precalculus

[jagged glade]

What do you mean by restricted domain?

[ocean temple]

This thing ?

Ahh no

Sorry it's for range

How do we proceed from here

[jagged glade]

Then using quadratic inequality we got

Smaller root ≤{x}≤ bigger root

[ocean temple]

Oh yeah got it

[jagged glade]

Nice

Next one then?

[ocean temple]

Yeah

[jagged glade]

For the equation [2cosx]+[sinx] = -3

To be valid

We can find the range of possible values of x

[ocean temple]

X belong to pi from 3/2pi roughly I think

[jagged glade]

We can get
-2 ≤ 2 cos x < -1 and -1 ≤ sin x <0

[ocean temple]

Ok

[jagged glade]

So, it's pi to something else

Because of the 2cosx<-1

[ocean temple]

Yes figured that out but no idea after that

It should be 4/3 pi

[jagged glade]

Yea

[ocean temple]

Pi to 4/3 pi

[jagged glade]

π≤x<4π/3

Now

Do you know how to simplify sinx +√3 cosx

[ocean temple]

+-√a^2 + b^2 right ?

[jagged glade]

I'm not sure about your method. But what I did is

2(½ sin x + √3/2 cos x)

[ocean temple]

Oh, I know this

2sin(x+pi/3)

[jagged glade]

Yea

[ocean temple]

And here we find max and min ?

[jagged glade]

Plug in this to check

[ocean temple]

Ok I got it

@jagged glade thank you so much , you are great

[jagged glade]

👍

[ocean temple]

I have a few more should I send them ?

[jagged glade]

Oh sure

Btw I made a mistake

[jagged glade]

Should be all <

[jagged glade]

,rotate

[ocean temple]

Ah

That's the command

,rotate

[jagged glade]

Question 16, completing square?

[ocean temple]

I don't how this helps but

,rotate

[jagged glade]

Denominator part is not correct

Can do 2(a²+4a+4) first

[jagged glade]

Yea

I bet you know what to do next

[ocean temple]

Partial fraction

Then sin 1 and 0

[jagged glade]

Since alpha is between -1 and 1

You can first plug in the extreme values of sinx into the function first

[ocean temple]

Yup got it

[jagged glade]

Secondly you have to check if the denominator will pass through 0

Or else it will be a bit uglier

So, next?

[ocean temple]

This maybe

[fringe raft]

You still want help ?

[jagged glade]

It won't pass through 0 this time

[ocean temple]

Which question

[jagged glade]

(alpha +2) is between 1 to 3

So it won't pass through 0

[ocean temple]

Oh sorry

[jagged glade]

Question 15 now?

[ocean temple]

Yup

[jagged glade]

2 cases

[ocean temple]

X>0 and x<0

[jagged glade]

x≥0

Or you can do 3 cases if you want XD

[ocean temple]

Which 3rd case

[jagged glade]

x=0

[ocean temple]

Let's proceed with 2 as we know x--->0 is =0

[jagged glade]

Yea

[ocean temple]

,rotate

[jagged glade]

Yea, I got that too

So the range will be?

[ocean temple]

Case 1

[jagged glade]

Nice

Case 2 is easier

[ocean temple]

Yeah case = 0

[jagged glade]

So done.

[ocean temple]

Yup

This was easy ,what was I doing wrong

Earlier

[jagged glade]

Question 12?

[ocean temple]

Yup

[jagged glade]

What does the P mean

[ocean temple]

Permutation I guess

[jagged glade]

Ah

Is it $^{7-x}P_{x-3}$?

[obsidian monolith]

Biscuit

[jagged glade]

Can't see it clearly

[ocean temple]

Yes

[jagged glade]

This one I don't know

If it's permutation, I'll just do it case by case

But the range of the choices doesn't seem right

Like

For x=3, f(x) will be 4!/0! = 4!

If P is permutation

[ocean temple]

I don't think so

[jagged glade]

Oh

[ocean temple]

Is this correct formula

[jagged glade]

Yea, I just checked online

[ocean temple]

Sorry to bother with this question . I did it

[jagged glade]

Ok

[ocean temple]

Thank you so much for help 💕 🙂

[jagged glade]

You're welcome🍪

[viscid thistle]

can someone help me understand this

how does this explain e as the limit

[willow bear]

A(e) = 1 essentially

[mild swan]

@viscid thistle you get that?

[viscid thistle]

ohhhhhh

[mild swan]

It basically deals with integration - finding the area under y = 1/x

Which is a #calculus topic actually

[viscid thistle]

the whole shaded region =1 then?

[mild swan]

Correct

Assuming x = t

[viscid thistle]

daaaaaaaaaaaaaaaaaaamn

i see now 🙂

[mild swan]

$\int_1^x \frac{1}{x} dx = \ln{\abs{x}} - \ln{\abs{1}}$

[viscid thistle]

i think integration is beyond the scope of my class rn im just in algebra

i assume ill just have to memorize it til later then

[mild swan]

Yeah lol

[viscid thistle]

ty @mild swan

[mild swan]

np

[mild swan]

Thank you, @narrow lichen

I have no idea what was going on

[obsidian monolith]

Clay Kirby

[mild swan]

THERE WE GO

[mild coral]

You used a double angle formula when you can't

[sick steppe]

You didnt apply a double angle formula properly, but also used one that made the expression worse

$2\sin(x)\cos(x)$

[obsidian monolith]

moshill1

[unborn blade]

so when solved for L and substituted it into the cost function

the roots of the derivative don't yield the correct answer (i even used an online derivative solver to verify)

but when I solve for h instead of L and following the same process, I get the correct answer

why is this?

[ocean temple]

<@&286206848099549185>

[fading token]

,rccw

[fading token]

Hint: think about the function x^ln(x)

[unborn blade]

do you mind helping with mine?

it's right above his question

[ocean temple]

,rotate

<@&286206848099549185> please help

[granite ice]

<@&286206848099549185> need help on this problem

[tender quest]

From

#❓how-to-get-help

Rule 4

If your question has not been answered for a minimum of 15 minutes, you may use the Helpers tag once. Please do not try to bump your question using this ping unnecessarily. Do not abuse this ping. Do not individually ping users with the Helpers tag without their express permission.

[granite ice]

u can make ln(e+h) into (log(e+h)(e))^-1 if u know what i mean

[granite ice]

<@&286206848099549185>

[ocean temple]

Yup did that one

[ocean temple]

Please help guys

[turbid fern]

Are there any good resources to prepare for precaculus?
And is it a bad idea to take a precalculus class over 8 weeks with a full time job schedule?

[fringe raft]

Divide by the highest powers both in the exp and base answer will come out 9

[gritty stratus]

do i plug in v=2w^5+1 before or after i derive

i solved it both ways but im not sure which is correct

[uncut mulch]

you could differentiate directly but your need to apply the chain rule ( which you didn't)

[ocean temple]

$\int_{0}^{1}\frac{\left(x^{2}+5\right)}{\left(x+1\right)^{2}\left(x-2\right)}dx$

please help

[obsidian monolith]

Pratham_GG

[civic sierra]

Use partial fractions

[ocean temple]

ok

[ocean temple]

<@&286206848099549185>

[tender quest]

From

#❓how-to-get-help

Rule 4

If your question has not been answered for a minimum of 15 minutes, you may use the Helpers tag once. Please do not try to bump your question using this ping unnecessarily. Do not abuse this ping. Do not individually ping users with the Helpers tag without their express permission.

[willow bear]

$\log_M(N) = \alpha + \beta$, where $a \in \bZ$ and $\beta \in [0,1)$; $M$ and $\alpha$ are twin primes and $M + \alpha = 8$, and you're asked to find $\floor{N}$

[obsidian monolith]

Ann

[willow bear]

did i get that right?

@ocean temple

[ocean temple]

Yes

[willow bear]

it sounds to me like there is not enough info

M and alpha are 3 and 5 (in some order) obviously but

we know nothing about beta

$N = M^{\alpha + \beta}$

[obsidian monolith]

Ann

[ocean temple]

See I know that this is a valid question

Because it was asked in viva of batch 1

[willow bear]

so N is either in [243, 729) or [125, 625)

so theres literally hundreds of possible values for floor(N)

unless you're missing some more details

$N$ is either $3^{5+\beta}$ or $5^{3+\beta}$

[obsidian monolith]

Ann

[ocean temple]

Yup

[willow bear]

it looks like that's all we can get

[ocean temple]

Wait I'll ask from students what they got

[willow bear]

are you sure you didn't forget any more details from the question

[willow bear]

,w log_3(672)

[obsidian monolith]

Results provided by WolframAlpha

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

Requested by Ann#0413

[willow bear]

,w log_3(250)

[obsidian monolith]

Results provided by WolframAlpha

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

Requested by Ann#0413

[willow bear]

these both result in M=3 and alpha=5

and yet N is different

so we can't find N

the question is messed up

[ocean temple]

Hmm maybe

[steel venture]

hmm definitely

[ocean temple]

Some idea on 4th one

[past meadow]

are those x's in both exponents?

[ocean temple]

Yes

[fringe raft]

Convert it in a and b form by assuming variable

[ocean temple]

Then

Fuck , I got it it's reciprocal

[willow bear]

yeah those roots are reciprocals of each other

[muted steeple]

where i'm going wrong here?

https://www.symbolab.com/solver/logarithmic-equation-calculator/\log_{8}\left(\frac{\sqrt{2}\cdot\sqrt[3]{256}}{\sqrt[6]{32}}\right) this should be my answer apparently.

excuse my handwriting.

[uncut mulch]

,rotate

[uncut mulch]

$\sqrt[n]{x} \neq x^{n/2}$

[muted steeple]

ohh yes you right, thank you!

[obsidian monolith]

ℝamonov

[uncut mulch]

also make your division line more slanted

it took me a while to see your intentions

looked like 112, 312, 612

[muted steeple]

the fraction exponent ones?

i see.

thanks again.

[dark sky]

i dont understand this

[uncut mulch]

what don't you understand about it?

[dark sky]

i dont know how to find the range of angles that work

[willow bear]

think carefully about the definition of arccos

[uncut mulch]

this question is mostly about applying/recalling definitions
do you know the definition of arccos and/or the range of arccos?

[dark sky]

well arccos is the inverse of cos

and range is 0 to pi

[uncut mulch]

and that's more or less what you need to apply here

(assuming you meant inclusive of both 0 and pi)

[dark sky]

o but idk how to apply it

help :(

is there only 1 angle that will work?

cuz i graphed cos x and arccos x on desmos and they only intersect once

[uncut mulch]

ur overthinking it

also you're graph doesn't represent what the question is asking for

well arccos is the inverse of cos
and range is 0 to pi
the answer is just [0, pi]

from the definitions

[dark sky]

wait what

really?

dang

idk how that works but thx

[uncut mulch]

that's literally the definition

[unborn blade]

so when solved for L and substituted it into the cost function
the roots of the derivative don't yield the correct answer (i even used an online derivative solver to verify)
but when I solve for h instead of L and following the same process, I get the correct answer
why is this?

[dark sky]

the range would be from -1 to 1 right? because that's the range of cos?

@uncut mulch

[weak needle]

Do you still need hep with this?

[dark sky]

am i right or wrong

[weak needle]

Wrong

[dark sky]

o dang

why

[weak needle]

Damn wait

I got the order of the equation wrong

Let me check again

[uncut mulch]

assuming ur including -1 and 1, yes

[dark sky]

thanks :)

[weak needle]

All real numbers I think

[uncut mulch]

it is implied that at their level they're only working with real numbers

where arccos(x) is only defined for x in [-1,1]

[unborn blade]

@grand remnant anyone mind helping with my question

[unborn blade]

this is the question

[uncut mulch]

what values are you getting from the two different routes?

[unborn blade]

like final answers?

i get the correct answer when i first solve for H and substitute it into the cost formula

but when I solve for L and substitute it into the cost formula, find the derivative, and solve for the roots

i dont get the correct answers

[uncut mulch]

what do you get continuing from what's on the page

because i used a calc and pretty much reached the same result

[unborn blade]

when I use any calc it keeps saying no real roots

i used symbolab as well

[uncut mulch]

the f(x) written there is a simplification of the original function
not the derivative

[unborn blade]

oops

thats what i meant to screenshot

[uncut mulch]

look at the disclaimer

in other words, the calculator sucks

,w 16/(sqrt(5x))-10/x^2 where x=1.25

[obsidian monolith]

Results provided by WolframAlpha

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

Requested by ramonov#0921

[unborn blade]

ahh

ok that clears things up

[ashen skiff]

quick question, what does [[ mean in a function? like [[x-1]]

[sick steppe]

Ive seen it used for greatest integer function

[sterile wedge]

how do I get the intercepts and graph this?

[dark sky]

how do i do this

[ashen skiff]

@sterile wedge the intercepts of a function are when x equals zero and y equals zero. When x equals zero is simple, just plug in zero for x and you get the y intercept. For the x intercept, set the numerator of the fraction to equal to zero, because if the numerator is zero then the whole fraction is as well. The numerator factors to (x+2)(x-1), so we can make that x+2=0 and x-1=0. Solve from there and you get your intercepts. Usually that's enough information. If not, plug in a few more points that seem relevant

[round quest]

how do i eliminate the parameter

[viscid thistle]

hint: sin^2(theta)+cos^2(theta)=1

@round quest

[fossil rock]

how do i find the total distance traveled by an object that is launched straight upwards when given acceleration and velocity?

[viscid thistle]

you use the laws of motion

[fossil rock]

🤔

[viscid thistle]

this is an extremely googleable question

[fossil rock]

ok

[honest shale]

Guys help please

I can't, I am about to cry, what is thiss??

I am super confused

And cant do it

[austere void]

@honest shale Do you still need help with this?

[honest shale]

Thank you so much, it took me a long time hut I figured it out!

[austere void]

Good job

[obsidian monolith]

*-*

[sick steppe]

,w graph sqrt(x^2-16)

[obsidian monolith]

Results provided by WolframAlpha

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

Requested by moshill1#9771

[sick steppe]

yeah you can only approach from the right side

[thin summit]

can I get help with identities?

I understand this muchbn

really

all I know how to do is apply the identity

but i don't know what steps to go from hyere

<@&286206848099549185>

also is cos(a-b) the same as cos(a) - cos(b)

[mighty pier]

$\frac{1}{\sin{x}} - \sin{x} = \frac{1 - \sin^2{x}}{\sin{x}}$

[obsidian monolith]

error 404

[mighty pier]

you should be able to do the rest

[mighty pier]

no, generally it's not true

[thin summit]

thank you

[muted steeple]

so according to factor theorem if x-1 is a factor of f(x)=x^n - 1 then x = 1 should mean f(x) = 0

right?

<@&286206848099549185>

[willow bear]

or to put that more concisely, f(1) = 0.

also, you should not ping helpers right away.

[tender quest]

From

#❓how-to-get-help

Rule 4

If your question has not been answered for a minimum of 15 minutes, you may use the Helpers tag once. Please do not try to bump your question using this ping unnecessarily. Do not abuse this ping. Do not individually ping users with the Helpers tag without their express permission.

[muted steeple]

15 mins, gotcha.

my question i guess is then more concerned with 2nd half.

i'm not sure how division here would work

[uncut mulch]

have you done polynomial long division before?

[muted steeple]

yes.

[willow bear]

try looking at some particular values of n, like 3, 4 or 5

see if you can notice a pattern in the quotients

[muted steeple]

alright.

only that the degree is one less than the dividend?

idk.

[brazen marsh]

did u carry out the full division?

[muted steeple]

okay so for n = 4 i get the following quotient

x^3-x^2-x-1

correct?

[brazen marsh]

correct

[muted steeple]

okay, so if i were to generalise this for n

[brazen marsh]

notice, that when u do this for n=5

wait a second

[obsidian monolith]

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

[muted steeple]

aight let me draw it out

[brazen marsh]

u messed up the signs actually

[muted steeple]

did i?

[brazen marsh]

yes

it will be (x^2+1)(x+1)

which means the signs will be positive

[muted steeple]

uhmm

i don't get it then.

x^5 - 1 / x-1

first quotient is x^4

[brazen marsh]

yea

[willow bear]

parentheses.

[muted steeple]

then i'm left with (x^5-1)-(x^5-x^4)

remainder is -x^4-1 right?

[willow bear]

+x^4 - 1

[muted steeple]

isn't my divisor x-1?

that -1 should invert my quotient.

[willow bear]

(x^5 - 1) - (x^5 - x^4) = x^5 - 1 - x^5 + x^4.

[muted steeple]

i see.

thanks.

i see the pattern now.

how do i express it in a general way then?

$x^{n-1} + x^{n-2} + ... x^{0} $?

[willow bear]

sure

$x^{n-1} + x^{n-2} + \dots + x + 1$

[obsidian monolith]

Ann

[willow bear]

have you seen geometric progressions before? bc this has an almost direct connection to the formula for the sum of a geometric progression

[viscid thistle]

Where did general channels go?

[willow bear]

you have the studying role

[slender quarry]

Hey so I’m a little bit confused about the answers given in my textbook

My question was cos^-1(1-x) and I solved it and graphed it

I checked it on a graphing calculator and it seemed to work

So I checked the answer and I am confused now

It’s the highlighted question

I got this though

While they got this

And I’ve ensured it’s the right question

(Not to mention their graph literally defies the range given)

[muted steeple]

not yet, but i see that the upcoming topics are to do with series, sequences, is that where i find geometric progressions?

[willow bear]

probably yeah

[viscid thistle]

Anyone know how to do this stuff

https://byjus.com/jee/important-solutions-of-triangle-formulas-for-jee/

there's all the formulas you'll need

[muted steeple]

i get to this point $(x-1)(x^2-5x+6)$

[obsidian monolith]

Vocal

[sick steppe]

what's preventing you from finishing?

[muted steeple]

not knowing what factor to use for the other one...

same one?

[willow bear]

@muted steeple your other factor is just a quadratic...

[muted steeple]

oh yeah...

i forgot i know how to do those.

i guess, gonna call it a day after this one...

thanks btw.

is there a need for me to know determinants for calc?

"2nd order determinants"

[willow bear]

not really

[viscid thistle]

in synthetic division if x=-3 then its beign divided by x-3 right

[sick steppe]

x=-3 means x+3=0, so the factor is x+3

[viscid thistle]

oh alr

[dusk oriole]

does anyone know how to do this

[viscid thistle]

this is the graph but where does it show that as the end behavior

[trim hemlock]

the graph of what function?

[novel oyster]

oh?

[mild coral]

What do you mean?

[woven ravine]

based on the graph we can tell that it's a negative even function, meaning that whether x approaches pos or neg infinity, f(x) will be negative

[proven sentinel]

helpppppp

[sick steppe]

with which?

[proven sentinel]

both😅

[sick steppe]

Ok well they're likely both just combine the fractions and simplify from there

[rough carbon]

Always start from the more complicated side to the other side I find that a lot easier

[proven sentinel]

thank youu @mosh & @rough carbon <333

[hearty bronze]

Can anyone help with this?

[sick steppe]

expand the left hand side and hope for simplification

[hearty bronze]

so expand using addition/subtraction formula?

[sick steppe]

yes

as opposed to the other way to expand sines and cosine. .

[dusk oriole]

does any one know how to do this, I'm a bit confused

[willow bear]

have you ever calculated projections of vectors onto other vectors?

[dusk oriole]

yes, but I'm confused how you would write it as a sum of the orthogonal vectors. would it just be the answer you would get after solving for projvu?

[willow bear]

one of the vectors is proj_v(u) and the other is (u - proj_v(u))

[dusk oriole]

ohhh ok thank you so much!!

[hearty bronze]

im confused on this, it says to list all possible solutions

[willow bear]

have you tried factorizing ths as sin(x) (1 - sin(x)) = 0?

[hearty bronze]

how would that work

[rough carbon]

If the sin’s are confusing you, just imagine it as a normal quadratic. If u= sin(x) for eg, u-u^2=0

[willow bear]

do you understand why $u - u^2 = u(1-u)$?

[rough carbon]

You know how to factorise that

[obsidian monolith]

Ann

[hearty bronze]

ah ic

y that works now

but im still confused

bc the question asks for all possible solutions

and i have sin(x)(1-sin(x))=0

[rough carbon]

Yeah so either sinx=0 which occurs at 0,180,360

Or sin(x) =1

Bc 1- sinx=0

Which occurs at 90

Or the equivalent in radians if you have to give it in radians

[hearty bronze]

oohhh

i get it

so then for

when i factor it down

should I split 5sin(x)

into 5 sin(x)

so like

2sin^2(x) - sin(x) + sin(x) + sin(x) + sin(x) + sin(x) + 2

[rough carbon]

Nah, I would think of it more like a normal quadratic, so let u= sinx( I always recommend this btw, makes things easier to see)

2u^2 -5u+2=0

And factorise that

[hearty bronze]

yeah thats easier to visualize and factor

[rough carbon]

Can you get it from there?

[hearty bronze]

yeah

i can factor it

down

so its (2u-1)(u-1) right?

making it

(2sin(x)-1)(sin(x)-1)

[rough carbon]

No bc that only gives you -3u

[hearty bronze]

o wait type

typo

(2u-1)(u-2)

[rough carbon]

2u^2 -4u-u+2=0
2u(u-2) -1(u-2)=0
So (2u-1)(u-2)=0

Yeeee

[hearty bronze]

(2sin(x)-1)(sin(x)-2)

[rough carbon]

So either sin x is 2 but that’s not possible so we ignore that bracket

Or sinx is 1/2

[hearty bronze]

ic

then what about

sin(x)=cos(2x)

i can rewrite cos(2x)

as

1-2sin^2(x)

so

[rough carbon]

Is this a different question?

Wait lol

[hearty bronze]

yeahh

[rough carbon]

Ah ok

Yeah that’s right, double angle formula

[hearty bronze]

yeah so it can be written as

sin(x)=1-2sin^2(x)

[rough carbon]

Yes

Then move everything on one side, and it’s the quadratic in sin again

[hearty bronze]

making it

2u^2+u-1

[rough carbon]

Yep

[hearty bronze]

which is (2u-1)(u+1)

[rough carbon]

Yep

[hearty bronze]

then u is -1 or 1/2

[rough carbon]

So this time, we take both brackets bc sinx=1 or sinx=1/2

Yes

[hearty bronze]

ic

thank you for your help @rough carbon

[rough carbon]

No worries

[faint nest]

am i doing this right

@rough carbon

[rough carbon]

Yeah so the modulus multiplied and then the args add

That’s correct I think

[faint nest]

@rough carbon

how do i put it in standard form

a + bi or whatever

[rough carbon]

Literally just multiply it all out

You’ll get a real number plus ki

Which is what you want

[faint nest]

@rough carbon

Do u know this

[rough carbon]

I think that’s A2 for me

I haven’t done that yet

[misty imp]

can someone help me with this problem?

sec(θ)=13/12
cos(ψ)=15/17
θ and ψ lie in quadrant 4

Find:
tan(θ+ψ)
tan(2ψ)

[cedar pawn]

For first one tangent addition formula

and second double angle tangent formula

[frail patrol]

I forgot some simple log rules how do I go from here

[obsidian monolith]

A regular kugelblitz

[sleek pawn]

is the same as:

[obsidian monolith]

A regular kugelblitz

[sleek pawn]

so x=3

@frail patrol

remember, logs are referring to an unknown exponent, so same rules apply

[frail patrol]

Oh I completely overlooked that lol thanks

[sleek pawn]

np

[flint rose]

hi i have a question on what relevance does the 3rd derivative have to stationary points on a graph?

[willow bear]

little if any

[flint rose]

this was the mark scheme

and im confused to why

the 3rd derivative was used

[viscid thistle]

a point of inflexion is a point where concavity of the function changes.

hence third derivative not equalling zero means that it is a point of inflection, (the sign of second derivative gives direction of concavity, and if the f’’’ is not zero, this means f’’ changes sign at that root of f’’

try thinking of it this way, we want a point where sign of f’’ changes

hence it has two conditions, f’’(x)=0

and f”’ (x)not equal to zero

[flint rose]

ok thank you

its still confusing but ill accept it

like im struggling to visualise back to f(x)

but thank you for the explanation :)

[flint rose]

what happens if f'''(x) is equal to 0

[austere void]

You don't have to check the 3rd derivative to find inflection points

[viscid thistle]

consider x^2 for example, x^2 is zero at x=0, but the sign of the function stays the same when we go near zero, this is because x=0 is a repeated root , or because the next derivative is zero too at x=0

[viscid thistle]

i think yes, for f=x^4 , 0 has first and second derivates as zero,,but is not an inflection point

[austere void]

You only need to find zeroes of the second derivative and look at its signs

If it changes, it's an inlection point, if it doesn't, then it isn't

[viscid thistle]

i just started pre-calculus and

i'm still learning linear

[thorn birch]

Linear what

[fleet yew]

Linear calculus

[viscid thistle]

henlo !
can someone help me with this problem : find two numbers whose difference is 68 and whose product is a minimum

[willow bear]

you're flipping only one component

instead of both, to actually multiply the prev answer by -1 as youre supposed to

[tender lotus]

wow thank you

[true mica]

Hey anyone know some great resources to learn derivatives? Im a little behind on that

[west crown]

Can anyone help with grade 10 per calculus?

Pre*

<@&286206848099549185> please I need help

[lethal oracle]

@west crown Pick a problem first

[west crown]

5 the rectangle

[lethal oracle]

let me see if I can help

so you have a rectangular

and it's volume is a function

what is the formula for volume of a rectangular prism

@west crown

[west crown]

Thanks

[lethal oracle]

You understand it now?

[smoky pagoda]

@west crown is this a test

[west crown]

No

[somber marsh]

are parallel vectors always opposite or equal vectors?

^Someone help please

[viscid thistle]

no thats not true

for example 3i^ and 5i^ are parallel

but neither equal nor opposite

[strong ermine]

how would i do 3b

[willow bear]

use your knowledge of the 68-95-99.7 rule, if you have been taught that

or failing that, use a z table

or failing even that, use a calculator capable of computing the inverse cdf for a normal distribution

[formal forge]

Idk how I am supposed to solve this system of equations with 4 variables

[sick steppe]

Matrix and Gauss-Jordan?

[formal forge]

I've never heard of those things :/ @sick steppe but I can look them up

[sick steppe]

https://matrixcalc.org/en/slu.html#solve-using-Gauss-Jordan-elimination({{6,13,3,7,162},{15,5,11,0,110}})

You get a plane of solutions due to having underdetermined system

[formal forge]

If I translate the question: Lina has four different kinds of coins - gold - silver - bronze - copper. (All coins of the same kind weigh the same, and all weights are whole grams) 6 gold + 13 silver + 3 bronze + 7 copper = 162 g and 15 gold + 5 silver + 11 bronze = 110g. I'm assuming that the answer is unique weight for each type of coin. I'm not sure however

[sick steppe]

yeah you need 4 equations to be likely in getting a unique solution

[formal forge]

Would it work if I created 2 equations from the 2 given ones?

[uncut mulch]

what's the question asking for?

[formal forge]

What the weight for each type of coin is

[uncut mulch]

see like it's gonna take a decent amount of effort if a solution exists

maybe you could consider adding your equations

and expressing 272 in the form
3m + 7n

[sick steppe]

what have you tried?

[fluid flame]

nothing I just dont know where to start

[sick steppe]

well which looks like a better place to start? LHS or RHS?

[fluid flame]

whats LHS and RHS?

[sick steppe]

left hand side

right hand side

[fluid flame]

oh i would say the right side looks better

since you would solve it backwards right?

[sick steppe]

I mean dont you think simplifying is easier?

[fluid flame]

well im learning it so the harder way is better for me

if that makes sense

[sick steppe]

you want to learn the efficient way, not the hard way

efficient way is simplifiying as that is what you're more use to in math

making something messier usually requires black magic steps of adding 0 or multiplying by 1

[fluid flame]

I see

[sick steppe]

So start with the LHS, cause it's clearly messier

$\frac{\cos^2(x)}{1-\cos^2(x)}+1 \=\frac{\cos^2(x)}{1-\cos^2(x)}+\frac{1-\cos^2(x)}{1-\cos^2(x)}$

[obsidian monolith]

Mosh

[edgy blaze]

so if the interval is -pi<θ<pi, how does that work? isn't -pi coterminal with pi. is that an an interval?

[obsidian monolith]

Goose on a Moose

[slender river]

@heady drift complete the square within the square root and square both sides and move terms around

as for determining if it's half of an ellipse you can restrict the result of 2x-x^2 to be nonnegative and bound x that way

[heady drift]

Yes thank you. This is exactly what I needed.

[willow bear]

$y = \sqrt{2x - x^2}$

[obsidian monolith]

Ann

[willow bear]

just for reference

[willow bear]

this screenshot is so low-res it is impossible to read

[granite ice]

sorry let me crop it

may i ask why only solving this problem (and integrating cosecx too) needs to be done like this? my teacher said that the domain won't change but i want more specific details about it

[willow bear]

no, it does not 'need' to be done like this.

[granite ice]

oh?

[willow bear]

there is another, although somewhat longer, way of doing it

[granite ice]

can explain in detail?

[willow bear]

sure...

[granite ice]

bc my teacher was teaching tanx and secx integration and other parts of it

[willow bear]

$\sec(x) = \frac{1}{\cos(x)} = \frac{\cos(x)}{\cos^2(x)} = \frac{\cos(x)}{1 - \sin^2(x)}$

[obsidian monolith]

Ann

[granite ice]

ahhhhh

[willow bear]

substitute $t := \sin(x)$ to get $\int \frac{\cos(x)}{1-\sin^2(x)} \dd{x} = \int \frac{\dd{t}}{1 - t^2}$

[obsidian monolith]

Ann

[willow bear]

then partial fractions

[granite ice]

yes yes the next section is that

ok thx ann

[granite ice]

but i'll still like to ask why can be done like this

dividing f(x) will change it's domain but how does the domain stays the same in this context

[willow bear]

tan(x)+sec(x) is undefined at the same points as sec(x) itself

?

no

[granite ice]

wait

[willow bear]

what are you talking about

[granite ice]

lol i misunderstood

i don't understand what u meant by undefined

[willow bear]

...you don't know what undefined means?

[granite ice]

let me translate that to chinese hahah

give me a sec

oh

same points as in?

[willow bear]

points as in values of x

[granite ice]

so this correlates to the fact that the domain won't change bc of this right?

alright i got it thx for ur help!

[viscid thistle]

hi off topic but can anyone help me solve this physics question like really help me understand the concept my teacher and i just don’t

[jolly raven]

@viscid thistle I can try

Haven't done physics yet but I know a good amount of math

[bitter trellis]

Im practicing for a test and this is an example question how would i do this

[willow bear]

you can't, because this question is screwed up

they give you a function named g but then ask about another function named f

[naive venture]

How would you do it even if they didn’t mess up?

[willow bear]

if the g at the beginning really was meant to be f,

then you would understand the pairs (1,2), (2,9) etc as (input, output) pairs

and, for example, to find f(2), you would look for an input of 2, and read off its corresponding output

[obsidian monolith]

waler

[trim hemlock]

the first expression was a result to an integral

but as i checked on wolfram alpha, there was another answer which is arsinh(x/2)

i guess i dont need to convert it, but im curious on how you would do that

i cant seem to figure it out

[willow bear]

sec(arctan(x/2)) = sqrt( (x/2)^2 + 1) doesnt it

[trim hemlock]

yeah

oh so

ah alright so we got sqrt((x/2)^2 + 1) + x/2

ahh right

[glossy cairn]

is that tuple notation actually something people use?

looks really weird

a function that is equal to some set of tuples lol

[viscid thistle]

isn't it just theta?

theta/2pi * 2pi = theta

here 2pi - theta?

[viscid thistle]

<@&286206848099549185>

[eternal gust]

The arc has length theta on a unit circle (that is when r=1). When r/=1 this won’t be true

[eternal gust]

For this question you are right, because the circle is a unit circle

[eternal gust]

Well actually you forgot to add the two radii that are now a part of the diagram, sorry

[viscid thistle]

what do you mean

[viscid thistle]

theta/r, then?

[eternal gust]

No r*theta

[viscid thistle]

why

theta/2pi * 2pi = theta
part of the circumference * circumeference = theta

[eternal gust]

The circumference of a circle is 2pir and we are taking theta/2pi of the circumference, so we get thetar

Make sense?

[viscid thistle]

no

[viscid thistle]

.

[eternal gust]

The circumference is 2pi*r, not 2pi

[viscid thistle]

ik

this is for the r = 1

[viscid thistle]

.

value of rad would be smaller?

[eternal gust]

The first question is for any r

[eternal gust]

In degrees, cos(3.14) is really close to cos(0)=-1, in radians cos(3.14) is close to cos(pi)=-1

[viscid thistle]

man

i'll paste it all again

theta/2pi * 2pi = theta
part of the circumference * circumeference = theta

so if the circumference is 2pi * r

then you have theta /(2pi * r) ( 2pi)

so theta/r

[eternal gust]

No it is (theta/2pi)2pir.
You said it yourself, fraction of circumference*circumference.

The fraction is theta/2pi, and the circumference is 2pi*r

[viscid thistle]

oh right

anyways shouldnt it be
theta/(2pi * r) * 2pi * r = theta

[eternal gust]

No, see theta is the angle, and in one rotation of the circle we go around by 2pi, so the fraction of the full circle we are rotating by is theta/2pi

[viscid thistle]

got it

[eternal gust]

As an example, if theta=pi, we go halfway around the circle, so the arc length will be 1/2 * (circumference)

[eternal gust]

Ok cool

[viscid thistle]

btw you haven't explained this

[eternal gust]

Here is an easy proof, using this idea. 3.14 in degrees is <90 and >0 , so cos(3.14) measured in degrees is positive. 3.14 in radians is >pi/2 and <pi, so cos(3.14)<0. You can use this to see who is measuring in degrees and who is measuring in radians

[viscid thistle]

where does this come from

3.14 in radians is >pi/2 and <pi, so cos(3.14)<0

give me a second

[eternal gust]

Pi=3.1415... so we get those inequalities

[viscid thistle]

ye i got it

what about this?

it makes intuitive sense

because it's in the 4 quadrant

right under the horizontal axis

but idk how to use the calculations to prove it

[eternal gust]

Use the approximation they give to give a justification, they aren’t asking for a full proof

I have to go now, sorry

[viscid thistle]

i don't know how to

[eternal gust]

$710=2113(350/113)$

[obsidian monolith]

saketh

[eternal gust]

This shows 710 is close to 2pi*113

[viscid thistle]

where does it come from?

cos 710 means the value of the x coordinate at the degree 710 from horizontal axis counter-clockwise

i don't see where you got that 2pi *113

or $710=2113(350/113)$

[obsidian monolith]

sine

[eternal gust]

This is just some arithmetic, the 113’s cancel out and then 2*350 is 710

[viscid thistle]

i get it but how have you come up with this specific figure

710 could be written in inf number of ways

[eternal gust]

I got it because i wanted the relation between pi and 710 in order to calculate the cosine. We were given pi=350/113 approximately, so all you have to do is see what time pi will give 710

[viscid thistle]

oh

so it was meant to be 355/113 and not 350/113

[eternal gust]

Oh yeah, sorry, that was a typo

[viscid thistle]

anyways i don't see how does it prove that cos 710 ~ 1

[eternal gust]

Cos(2pin)=1 for any integer n, (because it is like making n full loops around the circle). And we just showed $710~2pi113$

[obsidian monolith]

saketh

[viscid thistle]

got it, thanks

[eternal gust]

Cool

[ocean temple]

Pls help

<@&286206848099549185>

[blissful ridge]

<@&268886789983436800>

You might wanna inspect this

[slender river]

bro are all of the mods asleep or something

istg it's like

the last two days no one has been on

anyways @ocean temple is this an examination

it sure looks like one

[ocean temple]

No it is not

@slender river

@slender river

Now please tell me how do we do these questions

[ocean temple]

<@&286206848099549185>

[tranquil haven]

Hey what do these brackets mean ?

[uncut mulch]

here they're just grouping symbols

they could use () but using [] makes it clearer what's being grouoped

[tranquil haven]

Ah okay, thanks

[kind musk]

<@&286206848099549185>

[viscid thistle]

probably x = inf but idk

[ashen skiff]

if you're given 2^a + 3^b = some number, is there any way to see if a and b are integers? like if i'm given 2^a + 3^b = 1284, is there any way besides guess and check?

[limber prairie]

Maximum profit = Highest selling price - lowest cost, which means you are to find the value of a single x that will give you the highest selling price possible and lowest cost possible. 🙂

[rustic minnow]

fiitjee tab!?

[safe fiber]

yup

[ocean temple]

Yes

[rustic minnow]

nice

[brittle solstice]

,help

[obsidian monolith]

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

[brittle solstice]

,list

[obsidian monolith]

My commands!

Use ,ls to obtain a briefer listing, and use ,help <cmd>to view detailed help for a particular command, or ,help to view general help.

If you still have questions, talk to our friendly support team here.

Info

Display information about the guild, members, roles, channels, and more!
​ ​ ​ ​ ​ avatar: Obtains the mentioned user's avatar, or your own.
​ ​ ​ roleinfo: Displays information about a role.
​ ​ ​ userinfo: Shows various information about a user.
​ ​ guildinfo: Shows information about the guild.
rolemembers: Lists members with a particular role.
channelinfo: Displays information about a channel.

Utility

A handy toolbox of miscellaneous small but useful commands.
​ ​ ​ ​ time: Displays the current time for a user
​ ​ rotate: Rotates the last image.
​ ​ giveme: Request, list, and modify the guild's self assignable roles.
notifyme: DMs you messages matching given triggers.

Page 2/4

[oblique pulsar]

/For each of the following write, in point form, the transformations from the base
function y = x. (for example: stretched by a factor of 5, translated/shifted left 8
units, etc...)/

im trying to study for my test and im not sure how to do that

[gaunt mason]

@oblique pulsar What is the base function here?

[oblique pulsar]

/For each of the following write, in point form, the transformations from the base
function y = x. (for example: stretched by a factor of 5, translated/shifted left 8
units, etc...)/

y=x

[gaunt mason]

I think that would be the example base function

IN this case the base function is different

[willow bear]

are you sure it's y = x and not y = x^2

[oblique pulsar]

@willow bear my bad it is ^2 im stupid my bad

so do you know how to do that @willow bear

[gaunt mason]

Always:

• the thing next to x directly is the horizontal shift (translation) - to the right is negative, left if positive
• the thing in front of that but outside of square is the vertical stretching (if bigger than one)/contraction(if smaller than one)/or flip (if minus)
• the thing that stands alone is the vertical shift (translation) - upwards if positive, downwards if negative
  @oblique pulsar

These are applied in that order.

That's all you need to know 🙂

[oblique pulsar]

aight thanks @gaunt mason

[tranquil haven]

Heym,

I'm having trouble connecting the c(n, r) formula to pascals triangle ?

Can someone please explain how they link toghether

Already thanks

[willow bear]

from a formal standpoint, the binomial coefficients obey the same recurrence relation which generates pascal's triangle

namely: C(n,r) = C(n-1, r) + C(n-1, r-1)

[viscid thistle]

why can you do this

[willow bear]

$\sin(-x) = -\sin(x)$

[obsidian monolith]

Ann

[viscid thistle]

t.anks

[viscid thistle]

why?

[whole igloo]

You see the second line? They moved the a^2*cos^C over beside the first term and then factored out the a^2.

After that, sin^2(anything) + cos^2(anything) = 1. It's a trig identity.

[arctic kestrel]

hey

not understanding the tangent line

<@&286206848099549185>

wouldn't it be m, y-int?

and 0, -5 for its line

2, 11?

2,11 : 0, -5

[viscid thistle]

where does the multiplication come from

also

shouldn't it be
(n-1)!/[(k-1)!(n-k-1)!] in the first line?

hey

what happens when I have this degenerate conic?

occupied

go to some other channel

[viscid thistle]

The (n-r)! in nCr is (n-1 - k + 1)! = (n-k)! , not (n-k-1)!

nCr = n!/(r!(n-r)!)

what's nCr

n Choose r

what does it mean

same thing as $\binom {n}{r}$

[obsidian monolith]

duck34

[viscid thistle]

i don't understand why

well in this case, n is (n-1) and r is (k-1)

so n-r = n-k

You have to distribute the negative sign

ok i se

$n_C_r = \binom{n}{r}$

could you expalin the multiplication

[obsidian monolith]

duck34
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[viscid thistle]

which one

from 1st to 2nd

line

Oh, well we need a common denominator

to comine fractions

From there its just a bunch of algebra

could you elaborate

Well for example a/x + b/y

you need a common denominator of xy

to combine the fractions

i mean in this example

Yes, in this case, we can factor out (n-1)! since it is in both of them

then combine fractions

so k(n-k) is the common denominator

and we get

1/n-k + 1/k = (n~~-k+k~~)/(k(n-k)) = n/(k(n-k))

-k+k cancels

so its just n/(k(n-k))

then we can multiply back again

[viscid thistle]

what do you mean

between 1/k and 1/n-k

to find the common denominator you multiply both

if they dont have anything in common

like my earlier example

you multiply xy for the common denominator

what about this

ah

well that is simplifying the factorials, (n-k)! = (n-k)(n-k-1)!

by definition

same logic as 5!= 5(5-1)!

so you can factor out a (n-k-1)!

(k-1)!(n-k)! = (k-1)!(n-k)(n-k-1)!

alright thanks

i'm done with this shit

time to move on to limits

[unkempt pebble]

Help with optimation proplem please

A farmer wishes to fence a rectangular field alongside a river, and subdivided into 6 equal smaller fields by fencing two lengths parallel to the river and four widths perpendicular to the river. Fencing along the riverbank is not required. If the farmer has 1000 m of fencing to use, determine the dimensions of the field that will give the maximum overall area

[solemn flame]

What’s this radian

[sick steppe]

why do you have pi/3 as a right angle...?

[solemn flame]

@sick steppe my teacher is mixing things up

[sick steppe]

what???

your teacher isnt teaching you the right thing

[solemn flame]

@sick steppe my teacher wants us to find the next radian

[sick steppe]

well it's wrong regardless

[solemn flame]

I know what it looks like

But he’s making it into a whole other circle with different angles

[sick steppe]

ok.. well then it's pi/9 + pi/3

[subtle bone]

Hey anyone free to help

[lime rivet]

yo

wtf

I was in dms

[arctic kestrel]

how can I find tangent of line?

<@&286206848099549185>

[fringe raft]

Just take the derivative of the function and set x=-5

[arctic kestrel]

first point x=-5 and what's second point y=0?

and what's the second point of equation?

2, 11?

[fringe raft]

A point is defined using 2 coordinates (x,y)

[arctic kestrel]

yeah 5, 0?

to 11, 2?

[fringe raft]

Here after taking derivative we don’t care about the y coordinate(because there’s no y term)

[arctic kestrel]

w

i know but

it's hard because you see

@fringe raft

<@&286206848099549185>

[languid bolt]

I have a final for pre calc tomorrow and I forgot how to do a simple thing

how will you answer b?

I got x = 4pi/6 and x = 5pi/6 but I forgot what to do after that

<@&286206848099549185>

got it

[subtle bone]

Hi

What rule would I use to differentiate

[willow bear]

well this is several things multiplied together, right

[subtle bone]

Yeah

[gusty sundial]

Use product Rule

Differentiate anyone keeping other constant and continue this process for three times.

Cuz there are three variables needs to be differentiated for three steps.

[viscid thistle]

Shouldn’t it be a-bi?

[gusty sundial]

The whole line is just confusing lol

[viscid thistle]

Sixth edition

[willow bear]

i mean

the bad typesetting notwithstanding, isn't it nearly obvious that taking the conjugate of the conjugate brings you back to the original number?

[viscid thistle]

I think this should be only true when imaginary part=0

[viscid thistle]

I still don’t get it

[willow bear]

taking the complex conjugate means flipping the imaginary part (and only the imaginary part)

for example, the conjugate of 5 + 8i is 5 - 8i

the conjugate of -3 + 22i is -3 - 22i

the conjugate of 9.5234 - 420.69i is 9.5234 + 420.69i

and so on

do you understand that or not

[viscid thistle]

Then shouldn’t conjugate of z be a-bi?

[willow bear]

yes, the conjugate of z (=a+bi) is a-bi

your book has a lot of typos

i'm imploring you to look past that

and also you didnt answer my yes/no question

[viscid thistle]

Yeah I knew that

[willow bear]

yeah, so...

was the bad typesetting in the book your only point of confusion?

[viscid thistle]

Yes

[willow bear]

ok

$\overline{\overline{z}} = \overline{\overline{a+bi}} = \overline{a-bi} = a - (-b)i = a + bi = z$

[obsidian monolith]

Ann

[willow bear]

this is how it should've gone

[viscid thistle]

Thx

[misty imp]

did i find 4u*3v correctly?

[split glacier]

Can someone please help me

[viscid thistle]

no

actually yes u did find it correctly

but 4v should be 3v

[tranquil haven]

Hey, what are unit vectors used for ?

Just learned how to calculate them from a vector

[willow bear]

this property is not unique to unit vectors. the vectors [19, 2] and [-5, 44] can also form every other vector by linear combinations.

@tranquil haven from a 'practical' standpoint, you could say unit vectors are used when all you need is information about direction but not length

[daring eagle]

i know i shouldn't be asking this at my level, but suppose a function f(x) = 2x, does it make sense to say f=2?

a 2 function?

a function of 2?

[somber yew]

f(x) is not f multiplied by x. It is notation suggesting that the function f evaluates to 2x when the input is x.

So the bit inside the parentheses is argument of the function: f(y) would be 2y, f(3000) would be 6000 and so on.

[daring eagle]

yup. i get that

i ...trying to think of f without the x?

if that makes sense

[somber yew]

I think I see what you're saying, but again letting the x be there isn't harmful. If you just want to go by words, say "f is a function which doubles its inputs" or sth.

That's a valid but cumbersome way of expressing it.

[daring eagle]

oh that's nice. idk...this has been something, that i forget frequently, but when i see it, it eats away at me

f is a doubling function

thank you @somber yew !

[somber yew]

No worries.