it be this

I could be wrong tho

Nvm

I read it wrong

[torn lark]

Understood. Thank you very much.

[steady trench]

I was studying about the pascal's triangle and was wondering how to prove that the item x at the n-th row and k-th column of the triangle is equal to C(n, k), I searched a little more and found that it was because x is equal to the sum of both numbers above it, which are the numbers located on the ((n-1)-th, k-th) and ((n-1)-th, (k-1)-th) spot, but how can I prove that the numbers above it are C(n-1, k) and C(n-1, k-1)? Would I need to prove it recursively?

[frail sandal]

oh so no

[steady trench]

what do you mean?

[vapid plaza]

C(n-1, k-1) + C(n-1, k) = C(n, k) you can prove straightforwardly; then you can do induction and prove that each row implies the next row

the induction step can also be phrased recursively, as each cell depends only on the cells above it

[mint vessel]

Yes, you can prove that the numbers above the x in the n-th row and k-th column of Pascal's Triangle are C(n-1, k-1) and C(n-1, k) recursively using the formula for combinations.

Recall that the formula for combinations is C(n, k) = n! / (k! * (n - k)!), which gives the number of ways to choose k items from a set of n distinct items.

Now, consider the (n-1)-th row of Pascal's Triangle. The k-th column of this row contains C(n-1, k-1) items, which represents the number of ways to choose k-1 items from a set of n-1 distinct items.

Similarly, the (k-1)-th column of the (n-1)-th row of Pascal's Triangle contains C(n-1, k) items, which represents the number of ways to choose k items from a set of n-1 distinct items.

Therefore, the x in the n-th row and k-th column of Pascal's Triangle is the sum of C(n-1, k-1) and C(n-1, k), which represents the number of ways to choose k-1 items from a set of n-1 distinct items and the number of ways to choose k items from a set of n-1 distinct items, respectively.

Thus, we can recursively prove that x = C(n, k) by using the formula for combinations and the fact that the numbers above x in Pascal's Triangle are C(n-1, k-1) and C(n-1, k).

[viscid thistle]

It is necessary to put a mod everytime we open a root ?

[steady trench]

thanks

[tepid sorrel]

how can i hack a discord account using calculus

[tepid sorrel]

bruh its a rule

sqrt of x^2 = IxI

because we dont know whether x is psitive x or negative x
cux the square will give the same result

[willow bear]

no

[tepid sorrel]

okok i will be in limits
and show intergrity

[willow bear]

spose those jokes are funny to a 16 year old learning calculus for the first time.

[wraith sequoia]

lol

[tepid sorrel]

calculus didnt help me hack my crush's phone

im pissed

[willow bear]

why would you want to significantly worsen your crush's life by hacking their phone?

[gentle kettle]

id reject you if i was your crush

[tepid sorrel]

stfu ugly kitty

[tepid sorrel]

she said im cute

but i wanna see if im the only one

[gentle kettle]

probably not

[willow bear]

so, because your crush called you cute, you want to make her life worse.

is that what you're saying? @tepid sorrel

[gentle kettle]

😠

[tepid sorrel]

bruh i will not make her life worse

[tepid sorrel]

ohh scary

[willow bear]

well, you apparently want to hack her phone.

[tepid sorrel]

yes yes

[willow bear]

people generally don't like it when their phones are hacked!

[gentle kettle]

🙄

[tepid sorrel]

but i will not let her know 😏

hehe

[willow bear]

would you like your phone to be hacked?

[tepid sorrel]

hahahah
i have antivirus

[willow bear]

nice question dodge.

[gentle kettle]

💀

[tepid sorrel]

nobody can affect me

[willow bear]

but the reason you have an antivirus is probably because you in fact don't want your phone hacked.

isn't that right?

[gentle kettle]

in a nice way?

[tepid sorrel]

lmao im broke
and my chats arent that good
i have no probs getting hacked

[gentle kettle]

sure

[tepid sorrel]

yes
she said my pic looks great

she so cute too

[willow bear]

this convo belongs better in #discussion or maybe even #chill, by the way.

we have definitely not been talking about any precalculus here.

[tepid sorrel]

oh im talking about my limits
to hack

[gentle kettle]

yea she is talking to other guys behind your back

:3

[tepid sorrel]

there is calculus

[willow bear]

😑

[tepid sorrel]

💔 shut up silly cat

[willow bear]

you will do anything in your power to (a) be vile and (b) not take people seriously.

actually why am i arguing with a 16 year old on the internet lmao i have better things to do

[gentle kettle]

I feel like he isn't even 16 years old

more like 13

or 12

[tepid sorrel]

im 17

dude

🙂

[willow bear]

it is to be noted that i called him 16 twice and was not corrected.

perhaps because it's close enough

[tepid sorrel]

:// bruh
yall have to be so serious all the time

bruh
enjjoy ur life without rules
why so classy

alr ur wish

[floral ore]

Is there a way to do question 5 without looking at the options?

[summer crag]

How can I solve this equation ( f'(x) = 0 ) for x ??

It's turning out to be a quartic in x with weird coefficients

[summer crag]

Lol , it turns out I don't even need to solve f'(x) = 0 for real x , I just need to substitute sinθ₁ and sinθ₂ in the equation f'(x) = 0 to get snell's law.

But still I'm wondering whether snell's law can be derived without this substitution,i.e by finding the minima of the curve f(x)=t because desmos show that it has a absolute minima.

[summer crag]

Can someone help me with this , I really wanna know how can it be solved , it's turning out to be quartic equation in x with real coefficients (as h₁ , h₂ , v₁ , v₂ and l are all real constants) , I took lcm and then multiplied both sides by the denominator ( v₁v₂√(h₂²+x²)(h₁²+(l-x)² ) while stating the condition that this denominator can't be equal to zero ,

and finally got a quartic equation in x (of the form ax⁴ + bx³ + cx² + dx + e = 0) which I don't know how to solve for real values of x.

Should the general quartic formula be used in this case ? or maybe there is some "magical" factorization

[summer crag]

Here ,this equation.

[north remnant]

Anybody know where I went wrong? I was differentiating, Ln on both sides and used log rules so I didn’t need to do product and quotient rule.

[summer ruin]

already on the first line?

why is it ln(2x*2)

[north remnant]

Oh damn I took the derivative too early on that

Let me correct that and I’ll see if that does the trick

[north remnant]

Corrected but still not correct

[summer ruin]

it is correct

[north remnant]

Ok thank you 😊

I guess the answer key is wrong, my professor did say one of these questions answer was wrong and this is the last question

[soft cedar]

I think you could use change of base and then go from there

I could be wrong tho

its late so I did not want to write everything out

but I think i am right

[soft cedar]

Hey guys, do you have to do polar graphs and equations in your precalc class, I just wanted to know who does.

[floral ore]

Yeah I used change of base but I'm not sure how to go forward

I even tried taking bases like 2 or 3

[soft cedar]

Okay, give me one second. I am helping someone in help-16

Ill be there in a second

Iff you don't mid

is that okay?

[short saddle]

https://centerofmathematics.blogspot.com/2018/01/problem-of-week-1-30-18-recursion.html

how did they get the recursion for C_n

[solar olive]

yes

log_36(81) = m
log_(6^2)(9^2) = m
log_6(9) = m

log_18(16) = ?

log_18(16) = log_6(2^4)/log_6(18)
= 4 log_6(2)/log_6(18)
= 4 log_6(2)/(log_6(9) + log_6(2))
= 4 log_6(2)/(m + log_6(2))

find the value of log_6(2)
log_6(2) = log_2(2)/(log_2(2) + log_2(3))
log_6(2) = 1/(1 + log_2(3))
find the value of log_2(3)

log_6(9) = m = 2 log_3(3) /(log_3(2) + log_3(3))
m = 2/(log_3(2) + 1)
log_3(2) = (2 - m)/m
so log_2(3) = m/(2 - m)

log_6(2) = 1/(1 + m/(2 - m)) = 1 - m/2

log_18(16) = 4 (1 - m/2)/(1 - m/2 + m) = 4(2 - m)/(2 + m) ```

although theres prob a shorter way

[willow bear]

what is the goal here...?

"Take the derivative of this function, but deliberately screw it up by applying this specific incorrect version of the quotient rule"?

[uncut mulch]

vu’ + uv’/ v^2
where's that coming from

where did you see that

[floral ore]

It's (u'v - uv')/v² iirc

[floral ore]

Thank you

[faint pumice]

Is that the quotient rule for differentiation?

[faint pumice]

ye that

[abstract marsh]

Hello, my teacher gave a question that is for bonus marks. Can someone tell me that:
If the base of a exponent is negative, e.g. (-2)^x
What would be the domain?

[summer ruin]

what do you think

[abstract marsh]

idk

[summer ruin]

make a guess

[abstract marsh]

mmmmmmmm

idk

[hushed sphinx]

Find some values of x where you know the expression makes sense, and some where you don't know how to make sense with it.

Make an educated guess about what the values that make sense have in common.

[abstract marsh]

Okay

ty!