precalculus

[somber arch]

dumb question, we have like 2 ways to "decay something"
Pe^-kt
Pk^t

is this the same process? You adjust k and get one from another right?

[sick steppe]

yeah

cause e^(-k) is a constant

[somber arch]

woah

it is

thanks

[jolly raven]

does Pk^t just have an exponentially decreasing derivative tho?

bc then would that be decay?

if t<1 of course

[sick steppe]

e^(-k) and k>0 means e^(-k) in (0,1), so it will decay regardless

[somber arch]

the point is Pe^(-kt) = P(e^-k)^t

it's a power of a constant, everything should be the same

[viscid thistle]

yes

[obsidian monolith]

mechap

[lofty epoch]

sorry should be

[obsidian monolith]

mechap

[lofty epoch]

no

[obsidian monolith]

mechap

[lofty epoch]

yes that should be this

[remote jungle]

by the way addition to this @static linden the 10 here is not the percentage mutliply with 90g
its the percentage of 10% with 100%, which is 100/10 or 10
then we take 90g x 10

for example like 50% = 20g
then the percentage of 50% to 100% is 100/50 or 2
=> the full percentage is 20g x 2 = 40g if that make sense

[static linden]

LPV EU

[remote jungle]

what

oh haha

no problem

[static linden]

love u

[remote jungle]

npnp

[round hemlock]

quite a late response lol

[remote jungle]

yes

[teal plinth]

is this section being used rn?

[round hemlock]

you can ask your question

[round hemlock]

i mean, i should have put it as 0.1x=90

but idk

[remote jungle]

isnt this trio

[remote jungle]

sin(x+y)

man i forgot abvout trionometry

i think sin(x+y) is sin(x)cos(y)+cos(x)sin(y)

if thats what they were asking

[teal plinth]

thnaks

thanks*

[robust star]

can u find the value without a calculator?

[magic mantle]

Try to factor 1460

10•146=2•5•2•73

And 15=3•5

Now write as a fraction

log(1460)/log(15)

I dont see anyway to simplify further so you probably need a calculator

[late mica]

It depends on what type of answer you're trying to get. You could rewrite it as 15^x = 1460, and from there solve for x by taking the natural log of both sides and utilizing ln power rules. This would leave you an "ugly" answer, but an accurate one. So, it really depends.

[tepid cloak]

welp

[echo wagon]

This is not a question, so hard to help you

[high patrol]

Can someone help clarify this question

[full panther]

Which part specifically?

Domain: what values of x are included in function?

Do you know what g o f means cause I don’t

[high patrol]

Composition Function

H is the function that is made from f composed with g

[torpid kiln]

i dont understand what this is asking

[fleet yew]

perform the long division of f(x)/(x-k)

such that there is a quotient q(x)

and a remainder r

[torpid kiln]

i did synthetic division

since k is 2

I finished the problem now but im not sure if i satisfied the question

[fleet yew]

take an image of your work

[fleet yew]

,rotate

[fleet yew]

ok

q(x) is that polynomial to the left of the remainder

@torpid kiln

let's look at #10

the degree of the numerator is 4, the degree of the denominator is 1

[torpid kiln]

on the left?

[fleet yew]

the remainder polynomial will be the cubic

[torpid kiln]

oh okay

[fleet yew]

3x^3 + 1x^2 - 11x + 11 = q(x)

do you understand

[torpid kiln]

yes

well ik you're supposed to do that but not exactly why

but that doesn't really matter

so thats q(x)

[fleet yew]

do you actually want me to explain or are you comfortable just following that rule

[torpid kiln]

its fine

i just want to confirm my work now

see if i followed the right steps

[fleet yew]

yeah you did it right. just make sure you know what q(x) is

[torpid kiln]

ohh i get it now

i was dividing

so thats the quotient

man im just tired

[fleet yew]

yes.

exactly

let me illustrate with actual numbers

19/8 = 2 r3

so

19 = 8*2 + 3

19 is f(x)
8 is (x-k)
2 is q(x)
3 is r

[torpid kiln]

are you mimicking this bit with real numbers

oh yea

[fleet yew]

yes exactly

[torpid kiln]

okay cool

i was looking at my notes and had no idea what r was until i gave a little more thought into

it

but yea i feel comfortable with these types of questions now

thanks

[fleet yew]

yeah np

[viscid thistle]

Is X+Y^2=1 is y a function of x?

Answer says yes but I'm confused

[jolly raven]

its wrong

it isnt a function

[fleet yew]

A function is in the form y = something

[pseudo harbor]

how did this, turn into that-

[willow bear]

they factored out a constant factor of 36 from every term under the square root, by the looks of it.

[short pilot]

I have to find the distance between de lines

[viscid thistle]

$cos^4x*sin^2x$

[obsidian monolith]

codemonkey

[viscid thistle]

does anyone know how one should approach simplifying this using trig identites

I have tried to use reduction but that seems to just make it more complicated, the question wants it ina form with no exponents

[uncut mulch]

do you have the exact instructions

[edgy blaze]

for the sum and difference trig identities are they valid for cofunctions too? like would it just be 1/sin/cos/tan identity?

[sick steppe]

wdym

[edgy blaze]

like say

[sick steppe]

yes

cause you're just using compound angle on sin(a-b), which you know works

[edgy blaze]

i see thanks. I Wasn't sure with reciprocals worked that way since divide by zeroes or something

[uncut mulch]

your parentheses aren't placed properly

[viscid thistle]

When was the last time you guys used the inverse of an unknown function?

[willow bear]

in my bachelor's thesis

[viscid thistle]

Lol

you must have really had no idea what you were writing about

[willow bear]

??

what kind of incompetent crank do you take me for?

while the functions i was working with were not concrete, the existence of their inverses was very much known, seeing as they were strictly decreasing by construction.

[viscid thistle]

For the following exercises, rewrite the expression with an exponent no higher than 1.

[willow bear]

@tribal kestrel wrong server?

[shy kettle]

hello

anyone good with trig identities?

[mild swan]

@shy kettle Don't ask, just ask.

You can google a list of them online

[shy kettle]

well i have them, i just dont know how to do this one problem

ill ss it and post it

[shy kettle]

i got it guys!!

[viscid thistle]

Is it possible to lwarn calculus in 9th grade

[somber arch]

It's probably the norm in some communities

[warm forum]

One of the people I know have already learn calculus since 7th grade lol

[fleet yew]

Lwarn to spel first haha

[drifting dock]

https://gyazo.com/3b02475f6c5bf6cbdee5d23c292334aa

are these all the rules to log or are there more

[red tree]

Are you aiming to remember them all?

[copper vigil]

look it's real simple

just always remember that logarithms simply the operator by one degree

product goes to sum

exponentiation goes to product

[red tree]

Fun fact, 50% of math students don't ever learn the definition of log_b x

[drifting dock]

i already memorized them but i was wondering if theres anything missing

[red tree]

You forgot rule 0, what is log

The one rule to rule them all

[copper vigil]

look it's real simple. logarithms reduce operators by one degree

[red tree]

Is that the definition of logarithm to you?

[copper vigil]

yeah sure

it's the most important thing that logs do

[red tree]

At least it's better than the definition usually taught which is like "To find log base b of a number, set up an equation and convert it to exponential form"

wheeze

[copper vigil]

i mean what i said is not a definition

but it's the best way to remember how the rule works

[red tree]

Wonder what log of a tetration is

$\log({}^3x)\stackrel{?}{=}\log(x)^3$

[obsidian monolith]

Icy001

[copper vigil]

log(x^x^x) = x^x log(x)

no further simplification i think

what you need is a hyperlogarithm

[red tree]

So the rule breaks down after exponents? sadge

[copper vigil]

tetration is greater than the simple log

look up hyperlogarithm just do it haha

[red tree]

"one of the two inverse functions of tetration"

Ok I guess it's pretty to prove that an inverse of tetration is an inverse of tetration

yeehaw

[copper vigil]

another thing you might want to look into is commutative exponentiation

[red tree]

Is that the $x^{\log y}$ operation I saw in EpicMathTime?

[obsidian monolith]

Icy001

[red tree]

It is!

[copper vigil]

a ↑ b = b ↑ a = a^(log(b)) = b^(log(a))

haha you saw that same video

[red tree]

Yep

EpicMathTime is so good

[round hemlock]

to me, i just see it as if there is log of b to some base a, what exponent makes a become b?

[red tree]

That’s probably a lot closer to a good usable definition than the other ways of presenting it!

Hot take: every precalculus student should be able to derive the log rules from the exponent rules effortlessly, and the first step in doing that is to actually have a usable definition

[fleet yew]

To be pedantic

Defining the logarithm as the inverse of the exponential function is problematic

Because the definition of the exponential function sort of depends on the logarithm

[red tree]

Doesn’t have to! School math implicitly defines 2^x as the limit of 2^r for rational numbers r approaching x

The approach of defining logarithm first is first taken in calculus

[fleet yew]

Instead it's better to say that logx is the integral of 1/t from 1 to x

[clear venture]

can some one explain to me how to find the x whe nx^3/2=8000

[outer marten]

raise both sides to the power of 2/3

(x^3/2)^2/3 = x^((3/2)*(2/3))

which is just x^1

which is just x

[clear venture]

thnks

[quasi bolt]

how is the answer look like?

[jolly raven]

well use the quadratic formula to find it

[willow bear]

the answer looks like a pair of numbers, perhaps written like z = ____, z = ____ where the blanks are the roots you're asked to find

[red tree]

To be more general, the answer looks like a set of numbers 😎

(which should've been immediately visible if your teacher taught you precisely what "roots" means)

[quasi bolt]

ok thanks

[hoary grail]

hi so my upcoming exam is without a calcucator can someome help me understand how the squareroot of sqrt(-432) to 12*sqrt(3)*I

<@&286206848099549185>

[echo wagon]

432 = 144 * 3 = 12^2 * 3

[hoary grail]

what about the squreroot

[echo wagon]

Well, now you can apply the squareroot to both sides

[magic mantle]

Bro thw persin who wrote that really needs to use a different i

I thought it said 121

[viscid thistle]

WRONG, 'tisn't

[viscid thistle]

Tell your teacher that they dont know what "simplify" means

[tough vale]

can someone help me create a trinomial that has a GCF of 3 and has (x+3) as a factor.

[copper vigil]

GCF!

[red tree]

Also, most students who take precalculus and can solve logarithm problems, nevertheless probably could not tell someone what the definition of log_b x means without saying "to find log_b x, solve this equation..."

Do you remember how long ago was the first lesson on logarithms?

Do you remember if the teacher ever gave a precise definition of log

"kinda" is an interesting word choice

is that their words or yours?

Well "inverse of exponential function" is actually extremely precise if you are strong on "inverse of a function"

Well did you learn inverse functions?

What's the precise definition of the inverse of f, if f is a function?

That's just notation

but what's the definition

Ah-ha

The hole has been revealed

[tawdry wing]

Think about what "inverse" means

Then associate that with functions.

Ok

Lol

[red tree]

Everything in math builds on everything previous

that's the cold hard truth

wow, accelerated? 😮

was that because your school was impressed by you in 5th grade?

[tawdry wing]

Did you ever learn Math outside of school?

[red tree]

Interesting

yikessss

Did you care about math at all in 6th grade?

Hmm

ok

Did you have a good opinion of math when you were in 6th grade?

Hmm and you remembered most stuff from algebra 1?

Off the top of my head, maybe linear functions, graphing functions, what you said, proportional reasoning, and maybe that's it?

holy crap is that so slow when stretched over the entire school year

Was inverse functions taught in algebra 1 or 2?

Ohh...

So inverse functions is a relatively recent topic for you

yet you forgot about it hmmm

(until now?)

If the teacher was good, she would have used the fact that logarithm is the inverse function to exponential functions to derive all the log rules

and you know, showed the logic behind everything

But I have a good idea of how she (or he) presented it

just, here's what you do, solve these problems this way, practice

Hhhhhhh

canceling ln is pretty interesting in box 7

actually yeah holy hell, almost all the boxes show fundamental misconceptions

Gotta fix those asap

probably this order

1. learn the proper definition of inverse function from a good book or resource
2. learn the proper definition of logarithm from the same resource
3. ???
4. profit

oh yeah

0. learn the proper definition of function

based on your work in box 7

How are you with reading math books

ok

https://faculty.utrgv.edu/eleftherios.gkioulekas/OGS/Santos/santos-precalculus.pdf

I suggest this, it's one of the only precalculus textbooks I found that actually tries to make sense

although, since it is so logical, it may take more effort at the beginning to read it

Wow you read the table of contents fast

Binomial theorem is definitely accessible without calculus tools

I think starting at chapter 3 might work

It's not a quick reading by any means

Already huge misconception going from first to second line

I feel like the teacher is simultaneously going too fast in the unit while not teaching what anything means

How does this even constitute a math class

Hmmmm I've graduated from college

I'll say that

I'm not sure if I'm the one who can manage to sort everything out for you

There's lots of stuff to learn

1. learn the proper definition of inverse function from a good book or resource
2. learn the proper definition of logarithm from the same resource
3. ???
4. profit
   oh yeah
5. learn the proper definition of function
   based on your work in box 7

[tawdry wing]

Do you see how this wouldn't make sense?

[red tree]

It follows from logarithm rules that $\log_4 x-\log_4(x^3)=\log_4(x/x^3)$. But it does not follow from logarithm rules that $2\log_4 x-\log_4(x^3)=2\log_4(x/x^3)$

[obsidian monolith]

Icy001

[red tree]

That's why you have to precisely know the logarithm rules

and knowing why they're true helps a lot

like what they do and do not say

I don't see how anyone can learn logs properly with that schedule lol

So the logarithm rule that's relevant here is [\log_b x-\log_b y=\log_b(x/y)]

[obsidian monolith]

Icy001

[red tree]

That's what you (incorrectly) used so yeah

It doesn't say $2\log_b x-\log_b y=2\log_b(x/y)$

[obsidian monolith]

Icy001

[red tree]

Well let's see

$2\log_4 x-3\log_4x=-\log_4 x$

If $-\log_4 x=1$, then $\log_4x=-1$, which implies $x=4^{-1}=1/4$

[obsidian monolith]

Icy001

[red tree]

The first line is something that's always true, I only took the equation starting in the second line

explain

"the exponent" is vague (I assume the 3 in x^3?) and "the beginning" is vague

It's helpful not to think of it as moving anything

It's an application of a logarithm rule

namely the one that says $\log(x^a)=a\log x$

[obsidian monolith]

Icy001

[red tree]

The fact you're asking that question is because you were never taught what any of this means mathematically

Inverse of e?

Nawwwwwww that's a different kind of inverse

that's reciprocal

The fact that log is the inverse of e^x is in the functional inverse sense

You konw what

I'll give a precise definition of about 4 things

right now

First, a definition of a set

Set: a collection of objects (typically numbers)

Function: a function from a set X to a set Y is an assignment of an element of Y to each element of X. The set X is called the domain of the function and the set Y is called the codomain of the function.

Any questions so far?

ok next is inverse function

[frozen jetty]

excuse me fellas i know how to solve logs but i get stuck whenever i get logs on 2 different sides, whats the process of solving them?

logs are pretty easy lad

i really like them

[red tree]

the notation f: X -> Y says that f is a function from X to Y

Inverse: Given a bijective function f: X -> Y, the inverse of f is the function g: Y -> X such that g(f(x)) = x for all x in X

[frozen jetty]

i just dont know how you solve that

[red tree]

Another way to say that: Given a bijective function f: X -> Y, the inverse of f is the function from Y to X given by (y -> the unique x in X such that f(x) = y)

[frozen jetty]

anyone got a method for this then id appreciate it

[red tree]

Definition of logarithm: $\log_b$ is the inverse of the function $x\mapsto b^x$

[obsidian monolith]

Icy001

[red tree]

So if you apply the definition of inverse above

You find that $\log_b y$ returns the unique number $x$ such that $e^x=y$

[obsidian monolith]

Icy001

[frozen jetty]

hmm thats a legal rule?

is this legal? /3

[red tree]

jeez, teaching change log to exponential form without teaching the definition of logarithm is such a travesty

[frozen jetty]

so is it legal to divide by 3 ?

[red tree]

Am I suddenly a mathematical authority lol

Only logic is the mathematical authority

[frozen jetty]

ye thats what i mean by legal

is it legal within the logic of maths

[red tree]

Well logic says that

if $f$ is a bijective function then $a=b$ if and only if $f(a)=f(b)$

[obsidian monolith]

Icy001

[red tree]

I'm pretty sure that division by 3 is a bijective function

[frozen jetty]

why is @red tree mentioning functions

[red tree]

Every operation on a number can be thought of as a function

division by 3 for example

[frozen jetty]

sus

[red tree]

x->x/3

[frozen jetty]

its a instruction tho

sus

[red tree]

dividing by 3?

[frozen jetty]

ye its more of a instruction

[red tree]

Why wouldn't it be a function just because it's an instruction

Haven't you seen f(x) = x/3 as an example of a function

[frozen jetty]

why would you label every instruction as a function

[red tree]

Because it fits what a function is

:0

[frozen jetty]

waste of time

[red tree]

Lol

[frozen jetty]

thats such a weird format

[tawdry wing]

I just opened the picture in a new tab lol

[red tree]

Canceling all the ln's away and turning addition into multiplication would be such a chad move and I would call it that if you weren't having no idea what you were doing

[frozen jetty]

ok i managed to solve it lads

how do i do that cool format

with that bot

[red tree]

oh like $\log_4x$?

[obsidian monolith]

Icy001

[frozen jetty]

ye

thats weird

[red tree]

Dollar signs around math :^)

bot will make it automatically

I did say around

[obsidian monolith]

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

[frozen jetty]

lol

[red tree]

Compile errors are my favorite

[frozen jetty]

$\e^ln(24)/8 + 1$

[red tree]

$e^{\ln(24)}/8+1$

[obsidian monolith]

Icy001

[frozen jetty]

ye thats my solution

math is sooooo nice compared to mechanics

[red tree]

Wahoo

[tawdry wing]

3^3 is 27.

[solemn pollen]

nah its 14 ?

[jolly raven]

Uhh

No it isnt

3^3 is indeed 27

[solemn pollen]

nah cause 3 ^ 2 = 9 right

[jolly raven]

Yeah

[solemn pollen]

3^1 = 5

[jolly raven]

Lmao

[solemn pollen]

$3^3 = 3^2 + 3^1 = 9+5 = 14$

[obsidian monolith]

shriller44

[solemn pollen]

you have to use pythagoras and stuff

and the chain rule

[eager venture]

i know the answer is sin²θ i just dont know the process help will be appreciated as I'm studying for a quiz tomorrow

[jolly raven]

Genius moves 😎

[solemn pollen]

@eager venture so you know what sec^2(theta) is right

[eager venture]

uh

yea

[solemn pollen]

what

[eager venture]

1+tan^2

[jolly raven]

In this case it's more importantly 1/cos^2

[eager venture]

identities right?

[eager venture]

u talking to me?

[jolly raven]

Yeah

[glossy birch]

can anyone help me?

[solemn pollen]

yeah you want the form $\frac{1}{cos^2(\theta)}$

[obsidian monolith]

shriller44

[eager venture]

oh

ok

[glossy birch]

can we do a call>

?

[solemn pollen]

so you should be like do some algebra and manipulate into the correct form from taht sub

[jolly raven]

Why do people want to do calls so much

[glossy birch]

its easier

tbh

[solemn pollen]

just apply the compound angle formulae

[eager venture]

ok thanks a lot i think i got it

[jolly raven]

Nice nice

[glossy birch]

ok thanks

[jolly raven]

The identity must've helped you out

[solemn pollen]

just put these on top and do some rearranging

[jolly raven]

Also the cosine ones bc tangent

Most painful part of trig is that it's pretty applicable in engineering and physics

[glossy birch]

im doing business

[jolly raven]

Wanna do aerospace? Remember that stuff if ur designing

[glossy birch]

so hopefully

it doesnt apply

too much

[jolly raven]

Nah business wouldn't have that

Maybe if u do finance u will do calculus tho

[glossy birch]

yeah

i have cal next semester

im so stressed out

i have a unit test

[jolly raven]

Maybe the most trig you would do is just a trig regression line

[glossy birch]

friday

[jolly raven]

Join the club lol

[glossy birch]

what club??

[jolly raven]

The stressed out club

[solemn pollen]

calculus is calm

[glossy birch]

oh yeahh

frrr bro

[solemn pollen]

just watch khan academy hes the best maths teacher

or 3blue1brown essence of calculus series

[jolly raven]

If you understand that things change over time and more complicated stuff will be expressed by other complicated stuff then calc will be fine

[full dock]

to confirm the identity u need to use the trig identity sec^2(theta)-1= tan^2(theta)

then u need to know sec^2(theta)=1/cos^2(theta)

and u know that tan^2(theta)=sin^2(theta)/cos^2(theta)

now plug those back into the equation and simplify and see what u get

[solemn pollen]

calculus makes the most sense when you apply it imo

like all maths

[jolly raven]

In a physics simp bc I like calc

Trig is ok I guess

I've used trig for my robotics club tho last year in my freshman year

[solemn pollen]

shouldnt you have covered trig normally before calc

[full dock]

trig is pretty fun

[jolly raven]

It was one of those problems where u had to find the distance between a chord on a circle and the farthest point from it on a circle using the chords length and the diameter

[eager venture]

ok i get the first and third identity

what about the second

[solemn pollen]

yeah just recently did the fourier transform its acc so cool

[jolly raven]

I needed it to find at what point the ball would enter the chamber to the shooter for my robot

[solemn pollen]

3blue1brown video on this is amazing

[jolly raven]

3b1b is great

[full dock]

well u know that sec(x) = 1/cos(x) right

its the definition

of

secant

[jolly raven]

Just square both sides

[full dock]

yeah

[glossy birch]

@solemn pollen i still need help

[solemn pollen]

which question??

[eager venture]

isn't secant =1- tan^2

[jolly raven]

Lots of ways to define it

[full dock]

well thats also true, but the most basic definition is 1/cosx

[jolly raven]

That specific one isn't important here tho

[eager venture]

oh nvm yea i know that

[full dock]

you derive that from the identity

[eager venture]

cause cos is x/r

[tawdry wing]

That's an identity. The definition of sec(x) is 1/cos(x).

[full dock]

cos^2x + sin^2x = 1

[solemn pollen]

@glossy birch do you know the expansion of sin(x+y)

[glossy birch]

yeahh

i do

[solemn pollen]

so you write thse ontop of one another

[glossy birch]

ok i understand that

[eager venture]

ok those identities make sense ill plug them in rq

[glossy birch]

then what?

[full dock]

man to first solve these u gotta know the base definition

secx = 1/cosx
cscx = 1/sinx
cotx = 1/tanx

[eager venture]

yea i remember them now

they're the reciprocals of sin cos and tan

[eager venture]

uh im working it out right now and i dont know how to plug them in

[full dock]

mhm that's right

[full dock]

uhh

its not very hard

just replace

the

sec^2(theta) - 1 with tan^2 (theta)

and

sec^2(theta) with 1/cos^2(theta)

so

that will be

tan^2(theta)/(1/cos^2(theta)

[eager venture]

yea i have that so far

[eager venture]

i dont know what to do after this

[full dock]

then you just simplify that

oh

so remember

that when you divide by a fraction

its hte same as

[eager venture]

tan is sin/cos

[full dock]

multiplying by its reciprocal

ok so

[eager venture]

OHHH

[full dock]

it will be

[eager venture]

i got it

[full dock]

yep

yep that works too

you should have got

tan^2(theta)cos^2(theta)= (sin^2(theta)/cos^2(theta))(cos^2(theta)

and you can see

the cos^2(theta) cancels out

which leaves only sin^2(theta)

[solemn pollen]

$\frac{sinxcosy+cosxsiny}{sinxcosy-cosxsiny} \cdot \frac{\frac{1}{cosxcosy}}{\frac{1}{cosxcosy}}$

[obsidian monolith]

shriller44

[solemn pollen]

then just simplify

[glossy birch]

where did u get the multiplication side from?

[solemn pollen]

well we are allowed to multiply a fraction by same value on top and bottom

to get the same fraction

but the values we chose specifically break apart the cos and sin pairs

to form tans

sorry i fucked that

$\frac{\frac{sinxcosy}{cosxcosy}+cosxsiny}{sinxcosy-cosxsiny} \cdot \frac{\frac{1}{cosxcosy}}{\frac{1}{cosxcosy}}$

[obsidian monolith]

shriller44

[solemn pollen]

so thats just one example yk

[full dock]

i hate probability and combinatorics so much

[solemn pollen]

so you cancel out cos y and cosy

to get sinx/cosx = tanx

do the same thing for all others

[dim egret]

any ideas?

[unborn blade]

for basic related rates problems like these, how do we know to plug in y into the Pythagorean theorem and not x?

like what is it that tells us to relate x and z and not y and z

[fluid terrace]

scheese is good

roooo

yeah

[warped bronze]

I got a precalc class, so lame for me, the kid differentiating with respect to vectors and drawing graphs of (-1)^x

And trying to use matrices to solve quadratics. Funny: you can't invert the matrices, but they do work on solution to the equation.

[ruby oriole]

i am trying to read Apostol's Calculus Vol 1. I find myself getting stuck every single page for silly reasons (i don't pay close enough attention, don't realize altitude and kb/n was referring to x, etc.), and I'm only on page 7/8.
He is offering a proof of the Method of exhaustion for area under parabola, refers to Archimedes... anyway, i got confused at (1.11) because he has this inequality, and three possiblities for what A is, and then plays around with thw inequality under the assumption that A > b^3/3, and says that the inequality is obviously false when he inverts the sign of the inequality

so im just confused why its false if he inverts the inequality so that n > b^3(A - b^3/3)

iirc n was the number of rectangles we use so maybw that's why its obvious, because we can use an arbitrary number of rectangles whereas the area would just converge with more rectangles?

so infinite rectangles cant be less than the uhh statement he made, but i dont think thats what he was saying

i feel like i am missing the obvious

like what is he referring to?

[viscid thistle]

${\displaystyle \exists \mathbf {I} ,(\emptyset \in \mathbf {I} ,\land ,\forall x\in \mathbf {I} ,(,(x\cup {x})\in \mathbf {I} )).}

wrong chanle

[amber steeple]

I'm a bit confused on this, how would I do answer this?

[frank acorn]

What means fourth power?

[quartz inlet]

Something to the fourth power is for example $x^4$

[obsidian monolith]

Aslan

[quartz inlet]

Or in other words $x\cdot x\cdot x \cdot x$

[obsidian monolith]

Aslan

[hidden cobalt]

hey can anyone explain if this derivation is right

ples

[jolly raven]

this is precalc

not calc

[hidden cobalt]

it is? i kindof don't know calc

This is for modeling

[jolly raven]

no im saying

the chat ur in is precalc

what ur doing now is calc

[placid ledge]

looks good to me

[viscid thistle]

What do you think makes more sense: Writing the function infront of the variables, or writing it behind the variables?

[viscid thistle]

#calculus

[short pilot]

Does someone have the precalc seventh chapter 2 solutions?

[wispy dune]

is 10t mats easy

10th

[short pilot]

what?

[spark apex]

His question is a precalc one as well, I’m assuming by precalc you mean calculus that you take before uni

[spark apex]

Behind of course

[viscid thistle]

I think so too, I just wanted to know if anybody would have any objections to my opinions as is usually the gase

[spark apex]

I see

[spark apex]

I doubt that anybody could object because it’s human innate aversion to like the function behind

[viscid thistle]

I would have to agree to that

nice encouragemnt

[jolly raven]

no im saying derivatives is a calculus thing not a precalc thing

[spark apex]

I thought precalc had some derivatives and integrals

Yeah in our book the chapter is called precalculus

Sub chapters are: differentiation, integration, limits, application of diffrentiation, application of integration, application of integration in kinematics

[jolly raven]

what

huh

it might be different in the US

[viscid thistle]

really? this is surprizing

[viscid thistle]

this is literally mainstream calculus though

[timber wraith]

Nah, it has more of limits and trig identities, plus complex numbers and conic sections

it's really just a course that prepares you for calculus

[viscid thistle]

Precalc apparently= trigonometry and complex numbers

[quartz inlet]

Afaik is that "precalc" is a term that stemmed from the US. There's no official/systemic guideline of it in Europe, where I am sure there are some teachers that deviate from it.

[quartz inlet]

This to me sounds like high school calc in the US.

Correct me if I'm wrong!

[viscid thistle]

So precalc is sort of unnecessary

as in

precalc is not an officially recognized field on math

[digital cloud]

precalc is kind of a terrible name too

like it makes calc seem like it's the top of the hierarchy of math

when it's just another branch in a huge tree

[red tree]

Pre-calc does have things in common with the prerequisites for any higher level mathematics though

Like, being fluent with function language (although evidently most people leave precalc still not fluent)

[echo wagon]

I don't think it's because of the name. Why not have the same complaint about pre-algebra then?

[jolly raven]

It should be renamed to alg 3 imo

[viscid thistle]

ukw, these courses are trash descriptions

one should only focus on the math field

and not the course

it is discouraging from learning more math fluidly

[viscid thistle]

Hello

can somebody help me

please

[full pagoda]

don't ask to ask, just post a question

[graceful gulch]

Can I turn $\frac{1}{n\log(n)(\log(\log n))}$ into something simpler?

[obsidian monolith]

please request a new nickname

[shy kettle]

anyone know?

[sharp mist]

Is that like, the graph of cos^-1?

[shy kettle]

It might be, I don’t know though

But now that I think of it it does look like that

No it’s not, but I do think it’s a cos function because it starts at -1

[sharp mist]

Yeah, sorry, it just looked weirdly similar and I’m covering those

[shy kettle]

nah i know exactly what you mean

the graph is opening up, though, but i forget what that means i think that the coefficient is greater than 0

[crystal osprey]

That looks like a secant function @shy kettle

Although I'm assuming that you'd have to work through solving for it, given that it's worth 4

[shy kettle]

I gave up on it loool

Just a teeny hw no biggie I really appreciate everyone’s help though

[sharp mist]

Just wondering because I’ve heard some things. How easy is pre calc to self teach? Or learn over the summer or something

[cyan kernel]

If your high school algebra background is strong, not hard at all

[sharp mist]

Thank you

If it was weak, how bad would it be

[rocky nacelle]

@sharp mist Not going to be bad you'll just be reviewing the first few sections of Precalculus

then you'll go in depth with foundation and new ideas

[viscid thistle]

Let me try out some prototype new notation

(x)f

(x)f+g

(x)fg

[zenith solar]

Could someone help me with this, thanks

Question 1

[willow bear]

,rccw

[willow bear]

have you made any progress with this? (Y/N)

[zenith solar]

N

[willow bear]

are you familiar with the sum and product formulas?

[zenith solar]

Yes

[willow bear]

okay, so the roots of your equation are a and a^2

act accordingly

[static hawk]

Can someone please help me with c

I don’t know how to explain

[shy kettle]

honestly to me, precalc is pretty hard, but thats just because im not used to learning in big lecture classes im used to small classrooms

[robust quartz]

Supposed to manipulate left side only. I’m stuck at this part

[viscid thistle]

wait, are you learning precalc in a huge hall with a dude who attempts to explain it eventhough he is super boring?

why do you do this to yourself

?

[shy kettle]

i didnt choose my class, it was chosen for me, and it is only taught in lectures so there isnt a class available where you can get like 20 sutdents, its only taught in big auditoriums with like 70 other kids

[viscid thistle]

who decided that was a good idea?

[red tree]

That’s quite normal

In college

The biggest reason it turns out ineffective for most students is because most students lack the prerequisites to understand the lectures, for example fluency in mathematical language

And everyone has different gaps so each of them gets lost by a different thing

[viscid thistle]

Well, I personally think that the teachers are trying to be too general

and are more or less actually saying stuff rather than showing how you can use stuff

[red tree]

I have heard that criticism before

[lime sequoia]

you mean secant?

because that's definitely not arccos

[wise spoke]

cause sec = 1/cos

[sick steppe]

yes

[wise spoke]

thanks

[neon swift]

You would be surprised, i didn’t really even take pre calculus yet I’m in calc 3 in college, college is weird

[pseudo harbor]

how to find for the d,e, and f in (1,4) (-1,2) (4,-3)

[pseudo harbor]

so it is circle

i alr did the 3 equations

[pseudo harbor]

they are from the genform x^2+y^2+Dx+Ey+F=0

[lavish edge]

hmmm

(X-1)^2+(y-4)^2=r^2

isnt radius given

[pale talon]

someone help me as soon as possible PLEASE <@&286206848099549185>

[upper tundra]

I suppose they meant (1,4) lies on the circle and not the centre of the circle

[lavish edge]

oh ok

[shy kettle]

anyone know?

[odd rivet]

What have you tried?

[shy kettle]

i dont know how to do the parenthesis of the equation

i know the rest tho

but not the phase shift or period part

[oak drum]

if cos2b=2cos^2 b-1, would 20a cos(b(2))+d simplify to 2cos^2 b-1 =20? or further to cos^2 b=11?

[odd rivet]

Can't be , -1 =< cos x =<1

[lavish edge]

@shy kettle

its quite simple really

so when you move to the left or right the sign is opposite for example if you move 11 units to the right it becomes x-11 and x+11 for the left

so now you have tan(x+11) because if you have horizontal shifts it stays in the bracket

but now the x axis gets reflected right?

[shy kettle]

Yes

[lavish edge]

so try flipping it in your mind

[shy kettle]

The reflection is hard

[lavish edge]

it goes to the left

[shy kettle]

I ended up with tan(x+11)

[lavish edge]

so it becomes tan(x+11) since sign changes

yup

now

1 unit downward means -1 outside f(x)

so after tan(x+11)

just put tan(x+11)-1

did you understand

[shy kettle]

So now we have tan(x+11)-1

Is that after the reflection?

[lavish edge]

yea

[shy kettle]

The value in the parenthesis is the one that changes after a reflection across the x axis, correct?

[lavish edge]

yes

[shy kettle]

We have to refelect it across x axis, that was the part I’m not too sure on. I think my final answer before I got stuck was y=11tan(x+11)-1

[lavish edge]

how 11 tan

[shy kettle]

It was stretched by 11

[lavish edge]

o yea

[lavish edge]

then this ans is right

[clear torrent]

Anyone know the sol?

[gilded sapphire]

you know that every constant is differentiable right/

[willow bear]

@gilded sapphire and how exactly does that matter?

[clear torrent]

I have boards in 10 days and ik nothing 🥲

[willow bear]

boards?

[clear torrent]

Cbse board exams

[willow bear]

if you really know completely nothing then you are fucked no matter what

10 days is not nearly enough to prepare from zero

[clear torrent]

Ik the basics

Idk differentiation that much

[willow bear]

if you don't know differentiation then you are still fucked

[clear torrent]

🥲

chain rule, product rule, quotient rule that's all ik

[red tree]

Do you know what a function is?

[native hill]

how'd you find the unit vector of a 3d vector

[willow bear]

the same as in any other vector space

divide the vector by its own norm

[elder scaffold]

Everything was going great in precalc this semester than I got to fucking mathematical induction.

[red tree]

Mathematical induction can be stated in one line!
For a function $P$ from $\bZ_+$ to ${\text{true},\text{false}}$, if we know that $P(1)=\text{true}$ and that for every positive integer $n$, [$P(n)=\text{true}$ implies that $P(n+1)=\text{true}$], then it follows that for every positive integer $n$, $P(n)$ is true.

[obsidian monolith]

Icy001

[native hill]

are proofs really precalc?

[red tree]

Shouldn’t it be -pi/3?

[odd rivet]

Sin inverse exists on restricted domain

[clear torrent]

Yes

[grave tartan]

do you know how the graphs of inverse functions are just reflections over the line y=x

basically if you do the same with any trig function

you no longer have the graph of a function

[clear torrent]

Tbh I don't know

[grave tartan]

because for every input between [-1, 1] you will have infinitely many outputs

this is a problem because it doesn't follow the definition of a function

where every input has only one output

and therefore its not as powerful

as a way to manuever this problem, we restrict all the inverse trig functions to specific ranges

so

for sin(x), we can get all the possible outputs of the function in the domain [-pi/2, pi/2]

that is just fancy math talk for saying if we input all the numbers on the interval [-pi/2, pi/2] we will get all the possible outputs of sin(x)

like the output for 3pi/2, or -1, can be found by inputing -pi/2

this makes it better for us

tbh if you are so close to the exam date I would just grind videos on precalc

but also yeah

[red tree]

Can you elaborate on what you've been taught or what you remember the definition of a function to be?

[grave tartan]

$\sin^{-1}{x}$ has a restricted range of $[-\frac{\pi}{2}, \frac{\pi}{2}]$

[obsidian monolith]

vrang

[grave tartan]

as a result we have to add 2pi*n to the solution we obtain as well has find the same value on the other side of the unit circle to account for all the possible solutions

theres a bit more to it but I can't explain it all rn

[clear torrent]

Relationship btwn set of inputs to one output each

[red tree]

You know how you write sqrt(x) and such, is "sqrt" considered a function?

what about sqrt(sqrt(x))

[clear torrent]

Is that a function

[red tree]

yes, yes it is

[clear torrent]

Ooh

[viscid thistle]

can you only apply this formula when the lim x-> a is same as the constant a?

or can you apply this regardless of the limit and the constant match?

[willow bear]

...i mean the a's all have to be the same obviously

[viscid thistle]

i was having second doubts since i applied this to this question and still got the right answer

im guessing it was coincidence lol

[odd rivet]

No

[willow bear]

you have $\lim_{2x \to 1} \frac{(2x)^2 - 1^2}{2x - 1}$

[obsidian monolith]

Ann

[viscid thistle]

oh so that was why

ty i get it now

[wise spoke]

When I tried to prove this I got $(\sqrt{2} - 1)$ using both \frac{1}{\sqrt{2}} = \frac{1 - tan^2 22\frac{1}{2}}{1 + tan^2 22\frac{1}{2}}$$ and $$tan22\frac{1}{2} = \frac{sin22\frac{1}{2}}{cos22\frac{1}{2}}$$.

[obsidian monolith]

Inheritanc-e ♦

[wise spoke]

Could someone help me solve this, thanks!

[olive zephyr]

Sqrt(2)-1 is equivalent to sqrt (3-2sqrt(2))

[wise spoke]

ye got it

thanks

[eternal stone]

perhaps its more appropriate for me to ask here than in the multivariable channel

in the event of 0 = 2x(4y+7)e^((-x^2)(-y^2))

can i just... ln both sides?

[odd rivet]

No need e^x for all x in R is not equal to 0

[eternal stone]

👀

[odd rivet]

Is there anything else you type than that?

[eternal stone]

im a bit surprised if thats the case, i have a f(x,y) function and i took the partial derivative of x, and im trying to find the critical points by setting that Fx to zero... there's really no critical points then?

[odd rivet]

I just said about e^x

[eternal stone]

oh

[odd rivet]

What about 2x(4y+7) part

[eternal stone]

looks like for 0 = 2x(4y+7), x = 0 and y = -7/4

[odd rivet]

Not necessarily

[gilded sapphire]

f(x) at x=1 is a constat

[gilded sapphire]

boards don't have a very large portion if you practise 60questions(3hr) per day in maths it would be enough

try doing the solved examples of ncert

[clear torrent]

Ok

Thank u dude

[gilded sapphire]

also practise solved examples of rd sharma to grasp the basic concepts

[clear torrent]

Ok

[gilded sapphire]

are you not planning to give jee?

[clear torrent]

After this year ig

[gilded sapphire]

its not like that you still have 4months try giving mains, due to recent pattern change cbse board exams are mcq so level of board exams and mains will be same

they will not just ask SI unit this time its going to be harder in boards

[clear torrent]

Ooh

But I still have to crack advanced right?

[willow bear]

no shit, any function on a singleton is constant.

[gilded sapphire]

advance is for when you want to get into IIT you can get good NITs which have the same level of education but different brand value than IIT

[clear torrent]

Ooh

I thought of taking a gap year to learn the base concepts properly and for Iearning C

[gilded sapphire]

its not going to work like that, people who get good ranks in drop year are those who gave their efforts in the first but didnt get the cut off you still have time, for physics you can watch physics galaxy/wallah to get the concepts

[clear torrent]

Hmm I'll try my best

[gilded sapphire]

best of luck

[clear torrent]

Thanks

When exactly is jee mains

[gilded sapphire]

dk now probably march and june

[clear torrent]

Ooh

[gilded sapphire]

this channel is getting off topic so sorry

[clear torrent]

😌

[mossy hearth]

This is a trigonometric equation

im having trouble on what will be the process in solving for "a" since my precal teacher didn't quite teach us about this kind of equation including a fraction form

[meager kayak]

Alrighty, have you heard of the ArcTan function?
(also known as the Tan inverse function?)

[mossy hearth]

yeah

[meager kayak]

Do you know what it does, and if you do, how to use it?

[mossy hearth]

sadly no

[meager kayak]

Hmm.
Are you familiar with how the regular Tan works?

i.e. the tan that's in your question

[mossy hearth]

yes

[meager kayak]

Do you know why that condition was given in the first place?
i.e. Had the question not included that condition, is it possible to solve for alpha?

like what i mean is, is that condition even necessary to solve for alpha?
"yes/no/i don't understand your question"

[mossy hearth]

yes

[meager kayak]

Why is it necessary, then?

[mossy hearth]

oh wait I change my answer, i think that even if the condition is not included, only solving for the given is what Im processing

[meager kayak]

Okay but,
You know how like Tan(45 degrees) =1 right

[mossy hearth]

mhm

[meager kayak]

45 degrees isn't the only angle upon taking the tan, that will give you 1

for example

Tan(225 degrees) is also =1

or Tan(405 degrees) = 1

[mossy hearth]

oh yeah

[meager kayak]

or Tan (-135 degrees) = 1

in fact there are infinitely many such angles that'll do this

so we want to find the ones that are specifically between 360 and 720 degrees

[meager kayak]

or rather, there are alot of possible values of "a" that will work. We want to find the values of alpha that are between 360 and 720, yeah?

[mossy hearth]

ok ok

[meager kayak]

I'm having difficulty proceeding. Perhaps you could tell me what you know about these values

-135, 45, 225, 405,
all of these angles, upon taking the tan of them
they'd equal 1.
do you know why this is the case?

say something man
If you don't know don't worry

[mossy hearth]

the rotation?

like what you said there are many ways you could get equals by 1

with dif possible numbers that equals to 1

in tan

[meager kayak]

yeah. you're sorta right here

[meager kayak]

notice how all of these values are seperated by 180 degrees

the Tan function kinda has a cycle that repeats every 180 degrees

[mossy hearth]

ohh i get it

[meager kayak]

so tan (0 to 180)
then it repeats again
tan (180 to 360)
on and on and on

[meager kayak]

so our strategy is going to be, we're going to find one possible value of "a", and then add/subtract 180 over and over again until we get numbers that are in the 360 to 720 degree range

so like if i wanted an angle between 360 and 720 whose tan was 1,
i'd first look at 45 degrees, but i know the cycle repeats every 180 degrees
so i'd add 180 to 45 over and over again until i start getting numbers in the 360 to 720 range

[mossy hearth]

405,585

[meager kayak]

If your question is why exactly does the tan function have this cycle that repeats 180 degrees,
(a) your pre calc teacher sucks
and
(b) We can discuss that another time

[mossy hearth]

ill go with a

[meager kayak]

One thing about these cycles is that some other functions also have some kinds of cycles

like the sin and cos have cycles of 360 degrees (i.e. every 360 degrees they repeat)

but we digress

Ok.
Another thing you gotta know is the ArcTan function

You know like how Tan 45 is 1 right

[mossy hearth]

mhm

[meager kayak]

So we say ArcTan 1 = 45

it's kinda like the reverse thing

[mossy hearth]

yeah noticed that

[meager kayak]

ArcTan (1) basically asks the question, what number such that if I took the Tan of it, I would get back 1?

so 45 degrees would be the answer
because taking the tan of it = 1

[meager kayak]

if you've learnt about logarithms this kind of notion may be familiar

[mossy hearth]

never heard of it

or "haven't"

[meager kayak]

k nvm you'll see it in a few months ig

[mossy hearth]

yet

[meager kayak]

yeah but it's like asking the question in reverse

if you get what I mean here

so If i gave you the equation
Tan(x) = 2
And asked you to solve for "x"
you'd say, okay, x is such a number that if i take the tan of it, i'll get 2.
So x must be ArcTan(2)

technically this isn't precisely correct but do you get the idea?

[mossy hearth]

yeah i get the idea

[meager kayak]

calculating the arctan(2) btw would be a job for your calculator

Okay so we're kinda ready to solve this question now

[meager kayak]

firstly can you isolate "a" on one side?

as much as you can

[mossy hearth]

wait hold on

[meager kayak]

lmk when you get suck (if you get stuck)

[mossy hearth]

can i isolate 5?

like tan(a/2) = 30/5

[meager kayak]

yup that's allowed

that's valid

so you now have tan(a/2) = 6 right

[mossy hearth]

yeah

[meager kayak]

does this equation look familiar

[meager kayak]

perhaps to this

[mossy hearth]

uh huh

[meager kayak]

So, the equation is asking, what value, such that when i take the tan of it, gives me 6?

This is where the arctan function comes in
Can you try using it to write the next step?

[mossy hearth]

alright give me a min

[meager kayak]

btw it's -30/5

i.e. -6

you moved 30 to the other side, it becomes negative

[mossy hearth]

yeah my bad

[meager kayak]

ArcTan(-6) is basically shorthand for "The number whose tan is -6".
Actually calculating that value is a job for the calculator, at least for now.

[mossy hearth]

i got -80.54

[meager kayak]

yeah that's the arctan of -6

if you typed Tan(-80.54)
your calculator would return -6 again

or a number very close to -6

[mossy hearth]

yeah i got -6

mhm

[meager kayak]

https://cdn.discordapp.com/attachments/690488884747960331/914401192044560384/478125996885934082.png

So what can we say about "a/2"?

[mossy hearth]

i think "a" should be moved?

[meager kayak]

hold on

you know that if we take the tan of a/2, we get -6 right

so is it safe to say, that a/2, can be -80.54 degrees?

[mossy hearth]

could be

a/2 = -80.54 deg? but what about the /2

[meager kayak]

I think you can figure that one out :)

[mossy hearth]

alright, thanks for the help btw, really appreciate it

[meager kayak]

uhh

hold a sec we're not done lmao

we're half way there!

well no we're actually 90% done

[mossy hearth]

oh i thought ur gonna leave it to me

[meager kayak]

there's one crucial thing you gotta think about after you figure this out

[mossy hearth]

anyways

[mossy hearth]

what's that?

[meager kayak]

you remember how the question specifically asked for an "a" that's in between 360 and 720 degrees?

[mossy hearth]

mhm

[meager kayak]

when you figure this out, you'll realise that the value of "a" that we get isn't between 360 and 720

Remember that alot of different possible values for tan can give you the same answer

like how tan 45 = 1, tan 225 = 1 etc

so "a/2" COULD equal -80.54, but there also a lot of other options that it could equal

[mossy hearth]

i got -161.08

i multiplied both sides of a/2 and -80.54

by 2

so i could get a = -161.08

[meager kayak]

yeah so

that value WOULD work in that equation

but it's not between 360 to 720

so it's unfortunately useless

[mossy hearth]

yeah sadly

[meager kayak]

what you need to do is to now figure out what other values "a/2" could equal

like we know it could've been -80.54

[meager kayak]

but that lead us no-where

but there are other values whose tan also gives -6.

So if you can find those special values (there are alot of them!)
and then solve the equation, you'll get alot more possible values for "a",
Hopefully, some of which are between 360 to 720

And how exactly do we find those other values?
Well a good thing to keep in mind is that the Tan function repeats every 180 degrees, maybe try using that. :)

[mossy hearth]

ok ill try

[viscid thistle]

Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of 105°F occurs at 5PM and the average temperature for the day is 85°F. Find the temperature, to the nearest degree, at 9AM.

Not sure how to approach making a model of this, having trouble finding how to derive the period of the said function

so far I have

$85+30cos()$

[obsidian monolith]

codemonkey

[viscid thistle]

where I 5 PM is 0 on the graph

[grave tartan]

should be 20 not 30 I think

but if you already know that the coefficient is 20, you pretty much know how to find the period

wait

is this the whole question?

nvm its just annoying

so in a 24 hour day, we can only have 1 high temperature right

doesn't make sense to have 2 highs in one day

using that do you think you can figure out the period @viscid thistle

[viscid thistle]

yeah

literally just thougth that before you messaged

[viscid thistle]

professors word problems are beastly thank you

[grave tartan]

yeah all good dw

[viscid thistle]

also yes my amplitude is off thank you again

[grave tartan]

np

[wraith jay]

Hi, could anyone help me in #help-1 ? :)

[short pilot]

Can someone explain me how 0.01^x its 10^-2x?

[willow bear]

0.01 = 10^-2