precalculus

[brave hinge]

(The number from that interval should be taken)

[charred rapids]

I mean, if -x its not in the domain, clearly f(x) its not equal to f(-x) as it is not even defined, thats why I specified that the interval its not centered

[quick pebble]

and that's the issue

if f(-x) is not well defined then you cannot talk about whether it is equal to f(x) or not

it is actually neither

[onyx grail]

guys can any one help me with numerical integration

[thick breach]

got a specific question?

[onyx grail]

ineed some one to explain ut

[brave hinge]

What are integrals? To put it briefly, their functions are more precise

lim_x →a f(x) = L ∀ϵ> 0, ∃δ> 0 : ∀x ∈X (0 < |x - a| < δ|f(x) - L| < ϵ)
The answer is L here, right?

[nova dew]

he said he needs a method for numerical integration

[merry vault]

bro really put epsilon delta definition insead of a numerical method

[old iris]

Yo how in the world did you not manage to destroy the world after dropping that bomb

[viscid thistle]

NO EPSILON PLEASE

AAAA

[mighty kelp]

Integrals are so fun

[ancient anvil]

is this California

[hazy ivy]

What?

[thick breach]

Was not me who dropped

[brave hinge]

Я подумаю

[plucky rover]

Can anyone help me with derivatives pls

[willow bear]

!da2a

[tender quest]

No need to ask “Can I ask…?” or “Does anyone know about…?”—it’s faster for everyone if you just ask your question!

[willow bear]

@plucky rover

[plucky rover]

Yea

[willow bear]

send your question mate

[plucky rover]

How do we find the deriative of dx^2

[willow bear]

"dx^2"?

[plucky rover]

yea

[willow bear]

do you mean like x^2 * d or do you mean the nonsensical way

!xy

[tender quest]

Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.

[plucky rover]

X^2*d

[willow bear]

do you know how to differentiate something like 7x^2

[plucky rover]

No this is my first time

[willow bear]

do you know how to differentiate anything?

[plucky rover]

Nah

[willow bear]

https://youtu.be/rAof9Ld5sOg

but if you want the executive summary

[plucky rover]

Oh t

Ty

[willow bear]

the power rule says the derivative of x^n is given by n*x^(n-1)

and constant multipliers outside the thing stay as they are

so yours becomes 2d*x

however!!

i will say that you should definitely do more reading on differentiation from your textbook

also be very very VERY careful with naming anything with the lowercase letter d when doing calculus

[plucky rover]

Ok 2:19 of the vid it looks like to me that we have to understand the function as well is that true

@willow bear

[willow bear]

i mean

sure you do

if you don't understand your function then how can you hope to do anything with it

[plucky rover]

I mean it’s pretty useless in derivatives I don’t know y I asked that

I copied the link cuz I have to go and I’ll watch the rest later

It’s kinda interesting

[amber bison]

ur pfp matches up lol

[old iris]

"Father of the atomic bomb."

[fossil kindle]

can someone explain derivatives for me

like ik how to find the derivative of a function

but what does it actually mean

[bronze idol]

can someone help me with this integral my tutor put it at the end of my holiday homework ive been looking at it for like 20 mins

integral sqrt(tanx) dx

just give me a hint if u can

i dont want to know the full answer yet

[raw hill]

For future reference, questions on integration belong in #calculus.

Consider substituting $\tan x=t^2$.

[obsidian monolith]

Civil Service Pigeon

[bronze idol]

oh sorry i didnt know there was a calculus channel

i only have access to geometry and trig, precalc, comp math abd pre alg and algebra

[raw hill]

Go to the id:customize if you can't seem to find the relevant channel - a lot of subjects are in the weird right before uni/at the start of uni overlap, so they just got thrown into the early university. Alternatively, help channels work too ( #❓how-to-get-help)

[bronze idol]

ok thankyou

[raw hill]

[bronze idol]

wait i cant click more than one role

i want to be in the pre uni and early uni one

[raw hill]

you shouldn't need the role to see the channels

just follow the category in browse

at least I think

[bronze idol]

omg thankyou so much ive got them all now

ur a legend

[raw hill]

[loud heron]

ew mrbeast

[floral veldt]

hellppp

i dont see any mistakes in my method

thats the solution

they get arcsec in the integral i get arctan

but funnily we get the same constant

and i dont see any mistakes

any help would be good

my method 👆

[worldly jay]

i think everything until the second half of page 2 was correct

i dont think subbing w for sqrt(x^2 -1) was the right play tho

when i tried the problem it seemed like if we were doing it this way (or at least how i attempted it) was to expand everything after squaring

after adding one to both sides, everything except for 9x^2 and 3x^2 cancel, and the bottom turns into 3(sqrt(x^2-1)+sqrt(3))^2

then its simplification and u get the answer that is the one on the soln sheet

your method of getting arctan was correct tho, so dont worry about that

[low slate]

there is a giganomatrious amount of videos for this on youtube.

so if you have a function, f(x), by getting the derivative of this function, the outcome is a also a function! usually named f'(x) with that little '

so this new function, f'(x)

lets you find the instantaneous rate of change for any point on the f(x) graph.

all you have to do is plug the x of the point of which you want the instantaneous rate of change for into f'(x)

and the outcome is just- the instantaneous rate of change at that point.

[fossil kindle]

oh so its just

the rate of change of a function at that specific point

[low slate]

indeed!

[fossil kindle]

ok ty

[low slate]

and it can have many names as well. one of them is the "slope of the line tangent to that point"

if you want me to explain further just say so.

[low slate]

np.

[woven forum]

Can someone explain to me epsilon delta limit of functions definition

[raven marten]

💀

[woven forum]

What

[raven marten]

1st time hearing that word

[woven forum]

You dont study real analysis?

[raven marten]

nope

[woven forum]

Oh wrong chat then

[raven marten]

k mb

[floral veldt]

how am i supposed to see this 😭

this question is nuts

u need a rlly good intuition for it imo

[tame crane]

Is that a limit of a function question? @floral veldt

[floral veldt]

No, you should be able to answer it with basic limit knowledge

[static fiber]

Cant you just use the variable separation method?

[ebon cypress]

you use an integrating factor here don't you

cuz first order DE

take the sin 2y to the other side, and use an identity to turn the cos squared into a cos 2y term

4 cos^2 y becomes 2+ 2cos 2y I think

then use a substitution

u = -sin 2y + cos 2y

work from there

is that from A level further maths?

[rough leaf]

this cannot be precalc

if ts is precalc i’m fried

[jaunty sky]

Trigonometry 😭

No need to worry, this is from the hardest test for high school students.

[rough leaf]

what test

and yeah ik trig but i don’t know what harmonic mean is besides the name of it

[thick breach]

<@&268886789983436800>

nvm

[jaunty sky]

JEE Advanced

[rough leaf]

isn’t the jee a college entrance exam

idk

[thick breach]

eh

[jaunty sky]

Yeah...

[thick breach]

so what have you tried?

[jaunty sky]

I got scared seeing this

T_T

[thick breach]

same

[rough leaf]

wait is that question lile one of the normal quesyions on the jee or is it like a harder one

[jaunty sky]

I mean it is normal I guess.

[rough leaf]

holy crap

[thick breach]

hmm.

[rough leaf]

how would you even solve something like this

[thick breach]

then it is \big pi tan(pi/3( 1+ thetar))*tan(pi/3(1-thetar)) = k

[jaunty sky]

Alright lemme try

[thick breach]

its tuff maybe, let's see if some manipulations work

nah fok it's just like terms cancel

[rough leaf]

wait why di we have use rhe first part cuz don’t we get the solution already

i’m so confused

[thick breach]

the solution has unknowns to match them

[jaunty sky]

I am skipping this now

Will do later.

[warm cobalt]

um, what do you mean by manipulations??

[thick breach]

Conversions

[warm cobalt]

ok thanks mate

[iron ingot]

Took me 14 minutes but answer should be a = 8 and b = 3

So a-b is 5

[iron ingot]

I mean you can just brute force and observe the telescoping series

Thats one way

I managed to convert it into Pi(f(r)/f(r-1))

Solution is kinda lengthy but i'll post once im home

[jaunty sky]

😭 orz

[versed pumice]

cool problem

basically reduces to a telescoping sum of f(r+1)/f(r) from r=0 to 50 where f(r) = tan(3^r*pi/(3^50-1))

[versed pumice]

can confirm a=8, b=3

will drop solution

first let f(r) = 3^r*pi/(3^50-1)

pi/3(1+3^r/(3^50-1)) = pi/3 + f(r-1)

and pi/3(1-3^r/(3^50-1)) = pi/3 - f(r-1)

so the expression can be written, by moving terms over to the Left hand side as the following

$$\prod_{r=1}^{51} \tan(\frac{\pi}{3} + f(r-1))\cdot \tan(\frac{\pi}{3} - f(r-1))$$

[obsidian monolith]

vivdax

[versed pumice]

then use tan(a+b) and tan(a-b) fomrulas to get that

$$\tan(\frac{\pi}{3} + f(r-1)) = \frac{\sqrt3 + f(r-1)}{1-\sqrt3 f(r-1)}$$

[obsidian monolith]

vivdax

[versed pumice]

$$\tan(\frac{\pi}{3} - f(r-1)) = \frac{\sqrt3 - f(r-1)}{1+\sqrt3 f(r-1)}$$

[obsidian monolith]

vivdax

[versed pumice]

So multiplying these 2 values together gives $$\tan(\frac{\pi}{3} + f(r-1))\cdot \tan(\frac{\pi}{3} - f(r-1)) = \frac{3-f(r-1)^2}{1-3f(r-1)^2}$$

[obsidian monolith]

vivdax

[versed pumice]

now, look at the values f(r)

we have that the angle inside the tangent expression for f(r) triples every time, since its 3^r

therefore, we can use the triple angle identity to say that $$f(r) = \frac{3f(r-1)-f(r-1)^3}{1-3f(r-1)^2}$$

[obsidian monolith]

vivdax

[versed pumice]

since tan(3x) = (3tanx-tan^3x)/(1-3tan^2x)

and if u look at the expression we currently have, we see that $$\tan(\frac{\pi}{3} + f(r-1))\cdot \tan(\frac{\pi}{3} - f(r-1)) = \frac{f(r)}{f(r-1)}$$

[obsidian monolith]

vivdax

[versed pumice]

Therefore, the equation reduces down to $$\prod_{k=1}^{51} \frac{f(r)}{f(r-1)} = k$$

[obsidian monolith]

vivdax

[versed pumice]

which reduces to f(51)/f(0) = k

we see that f(51) = tan(pi*3^51/(3^50-1)) = tan(3pi + 3pi/(3^50-1)) = tan(3pi/(3^50-1)) = f(1), since tan(x) = tan(x + kpi) for integers k

therefore, our equation is f(1)/f(0) = k

However, we have the formula for f(r)/f(r-1) above, so plugging in x = f(0) = tan(pi/(3^50-1)) gives the following:

f(1)/f(0) = (3-x^2)/(1-3x^2) = k

from here, we can rearrange to get (3x^2-9)/(3x^2-1) = 3k

so 1-8/(3x^2-1) = 3k

and thus $$1-3x^2 = 1-3\tan^2(\frac{\pi}{3^{50}-1}) = \frac{8}{3k-1}$$

[obsidian monolith]

vivdax

[versed pumice]

therefore a-b=5

[jaunty sky]

HOLY MOLY

ORZ

[jaunty sky]

THANK YOU SO MUCH

[iron ingot]

Ironically this is a question I wont recommend Pursuing in exam

Because there is atleast 3 steps you can get stuck at

The much more obvious sin and cos summations can be solved during the exam but something like this needs time

[versed pumice]

JEE is like 6 problems in an hour right?

[iron ingot]

Mains and Adv are different

For Mains you need to get 15-16 of 25 questions in an hour(For maths atleast)

For Adv. You want a score of 180+/360 in all 3 subjects combined

Pattern changes for jee advnaced every year but the total number of questions stays around 18 per subject

[versed pumice]

oh alr

[iron ingot]

Its in 2 papers so total questions are 18 x 3 x 2 = 108

[jaunty sky]

Do you think such problems are worth solving in an exam?

[iron ingot]

Not in the first go nope

I used to teach for JEE and generally if i had made them solve such a question in the class and it instantly clicks them

Then and only then they should attempt it when they see it

Otherwise leave it or do it later

[versed pumice]

what concepts do JEE advancec math go up to

is it like up to Calc BC or does it go further

[iron ingot]

Im not familiar with the AP class systems what kinda stuff is in Calc BC

[versed pumice]

its like integration techniques (integration techniques, trig sub, more advanced u-subs), sequences, series, and that stuff

its basically calc 2

[iron ingot]

I'd say thats in The syllabus Yes

Is there stuff like LMVT?

I think Integral inequalities and the mean value theorem is about where calculus goes to its height there

There is stuff like

Derivative applications and Limits continuity and Derivability

[jaunty sky]

Yes. I am familiar with both and JEE goes much beyond.

Though it skips over much of Unit 10 of Calc BC

JEE has Calculus in addition to Probability, Permutations and Combinations, Vectors, Matrices, Coordinate Geometry and Algebra.

Also, ofcourse Trigonometry such as this.

[iron ingot]

Combinatorics

I mean you can basically break it into like 4 big chunks ig

[jaunty sky]

Yeah

You used to teach for JEE?

[iron ingot]

Algebra
Combinatorics
Two Dimensional Geometry
Three Dimensional Geometry
Calculus

5

[jaunty sky]

Indeed

[iron ingot]

Until 2024 yes

I mean

[jaunty sky]

Oh in a cram school/coaching?

[iron ingot]

I focused more on RMO and IOQM

[jaunty sky]

Oh understandable

[iron ingot]

I also taught Physics but only for InPhO and its pre quals

[jaunty sky]

I think we should shift to #chill

[thick breach]

Indeed

[icy verge]

Guys

Can u help me with proof of upper and lower boundary theorem

[round geyser]

Show me an example question

I’ll try to help

[viscid thistle]

You are indian sis?

[west ice]

<@&268886789983436800>

[winter comet]

<@&268886789983436800>

[west ledge]

which alternatives for long devision work for y'all?

[lyric patrol]

Synthetic division

[west ledge]

Oh #### naw

[lyric patrol]

It's easier than long division

Or you can use factor therom

To check for factors

And remainder therom for remainders

[west ledge]

what do the people in germany teach

[lyric patrol]

Uhh idk I'm from Australia lol

[west ledge]

Nice so next to allmange

[graceful scarab]

Horner table

only when divisor is x-a though so

[west ledge]

finally

[neon dagger]

isnt ths just synthetic division then

[graceful scarab]

uuhhh could be

not sure about the english names tbf

[neon dagger]

since it only works with x-a factors it kinda suks

[graceful scarab]

ones you get to uni you can just use maple to solve everything👍

or whatever math software they use

[neon dagger]

tru

thats wat my dad says

and everyone else

[graceful scarab]

yeah i fear my basic math skills are already deteriorating

[neon dagger]

eh ur prolly better than me at everything considering ur undergrad math tag

[graceful scarab]

maybe i can solve a partial differential equation or two, but don't ask me to count past twelve without my fingers

[toxic nebula]

So just euclidean division

[neon dagger]

remind me what that is again

[toxic nebula]

Uhhh

So you have some polynomial or so and so equation let's call it P(x)

And you have x-a

You have to input constants or variables, multiply them by the x-a and then deduct that from the polynomial

And if the remainder is zero

[neon dagger]

ohhh

ik this one

i forgot the name tho

i call it the same thing as long division

[graceful scarab]

Yeah I guess, horner is just a streamlined version that is more efficient when the divisor is x - a

[toxic nebula]

You know that the polynomial P(x) is equal to (x-a)(constants and variables you used)

[graceful scarab]

So it's only useful in special cases

[toxic nebula]

I see.

Thanks for the clarification

[graceful scarab]

The op didn't really specify too much

What they want to use it for

[neon dagger]

anyways is it that hard to js bash out long division

[graceful scarab]

yes!!

i dread long division

[neon dagger]

dang

[toxic nebula]

I mean

Is there any case you really need it to be much faster than euclidean division??

[graceful scarab]

long division is just slow and messy

better to leave that stuff to the computers

for like high degree polynomials of course

[toxic nebula]

I get it

[west ledge]

Yes

[viscid thistle]

Use synthetic division

[analog flower]

the moderators agree synthetic division is good

was there some reason you pinged us though

dm modmail

[vague totem]

Horner's method/scheme is for polynomial evaluation. Like you can evaluate p(a) for any p(x) through the polynomial remainder theorem which states p(x)/(x-a) gives p(a) as its remainder (not quotient). So if you want to find p(5) for some p(x), just do p(x)/(x-5) and the remainder will be p(5).

Also, (I think), you seem to have a misconception just like @toxic nebula and @neon dagger, that synthetic division only works with linear monic divisors like (x-a) but that's not true. Ruffini's rule is a special case of synthetic division for linear monic divisors, but synthetic division in general is capable of dividing any two polynomials, including non-linear, non-monic divisors. It just happens to be that most schools teach only Ruffini's rule without fully explaining that synthetic division is fully applicable in general to any arbitrary polynomials.

Read https://en.wikipedia.org/wiki/Synthetic_division#Expanded_synthetic_division

[graceful scarab]

I just meant Horner is a quick shortcut for the x−a case, not that synthetic division only works there. I wasn’t really talking about the full general version of synthetic division

[lone temple]

my friend got this question, i am interested but idk what to do, any ideas?

"Cho" is Given
"Tính" is Calculate

[west ice]

Ehh, you can just pick a function with f(3)=8 and f'(3)=6

It should work perfectly

[lone temple]

how

[west ice]

If $f(3)=8$ then $\lim_{x \to 3} \frac{f(x)-f(3)}{x-3}=6$

[obsidian monolith]

Fionna The Unemployed

[west ice]

$\lim_{x \to 3} \frac{f(x)-f(3)}{x-3}=f'(3)$

[obsidian monolith]

Fionna The Unemployed

[west ice]

You could pick a linear $f(x)=6x-10$

[obsidian monolith]

Fionna The Unemployed

[lone temple]

ooohhhhh

f(x)=f'(x0)(x-x0)+f(x0)?

didnt think of that

[west ice]

No what

It looks like the tangent equation

But no

[lone temple]

oh well idk whats it called but ur right

ummm

what is it then

[west ice]

This's the definition of f'(x)

[west ice]

That's it

[lone temple]

AH

ok i see it now lol

[west ice]

We choose a function satisfy both f(3)=8 and f'(3)=6

ideally f(x)=6x-10

Cuz it's linear and easy to work with

[lone temple]

this might be a dumb question but do i have to make it as y=ax+b like what if its quadratic or somewhat, that sounds like guessing

also the question is calculating I btw

[west ice]

I mean, any equation work just fine, as long as f(3)=8 and f'(3)=6

Take what I said as a trick to solve it

[lone temple]

no i mean like f(x)=6x-10 satisfies and put into I then f(x)= something idk but have f(3)=8 and f'(3)=8 other than f(x)=6x-10 will have the same result? or am i getting the wrong idea

[west ice]

Yes it would be the same

[lone temple]

how can you sure about that?

[west ice]

In a lazy way, it has all the restrictions that're given in it

So it works

And since there's only 1 answer

You should have the same

[lone temple]

bruh

but thanks!

[blazing oak]

Pls send solution of 58

[burnt sundial]

ı solved it bro

page 2 on the left page 1 on the right

[blazing oak]

Thanku so much brother 🙏🙏🙏

[burnt sundial]

np

[jaunty lake]

anyone done de precalc?
my de precalc class starts j28 and i would like to hear from anyone who did precalc how it gone

[heavy dome]

wait so why do u add the 0??

in the video

[heavy dome]

someones too "tanned"
💀 😂

[west ledge]

I am guessing the majority has done pre calc

[jaunty lake]

yea prolly
how was it?

[west ledge]

Easy? Just some polynomials and how to navigate on the coordination system

I had a b or c as far as I can remember

[jaunty lake]

oh ok

[heavy dome]

isnt a b or c bad..?

[west ledge]

It Is

Never felt the need to turn in tests though

[vague totem]

because you need to the write ALL the coefficeints of the numerator (I think the article uses the term verbose coefficients), which is actually going to be $$2x^4 + 0x^3 + 3x^2 + 5x^1 + 1x^0$$

[obsidian monolith]

Flax.

[heavy dome]

oh oops
yea i didnt realize there was no x^3 term

thanks

[heavy dome]

isnt synthetic division algebra 2

[silver mason]

verifying trig identities is the one math concept that should be abolished

[heavy dome]

whats trig identities?

having to know sine, cosine, arcsin, arcosine, tan, arctan, secant, cosecant, etc.?

[raw hill]

Equations involving trig functions that are always true (in the domain)

[heavy dome]

🤪 HALP

[light tangle]

👋

So are derivatives a prereq for integration?

[raw hill]

yes definitely

Also this belongs in #calculus, not #precalculus

[light tangle]

Someone told me to message in here, my bad

[raw hill]

[sonic hill]

That depends what you mean. I wouldn't say you necessarily need to understand what a derivative is to understand what an integral is, but you typically use derivatives in integration techniques so if you're talking about actually solving for antiderivative functions it's recommended you know how to take derivatives.

[round geyser]

Hey I am doing exponential functions

Do you know what’s critical pt

[fallen stratus]

yup

wait wait

i got it a bit wrong

basically critical points represent local maximas and local minimas

[round geyser]

I have this

The teacher highlighted critical pt

[fallen stratus]

how did you understand this handwriting 💀

[round geyser]

That said “label critical pt key pt”

I think she meant y-intercept

[fallen stratus]

y intercept is the point after which the graph is increasing

[round geyser]

But then here

[fallen stratus]

What is the question can you please tell?

are you plotting graphs?

[round geyser]

Now I think critical pt is the exact value

[fallen stratus]

i mean im confused

[round geyser]

So now

Someone said that exponential doesn’t have a critical pt

Like on Google

[fallen stratus]

the base doesn’t have it

as it can never be zero

[round geyser]

Asymptote: y=0

[fallen stratus]

idts y can be equal to zero

in an exponent function

a^x

can never be zero

where a is an integer

[round geyser]

What if it’s a* b^x

[thick breach]

,w plot e^x

[obsidian monolith]

Results provided by WolframAlpha

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

Requested by ihavebecumdeath

[raw hill]

Seems like an abuse of terminology. I assume you're using "critical point" to (informally) mean some point of importance/usefulness. However, in calculus, one of the common definitions of a critical point is where the derivative is either zero or undefined (this is one of the most common use cases of the term "critical point", so I'm assuming this is what Google was referring to). And since exponential functions are monotonic and always differentiable, the derivative is never zero/undefined, and hence there are no critical points.

[round geyser]

Thanks

[nimble ridge]

whats wrong with method 2

[willow skiff]

method 2 involves a 0/0 - 0/0 form

well, you've turned it into an infinity - infinity form at the end, but that's also an indeterminate form

[nimble ridge]

yeah i was wondering if i could solve it further

[willow skiff]

you couldn't

[nimble ridge]

ok

[round geyser]

Hey I am now doing solving exponential expression

Is there three properties
a^x=a^m; x=m
x^a=m^a; x=m

I dont recognize the third one

[subtle pond]

you can google..

[round geyser]

But first of all am I correct on the first two

[subtle pond]

yes

btw you dont need to put much load on memory

you can also 'compare'

[round geyser]

I learned different ways to solve these

[round geyser]

First one is same exponents second one is same base

[subtle pond]

yes

[subtle pond]

wdym

[round geyser]

Ill talk to u later

[round geyser]

Okay

[round geyser]

Like taking their reciprocal on fractional exponents

[subtle pond]

example

[woven fulcrum]

guys

i want to relearn all of precalculus as im going to take calculus bc

could anyone send me a good full course on everything

[long ruin]

https://www.greenemath.com/Math-Practice.html

This website is gold mine for math

U can generate infinite problem's to solve

[round geyser]

Yes

x^3=8^2

[raw hill]

<@&268886789983436800>

[solar ferry]

can someone help me with this

[raw hill]

For future reference, this belongs in #calculus , not #precalculus. Also, when asking for help, show what you've done/thought so far - it gives us more context and saves time from explaining things unnecessarily.

If we had either of the functions individually for the entire real line [so either arctan x or 1/2 (|x|-1)], then they would be differentiable everywhere. So, we really just need to check where the domain boundaries are (where the function "switches" from one rule to another).

To be differentiable at a point, both the original function and its derivative have to be continuous there (left and right hand limits have to be the same).

[gray jungle]

What’s a good resource to study precalculus (textbooks,youtube channels,professors ,whatever)

[slim bramble]

Khan Academy is the cleanest free precalc path and it builds skills really smoothly. So I suggest khan academy

[shrewd sail]

UH, when people confuses precalc with calc, uhhhuuuhh

[round geyser]

Hi so according to the ans it’s x=2

[raw hill]

<@&268886789983436800> y'all know the drill lol

[flint parrot]

why is there no calculus in pre uni?

[river drift]

#calculus is for calculus in both pre-university and university classes

[subtle pond]

where are you taking reciprocal

$$ x^3 = 8^2 $$

[obsidian monolith]

fysch

[subtle pond]

$$ x^3 = 2^6 $$

[obsidian monolith]

fysch

[subtle pond]

$$ x = 2^2 $$

[obsidian monolith]

fysch

[round geyser]

The x^3

Aka 1/3

[subtle pond]

nah

you raise both sides to power 1/3

[round geyser]

Yea that’s what I did

[round geyser]

And look

[subtle pond]

yeah thats not called reciprocal

[round geyser]

What’s it called then

[subtle pond]

its js wording, u dont take 'reciprocal'

[round geyser]

Oh

[round geyser]

I’m doing 3 cases: evaluating,variables in the exponents, and variables in the base

Which one is easier

[subtle pond]

all of em

[round geyser]

Good

[subtle pond]

its upto you

[round geyser]

What’s this considered

[subtle pond]

i dont remember methods

[round geyser]

I assume it’s variables in the base

[subtle pond]

i js do

i dont think they taught me names anyways

[round geyser]

Like the wording

[subtle pond]

yeah thats better

remember like this

[round geyser]

“Free them out”

[subtle pond]

free the unknown from exponent

[round geyser]

Yes

And I wonder why the denominator matters in how many solutions

Like x^1/2=# then it’s +/-

X^1/3=# is not

[subtle pond]

cuz duh

[round geyser]

For example

[subtle pond]

its odd even

when you got even base

x could be either +/-

cuz raising by even would make it positive

but raising by odd always makes it retain its original sign

odd = even + 1

[round geyser]

x^1/2=4
(x^1/2)^2=(2^2)^2
x=+/-2

You know I mean right

For taking its sqrt

[subtle pond]

if its sqrt then x = 2

[subtle pond]

cuz thats how the function is defined

[round geyser]

Good

[subtle pond]

but normal

x^1/2

thens its +/-

x^1/2 is not a function

sqrt(X) is a function

[round geyser]

I think that’s how fractional exponents works with radical

[subtle pond]

they both do the same work

but you will learn the difference between functions in higher classes

[round geyser]

Yes

[subtle pond]

for now think like a function CANT take two inputs and give same output

so sqrt x cant take x as +/-

but x^1/2 aint a function

so it takes both

in the end they do the same work

[round geyser]

Another one where fractional exponent is related

I have another way

On -2(x+2)=4/3

[subtle pond]

listen

i see in green you didnt divide by 2 on both sides

it makes stuff faster

[round geyser]

Multiply by -1/2 on 4/3 and 4 and 2 cancels to get

[subtle pond]

yeah thats how you do it

[round geyser]

x+2=-2/3

Subtract the 2

[subtle pond]

yes bro

yes

algebra

iget it

[round geyser]

-2/3 - 6/3=-8/3

Different way to approach it

[round geyser]

Omg I love algebra

[subtle pond]

this aint integration bro 🥀

you are doing the same stuff

[round geyser]

Getting to the same ans

But different ways

[subtle pond]

but disguised as different

[round geyser]

I also did limits

That as limits that x approaches #

[subtle pond]

yeah

why are you doing exponents then

thats supposed to be easy

[round geyser]

That’s like 3 units after

Exponents is Unit 6

Limits is in Unit 3

[subtle pond]

dang they teaching you stuff in the same year

you never learnt exponents before

?

[round geyser]

Then I did U4 about Long Division and etc

[subtle pond]

bro thats like 8 grade stuff

then you skip onto limits

11 12 grade stuff

[round geyser]

That’s after

[subtle pond]

goofy ah book

[round geyser]

Then U4B Min/max word problems

I get to use Desmos

[round geyser]

They add on Rational root theorem and Complex Zero theorem

[subtle pond]

thats also 11 stuff

[round geyser]

After that Unit 5 I did rational functions like graphing

With out Desmos

[round geyser]

Cuz last year I took Algebra 2

[subtle pond]

they make you redo exponents 🥀

[round geyser]

I’m so smart that I jumped from Algebra 2 to directly to Pre-Calc

[subtle pond]

aint precalc same

[round geyser]

Also I have to graph it’s exponential functions

[subtle pond]

[round geyser]

Like 2^x, 3^x, e^x

[subtle pond]

move onto calc bro

[round geyser]

Also transformations

I’m 12th grade in HS now

[subtle pond]

wait you are abt to end school and you are doing exponents

😈

[round geyser]

Oh wait that made me thought of i redid factoring with difference of squares and sum and difference of cubes on U1

[subtle pond]

u should be doing calc broski

[round geyser]

Btw my teacher said by next semester I am doing trigs

[subtle pond]

broski u aint done trig

[round geyser]

And gave a spoiler that U7 is logs

[subtle pond]

🤒

[round geyser]

logarithm

That has VA and Exponents gas HA

[subtle pond]

they nerfing students nowadays

[round geyser]

What

[subtle pond]

no wonder college gets them cooked

thank god i didnt get school nerf

[subtle pond]

they weakening you

[round geyser]

But it’s soo easy that I get 3 100s on quiz

[subtle pond]

yeah cuz they are teaching you this stuff in grade 12

thats the ripe age to learn integration

[round geyser]

It’s the 2 quizzes on U3 and 1 on U4

[round geyser]

Somehow I didn’t

Also derivatives

Cuz I’m in a basic level

Of pre-calc

The teacher avoided it

[subtle pond]

teacher want you to be soft

[gray jungle]

Jake what’s your resource to study precalc?

[astral apex]

derivatives and integration are calculus, not precalc

[icy verge]

Is it correct

[woven basalt]

how to be doing coordinate geometry

[willow skiff]

just practice more problems and if you're stuck, go back to the textbook chapter and the worked examples

[woven basalt]

how about practicing for competitive exams, like the textbook chapter and their worked examples are easy but the previous year questions are really different from usual

[willow skiff]

oh maybe try asking in #competition-math?

[woven basalt]

thankss 😄

[sick karma]

How does pascal’s triangle actually work?

I remember learning it once but now I completely forgot

[willow skiff]

the borders of the triangle are all 1s

   1
1    1

1 ... 1
1 ... .... 1

then you add the two numbers on the top to make the number at the bottom

that's how the pattern begins:
1
1 1
1 2 1
1 3 3 1
....

[sick karma]

If you have a problem like (x+2)^5, how would you solve that

Would you go to the 5th row?

[willow skiff]

you'd look at the 5th row of Pascal's triangle, so that's 1 5 10 10 5 1

yes

so 1 * x^5 * 2^0

• 5 * x^4 * 2^1
• 10 * x^3 * 2^2
  ....
• 1 * x^0 * 2^5

[sick karma]

Thank you!

[willow skiff]

no worries!

[random swallow]

I'm guessing if i have more than 2 variables i have to use the big general one

[willow skiff]

yeah, if you have (x + 2)^n

[round geyser]

Idk

[round geyser]

It’s a^5+5a^4b+10a^3b^2+10a^2b^3+5ab^4+b^5

[granite pumice]

anyone care to explain to me how this relates to combinations tho 😭 i didn't get to attend class when they were taking the lesson on this and the binomial theorem

[astral apex]

that's kind of tricky to see. It comes from considering how to calculate a given entry in pascal's triangle, without just filling in the entire triangle above it

the number that's k places along the nth row is the binomial number (n k), which is also n choose k from combinations

[short delta]

i think khan academy and various textbooks are very helpful for precalc

[astral apex]

(using 0-indexing, so the very top 1 is row 0, and the left-most 1s are in place 0 along their row)

woah

apparently each entry in Pascal's triangle is equal to the number of distinct paths going down from the top 1 to that entry

that's the way to connect it to combinations, I think

[granite pumice]

eesh, sounds annoying. thanks for the explanation

ill look into it

[astral apex]

this is actually quite nice

consider that at each step, a path down can either go left or right

consider the 6 that's at row 4, column 2

for a path to reach there, it has to go down 4 levels, and the number of lefts and rights have to be equal (because it's in the middle of the row)

[round geyser]

Am I right

[astral apex]

so there have to be two lefts and two rights, but they can happen in any order

[astral apex]

the possibilities are LLRR, LRLR, LRRL, RLRL, RLLR, and RRLL

so (4 2) = 6

[radiant forum]

Does anyone has this pdf whom could send for?

[rough leaf]

yo do you need to know binomial expansion theorem or what

cuz i js likr lwk use pascal’s triangle instead

[round geyser]

Can we cancel same exponents

And move the 45

To get x=117-45=72

[rough leaf]

wait

doesnt 117^2 -45^2 become

(117-45)(117+45)

i maybe wrong tho

[rough leaf]

i don’t think so

you gry lile x=+-108

i think

[celest tulip]

^ Indeed the answer is no. This is because exponents arent distributive over addition.

I.e what you are basically proposing is to take the square root of both sides, but: sqrt(a^2 + b^2) is not a + b.

[round geyser]

What if it’s x^2=117^2

Can we say it’s 117

[rough leaf]

yes

square root both sides

[celest tulip]

In that case then yep, but as ILikeFood mentioned, you need to take the plus minus case as well.

x^2 = 117
then x = 117 is not the only solution.

[rough leaf]

it’s also -117 right

?

[round geyser]

Okay

Yea I’ll see that

Same exponents can cancel also same base

The one I showed u

[rough leaf]

the property i used is called difference of squares i think

if you wanna look into it

[round geyser]

“x^3=8^2” that should be “x^3-8^2”

My teacher went over it

for “x^3-8^2” set it =0 and move 8^2 over

[rough leaf]

uh

there’s another way

for that

i think

cuz you can multiply both sides by the exponent of 1/3

and aolve

[celest tulip]

Well it's not wrong I suppose. If you are trying to solve for x, then you can ||factorise it||, which is probably the setup here.

[celest tulip]

Exponentiate :)

[rough leaf]

is that what it’s called

[celest tulip]

Yeah

[rough leaf]

i don’t know the bame

i just use it

[rough leaf]

oh

[celest tulip]

Exponentiate both sides by power of 1/3

[rough leaf]

we all learn something new everyday

[celest tulip]

Lol fair enough

[rough leaf]

that’s uh more complicated

[round geyser]

Then the ans is 4

[rough leaf]

imo

[rough leaf]

yeah

[round geyser]

My teacher coincidentally went over that question

That’s I showed u

[celest tulip]

Wait actually it might just be called 'raising both sides to the 1/3rd power', exponentiation is viewed in a slightly different way.

Like a = b
e^a = e^b

[round geyser]

When I first saw that question I thought I have to factor

[round geyser]

My teacher used that term

[rough leaf]

thatsbwhat i thought

[round geyser]

Raising both side by its reciprocal

[rough leaf]

whatbis exponentiation

[round geyser]

?

[rough leaf]

what math class are you in btw

[round geyser]

Exponents differentiation

[round geyser]

pre-calc

[celest tulip]

Its basically when you have some equation like:
a = b
And you put them all over a base.
e^a = e^b

[rough leaf]

oh

[celest tulip]

This is the correct term for it, mb.

[rough leaf]

wait rhat has a bame😭😭✌️

[celest tulip]

But this is just raising both sides to 1/3 power

[rough leaf]

i see

imma be taking this in the summer

[round geyser]

Are u the US

[rough leaf]

yeah

[celest tulip]

Yep

[round geyser]

then when I did it

I corrected it to x^3 = 8^2 then start solving

[rough leaf]

wqit

may i ask which part of the us you are in

cuz i’m in alg 2 and i’m doing similar stuff

[round geyser]

NJ

[rough leaf]

oh

[round geyser]

East coast

[rough leaf]

not too far from me

[round geyser]

What state are u in

[rough leaf]

CT

[round geyser]

Omg

We are near

[rough leaf]

yeah

what unit of precalc are you in

?

on*

[storm violet]

does anyone know where to start for this

[rough leaf]

uh

i think i do

inwould try to make exponents equal

so 70^71 = 70(70)^70

so inwould do 70 x 70

and get 4900

so now you have 71^70

and 4900 ^70

but i could be wrong tho

[storm violet]

the answer was that 71^70 was larger

[rough leaf]

oh

[celest tulip]

Remember BODMAS, you have to do exponents first :)

[rough leaf]

yeah i’m geeked

[rough leaf]

yeah

i forgot abt this

never listen to me again

[celest tulip]

So you can't multiply 70 and 70 since it's 70(70)^70

[storm violet]

wait i think u have to make the bases the same first

[celest tulip]

Oof

[storm violet]

but idk like how to start with that

[rough leaf]

holy

my fault bro

[storm violet]

np

[rough leaf]

are you sure

i looked it up

[storm violet]

wait maybe the answer key is wrong

[rough leaf]

it says 70^71 is larger

yeah

[storm violet]

yea

i still dont know how to show it tho. i forgot how my teacher did it but ik it was with logs

[rough leaf]

oh

wait

[round geyser]

U6

[rough leaf]

i need to do an assignment rq

but i may jnow jow to do this

[round geyser]

Show me

If you mind

[rough leaf]

which unit is this

[rough leaf]

?

what

[round geyser]

Exponential function

[rough leaf]

it’s a hewlth assignment😭

[rough leaf]

oh

i’m doing this too

[round geyser]

Sorry I thought it’s pre-calc

Easy to compare exponential function with logarithm function just by their domain and range

[rough leaf]

it’s like

uh

70 x log(71)

and

71 x log(70)

[round geyser]

Exponential D: All real numbers, R: Y<0, Log D: x0, R: All real numbers

[rough leaf]

idk how to evaluate farther

[storm violet]

alr thx

[rough leaf]

you can do log

or

ln

dyk what ln is?

[storm violet]

yea

[rough leaf]

i would use that

its easier

but idk

[storm violet]

idk because i feel like 71^70 seems larger

[rough leaf]

you like end up gettin ln(70)/70 and ln(71)/71

[rough leaf]

idk

you gotta ask someone esle

i havent unlocked this stage of math yet

im not very good with logs and stuff

i js know basics

[storm violet]

my notes literally cover the other 2 questions and just skip this one bruh😭

[rough leaf]

if you get the solution tho

share it wit me too

[storm violet]

alr srue

[rough leaf]

cuz im lwk interested in this

yo

i lwk have a way to solve

but its lwk stupid ash

[storm violet]

idk gemini brought me to derivatives but ik u can definitly do it without calc

i think

[rough leaf]

i can explain my way

idk if its gonna make sense tho

[storm violet]

sure

[rough leaf]

im think of it as like

lets js use log with a base of 10

log wit base 10 of (70) is saying 10^x = 70

and same thing for 71 but its saying 10^x = 71

the x in the second equation gotta be larger

but

[rough leaf]

if wee go back to this

its being divided by a bigger number as well

so it ends up being smaller

idk

it makes sense in my head

i have zero clue on how to put that on paper

[storm violet]

idk because ur dividing ln71 instead of ln70

[rough leaf]

wait

no cuz i did like

70 x ln(71) and 71 x ln(70)

js divide both sdies

sides*

[storm violet]

u gotta divide both sides by the same number tho

[rough leaf]

yeah you do

divide by 70 first

you get

[storm violet]

because if u ln both numbers u get 70ln71 and 71ln70

[rough leaf]

ln(71) is 71 x ln(70)/70

then divide by 71 both sides

[celest tulip]

The only other way thats coming to me is a graphical method. Consider the functions y = x^70 and y = 70^x. Ones a parabola, and ones an exponential. There will always be three intersection points then: One where x is negative, and two where x is positive.
The goal is to find where these two intersections are when x is positive.

The second point of intersection should be a little obvious.

To find the first point of intersection, what happens to this function at x = 0 and x = 35?

[rough leaf]

idk i might be eeked

my wya js made sense to me in my hedad

head*

[storm violet]

yea idk

[rough leaf]

you might need to put in a ticket for this one

[storm violet]

yea probably

[celest tulip]

Are you guys like ignoring me 😭

[storm violet]

this is my first time like asking a question here idk how u put in a ticket

[storm violet]

no i saw

i just dont know how to solve that if i set those equal

or do i just do like change of bases

[rough leaf]

i dont understand your method

[celest tulip]

You just post a message on any of the available math channels, that will open up a help channel

[rough leaf]

its js too advanced for me

[celest tulip]

You don't need to solve anything

[rough leaf]

sigh

[storm violet]

alr thx

[rough leaf]

one day

ill get it

[celest tulip]

Wait let me explain

First of all

Do you guys know what parabolas and exponentials look like

[rough leaf]

yeah

[storm violet]

yea

[celest tulip]

Okay, so

Do you firstly agree that

A parabola and exponontial always has three points of intersection?

[heavy dome]

They do?

[celest tulip]

,w graph y = x^2 and y = 2^x

[obsidian monolith]

Results provided by WolframAlpha

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

Requested by redfernstation

[celest tulip]

Well not always per se

I meant to say, this particular set of equations do.

As in,

[heavy dome]

what about y = -x^2 and y = -2^x?

[celest tulip]

Two functions in the form 2^x and x^2 will intersect three times.

[heavy dome]

,w graph y=-x^2 and y=-2^x

[obsidian monolith]

Results provided by WolframAlpha

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

Requested by .8596945

[celest tulip]

I'm not quite interested in that form but pretty sure they do too

[storm violet]

the problem is i need to do it algebraiclly

[heavy dome]

Isnt that wrong

[celest tulip]

Eh why?

[heavy dome]

shit i forgot parentheses

,w graph y=-x^2 and y=(-2)^x

[obsidian monolith]

Results provided by WolframAlpha

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

Requested by .8596945

[rough leaf]

where do you get these problems

?

does your teacher give them?

[storm violet]

math class

yea

[rough leaf]

dyk fromwhere

[heavy dome]

what grade u all in

[celest tulip]

Well obviously those don't have three points of intersections, and the right function you gave is not a well defined function in the reals.

[heavy dome]

im in 8th

[rough leaf]

like a certain textbook or smth

[storm violet]

no idk

[rough leaf]

im in 9th

[storm violet]

i think she makes them

oh

[rough leaf]

oh

i see

[storm violet]

im in 11th

oop

[rough leaf]

i lwk started math too late

i shouldve started ts from the womb

[celest tulip]

Hmmm

[storm violet]

anyways my friend said to divide both numbers and check if it is greater or less than 1

[heavy dome]

can i kill ur graph machine

[rough leaf]

wait

so like

divide 70^71/71^70

?

[storm violet]

yea

and like manipulate that

[rough leaf]

that does make sense

[storm violet]

so change it into a way where it like has the same base or something

[rough leaf]

your gonna need the lcm for that no?

since 71 is a prime number

or am i js stupid

[storm violet]

its gonna be really big

i think it was done with logs

because like the unit we are on kinda focused on logs and inequalities

[heavy dome]

,w graph x^{2}+y^{2}=3^{s}, \left|\left|y-\sqrt{3}x\right|+\left|y+\sqrt{3}x\right|+\left|\sqrt{3}y\right|\right|=3^{s}, \left|\left|y-\sqrt{3}x\right|+\left|y+\sqrt{3}x\right|+\left|\sqrt{3}y\right|+\left|\sqrt{3}x\right|+\left|x+\sqrt{3}y\right|+\left|x-\sqrt{3}y\right|\right|, \left|x\right|+\left|y\right|=3^{s}, \left|\left|y-x\right|+\left|y+x\right|+\left|y\right|+\left|x\right|\right|=3^{s}, and \left|x+y\right|+\left|x-y\right|=3^{s}, where s=s=\left[-3999,-3998,-3997...\ 1100\right]

:)

[obsidian monolith]

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

Requested by .8596945

[heavy dome]

damn it

its 30,600 shapes

crashed desmos

[rough leaf]

at this point

js use a calculator or smth

or js use derivatives

idek

[storm violet]

im so cooked for my test

[rough leaf]

i would js learn the derivatives part necessary to do these type of questions

[heavy dome]

R. I. P. desmos

[storm violet]

a lot of stuff was taught before break😭

[rough leaf]

i dont know much about derivatives so idk how to use them here

[storm violet]

we did learn derivatives but our math teacher doesnt want us to use them

like for this question

i remember solving it with like a much easier way

[rough leaf]

yo

i lwk copiloted the question

and i was kinda on the right track

its saying like for f(x) = ln(x)/x for all x>e the function is decreasing

anf since 71 > 70

ln(71)/71 would decrease more than ln(70)/70

idk

so the 70 one is greater

[storm violet]

yea but idk because ur dividing the two numbers by different numbers

so the comparison kinda like fails if u do that right

because the numbers u get have to stay like proportional

[rough leaf]

wait i dont get your confusion

cuz ln(70^71) is equal to 71 times ln(70)

and same thing for 71^70)

so i js divide

[storm violet]

lets say ur comparing like 2 and 1 and seeing which one is greater. u cant divide 2 by like 4 and 1 by like 3 and see which one is greater right

[rough leaf]

im not though

ill walk you trhough what i did

[storm violet]

ik u divided the 71 from taking it out of the exponent

[rough leaf]

yeah

so you get

[storm violet]

but thats not really changing the number, ur just expressing it in a different way

[rough leaf]

ln (70) = (70 x ln(71))/71

then js divide by 70 both sides

[storm violet]

wait the problem is we dont set these two numbers equal

we cant because we know they arent equal

wait lemme put in a ticket

[rough leaf]

wait

i get wait your saying

but

we js multiplied both of them by 1/ 71 x 70

we never set them equal

we js manipulated them both the same way

[storm violet]

wait

[rough leaf]

alright

[storm violet]

is it possible to put it into like latex form

like so the bot can make the equation