precalculus

[clear berry]

The object is just moving in a circle with constant angular velocity, you would usually use that in like a pendulum with changing angular velocity

[frail lion]

Yeas you're right

I hadnt paid attention to the constant angular velocity that's why I used work energy theorem and missed this too

Basically one mistake accounted to both

Hmm yeah now using equations of motion angle of projection and time both will come , so we will get every information about the projectile
Then it's just basic kinematics and all

[clear berry]

yeah

[livid raft]

Hi! Does anyone know if it is possible to represent w = 6e^π/3 in form of x + yi WITHOUT using a calculator? It is from exam drill, and we won't be having calculators

[modern wedge]

thats a real number though

theres no imaginary part

[livid raft]

yeah :( Do u know what is a way to represent it, with a complex part being 0i? (and without using a calculator)

[modern wedge]

im confused

itll just be 6e^pi/3 + 0i

do you mean $6e^{πi/3}$

[obsidian monolith]

ashy!

[livid raft]

nope (sorry its in finnish, says "write the point... in a form...")

[modern wedge]

probably a typo

should have an i

[livid raft]

I see!! Thank you!

[livid raft]

It's not a typo : (

[modern wedge]

strange

alright

[frail lion]

Yeah man honestly i was thinking same too

[lone temple]

hey everyone, i would love to understand more about second derivative, i met a lot of questions using it but i still dont know how it worked
thanks in advance!

[modern wedge]

are you asking how to find the second derivative

or what are its uses

[lone temple]

the uses

[modest bolt]

if you have a function f(x), its first derivative tells its slope at any point and second derivative tell the rate of change of it's slope.

So like suppose you get f'(x)=0 at some x, that means either x is local maxima or local minima but we don't know which one is it so we can take second derivative f''(x).
If f''(x)>0 that means the slope is increasing that means the x we found is the local minima.

[golden harbor]

Hey guys, if you need help of precalc you should check out voluntors.org, its a website I made for students to tutor and get volunteer hours (or get tutoring)

[terse surge]

would this be exponential decay or growht

[frail lion]

Looks somewhat like y=-(2)^x
Pretty confusing, definitely not growth but not standard decay either

[fluid sorrel]

A function exhibits exponential decay only when its growth factor (base) is between 0 and 1.

So it's just a negative reflection of ig 2^x graph

It shows exponential change in magnitude

[patent barn]

It doesn’t have the right range to be decaying

[frail lion]

Yes

[fluid sorrel]

Yeppp

[willow skiff]

what have you tried?

[frail lion]

Yes

Obv

Bro what why its easy

Peasy ?

[willow skiff]

you didn't answer my question

[frail lion]

Oh ok i misunderstood

Look first i found meeting points of the circle and line

The circle passes through origin

So one point Is origin

Other is (0.5,-0.5)

Now the center of circle is 1+a/4 and 1-a/4

And the point 1+a/2 ,1-a/2 passes through the circle

So joining origin center and this point

We get a diameter

So from there I'm trying to do things

I tried putting a point on y axis

[willow skiff]

right, the locus of the midpoint of the chord on the circle

that's another circle

[frail lion]

And got a as 3

[frail lion]

Ok ill check for it

[willow skiff]

it has centre being the midpoint of ((1+a)/4, (1-a)/4); ((1+a)/2, (1-a)/2)

and half the radius of the original circle

[frail lion]

What bro how i don't understand

Can ya give solution written somewhere

[willow skiff]

here's a diagram

[frail lion]

No I get that

[frail lion]

.

[willow skiff]

I don't understand how you got this

[frail lion]

Wait I'll

Send

[willow skiff]

the equation of the locus should be $\left(x-\frac{3\left(1+a\right)}{8}\right)^{2}+\left(y-\frac{3\left(1-a\right)}{8}\right)^{2}=\frac{1+a^{2}}{32}$

[obsidian monolith]

south

[willow skiff]

this is probably not the most efficient way but it works

[frail lion]

Comes out as one of the circles following every criteria i guess

[willow skiff]

I believe you've totally misunderstood the problem

have a look at this sample problem: https://www.youtube.com/watch?v=DsAM9IWW6vo

[willow skiff]

you can't make another chord passing through the green point that bisects x + y = 0

[frail lion]

Yeah that's that's right you're right

[frail lion]

Ok Wait hol on lemme see it

Ok I'm done

Yes I was going completely different

[willow skiff]

this question appeared on the 2023 JEE Mains, so everyone on YT is using the same method

I have a different method: write the circle equation into $(x - h)^2 + (y - k)^2 = r^2$, where you need to find $h, k, r$ in terms of $a$

[obsidian monolith]

south

[willow skiff]

if you then take the midpoint and halve the radius, you should get this

then you just need to sub in y = -x and set the discriminant > 0

it's a lot of work yes but I think it's the most clear method

[frail lion]

Hmm

[willow skiff]

I didn't use $xx_1 + yy_1 - h(x + x_1) - k(y + y_1) = 0$

[obsidian monolith]

south

[willow skiff]

(note that $x^2 + y^2 - 2hx - 2ky= 0$ is the equation of a circle which always passes through the origin)

[obsidian monolith]

south

[frail lion]

Bro look one thing I wanna ask

[frail lion]

Like we use this method

IF we use this method

Then how will we take a general point

On circle

Like theta and all typo shi gets in

[willow skiff]

you need this locus principle

[frail lion]

What's this principle ive never seen it

That's the principle only ?

This statement

[willow skiff]

yes

it's another circle that passes through the given point and has half the radius

from that you can deduce the location of the centre as being the midpoint

[frail lion]

Ok

[lone temple]

f"(x)>0 means the slope of derivative is increasing alright but why is it minimum value by that so?

[frail lion]

Beacuse it lowkey means the graph will climb up from there

Yk increasing

The graph climbs

So imagine you climb from somewhere than your starting point is minimum point

That's why minimum

When f''(x) <0 , means graph falls , so same logic , it's local maxima

[modest bolt]

you should watch some animation video explaining how it works, it's very hard to understand if you can't imagine the slope and the graph in your mind

or i can send a video link if you want

[lone temple]

i would love to

[fluid granite]

dis is tuff

[modest bolt]

check dm

[broken wigeon]

<@&286206848099549185>, can smbdy help check if what im doing is right😭 the equation is y=3secx. also, how do i graph this im still confused graphing secants 😭😭😭🙏🏻

[tender quest]

Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.

[trim shard]

when it comes to graphing secant or cosecant it helps to draw the sine or cosine of the same function

graph 3cosx and place your asymptotes, then you should be able to sketch it out

(just for visual representation)

[fluid sorrel]

Draw a graph of cosx for ur reference to draw secx graph

But remember that it's 3 secx

So like u need to draw graph of 3 cosx for reference

How will u do that

Just remember that range of cosx is between -1 to 1 including 1 and -1

So for 3cosx it would be -3 to 3

As shown in above graph

[fluid sorrel]

Here

N for y=3secx

Draw y=x graph

Then the point of intersection for a given interval would denote the number of solutions

[soft plover]

A

[frail lion]

Yeah

[river drift]

<@&268886789983436800>

[heady bluff]

Can someone give me a brief explanation of exponential functions?

And like what can they represent?

[frail lion]

Exponential functions "represent" the traditional y=f(x) functions only , we throw in some value of x in f(x) , we get some value on y

Like for example, y =2^x
Throw in some value of x
X=1 ,y=2(2^1)
X=2,y=4(2^2)
Like that
It's just with exponents in it , BUT make sure x is in the exponent , coz x^2 is not an exponential function
General form = a.b^x

[heady bluff]

Ohh ok, thank you!

[median spear]

Exponential functions always gives positive values, base can't be negative or zero or one

[cyan quiver]

where can i find materials for pre calculus honors

[rich elbow]

Also wy istn lateX

[river drift]

remove the spaces between $ and equation

[rich elbow]

I feel so washed, but is $| v_{1} + v_{2} | = sqrt{|v_{1} |^{2} + |v_{2}| ^{2}}$ ?

[obsidian monolith]

Jakexz

[rich elbow]

wat happened to my sqrt 😕

[low slate]

| v + u |^2 = |v|^2 +|u|^2 + 2|v| |u| cos(x)

where x is the angle between the two vectors

[low slate]

you're missing this

[rich elbow]

so only if they're orthogobal?

[low slate]

no, that works with any two vectors

[rich elbow]

Ah I see

[low slate]

oh wait

wait wait

you meant the angle between them is 90

[rich elbow]

ya

[low slate]

so yeah cos(90) is 0

[low slate]

and yeah that would be correct

[river drift]

that's basically the statement of the pythagorean theorem

which applies only to right triangles

[rich elbow]

oh my

[low slate]

orthogonal means that multiple vectors make 90 degree angles with one another, and by multiple i mean more than two. like the x y z axes are orthogonal.

the right phrase to use here is "Only if they're perpendicular" then yes, it would look like this | v + u |^2 = |v|^2 +|u|^2

I also think that orthogonal means that a system of things are perpendicular on one another, not only with vectors

[quick pebble]

you can have just two vectors be orthogonal too

perpendicular is a geometrical term, orthogonal is more general, but in this case they are the same

nothing wrong with what he said

[frail lion]

In "this" case they're the same ? Are there any cases where they aren't ?

[quick pebble]

yes, when you are not working with vectors that have a geometric interpretation

for example you can have polynomials that are orthogonal

but "perpendicular" would not really make sense in that context

[frail lion]

Ah yes in that way ok

[low slate]

understood

(in arabic, orthogonal translates to (متعامدين مثني مثني) which suggests multiple objects)

ty for making the english meaning clear

[west ice]

https://tenor.com/view/whyareyoublue-walterwhite-breakingbad-blue-jesse-gif-12242572209403578115

[low slate]

https://tenor.com/view/mean-girls-why-are-you-white-amanda-seyfried-gif-3551925

[west ice]

Cuz I'm not active

[low slate]

L fionna, L.

[west ice]

HOW DID YOU GET BLUE

ARE YOU ABOUT TO BE HELPFUL?

[low slate]

scroll up, i misinformed someone like 2 hour ago ❤️

[sullen gull]

I woke up now and forgot what the integral of lnx is 🙏😭

[frail lion]

Guys i suggest we focus on actually something informative than this bs going on

[west ice]

You should open a help channel if you want to get help quickly

[sullen gull]

Pls help

[frail lion]

Yeah mate fax continue with your thing

I mean mam

I guess

[frail lion]

xlnx -x

[sullen gull]

Thx dude

But u forgot the +C

[sullen gull]

Jk

[frail lion]

Yeah man +c

[fossil kindle]

hi

[frail lion]

https://tenor.com/view/alone-karu-cute-frog-karu-love-gif-9073542059226148631

[fluid sorrel]

https://youtu.be/fYfbyrnOmT8?si=s5WjOQquiPHrffHh

😃

Level advanced hai

[frail lion]

Bruv I want to discuss it first with some people rather than just

Jumping on the solution

[carmine orbit]

graphically

would be tedious

[carmine orbit]

what approach are you thinking

[frail lion]

Man graphically yeah we are getting pretty much every case here

I actually got p(p(x)) but it's not what the question asks

It asks about conditions

And about that my brain is like not able to think anything

[carmine orbit]

i see

we can confirm d option by using curve sketching

[carmine orbit]

c looks trivial but i cant assure both is negative

anyways ping me when you get the answer

[frail lion]

Yeah both is negative i

Also think so

[frail lion]

If both roots negative then c automatically positive

[carmine orbit]

true

[frail lion]

Here is the answer what am I supposed to ping

[carmine orbit]

i aint watching a 10 min vid🥀

[frail lion]

Ok bro I'll tell ya

[carmine orbit]

yeah

all the besht bro

[frail lion]

@carmine orbit yeah mate i deduced that actually this was not correct

It was c<0 and alternate roots

About d idk

I will still wait one more day to think and also if anyone else wants to yk

Try

[fluid sorrel]

Bro u can't discuss in exam

Theta the problem

I tried it yesterday

N i got it

But i liked the way the person explained his approach

So i sent the video

[frail lion]

Ok bro

[carmine orbit]

yeah but he is trying to exhaust his thought process even more

[carmine orbit]

but how

[frail lion]

I yk did some calculation and stuff but I can tell the observation part

Not some actually pretty much

Look man p(p(x))=0 basically means that we want an "X" on which p(x) comes out to be a "negative root" of p(x) only

Basically value of y which when plotted on x is negative root of equation

[carmine orbit]

makes sense

[frail lion]

Or just root not important to be negative the value of "X" that we put should basically be negative

Basically just a value of y on negative x which is root of y

When we take c<0 , we can actually surely get such a

Value of p(x)

On a negative X only

[frail lion]

This is by some ugly ass inequalities

[carmine orbit]

yeah i get it by intuition

[frail lion]

Yeah

The thing is I do not have any idea about the D option

Or probably I need to think more

[carmine orbit]

youll get it by maniuplating actual p(p(x))

[frail lion]

How

Elaborate

[carmine orbit]

try it

first

[frail lion]

Ok yeah you're right

Lemme try first

Nah mate nothing

I'm done

I can't think anymore

[carmine orbit]

aight

y=x^2+bx+c

p(p(x)=y^2+by+c

for the p(p(x)) to have real solution y must have atleast one real solution

now look at the option D

assume c-(b^2/4) to be k

so it becomes k^2+bk+c

from this we can say k=y

c-(b^2/4)= y

so get get y^2+by+c<0 and b>0

y is the minimum value of p(x) by x=-b/2a a=1 we get y = c-b^2/4

hence we can conclude one root of p(x) is negative which implies p(p(x)) has atleast one neg root

[carmine orbit]

yo

[frail lion]

Hmm yeah ok

[still brook]

hey gyts

how is this wrong

the highlighted box is wringbut idk how

[heady bluff]

Ohh alr I see kk!

[frail lion]

I respectfully do not understand your solution honestly

[still brook]

what was the point of saying ts

[frail lion]

The point is that because I can't understand your solution, i cant find mistake in your solution, i can only provide you with solution, mistake you'll have to find yourself

[still brook]

hold on

help me pls

[frail lion]

Ok so you're using synthetic division method

Hold on just give me some seconds to learn ts and answer ya

Man you need to take the 4 common out of the quotient

That you've got

12x+20

Reduce it to simplest

3x+5

[crystal idol]

im doing honors pre calc as a hs freshamn

i migth be cooked 😭

[cyan quiver]

its not bad

(i have a 73 average rn)

[soft plover]

what's the answer?

[frail lion]

Hol on lemme check

@soft plover A and D

[still brook]

help

[modest bolt]

l+b = 280
Area = l×b
You can use first derivative to find maxima

[still brook]

be so fr

[limpid idol]

-23 can be written as -25 + 2
So after that. it becomes.(12x^2 + 5x -25 + 2) / (4x-5)
Now. it becomes.
(12x^2 + 5x - 25) / (4x - 5) + 2 / (4x - 5)
now you divide (12x^2 + 5x - 25) / (4x - 5) and that should be the answer.

which comes out to be

3x + 5

[frail lion]

Do Ya know what is first derivative?

[still brook]

no

[frail lion]

Yeah man thats what I was thinking

[limpid idol]

i got an answer of 70 and 70 for this

[frail lion]

You in 8th grade

[limpid idol]

is it correct?

[still brook]

what

[still brook]

what

[still brook]

yes

[limpid idol]

oh

[limpid idol]

alright so it does require you to take the derivative of the function

[still brook]

what is derivative

[limpid idol]

slope of a graph at any given point

[still brook]

ih

[frail lion]

Oi listen here just put the total length of fence in perimeter of rectangle formula and do l=b

[frail lion]

For rest of ma folks it's AM GM inequality

[still brook]

gng what

[frail lion]

This

Is for ya

Mate

2(l+b) = 280

L+b =140

Do l=b

70 70

[limpid idol]

oh also

should the most area come from a circle?

because a circle has the largest area to perimeter ratio

just asking

[frail lion]

Yeah

[limpid idol]

yeah but for a quadilateral the most area to perimeter ratio comes from a square.

so just putting l = b works

[frail lion]

Yk the question is WHY?

Why this

Why from a square

[limpid idol]

lemme try it

brb

[frail lion]

.

This is why mate

[limpid idol]

oh

[frail lion]

Yeah

[limpid idol]

just a random question. What will be cbrt of a negative number

[deft jetty]

a negative number

[limpid idol]

oh yeah

[deft jetty]

since (-a)³=-(a)³

[limpid idol]

yeah odd powers dont change the sign

[ruby lava]

what is a cbrt?

[river drift]

cube root

[ruby lava]

ok ty

[soft plover]

oh

[bitter pier]

yo guys i need help to understand this, my teacher did not send the solution

[broken wigeon]

thank you soo muchh, @trim shard & @fluid sorrel !!! 💗 i figured it out

[fluid sorrel]

Welcome 👍

[viscid thistle]

My notes never looked this clean

[viscid thistle]

Did u do it

Or u still want help

You see angles in a unit circle

Whole thing is 360 or 2pi

finding reference angle of a angle given is matter of finding which x axis part it is closest to

At the given position

Then subtracting that angle from this

And u get the reference angle

[bitter pier]

so its 315-360? or 360-315?

[viscid thistle]

315, closest is 360-315

[bitter pier]

315 is quadrant IV right?

[viscid thistle]

Yeah

You did it right

[viscid thistle]

Yeah

[bitter pier]

oh

thankyou

[viscid thistle]

270 is 3/4

[bitter pier]

in B can i convert it first into degree and convert it later to radian?

[viscid thistle]

5pi/6 that’s 0.8 pi ish which means that after the 0.5 pi that is 90 and but it’s before pi

Which means second quadrant

Pi - 5pi/6

[viscid thistle]

No angles in degrees are harder to

Work with

Rather just convert angle into radians

And do the calculations

I mean if angle is convinent to work with do it just straight but

Never try converting radians into degree to do it

[viscid thistle]

See how I did it

Just need to check first number before pi

That’s all can tell position

0 is 0, 0.5 pi is 90, 1 pi is 180, 3/2 pi is 270, 2 pi is 360

Easy

[bitter pier]

oh

I think i understand it now

I'll send my answer later

the given is different

it's like i need to find the 6 circular function

[viscid thistle]

Yeah find reference

Number

Then find the functions

I am aware of those problems

Trick is to see, sin x = y and x = cos x
so every point in the circle first is cos x and second is sin x

Remember after finding the reference angle it’s easier to find these functions and others BUT crucial detail is the sign of these changes and its same as x and y at that position

Try digesting this information

@bitter pier

So if the reference angle came in second quadrant

y is positive and x is negative

That means

sin x is positive

And cos x is negative

And other functions are just ratios of these

I hope it makes sense

I think main idea is to think sinx and cosx as just the points on the unit circle

[soft plover]

I am curious about the solution

[bitter pier]

@viscid thistle correct?

sorry for my handwriting

[viscid thistle]

Gang

Gang gang

[viscid thistle]

Read from here

Real quick

[viscid thistle]

You are almost there

But missing crucial detail

The signs of cosx sinx are wrong hence every functions sign is wrong

You gotta fix them

[viscid thistle]

This is quadrant two where

cosx is negative

Sinx is positive

[bitter pier]

OHH THAT

[viscid thistle]

because cosx can be taken as x coordinate and sinx is taken as y

lol

[viscid thistle]

So by putting signs u basically went from cos45 to cos135, whole idea of doing reference angle is to be able to do big angles easily

[frail lion]

Damn bro ok

[viscid thistle]

so that u can compute known angles

So u complete the reference angles cos and sin then you put right

Sign for them

That’s how u get the original angle’s trig functions

I hope that makes sense

So someone can say compute
cos 120

You won’t do it directly lmao

Rather find reference to 120 it’s 60

Cos 60 is 1/2

But that’s not done yet

[viscid thistle]

This is not cos 120

Yet

120 is in second quadrant

So it’s negative

cos 120 = -1/2

[bitter pier]

all of them are negative except sin and csc?

[viscid thistle]

Yep

You got it

Mate

[bitter pier]

[frail lion]

@soft plover nah I'm jus playin with ya
Some body has already provided it , the video solution

I guess sherdil

Hol on lemme get it for ya

[frail lion]

Here

@viscid thistle i jus read allat stuff ya taught
hats off to ya bro I could never

[soft plover]

okay thanks

oh that video I have watched it but I think it's in hindi

it's not clear too what the person is saying

[frail lion]

Hmm you're not from India

Thats a problem

Did ya try the language change

Thing in yt

?

[soft plover]

oh not yet

[frail lion]

K then go ahead try it

Once

[fluid sorrel]

Ping me 😃

[frail lion]

Bro btw whats ping

Wym ping

What is pinging someone

[viscid thistle]

when replying someone

they get notification

thats ping

u can also do it with

@frail lion

using @ followed by the username

u can also turn off the ping when replying

[frail lion]

Ok bro 👌

[regal tree]

cool question i found if anyone wants to try!

[west ice]

$f(x)=x+1$

[west ice]

I'm not even kidding I have seen these kind of problem so many times already

it was fun the first time I saw it

[obsidian monolith]

Fionna The Unemployed

[regal tree]

oh damn

[west ice]

sorry for disappointing you, someone will find the problem interesting really

[regal tree]

yea its cool imo

also you didnt need to find the function lul

[west ice]

Yeah but the $f(2x)-f(x)=x$ is overused

[obsidian monolith]

Fionna The Unemployed

[regal tree]

ohh

[west ice]

So I spotted the funtion instantly

[regal tree]

ahh oki

[trim shard]

so unrelated but ur notes are so cute 😭

i’m glad i could help in any way

[fluid sorrel]

To tag someone by using @ symbol

[fluid sorrel]

We helped by explaining the 3secx graph

[fluid sorrel]

385??

[regal tree]

correct!

[fluid sorrel]

It's jee mains pyq right?

[broken wigeon]

HAAUSHSHAH

[fluid sorrel]

2025

[regal tree]

yep

yee

[fluid sorrel]

Yaaa

I tried it last to last week

Got it in 4 minutes

😅

[regal tree]

ohh nicee lol

[fluid sorrel]

That's alot

[fluid sorrel]

Really??

[regal tree]

you in 12th?

[fluid sorrel]

Yeppp

[regal tree]

ooo me too

4 minutes isnt that bad but could be better

[fluid sorrel]

Jan 🗿

[fluid sorrel]

Yepp

[regal tree]

scared af

[broken wigeon]

likee youu🤭hihi thankuusmm

[broken wigeon]

guess thats my last call for help since precal will be over next semester 😓 (THANK GOD)

[willow skiff]

time for calc after that?

or are you graduating by then already

[broken wigeon]

im still in 11th grade:3 next week semester break then 2nd sem

[broken wigeon]

bas cal in 12th

basic calculus*

[trim shard]

goodluck queenie, its basically smooth sailing afterwards

precalc was harder on me than calculus

[fluid sorrel]

Not being rude but plz use #chill

[broken wigeon]

ughhrhrh i hope so. i almost died earlier from my midterms exam hshshah I only understand the unit circle and forgot how to convert the 10°5"4' blabla to radians and degree😭

[broken wigeon]

sorry😮‍💨

[fluid sorrel]

Okii

[willow skiff]

10°5"4' is downright archaic

it was useful for sailors and astronomers centuries ago

arcminutes and arcseconds are just a totally unnecessary part of trig these days though

[broken wigeon]

yeahhh but my prof still included itg

[regal tree]

IKR why is it still taught

[river drift]

any decent scientific calculator can convert between dms and decimal degrees

[willow skiff]

yeah exactly

oh wait if you were allowed a scientific calc on your test, that's kind of on you

[regal tree]

we are not allowed any calculators on any tests 😭

[willow skiff]

(referring to this)

[river drift]

just try to remember the conversion between degrees and minutes and seconds is the same conversion as between hours and minutes and seconds

[regal tree]

wait is the degree related to the angle the minute hand of a clock moves in a given time?

like 12 hours is 180 degrees? thus 1 hour is 180/12=15 degrees?

[river drift]

1 revolution = 360 degrees = 1 hour for the minute hand = 12 hours for the hour hand

but the point was more that:

1 hour= 60 minutes
1 minute = 60 seconds
1 hour = 60 * 60 seconds = 3600 seconds

1 degree = 60 arcminutes,
1 arcminute = 60 arcseconds
1 degree = 60 * 60 arcseconds = 3600 arcseconds

these are exactly the same formulas

[viscid thistle]

chad

easiest way to remember these is

how close to time system this is

1 mins has 60 seconds , 60 mins has 3600 seconds

thats how i remembered it

💀 🙏🏻

never forgot it

but yeah if i see the use of arc seconds in my questions i go nuts

[frail lion]

Bro how did you get this explanation please

Anyone anyone can explain who's solved this

[bold carbon]

f(x) = x/2 + f(x/2)

Take an arbitrary r

then f(r) = r/2 + f(r/2) = r/2 + r/4 + f(r/4) = r/2 + r/4 + r/8 + f(r/8) = …

if we take the limit of the right, the f(r/2^n) term limits to f(0) =1 (because f is continuous) and the r/2 + r/4 + … limits to r, so f(r) = r + 1, for any real number r. Then f(x) = x+1

[frail lion]

Yeah ok I understand

But how did it strike you

What thing does one "see" in such questions so that these things strike

Like probably seeing limit tending infinite or something?

[bold carbon]

I started by plugging in numbers for x to understand what the functional equation was saying

so like when x = 1, f(2) - f(1) = 1

when x = 2, f(4) - f(2) = 2

this was getting more sparse which wasn’t helpful, but then I thought of going in the reverse direction, so that the points get closer together instead. since f is continuous, points being close together should tell us something about the behavior of f at certain points

that’s why I rewrote the function equation in the reverse direction, from x to x/2, and as I wrote out terms and repeated the expansion, I saw that the non-function terms formed a geometric series, and the functional term’s argument got closer and closer to 0, at which point we know what the behavior of f should be

that was my process

I didn’t realize the terms would form a series until after I started writing a few iterations, it’s not like I magically saw that strategy immediately

[frail lion]

Yeah I understand that that's why I asked for the striking point

Ok mate thanks

[bitter pier]

@viscid thistle am i still missing something?

im struggling with negative angles

like

how do i illustrate it

in the example i need to illustrate both angle and reference

[trim shard]

since positive angles form from clockwise rotation, negative angles will form if you go in the counter clock wise rotation

if you need to find the ref angle for a negative angles, you need to find its positive coterminal

smallest positive coterminal at that* then you can find ur ref angle depending on what quadrant ur terminal lies in

[median spear]

C. right?

G(x) is x basically

[regal tree]

correct!

[median spear]

it was easy dude

Js try putting x/2 in place of x

You will get a GP

Put the limit and you will get G(x)

[regal tree]

yep

[frail lion]

Guys I solved this using the special case of an equilateral triangle coz it didn't mention what sorta triangle
Can there be a general method too ?

[sturdy flume]

That's the only way as long as I know

[fluid sorrel]

Did u get the correct answer??

[frail lion]

Yeah

[frail lion]

That's a bit wierd coz otherwise they should've mentioned it

[fluid sorrel]

Which book is it

Bcz even in jee pyqs like old ones if they don't mention taking equilateral triangle works

I guess this would be a property to equilateral triangle

[frail lion]

Black book

[fluid sorrel]

😅 did u check the alternative solution else where

[frail lion]

Matter of fact it's a question in Inverse trigo

[fluid sorrel]

Can u send ur work bro

[fluid sorrel]

Ayee

[frail lion]

Yeah man i am confused too

[frail lion]

Wym "cash"

[fluid sorrel]

If u search the solution plz tag me

[vivid lintel]

this can be done by trigonometry too

[fluid sorrel]

Can

Typo

[frail lion]

Yeah man sure I can

[fluid sorrel]

He did it alr

[frail lion]

I did it by trigo only

[fluid sorrel]

Okkk

[median spear]

Isn't the 1st bracket incorrect

[frail lion]

Not very vigorous trigo but some basic observation and sin 30 cos 30 values

[median spear]

What is js I

[frail lion]

I is incenter

It's mentioned broski

[vivid lintel]

there is this property II1= 4RsinA/2 and I2I3 = 4RcosA/2

so add both

after squaring

[frail lion]

Damn bro what property is this where is this taught

[median spear]

Oh I read the half part

[vivid lintel]

and then you get (4R)^2 into (sin^2 A/2 + cos^2 A/2)

in RHS

[frail lion]

For any triangle ?

And what's A

[vivid lintel]

It was in my module

[median spear]

Thought I1 , 2,3 are also in centres

[fluid sorrel]

Bro where is this property

[vivid lintel]

yeah

[fluid sorrel]

Damm

[median spear]

Ahh 😯

[vivid lintel]

even i didnt remember, had to open the module and search

[frail lion]

.

[vivid lintel]

A is the angle

[frail lion]

Ok bruv

Does your module have it's proof ?

And which module is this bro

[vivid lintel]

nah ofc not

[vivid lintel]

Naryana

[frail lion]

Damn bro narayna cookin some crazy shi

[vivid lintel]

Its rare to find proof in modules

[vivid lintel]

nah one of the worst modules i have ever seen

[frail lion]

You've seen a lot of modules?

[vivid lintel]

they literally have solutions form different chapter, in one of the Sequence and series question, they gave hint that use L'Hopitals rule💀

[vivid lintel]

only naryana ones

[frail lion]

Then why say worst

Bro

Worst and best are relative things

[vivid lintel]

eod i use cengage and salas for mathematics

[frail lion]

Ok leave I'm just playin this is bs

[vivid lintel]

?

[frail lion]

This philosophy ahh

Is bs

[vivid lintel]

ohhh

[frail lion]

What's salas

[vivid lintel]

Its a book

not related to competitive exams btw

[frail lion]

Oh

But still bro i suggest trying this

Once

[vivid lintel]

trying what>

?

[frail lion]

The question bro obviously

What else you gon try

[vivid lintel]

ohh

um idk im mostly doing unreleated stuff

we have 1 week off so yeah

[frail lion]

Ok bro your choice

[vivid lintel]

learning new things mostly

[frail lion]

Ok brocochip cookie

I mean bro

[vivid lintel]

They should add a calc section too for pre university

[frail lion]

What

I don't understand

[vivid lintel]

I mean for the pre university section on the server, there should be a calculus channel too

like people do have a good amount of calculus pre uniersity

[frail lion]

Oh
Honestly mate it doesn't make a difference i guess

[vivid lintel]

a little bit of real analysis if elipson-delta definition is emphasised

[frail lion]

https://tenor.com/view/sweating-nervous-wreck-gif-12986085083782148824

Yeah mate i totally understand and agree with ya

Bro in lowkey think we should stick to

Jee only

Rest is your choice obviously i can suggest only

[shell bobcat]

Any tips on how to start studying Quantitative Reasoning from now? I need it as a course for Business Major for next semester..

[frail lion]

Guys anyone got any idea what I'm doing wrong ?

[snow fulcrum]

Your handwriting for starters

[west ice]

This's the proof for that II1 and I2I3 formulas

@frail lion

[frail lion]

Hmm ok

[frail lion]

And for the main course?

[frail lion]

If the proof is this long then I assume it's pretty irrelevant

[west ice]

Yeah I had't even seen that formula in my life

That was teh first time

It has a reason for being forgotten, that is the proof way too long ig

[fluid sorrel]

That's why we must memories it

[frail lion]

Yk actually the formula is pretty specific that's why it's probably not that much important

Like we barely deal with excircles i presume

[west ice]

I looked it up and I found absolutely nothing

Not even that formula

[frail lion]

Looked what up the question?

[west ice]

No the properties of excircle

I haven't looked up the question tho

[frail lion]

Yeah you wouldn't have probably coz it includes incenter too

Not only excenters if that's the name

[fluid sorrel]

The property is valid for x belonging to [-1,1]

[west ice]

Yeah I guess

That was annoying to prove

[fluid sorrel]

Yepp excentres and nine points center

[frail lion]

Broski i guess the property is always true

[fluid sorrel]

That arcsinx and arccosx property

[frail lion]

Yeah bro arcsin+ arccos = π/2

With x

[fluid sorrel]

For both arcsinx and arccosx the domain of x is range of cosx

And sinx

[fluid sorrel]

It's an inverse function 😭

So range becomes domain

[frail lion]

The principal domain thing yeah

I agree bruv

Im assuming these things are already in principal domain coz if not then how will we bring them in the P.D?

?

[frail lion]

Yeah mate the proof tells it

I mean mam

Ok leave

[frail lion]

?

[fluid sorrel]

I think

The inner value satisfies the domain

So we can do it

But i think the answer is 0

Bcz what u have done seems right

If we take mod as both negative and once 1 positive

Than we get 0 solution

Bcz 6^√3 is not in domain

Of cosx

[frail lion]

I tried it by putting the inner values between -1<= the thing <= 1

Also

[fluid sorrel]

I only get this using calculator

[frail lion]

Still no result

I'm getting

[fluid sorrel]

I am getting cosx = 6^+-√3 and 6^+-√2

[frail lion]

Yeah I am too

[fluid sorrel]

So answer is 0??

No solution

[frail lion]

No brotato chip

The answer is 4

[fluid sorrel]

So

[fluid sorrel]

This is the solution 😃

But np

[frail lion]

Yeah man pretty wierd thing

How can they be this far

Probably some printing error theyve done in the question

[fluid sorrel]

Yepp

[sullen gull]

Find alpha

How do I solve dis

[echo sinew]

Taylor-McLaurin expansion of log(1 + t), where t → 0

[hushed sphinx]

Or alternatively L'Hospital -- though it's essentially the same actual computations it leads to.

Or (yet a third way to justify the same computation) define f(x) = log(1+5x)-log(1+alpha·x) and recognize the fraction as the one that appears in the definition of f'(0), so differentiate the function symbolically and solve for alpha afterwards.

All of theses seem to depend essentially on calculus, and I'm not sure how much hope there is of approaching the problem with PRE-calculus tools.

[carmine orbit]

simplify the limit when x not equal to zero and equate it with 10

[hushed sphinx]

Yes, but how do you have in mind to simplify log(1+5x) and its neighbor term? Without calculus?

(Or how to simplify log((1+5x)/(1+ax)) if you prefer to rewrite it like that first).

[carmine orbit]

limx->0log(1+5x)/x=5 + logx->0log(1-ax)/x=-a which equals 5-a

[carmine orbit]

nah there is no need to do that

[hushed sphinx]

Where do you get those limits from?

[carmine orbit]

derived from standard result

we were taught in limits

[carmine orbit]

limx->0 log(1+x)/x =1 similarly limx->0(1+ax)x=a

[carmine orbit]

i have a question

no?

[hushed sphinx]

What?

[frail lion]

Bro i guess put the limit of above function to 0 and equate to the lower

It will be 5-a i think

So a = -5

[astral apex]

some knowledge of physics and calculus is needed to do that problem

[west ice]

This's more physic I think

The tangent is just basic calculus eh

Why don't you ask in Physic discord

[frail lion]

What where

How do I join

[frail lion]

Yeah mate understandable

[west ice]

#old-network

Tbh the Physic discord server is quite inactive, at least compare to this Math one

So you might still have to wait

[frail lion]

I actually posted it here coz somebody had

Posted one physics question before too

[west ice]

Yes

You're allowed

[frail lion]

https://media.discordapp.net/attachments/363224154469826562/1429700440471375982/image.png?ex=68fd068b&is=68fbb50b&hm=ba00c963a4fd9b1486277fbe3086b679c3360d6a4f3c38770c915d85fdea2f00&

[frail lion]

This

Somebody had shared this

[west ice]

But not many people know physic well enough

So sometime your question won't be solved

[frail lion]

I mean

What's his name i don't

Yeah sherdil

Hes preparing for jee

But yeah some only

[frail lion]

Ok

[west ice]

I send back the question so everyone could see it

[frail lion]

Ok thnks

[soft plover]

the language for the video can't be changed

I have actually attempted to solve it myself

the only thing I struggle is to find option D

that if you sub in x=c-(b²/4) you get negative

I do by casework

I have difficulty finding the condition when there are exactly 2 roots

[frail lion]

Here is the d option solution

[frail lion]

Ok unfortunate i guess

[soft plover]

how did they directly conclude this?

yea I also noticed that c-(b²/4) is the y-coordinate of the vertex of P(x)

@frail lion

[frail lion]

Yeah yeah I'm here

Man theres this thing I'm not able to explain it to ya

Wait hold on lemme think more to how to explain it

[frail lion]

Here this

Read this once

C- b^2/4 is minimum value of p(x)

And p(p(x)) has y as its variable

Oi @soft plover

[soft plover]

wait I may misunderstood sth

"P(P(x))=P(y)=0 has at least one negative root" does it mean at least one of y1 or y2 is negative or at least one of x1, x2, x3, or x4 is negative?

[frail lion]

Atleast one root is negative

X1 x2 x3 x4 not important it has 4 roots

It's not important it has 4 roots

Depends on b and c

But atleast one negative value of x exists on which p(p(x)) is zero

[soft plover]

I have read it but still couldn't understand

[frail lion]

Bro from which country are ya

Tell me once so I can probably help

[soft plover]

Indonesia

[frail lion]

Bro tell me what don't you understand

What is the problem

[soft plover]

I am having difficulties relating P(x) to P(P(x))

if I were to solve it I would probably expand P(P(x))

[soft plover]

I am trying to visualize graphically

[vivid lintel]

We need A little knowledge of hyperbolic trig here

[soft plover]

btw are the conditions A and D in the form of OR or AND?

i.e. must fulfill all of them or just at least one of them?

cuz for example P(x)=(x+2)(x-4)=x²-2x-8 have opposite roots and -b/2=1 and c-(b²/4)=-9 but P(-9)=91 >= 0 and b=-2 < 0

and P(P(x)) has 2 negative roots

[median spear]

What's the actual question @soft plover

[soft plover]

this one @median spear

[fluid sorrel]

Bro can u show ur work

Also tell me what's the tension of P given in solution

[fluid sorrel]

No

It's completely based on rope tension

[vivid lintel]

maybe for your approach

[frail lion]

D is a condition bro
It is asked "on this condition is one root negative"

This is how the question is asked

[frail lion]

Bro I did not do any work I have no idea how to proceed

[frail lion]

Hold on

Idk bro

Tell your answer

[soft plover]

I mean the answer is A and D

but are those 3 conditions all must be fulfilled for P(P(x))=0 to have at least one negative root or only fulfilling any one of them is enough?

[fluid sorrel]

Can u share the question in book

I am not sure

Is it from hcv?

[frail lion]

Any one enough

[frail lion]

I actually have got it from somewhere I don't have the book

[soft plover]

ah okay

I think I found my own method using visualization

not sure if it's valid tho

[frail lion]

K then share it

[soft plover]

it's kinda hard to explain but the idea is to visualize in 3D

like this

it's easy to see that if P(x) has opposite roots then so is P(P(x))

so option A is an answer

[fluid sorrel]

I think the tension force will be only bcz of the centre of mass

Bcz its not an ideal string

I am still thinking

[soft plover]

then if b>0 then the x-coordinate of the vertex -b/2 is negative which makes P(x) will have at least one negative root and so is P(P(x))

[west ice]

I can yap about why I hate Math for 2 hours straight if you love to

[gritty roost]

Ah, me too

That's why I'm looking for someone to oppose me😭🤚🏻

Make me love math, I swear I'll try💔

Jokes, aside. I need help with math combined physics

[west ice]

Math is bad enough, now with Physics

[soft plover]

me but for talking abt it not now cuz I am kinda busy

[gritty roost]

Sure, I'll add you

I'm preoccupied too so np

[soft plover]

ah okay

[gritty roost]

Yeah... It's for school, I'm a fast learner, but I prefer having fun with it rather than working on it, that's why I need🥹

[frail lion]

Ok bruv

[frail lion]

Hmm yeah

[gritty roost]

It'd be nice to live not hating it too

[west ice]

I would love to yapping about how much I hate Math

[frail lion]

Yeah ok mate I like yapping about maths too

[soft plover]

I think option D the condition where P(c-(b²/4))<0 is redundant

[gritty roost]

Woman, I think I can yap enough to myself about that. But I'd love to yap about adventure time too

[frail lion]

Wym redundant

[west ice]

Adventure Timeeeeee

btw what's your timezone

[frail lion]

@soft plover bro wym redundant

[gritty roost]

I need help, in terms of relating basic functions to the motion of a car. This is for pre-calculus. I've been lost with kinematics for hours.

I found 3, I just don't know how to explain it

[frail lion]

It's an option bro it's asked if this condition exists then is there a negative root

[gritty roost]

(GMT) +8

What about you?

[west ice]

I thought you're in Western really

[west ice]

around that

[frail lion]

What you mean deriving equations of motions by integral and all ?

[gritty roost]

Linear functions, Quadratic functions, Constant functions. The real life application of those to a car in motion

[gritty roost]

Lol why?

[west ice]

Cuz Asian usually watch anime, not cartoon and the chance of you being Australian is kinda small

[frail lion]

Honestly I don't understand what are you trying to imply here

A car can move in like , 3 ways

No offence btw

[gritty roost]

So, basically. I have to explain how I can apply basic functions to the motion of a car. In real life and the functions I've found related to that are those three

[soft plover]

oh nvm

yea I thought about it wrongly it's not redundant

so basically you just need to think of the properties of P(x)=0 which makes it have at least one negative root cuz then it will automatically map the same properties to P(P(x))=0

[gritty roost]

None taken

I'm not easily offended, so I can work with the latter

[gritty roost]

I've been watching cartoons since I was a child, that's why my English has been better than my native tongue

[frail lion]

Yes I mean what else could've caused tension

[frail lion]

Ok

[frail lion]

Yeah you got it brotato chip

[gritty roost]

I understand the basic functions, I understood the physics, but separately. I don't get how they're connected entirely

[frail lion]

Ok so what do you mean when you say you understood the physics seperately

[gritty roost]

Motion, Speed, Distance, Time, Displacement, Position. I know how that works...

[frail lion]

Great , now connecting all those stuff together is physics

And stuff are connected together by what ? Maths and different functions

[west ice]

Physic is Math in disguise

But more annoying

[frail lion]

Exactly that's what I said just now

[gritty roost]

The subject in question that I'm working on is pre-calculus

[gritty roost]

The basic functions I told you about. From pre-cal

[gritty roost]

.

[gritty roost]

https://tenor.com/view/math-is-math-incredibles-the-incredibles2-i-dont-know-that-way-why-would-they-change-math-gif-15024041

Math in Physics was easier..

[soft plover]

yea I am just having difficulties relating them 2 and I finally figured it out by visualizing

[frail lion]

I guess Fionna must have a better advice coz

I have just entered 11th grade

[gritty roost]

I'm in 11th grade too, mate

[west ice]

[gritty roost]

Can you help? Lol

[west ice]

I'm in year 12

just one above both of u

I'm not sure

[frail lion]

That's why I am saying you must have a bettee advice

[gritty roost]

You must've been through this already

[west ice]

Im deadass stupid, don't look at me

[frail lion]

In the name of calculus i just know about functions , a bit aod and inverse trigo

[west ice]

Ehh, the program is different between countries

[frail lion]

Nah you proved that shit that long ass property bro

[west ice]

And also school

[gritty roost]

So you can't help?

[west ice]

just pure luck

It depends on the material

In detail

[gritty roost]

This is supposed to be just basic functions... I know functions. What the heck is this..

[frail lion]

I mean yeah I can most obvious ly

Try asking something

[gritty roost]

How do the three given functions apply to a motion of a car?

How does it work?

[frail lion]

How do these three apply to a car

[gritty roost]

Yeah

[frail lion]

Ok look

1. Constantfunctions : help in constant speed
   2.linear speed: fuel use at constant rate i guess
   3.quadratic ': most obviously accelerating or curved path

[gritty roost]

Visually why?

[frail lion]

Visually why

Wym visually why

[gritty roost]

Graph

[frail lion]

The graphs

[gritty roost]

Mhmm

[frail lion]

Yeah constant function horizontal line
Linear must be straight line
Quadratic is parabola

[bitter pier]

lmao

[frail lion]

What bro why

[gritty roost]

Obviously....

Now, how does that connect to it?

[bitter pier]

it's just that its common knowledge lol

[frail lion]

Ok ok I'll tell that too

Like take constant function, y=c
Now the output which comes out does not change
Like v(t) = something
Then that is not going to change
Now take linear function
It's like y=mx+c typo
There's a constant rate change
Like for example if my car ccelerates uniformly , then distance increases linearly with time , not a constant quantity
Like d(t) = v(t) + d

Something like that

Basically rate of change constant and uniform

That's the difference between the two

And the quadratic functions have acceleration in them

Ok enough

[bitter pier]

wow

i learned something new

[frail lion]

For example the equation s=ut+1/2at^2

[bitter pier]

so in linear function the distance traveled is directly proportional to time?

like you said

y=mx+c

but

idk what for c

[frail lion]

Yeah I guess

Yes

No guess yes

[bitter pier]

like im thinking y will be the distance, m will be speed and x will be time, so what is the c?

[gritty roost]

Thanks

I think c is for constant..

[bitter pier]

oh

[frail lion]

It's inital distance