precalculus

[dense oriole]

i have no lue of what to do

won't be able to get phoyos

photos

but the question is

"There is a pendulum swinging near a wall. The swinging of the pendulum is defined by the sinusoidal function:
H(t) = 7 - 7 cos ([2pi{t-2}]/20)
Note: H(t) is in centimeters from the wall and t is in seconds.
How often will the function cross its midline?
Once every [ insert answer ] seconds"

Its from my precalculus course

[summer ruin]

clearly the pendulum is at the midline when the distance to the wall is maximal

[dense oriole]

My teacher told me to try inversing the function, but it didn't help

[summer ruin]

if the question implies it's between two walls?

[dense oriole]

No

just a pendulum near a wall

And a cosine function that determines its distance from the said wall

I figured that the func would be at the midline whenever the pendulum was in the middle

But i don't know what to do

What fumbles me is the "7 - " at the start

Hmmm

I should probably get a help channel

brb

[rain scroll]

Who ping me

[broken swan]

can someone help me with q8

[viscid thistle]

i need help

plz

[uncut jay]

is blue the f^-1(x) of the red function

excuse the bad drawing

just wanna know if i did it right

[crude umbra]

how would i solve smth like lim x->0 1/x

or like

some limit where the variable is in the denominator

and the variable is approaching 0

[river drift]

provided the numerator is not also approaching 0, then in general limits where the denominator approaches 0 will approach +infinity or -infinity

(or both, from different sides)

[crude umbra]

so in the case of lim x->0 sin(x)/x the answer would be 1 right

[river drift]

yes, when you have a limit where both the numerator and denominator approach 0, we call that indeterminate because it can approach any value depending on the specific functions involved

[crude umbra]

so smth like x/x as x approaches 0 would be indeterminate right

[river drift]

yes, it is indeterminate, so we can't just consider where the numerator and denominator go individually when solving the limit

we have to consider both at the same time

[crude umbra]

ah alr

i see

tysm

[halcyon yew]

i mean

x/x = 1

💀

[river drift]

indeterminate does not mean difficult

[halcyon yew]

yes

but what does indeterminate mean?

i mean if the answer is always 1

wouldnt that make it determinate

[river drift]

if you have two functions, f(x) and g(x), then most of the time, you can split the limit like this:
[ \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a}f(x)}{\lim_{x \to a}g(x)} ] As well, if 1 out of the 2 of them approaches $0$ or $\infty$, that also tells you where the overall limit goes. In other words, knowing the limits $\lim_{x \to a}f(x)$ and $\lim_{x \to a}g(x)$ is enough to know what the overall limit $\lim_{x \to a} \frac{f(x)}{g(x)}$ in most cases. The case where it's not enough is when both limits approach $0$ or both limits approach $\infty$, in which case we call it indeterminate, because just knowing the individual limits does not tell you what the overall limit has to be.

[obsidian monolith]

cloud

[novel mica]

6(2a+11d) = 186
a+19d= 83

Solve the the two equations to find a and d and then sub in : 20(2a+39d)

[night surge]

is it 20secs

[icy sparrow]

Can someone confirm if this is correct?

[vague hearth]

turn lights on

[icy sparrow]

Turn brightness on 🤣

[burnt wolf]

It’s all the way up, turn the light on 💀

[wind warren]

yes;)

[icy sparrow]

Thank you!

[loud snow]

How to verify that Intermediate Value Theorem is correct?

[river drift]

if you want to demonstrate for a particular case with some function f(x), you can just pick a number n such that f(a) < n < f(b), and solve for the value of c such that a < c < b and n = f(c) that the function says must exist. proving it's correct for all continuous functions, all the time, is probably a bit beyond your class' level

[loud snow]

I'm talking about general functions.

I mean I pefer to know how to prove this theorem that it is correct for ALL functions

[river drift]

proving the intermediate value theorem usually isn't done until a real analysis class

[willow bear]

are you taking real analysis

if not, then you will see a proof of IVT there.

[vale shard]

Can I get help with this please 🙏

[viscid thistle]

In what order should I take the following math classes:

Differential calculus (Calc 1)
Integral (Calc 2)
Discrete mathematics
Linear Algebra and Vectorial geometry

[hallow tide]

in my opinion it genuinely does not matter as long as you take calc 1 before calc 2

[loud snow]

I think general Discrete Math is the most easiest, but it depends on yourself.

[floral wigeon]

Desmos

[midnight root]

guys help

[willow bear]

are you gonna send us the problem you need help with or did you decide that we need to read your mind first

[ivory mango]

probably sent something in the help forum

[onyx lotus]

i understand how to get to this but why are we are u able to write:
sqrt(x)/x
as
1/sqrt (x)?
is it because its the same as 1/1?

[robust salmon]

can anyone explain this

[chilly bone]

in the first one the denominator is just rationalized

[robust salmon]

if x is 3 for example root 3 X root 3 is root 9 so therfore its 3 on bottom (x)

[brisk adder]

like the hyperbolic tangent or like the questions?

[robust salmon]

the questions

[onyx lotus]

oooooooooooooooooo ok thank you!!!

[brisk adder]

What work do you have so far

[robust salmon]

wdym

working out

[brisk adder]

ok
so for a. Can logarithims have a negative or a zero value?
and for b. I would try and work it out by substituting x for (2x/(1+x^2)) and simplifying and applying log laws afterwards

[robust salmon]

when its -1<x<1 what vlaue do i substitute

[brisk adder]

I wouldnt substitute random values between -1 and 1. Part b is asking you to make a composite function. Like if I have $f(x) = 5x^2+4x$ and I want to subsitute $g(x) = 4x^2$, I would go $f(g(x)) = 5(4x^2)^2 + 4(4x^2)$. You can apply this same idea to the ath function

[obsidian monolith]

Toasted Bread

[robust salmon]

are you able to send working cause the textbook doesnt have the answer for some reason

[brisk adder]

$$\log _{2}\left( \dfrac{1+\dfrac{2x}{1+x^{2}}}{1-\dfrac{2x}{1+x^{2}}}\right) =\log _{2}\left( \dfrac{1+x^{2}+2x}{\dfrac{1+x^{2}}{\dfrac{1+x^{2}-2x}{1+x^{2}}}}\right) =\log _{2}\left( \left( \dfrac{1+x}{1-x}\right) ^{2}\right) =2\log _{2}\left( \dfrac{1+x}{1-x}\right)$$

[obsidian monolith]

Toasted Bread

[robust salmon]

thxx

[neon steppe]

Hmm~
Differentiating x^2 we get 2x
What exactly would the 2x graph show us in relation to the x^2 graph?
2x is referred to as the tangent, the slope or the instantaneous rate etc etc right?

How do these two graphs relate to one another~
What aspect of x^2 does 2x show? 👀

[summer ruin]

the slope of x^2

[neon steppe]

Hmm 👀

How to relate the slope with the tangent and instantaneous rates? 🤔
Do all these mean the same thing?
Why is that? 👀

[summer ruin]

because people using synonyms doesn't change the object in question

[neon steppe]

Hmm
Still a bit difficult to wrap my head around it though~
I understand that they're all synonyms but in reality it sort of feels difficult to intuitively understand what's happening? 💀

[summer ruin]

well of course it's going to be hard to understand it by staring at the graphs alone

which is why you should learn the definition first

[neon steppe]

Hmm

[torn osprey]

you can imagine x^2 being the position graph of some particle, and 2x being its velocity graph

agree w elemental tho js looking at the graphs wont help u

[neon steppe]

Hmm

[willow bear]

@neon steppe are you asking about the connection between the defn of derivative as tangent line slope and the fact that like

the graph of 2x is a straight line

[neon steppe]

Hmm
I think I've sort of got it now though~
Wasn't entirely sure of how they were defined and stuff initially

[willow bear]

you... didn't answer my question.

[neon steppe]

Sorry, Though meant to say yes with the hmm
Though I didn't quite realise the purpose of the tangent until now either so 💀
It's just my lack of understanding

[willow bear]

ok right

so to follow up on that: no, there's no connection to 2x being a straight line

it's only a consequence of the fact that the original function was quadratic

[neon steppe]

Hmm

Thanks

[solar olive]

alternatively, sqrt(x)/x = x^(1/2)/x^1 = x^(-1/2) = 1/(x^(1/2)) = 1/sqrt(x)

[onyx lotus]

if sqrt x is in a fraction why is it:
x^(1/2)
and not:
x^-(1/2)?

[willow bear]

you can write $\frac{e^x}{\sqrt{x}}$ either as $\frac{e^x}{x^{1/2}}$ or as $e^x x^{-1/2}$ according to preference, but $$\frac{e^x}{x^{-1/2}} \quad \mbox{would be} \quad \sqrt{x} \cdot e^x.$$

[obsidian monolith]

Ann

[willow bear]

does this answer your question? @onyx lotus

[onyx lotus]

so that would mean sqrt(x) d/dx =

-(1/2)x^-(3/2)?

[willow bear]

the "d/dx" is supposed to go before the function that you are differentiating,

and no.

[onyx lotus]

sorry i mean -(1/2x^3/2) or 1/2 (sqrt x^3)?

[willow bear]

neither of those is the correct derivative for sqrt(x).

to hit you with a barrage of alternative forms for the same two things,

$\dv{x} \sqrt{x} = \dv{x} x^{1/2} = \frac{1}{2} x^{-1/2} = \frac{1}{2 \sqrt{x}} \ \ \dv{x} \frac{1}{\sqrt{x}} = \dv{x} \frac{1}{x^{1/2}} = \dv{x} x^{-1/2} = -\frac{1}{2}x^{-3/2} = -\frac{1}{2x^{3/2}} = -\frac{1}{2\sqrt{x^3}}$

[obsidian monolith]

Ann

[onyx lotus]

ohhhhh, is it because ur seperating the functions? e^x/sqrt x

i was confusing sqrt x as 1/sqrt x

[willow bear]

i did not even touch e^x

your question was to do with the root so i answered your question about the root

[onyx lotus]

ok i seeee, i subtracted twice from x^1/2

[onyx lotus]

ty this cleared it up

[onyx lotus]

can someone explain to me what happened between 3 and 4?

[leaden wind]

this is the only time where someone sends a question here and i know how to solve it

[onyx lotus]

[willow bear]

out with it then

what did happen from 3 to 4

[leaden wind]

yeah ngl im stuck there too

[willow bear]

didnt you just say you knew how to answer dog's query

ok like whatever

[leaden wind]

i meant the derivative

[willow bear]

they multiplied top and bottom by 2sqrt(x)

to get rid of the nested fractions

[leaden wind]

how did they not teach me that at school

[onyx lotus]

ohhhh is that where the 2x came from cuz sqrt x *sqrt x = x , 2x times sqrtx = 2x

[willow bear]

at no point is there a 2x being multiplied by sqrt(x).

also you should not drop the brackets around x in sqrt(x).

[onyx lotus]

so would it be sqrt(x)sin(x) * 2sqrt(x) = 2xsin(x)

[willow bear]

yes

[onyx lotus]

thank u!

[wanton isle]

Would the value of c just be -4? And how would I find b? for parts a and b

[viscid thistle]

+c

[wicked roost]

better just do it by f(x)/g(x) the square root will be removed

[neon steppe]

Sorry~
I feel like I am being dumb here

Though in accordance with what chat gpt answers here~
Would it not mean
Sin (a+b) = sin a + sin b?
It's saying the results are equal right

[summer ruin]

you already posted a picture of what sin(a+b) is equal to and it's not sin(a) + sin(b)

[neon steppe]

Third image implies sin (a-b) = sinacosb - cosasinb = 2 cos(a+b/2) sin(a-b/2) right? 🤔

[willow bear]

!nogpt

[tender quest]

Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).

[willow bear]

what 3rd image?

[neon steppe]

Then again
I have very little understanding of trigonometry atm

[willow bear]

is this from chatGPT?

[neon steppe]

Hmm it is

[willow bear]

... i think a lot of people won't understand "hmm" as meaning "yes"

"hmm" means more than ✅

[neon steppe]

Yes*

[willow bear]

anyway

!nogpt

[tender quest]

Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).

[neon steppe]

Is this not correct?

Then? 👀

[willow bear]

the first line is correct. the second isn't.

[neon steppe]

Hmm

[willow bear]

show the full input and output for GPT i guess

maybe that can tell us why it fucked up

but maybe it won't, no way to find out without looking.

[neon steppe]

One second, It's from another individual though and they were trying to find a certain result one second

[willow bear]

oh god XYproblem

[willow bear]

this looks like somebody's work maybe.

but what problem were they solving?

[neon steppe]

sin (a-b) turns into 2 sin (a-b/2) cos (a-b/2)~
They were confused as to how that happens

[willow bear]

you mean 2 sin((a-b)/2) cos((a-b)/2).

and that comes from the double-angle identity.

sin(2t) = 2 sin(t) cos(t),

except you have (a-b)/2 for t

[neon steppe]

Hmm 👀

[willow bear]

it would be nice if you got straight to the point instead of walking us down a kilometer-long path to the actual doubt/confusion/question/whatever-else that your friend had.

[neon steppe]

Thanks

[willow bear]

your point was "My friend was confused how sin(a-b) turned into 2 sin((a-b)/2) cos((a-b)/2). We tried asking ChatGPT about it [but we won't show you our input], and it produced this as part of its output. Is this correct or what?"

[solar olive]

if u really wanna use hmm that much i think u should add an "m" so it would be like mhm

[dusk flame]

Bro can someone pls explain to me what tf my teacher is trying to write, just the steps of the entire equation. I understand the Pythagorean relationships, but I hate how my teacher writes her fractions.

[manic ledge]

For the first step, she multiplied $\cos \theta$ and $1+\sin \theta$ to the numerator and demoninator of both fractions and added the two fractions.

For the second step, she used the distributive property.

For the third step, she simplified.

[obsidian monolith]

Primordial

[dense trench]

in a geometric sequence the sum of the first 6 terms is 9times the sum of the first three terms. if the first term is 5 what is the third term

this is precal 11 and my teacher refuses to upload how to do the work online so i’m lost, he didn’t explain it in class either

this is what i had and it’s unsolvable

[dense trench]

(solved)

[willow bear]

which one of these? @solar river

[warped pike]

How do you all go through a book and make notes? I tend to just solve problems and not make notes, but I feel like I should do it just for fun. the issue is, I dont know how, do I make detailed explanations or short ones? What things should I include? is just highlighting the text and bookmarking not a better option?

[fading monolith]

Take notes for fun, lol

[shadow summit]

There are few different ways.

The first way is to just read the explanations and definitions and paraphrase them in your book.

This is what I typically do when I’ve just started learning a topic.

Once you have learnt a topic and just need to know the formulas/very specific details I like to buy a packet of palm cards and write the important info down on them.

You should know that taking notes doesn’t necessarily mean you learn the topic.

[open plank]

I just write down all the definitions and important theorems and call it a day

[shadow summit]

I used to do that, but then I ended up only kind of learning the topic.

[thorn creek]

i generally explain things to my wife after studying them. if i can't explain it, i didn't learn it well enough and have to go back and study it again

[willow bear]

that requires having a wife

[fiery creek]

sounds like feynman's method

explain to a child

this is your sign to have a child now

[wooden brook]

There's a small magnetic cow that I had as a kid that sticks to radiators, now stuck to my whiteboard for me to explain stuff to when I don't understand something. Makes my logical flaws much more apparent somehow

[thorn creek]

my wife has a degree in mathematics so not really the same

[fiery creek]

yes, but that defeats the point, explaining to a child is harder

ok nvm

i don't know anything

i'm stupid

i'm worthless

[viscid thistle]

what is the best way to get good at math?

read a lot of books?

[thorn creek]

do lots of math

you learn from doing, not merely from reading

[warm vault]

Anybody understand Permutations? I'm genuinely puzzled how this makes sense

[willow bear]

which part confuses you exactly?

@warm vault

[warm vault]

I essentially did the equation mentioned, but I added x 2! since I thought it would go 2! ways

Like Girls first or Boys first

[willow bear]

you were wrong to do that. the choice of seating the boys first or the girls first doesn't result in new permutations.

would you like a different example to show why your logic is bullshit

[warm vault]

Umm okay?

[willow bear]

find the number of possible 2-digit codes if each digit can be anything from 0 to 9

by your logic, you have:

• decide on the left digit first:
• 10 options for the left digit
• 10 options for the right digit
  => 10 * 10 = 100 codes
• (?!) decide on the right digit first:
• 10 options for the right digit
• 10 options for the left digit
  => 10 * 10 = 100 codes
  ==> total = 100 * 2 = 200 codes ?!

[warm vault]

Tenths cant be 0 tho, cause itll just be 1 digit

so it's just 9 * 10

[willow bear]

yes they can, this is a code on a combination lock.

it CAN begin with a zero.

[warm vault]

Oh ok

[willow bear]

do you understand why your logic is wrong tho?

[warm vault]

Yeah, I'm pretty sure I get it now

[warm vault]

Just to clarify, 00 can be a code too?

[willow bear]

yes of course.

[royal breach]

Can anyone help me with some questions?🙏🙏

[willow bear]

http://dontasktoask.com/

ie you're supposed to post the questions you need help with.

upfront.

[quick cove]

does someone know how to factor this?

[open plank]

yes

[warm vault]

Lemme try to help you

I guess for the first part, -27y^6 could be simplified as (-27y^4 * y^2)

That way, you can add that value with -9y^4

And then add x^3 with x^2 by simplifying:

2x^2 * x

But try not to rely on me only, you need a second opinion

[plucky plinth]

hint: use a³-b³ = (a-b)(a²+ab+b²)
and a²-b² = (a+b)(a-b)

[viscid thistle]

Solve in $\mathbb{R}$ the following equation : $$(3^x+2)^{\log_5{3}}+2=(5^x-2)^{\log_3{5}}.$$

[obsidian monolith]

ilovepizza1344

[willow bear]

are you sure it got typed correctly?

it looks awful as-is.

i mean the solution x=1 can be guessed, i suppose.

[viscid thistle]

yea

[hallow tide]

in all honesty this seems like a valid approach

and u can reasonably prove that's the only real solution

[fiery creek]

,w log_5(3)/log_5(5)

[obsidian monolith]

Results provided by WolframAlpha

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

Requested by aguaman5355

[fiery creek]

,w log_3 5

[obsidian monolith]

Results provided by WolframAlpha

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

Requested by aguaman5355

[fiery creek]

@viscid thistle can you do a change of base to make log_3(5) base 5?

[hexed apex]

I have to make an equation following the transformations but I got it wrong. Can someone help?

[willow bear]

ii: multiply the WHOLE thing by -1 not only the first term

likewise with iv: multiply the WHOLE thing by 1/2

you would have had -(f(x)+7), theen -(f(x/3) + 7), then -1/2 (f(x/3) + 7)

also make sure that step i has you shift the graph UP and not any other direction

[hexed apex]

Ohhhh I see, thank you so much

[hexed apex]

Thank you for your feedback. So in the end, my expression is -1/2 f(x/3) - 7/2 ?

[willow bear]

so it would appear.

but also wow way to sound like a business email with "Thank you for your feedback."

[strong epoch]

Can someone help me I don’t understand this

Does anyone recognize a pattern in this sequence ?

I only see it being multiplied by 1 more than previous

First 2 then 3 then 4 so it’s factorial right

[fleet ember]

No

[strong epoch]

But what the heck is the 15 and and the 120 doing

[strong epoch]

Do you understand it?

[fleet ember]

I think that’s supposed to be 15 thousand 120

[strong epoch]

Holy

How stupid am I

[fleet ember]

And it’s not factorial cuz it would increase at a much higher rate

[strong epoch]

💀

Bro

I’m so dumb

It was so obvious

Thanks bro

Appreciate you

[fleet ember]

Np

[strong epoch]

This 10/10 wouldn’t be possible without you

Thank you

That was the only thing I was missing it was 7 on the top

Also couldn’t this just be 4n

Oh nevermind it is that I thought it was a factorial for a second which wouldn’t make sense

[wheat carbon]

i kind of dont get it :( for f(-x) does the bottom of -x cancel out the top which makes it -x^4+2x^2

for this 1 chat said neither but, how i got even

[jolly dew]

Hey guys!
I really need help on something
cos4x-cos2x=0
Find all solutions in the interval [0,2pi)

[harsh basalt]

anybody taking or taken ap precalc and has the frq progress checks

in college board

my teacher didn’t assign me those

[pine geyser]

Hi, let's consider 2x as alpha, you would have this:

Now I want you to write cos(2alpha) based on cos(alpha),
What do you have now?

[mild swan]

maybe look online?

[severe pond]

Last year I decided I wanna get into computer science, but I got real bad grades in high school in math and tested like crap for my college placements 10 years ago. However, I watched a bunch of YT vids to get me thru algebra 1 and was able to skip it on my placement test. I took algebra 2 / trigonometry in June of last year (it was a 1 month long summer class that covered both courses). After that class ended, I got a dog and it ruined all my momentum. I haven’t done a single math since I finished that class, and now I’m getting into it. I thought I’d try spending a month to relearn trig, but I’m worried that all the alg2 stuff has been erased from my brain. The big ass trinomial factoring and huge log problems and all that. Is there like a fast paced review before taking precalc you guys know of?

[summer ruin]

isn't that pretty much what precalc is anyway

[severe pond]

I have no idea

[summer ruin]

it's most likely will be easier if you have solid foundations than anything else

factoring, logs, trig aren't going anywhere in calc or other classes after that point

so you better have a solid grasp on all of these

[severe pond]

I feel like when I spend time relearning these things, by the time I’m comfortable with it, I run out of storage room in my brain and I forget the thing I learned 2-3 topics ago

[summer ruin]

it's either because you're learning it for the first time or that you never truly learned it, but just tried to remember it somehow

you should be trying to understand the big picture of what you're doing while solving exercises, that way you can easier identify what exactly you were doing and how that's relevant to material you're learning

instead of focusing on specific nuances of particular exercise (e. g. "raise this to this power", "use this identity", "combine terms")

[shadow summit]

An easy solution is to find when cos(a)=0, then make x and a equal.

[wind linden]

do i multiply it by the angle or by 3/4

[willow skiff]

By cos theta

So multiply 12 * 10 by 3/4

[wind linden]

so its 90 then

[winter comet]

don't you multiply 12x10xcos(3/4)?

wait

im stupid

lol nvm

yea u can just multiply by 3/4

[willow skiff]

Yeah exactly

Yep

[willow bear]

jesus fuck that's some disgusting typesetting

[open plank]

lmfao

[leaden wind]

What happened to the 4

[stoic tapir]

got multiplied on both sides :p

[leaden wind]

Dang i forgot

Thanks bro

[dull tendon]

what exactly is the partial derivative of f(x+y)?

let's say w.r.t x keeping y constant

[willow bear]

is f itself a single-var function

[dull tendon]

no

[willow bear]

!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.

[dull tendon]

what exactly is f''(x-vt) in this?

is it the partial derivative? cause they've used it in two different contexts, one in which x is constant another in which t is constant

and then equated both

[willow bear]

f is a singlevar function

[dull tendon]

yeah messed that up sorry

[willow bear]

into which the expression x-vt is inserted

[dull tendon]

yes

but since x and t are independent, what does f'(x-vt) mean?

do we just take dx=dt while finding the derivative?

[summer ruin]

it's just a regular single variable derivative

[dull tendon]

oh so it's just df(x-vt)/d(x-vt) ?

but normally f'(x^2) is df(x^2)/dx

instead of df(x^2)/d(x^2)

[summer ruin]

this is ambiguous to write this way

f'(x^2) is whatever the derivative of f is, evaluated at x^2

[dull tendon]

that makes sense

[modest oasis]

whats the formula for iroc

[modest oasis]

instantaneous rate of change

[open plank]

u mean the definition of the derivative

or

[modest oasis]

i mean the fomula to calculate it

this is my question

[open plank]

differentiate sin x and plug in pi

google derivative of sin x

im sure you can figure it ut

[modest oasis]

the course isnt calc btw

i need to use methods from advanced

[open plank]

this is calculus though

[willow bear]

from advanced what, exactly

[modest oasis]

functions

[willow bear]

the "instantaneous rate of change" is the derivative in all but name

google the definition of the derivative. apply it here. you want sin'(pi)
or d/dx sin(x) at x=pi
or whatever other notation you can use to phrase that.

[modest oasis]

derivatives arent taught in advanced functions

https://cdn.discordapp.com/attachments/1205266950129713245/1213600372246183946/image.png?ex=65f6106c&is=65e39b6c&hm=8edf4a9bb7dd2a9fdcff9697e02c0069966bcf407dedf4b804ac948680197ac7&

this is the fomula they used

but i want a non specific one

[willow bear]

well you do know how to find the avg rate of change over an interval, yes?

[open plank]

oh lord

[fading monolith]

[modest oasis]

ik how to find average rate of change

[willow bear]

right

so do it on two x values that are both close to pi but not equal to each other.

pi itself and 3.15, say.

your picture suggests that your teacher is fine with a 0.01-wide interval as being "small".

so follow that.

[open plank]

no, go smaller.... 10^-777777777

[willow bear]

don't put jokes where op might interpret them as actual instructions.

[open plank]

sowy..

[willow bear]

if you want, you can push the other point closer to pi. like instead of x=3.15 you could go for x=3.142, or something.

[open plank]

but to be fair the smaller the better

[willow bear]

@modest oasis do you understand what i am suggesting you do?

[willow bear]

of course, but op's going to be using a calculator.

and that doesn't have infinite precision.

[modest oasis]

doesnt the first number have to be smaller than the number they speak about

[willow bear]

no, it does not.

it just has to be nearby.

[modest oasis]

like this

does the order not matter

[willow bear]

of course it doesn't

a/b = (-a)/(-b)

[fading monolith]

You are multiplying a -1 up and down

[quick pasture]

I am having trouble understanding rational functions, I am missing a lot of math before I enrolled for this class in my community college they have a support class with the regular class but its not enough time for me to understand it and the teacher is not very good at simplifying things down. I am watching a video and am confused to no end as to how the x intercept of (2x+1) is equal to negative 1/2 I feel very stupid right now and was doing ok until we got to this new section. I am terrible at factoring.

[willow bear]

@quick pasture do you know in general what the x-intercept of a curve is?

like geometrically

[quick pasture]

No i am missing too much math and honestly feel like i should drop. I took trig last semseter and did all right but we had the winter break that makes me feel like i just forgot everything.

I never got to take geometry either

[willow bear]

the x-intercept of a curve is a point where the curve meets the x-axis.

do you remember now

[quick pasture]

not really? I still dont understand how (2x+1) is equal to negative 1/2

That makes sense yes but I dont know beyond that

[river drift]

x = -1/2 is the value of x that makes the equation
2x+ 1 = 0
true

[quick pasture]

Sorry I still dont follow

I think i am missing too much

[willow bear]

ok let's take it one step at a time

cause you just tried to rush all the way through it

[quick pasture]

Im going at the pace the class is going

[willow bear]

well if you wanna let me explain your confusion then we're gonna have to slow down a bit.

ok?

[quick pasture]

ok

[willow bear]

the x-intercept(s) of a curve are the point(s) where that curve meets the x-axis.
are you good with this Y/N

[quick pasture]

yes

[willow bear]

the x-axis consists of precisely those points whose y-coordinate is 0.
i.e. a point on the x-axis looks like (__, 0)

are you good with this Y/N

[quick pasture]

no, how is y 0?

[willow bear]

do you know how the cartesian plane works?

and like how coordinates work

[quick pasture]

like i said i feel like i forgot everything from trig cuz of the break

[willow bear]

this is not about trig

[quick pasture]

thats the only version of it i vaguely remember

[willow bear]

alright, hold on, let me take a bit of a potshot here.

do you play minecraft?

[quick pasture]

nope

[willow bear]

damn, ok.

i was gonna try to reference minecraft's coordinates.

but that fell through.

ok, hold on.

<@&268886789983436800> troll

@quick pasture here's a point. can you tell me its coordinates?

(it's the red point)

[quick pasture]

(2,3)?

[willow bear]

ok great

here is another point, in blue. can you tell me that point's coordinates?

[quick pasture]

(-2,0)?

[willow bear]

right.

so you DO actually know how coordinates work, by the looks of it.

[quick pasture]

i have very low confidence in math from past trauma

[willow bear]

you could have said that earlier i think

but ok

the coordinates of a point tell you how far you need to go horizontally (x) and vertically (y) to get to that point from the origin.

for example (2, 3) means go 2 units to the right (horiz) and 3 units up (vert)

negative numbers mean going left or down, respectively.

for a point that's on the horizontal (x) axis, no vertical movement is needed, so the y-coordinate is 0.

similarly, for a point on the y-axis, the x-coordinate will be 0 instead.

does this make sense to you?

[quick pasture]

yes

[willow bear]

alright, cool.

coming back to this:

the x-axis consists of precisely those points whose y-coordinate is 0.
i.e. a point on the x-axis looks like (__, 0)

and this:

the x-intercept(s) of a curve are the point(s) where that curve meets the x-axis.

[quick pasture]

ok

[willow bear]

to find the x-intercepts of a curve, you take the curve's equation and set y = 0 in it.

are you good with this y/n

[quick pasture]

no, i didnt see a y in the equation so that throws me off

[willow bear]

because you've dropped it.

the way i understood it, you were talking about the line y = 2x + 1.

but also, i was still saying general things.

not for your problem specifically.

but for any equation where you might want to find an axis intercept.

[quick pasture]

f(x) confuses the hell out of me, i get easily lost when i see that instead of it just saying y

[willow bear]

for our purposes, these are kind of interchangeable right now.

but also like

again

im saying general shit

i want you to think in general

[quick pasture]

general is good it will help me better understand the concept

[willow bear]

tell me if in general the idea of "find x-ints <=> set y = 0" makes sense to you

[quick pasture]

the video was being too specific

[willow bear]

what video?

[quick pasture]

https://www.youtube.com/watch?v=GRZDhCdBwYY&ab_channel=Mathispower4u

[willow bear]

hm ok

right

so then. right. what they're doing has more moving parts.

but the idea of "find the intercepts on an axis <=> set the other coordinate to 0" still applies

and then you have the single-var equation $$\frac{(2x+1)(x-5)}{(2x-1)(x+3)} = 0$$ that they go on to solve.

[quick pasture]

not sure i know what those notations mean is that greater then or less ?

[obsidian monolith]

Ann

[willow bear]

what notations?

<=>?

[quick pasture]

yes

[willow bear]

no, that's just a double-headed arrow that means "is equivalent to"

[quick pasture]

oh sorry im bad at notations too

[willow bear]

well it's a good thing you asked instead of just letting that non-understanding sit unaddressed.

anyway right

so yeah as i was saying there's a lot of different steps

once we have that equation in x, we (temporarily) forget anything about graphs and just focus on it as an equation

which they go on to transform into (2x+1)(x-5) = 0

and then from that, they apply the zero product property (or null factor property, or null factor law, or however else it's known at your school -- it goes by a few names, iirc)

i.e. they go from (2x+1)(x-5) = 0 to [2x+1 = 0 OR x-5 = 0]

is this familiar to you

[quick pasture]

no

[willow bear]

have you solved quadratic equations by factoring before?

[quick pasture]

can i get an example?

[willow bear]

solve this equation: $(x-4)(x+19) = 0$

[obsidian monolith]

Ann

[willow bear]

it is already in factored form and i do NOT want you to expand it

[quick pasture]

I think im wrong but here it is x^2+15x-76=0?

[willow bear]

and i do NOT want you to expand it

[quick pasture]

yea i dont know what that means

[willow bear]

you did exactly the thing i didn't want you to do

:p

ok right hm

i have a call to attend rn, so unfortunately i can't continue with this impromptu refresher section

ig i could like, refer you to khanacademy's videos

for quadratic equations, among other things

[split pelican]

i could try, i suppose

although i dont have much experience helping

[quick pasture]

its ok, like i said i feel like im missing too much

so maybe i should drop it

i appreciate the help

[spring tiger]

@quick pasture do you know what an equation’s zeroes are?

[quick pasture]

No

Im backtracking right now

[spring tiger]

That’s alr

[spring tiger]

This equation is equal to zero yeah?

[quick pasture]

so it says

[spring tiger]

Yep

How do you multiply two things and get zero?

One of the things you multiply together needs to be zero

[quick pasture]

so x is 0?

[spring tiger]

So in that equation, either (x-4)= 0 or (x+19)=0

Because when you multiply them together, you get zero

[quick pasture]

so just -76?

not sure im following

[spring tiger]

Do you understand that you are multiplying (x-4) by (x+19)?

[quick pasture]

-15?

errr 15?

[spring tiger]

Nah hang on

[spring tiger]

Do you get this part

[quick pasture]

yeah im sorry im super behind, i never made it past alg 1 in hs cuz i was working full time

[spring tiger]

That’s ok

[quick pasture]

I dont really get it

[spring tiger]

Ok. In the equation Ann sent before, (x-4) multiplied by (x+19) equals zero

And we don’t know what x is

Since when you put two brackets together, it just means you’re multiplying them

[quick pasture]

yea i got pemdas

i just dont get this

and i struggled thru the first section of the precalc class i am in too, CA doesnt let you take anything below precalc or trig in CC anymore.

so im kinda screwed in the class without the foundations even with the support classes since i work full time

I appreciate the help i just dont wanna frustrate anyone

[spring tiger]

Nah it’s alright

[split pelican]

yeah its chill

maybe try substituting in a value for x and seeing what happens

[spring tiger]

It’s important to understand the basics because maths just builds on previous maths

[split pelican]

that might help you get a better grasp

[quick pasture]

(x-4) is x just 4?

i dont really understand here

[split pelican]

x = 4 is a solution to the equation yes

[spring tiger]

Yeah your onto something there

[split pelican]

since 4 - 4 = 0, and anything multiplied by 0 equals 0

[quick pasture]

so is (x+19) is x -19?

[spring tiger]

Yeah 👍

[split pelican]

x = -19 is the other solution yes

[quick pasture]

I didnt know i could do 2 differnet values i thought x could only be one value for both

[spring tiger]

Since we know that the thing in the brackets equals zero

[spring tiger]

With this type of equation it can be two

[split pelican]

x can only be one value, but the equation can have multiple values of x that work

[spring tiger]

Yea

[split pelican]

to get back to factoring, factoring is the process of taking a quadratic in standard form (ax^2 + bx + c = 0) and turning it into a form like that, to make it easier to solve

its pretty tricky and to be honest im not very good at it either

[spring tiger]

I know how to factor most stuff and I’m in the process of learning how to factor polynomials but that’s beyond this

But yea we don’t need to worry abt it rn

[quick pasture]

my precalc class is rushing thru everything rn, the teacher tried to cover 3 sections i 2 hours

[spring tiger]

That’s not helpful

[split pelican]

that kinda thing is why im glad to be self taught lol

[quick pasture]

and she wont slow down

cuz she has to get thru the material

[spring tiger]

My teacher is kinda like that too so you gotta spend some time out of class making sure you understand

It’s annoying

[quick pasture]

Yeah i work full time 10 hour shifts i dnt have the energy for much after that

[spring tiger]

Damn

How old are you?

[quick pasture]

35

[spring tiger]

Ok

Yea I’m not up to that quite yet

[split pelican]

same

[spring tiger]

Anyway back to maths

So with the equation from before, x = 4 or -19

[split pelican]

yeah, factoring quadratics is generally just something you need to practice a lot and gain an intuition for

[spring tiger]

Both values of x will end up equaling zero

[quick pasture]

factoring is my weakest point

ok

that makes sense

[spring tiger]

Yea

[split pelican]

its a lot of peoples

[spring tiger]

Cool

[quick pasture]

so its (4-4)(-19+19)=0

without expanding

[split pelican]

not exactly

x can only be one value at a time

but either x = 4 or x = -19 will satisfy the equality

[quick pasture]

ok ann wanted me to solve it without expanding it, i didnt understand what they meant by not expanding it

[spring tiger]

Either works since one of the terms in the brackets will be equal to zero

[split pelican]

like if x = 4, (4 - 4)(4 + 19) = 0 * 23 = 0

[split pelican]

yeah, thats what im saying

but x cant be both 4 and -19 at the same time

it has to be one or the other, its just that no matter which you pick the equality will be satisfied

[split pelican]

and same thing for x = -19

[spring tiger]

Yea

[split pelican]

if you apply the distributive property it becomes x^2 + 15x - 76

she didnt want you to do that and then just use the quadratic formula or something

since you were learning factoring

[spring tiger]

@quick pasture with the expanded form (the one Aradia sent just then) it behaves the same

[quick pasture]

Im not sure what i did wrong then?

[spring tiger]

X can be 4 or -19 still

Where?

[quick pasture]

I gave the answer with the distributive property

[spring tiger]

Yeah but you didn’t solve to find the value of x

You just expanded it

[split pelican]

because thats not an answer

[quick pasture]

I wasnt sure how it was wrong since i dont know what expanding means i thought i shortend it

I didnt really know how to answer it

[spring tiger]

Expanding means to go from (x+a)(x+b) form to x^2+4x+8 type thing

[split pelican]

theoretically you could expand it and do all the plugging in of values with the quadratic formula but theres no reason to do that when you can find the solutions with some basic arithmetic since you already have the factored form

[spring tiger]

There was nothing wrong with what you sent, it just wasn’t what Ann was looking for

You went from factored to expanded form, Ann wanted you to find the value of x

[quick pasture]

ohhhhh

i used the wrong tool

I thought it was more complex then it was

[spring tiger]

Ok

[quick pasture]

I didnt see the question asking to find the value of x, it was just an equation with wording i wasnt familiar with. thats my bad

[spring tiger]

Ok that’s cool

Do you know how to go from expanded to factored form?

Basically to undo what you did before

[quick pasture]

no?

or maybe?

[split pelican]

let me see if i can find a basic example

[quick pasture]

thank you for the help

I truly appreciate it.

[spring tiger]

Actually it’s kinda complex and I gotta go for dinner rn

I’ll be back soon

[quick pasture]

no worries

[spring tiger]

No problem 👍

[split pelican]

alright let me see if i can word this in a way that makes sense

so like lets say we have (x + a)(x + b)

if we expand this, we get x^2 + xa + xb + ba

[quick pasture]

so what i did earlier basically?

[split pelican]

yes, but were gonna do it in reverse

lets take the quadratic x^2 - 9x + 8

[quick pasture]

ohhh hold on

[split pelican]

if we look at this, we notice the coefficent of b in the standard form of the quadratic ax^2 + bx + c, is a + b in our factored (x + a)(x + b)

and c is a * b

[spring tiger]

Damn you learnt this in a very different way to me

[split pelican]

so we need 2 numbers that add to -9, and multiply to 8

can you think of any?

[spring tiger]

Wait nvm thats how I do it

[quick pasture]

-1,-8

[split pelican]

correct!

so (x - 1)(x - 8) = x^2 - 9x + 8

you can double check that if you want but it is correct

so if we want to find the values of x that make x^2 - 9x + 8 = 0, we can just check the values that make (x - 1)(x - 8) = 0

which we can do in the same way we did that first one

[quick pasture]

got it

[split pelican]

it gets a bit more complicated when the x^2 coefficient doesnt equal 1, but its the same concept

[quick pasture]

yea i think thats where im stuck at, the teacher rushed thru Rational functions, inverse and radical functions and polynomial inequalities and we only have one class with her per week and i got 3 assignments due tomorrow at midnight then probably a test next week....

[split pelican]

sounds rough

[quick pasture]

yea i sent her an email telling her i might drop the class. I dont feel confident at all with the way shes doing things.

I appreciate the help though

[split pelican]

im not an expert by any means (im a 16yo so thats probably obvious) but maybe try self teaching? its worked out well for me so far

[quick pasture]

thats how i passed trig

had more time then tho

[split pelican]

what are you attending the class for exactly

[quick pasture]

Ill try to make more time thanks for the help tonite though

Prereqs for engineering degree

[split pelican]

ah

im not in uni yet so i dont really have any insight to give there

talk with an advisor maybe?

[quick pasture]

the advisors here suck lol

Thank you again for the help

[split pelican]

np

hope i was helpful

[quick pasture]

yes it was helpful ill see if i can unstuck myself with some extra videos

i gotta get up early for work tomrrow later

thank you for your patience as well

[vivid thistle]

32a

thanks 🙏

[spring tiger]

Yo I’m doing that too

[summer ruin]

integration by parts

[willow bear]

!status

[tender quest]

What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above

[spring tiger]

I’m loving the pre calculus in those questions

[quick pasture]

One last question so I now see the work for 2x+1=0 and the steps make sense to me but I dont see how it makes it equal 0 still. This is for rational functions, I am just trying to understand how the equation works when pluggin in -1/2 for x. I feel like im overthinking it and the steps make total sense but the answer just doesnt, im writing it out like 2*-1/2+1=0 and i just wind up with 0/2?

Oh wait

I see it now

nvm

So I got the correct answer here kinda, I had -4 and 2 instead of -2 and 4, I just wanna figure out where i got mixed up:

x^2-2x-8= 0

(x-4)&(x+2)=0

x-4=0{or} x+2=0

x = 4 or x=-2

[split pelican]

im not sure what you mean

did you get your signs switched up and factor it as (x + 4)(x - 2)?

[quick pasture]

I had (x-4) (x+2) when i wrote it out myself

errr hold on

lemme look at my notes again

Nvm i am second guessing myself here I think i mistakenly saw the subtract sign as -2x and i was thinking of what adds up to negative 2 and multiplies to negative 8.

[split pelican]

that is what youre looking for

[quick pasture]

i do this a lot im not sure why i mix them up

ignore me, i will figure it out, your earlier help really cleared this up for me.

[split pelican]

im glad

[spring tiger]

@quick pasture have you done graphing?

[quick pasture]

yea

i have to force sleep now or ill regret it but thanks for the unblock Ango and Aradia and thank you as well Ann, you were all very helpful.

[proven void]

!volu

[tender quest]

Helpers are just people volunteering their time to help you. Be polite and patient.

[scenic galleon]

where can i get free resources like textbooks for self studying precalculus😭

[dark sorrel]

youtube is great, if you want textbooks it might be worth looking at some library websites to see if they have pdf versions

[vivid thistle]

isnt it precalculus

[vivid thistle]

yea no shit but i couldnt prove it

[summer ruin]

did you use integration by parts?

[vivid thistle]

yeah

u want steps?

[summer ruin]

yes

[summer ruin]

you should be applying integration by parts to the integral on the left

[vivid thistle]

its actually easier to solve it from the right

[summer ruin]

what you have here is 8 lines longer than doing integration by parts on e^x * e^(x^2)

[vivid thistle]

hmm

which one should i substitute then

[summer ruin]

well you already have a hint in the right hand side

you have e^x times the derivative of e^(x^2)

[vivid thistle]

oh alright thanks

[vivid thistle]

it doesnt work

[summer ruin]

$\int_a^b u\ \dd v = vu\vert_a^b - \int_a^b v\ \dd u$, choose $u = e^{x^2}, \dd v = e^x \dd x$

[obsidian monolith]

Transparent Elemental

[open plank]

is this precalculus

goes apeshit

[stable gull]

im a cat

can sm1 explain this i'm having trouble with my classes new unit

1. Aproximate the area under the curve

f(x)=x^2+2, -2 ≤ x ≤ 1

[fading monolith]

That is taking the integral of f(x) from -2 to 1

Or need a numerical aproximation using squares?

[stable gull]

numerical aprox

[split pelican]

do you know algebra

you need a very strong grasp on algebra and ideally trig and some geometry to do calc

[willow bear]

are you fluent in algebra?

if you aren't, calculus is going to be full of pain and suffering.

ok then good luck

[split pelican]

get a textbook or use khan academy or something

[willow bear]

khanacademy's probably a good start.

[spring tiger]

Yo fonzeysifu is back

[quick pasture]

Im tryin to understand polynomial function i follow all of it until solving for the least possible degree about where the function has at most (n-1) turns, I dont understand how if something has 3 turns it equals 4?

[split pelican]

turns?

[spring tiger]

What’s the equation

[quick pasture]

finding the least possible degree for graphed polynomials

Theres no equation its graphed

[split pelican]

oh i see what you mean

[quick pasture]

how does 4-1 equal 5?

That is drving me insane

[split pelican]

well like think of a parabola

it only has one turn, but it is a graph of a degree 2 polynomial

i dont have much experience graphing higher degree polynomials but that might give you some insight

[quick pasture]

not really sorry

[spring tiger]

I can help you a bit

[quick pasture]

Im stuck on how subtracting turns into addition here

[spring tiger]

It’s like a parabola

(x-4) means positive four

[quick pasture]

none of these videos mentioned that hidden rule...

[spring tiger]

Wait I’m confused

What is being subtracted

I can do the question but with a different method which would be good to know

[quick pasture]

https://www.youtube.com/watch?v=LU1nLawYyH0&t=65s&ab_channel=Mathispower4u

[spring tiger]

Wait are you talking about how two turns means cubic

[quick pasture]

I dont understand how subtracting becomes addition here or why

[spring tiger]

What is the subtraction

[quick pasture]

The function has at most (n-1) turns

how does 4 turns minus 1 equal 5?

[spring tiger]

Yea so a cubic (degree 3) has at most 2 turns

[quick pasture]

my brain is saying no

[spring tiger]

The degree is five, so n-1=4

[quick pasture]

is there some general rule of thumb here im not seeing?

[spring tiger]

Probably

Are you confused with how the line with four turns has a degree of five

[quick pasture]

yes

[spring tiger]

Ok cool

[quick pasture]

it never touches 5 so i dont see how it got there

[spring tiger]

Let’s start simpler: a parabola is a polynomial with degree 2 right?

It has one turn

A cubic has a maximum of two turns

A quintic (degree 5) has a max of four turns

They can have less tho

[quick pasture]

ok i think i need to go back one more section i dont remember what they look like

[willow bear]

wait this is in a PRECALC context?

boy is that just a blackbox

[quick pasture]

I mean yes? this was one of the sctions , power functions and polynomial functions.

I get power functions

[willow bear]

ok right so like

hm

how should i phrase this

are you familiar with the following property:

a polynomial has at most as many roots as its degree

or to word it differently, a polynomial of degree n has ≤n roots

[quick pasture]

you mean turns?

[willow bear]

no, i mean roots.

talking about a different but related thing here.

[quick pasture]

no i dont follow

[willow bear]

a quadratic polynomial (degree 2) has at most 2 roots
a cubic polynomial (degree 3) has at most 3 roots
a quartic polynomial (degree 4) as at most 4 roots

and so on

[quick pasture]

ok yeah that makes sense

[willow bear]

ok right

[quick pasture]

so for example like 8x^3 is cubic and only has 3 roots am i right or no?

[willow bear]

so as far as i understand, you were confused why the max possible number of turning points is n-1 ?

[junior sequoia]

Need help here...

Can anyone help ...

[willow bear]

depends on if you count them with multiplicity or not -- 8x^3 has only one root (x=0), but it is a triple root

[split pelican]

we dont know if we can help if you dont tell us your problem

[willow bear]

you might want to actually post your question, otherwise we can't

[junior sequoia]

Can anyone else confirm the answer

[willow bear]

,rccw

[willow bear]

i, for one, do not see an answer to confirm. you haven't written one.

[quick pasture]

Err let me try again x^2+4x^3-5 is cubic root? I am trying to see if i actually am seeing this right or off by a lot

[junior sequoia]

My answer is A ...

[quick pasture]

just throwing out random polynomial to try to understand if that is cubic?

[willow bear]

4x^3 + x^2 - 5 is a cubic, yes.

[quick pasture]

ok

[willow bear]

as it happens, it has only one root.

[junior sequoia]

Why is it wrong?

[quick pasture]

wait... i think im not understanding what root means then.

[split pelican]

im actually confused on why thats wrong as well

[willow bear]

lemme make a picture

[quick pasture]

you can help themm first im not in a rush im doing practice work to build understanding

[willow bear]

@junior sequoia @split pelican

and what am i looking at, exactly

i think you've got it backwards?

[junior sequoia]

Red ones are the original and green are the new origin and translated point

[willow bear]

(4,5) are the OLD coords of P and we're asked to find the NEW coordinates of the same P
while you are taking some other point whose NEW coordinates are (4,5) and finding its OLD coordinates to be (2,8)

[willow bear]

the point doesn't go anywhere

the coordinate system does

but P stays where it is

[split pelican]

ooooh

that makes more sense

[junior sequoia]

Oooh, got it... So the P wasn't translated then...

[split pelican]

also if its not too complicated, how would you prove this?

[willow bear]

factorization

each root gives rise to a linear (or higher) factor

[junior sequoia]

Thanks @willow bear

[split pelican]

ooooh that makes sense

yeah that makes a lot of sense thanks

[spring tiger]

Ight I’m down to explain another important thing abt polynomials

Probably in another channel tho

[quick pasture]

whch one?

[spring tiger]

Idk

Just dm

[split pelican]

#prealg-and-algebra perhaps?

[spring tiger]

Yea sure

Ok @quick pasture

[quick pasture]

ok

[fading monolith]

Fundamental theorem of algebra

[quick pasture]

this is what is confusing me rn

[fading monolith]

How many times the graph crosses 0X axis?

[quick pasture]

2

[fading monolith]

So at least the degree is 2, because you have 2 roots

[quick pasture]

yea that i follow

[spring tiger]

The line doesn’t have to pass through the x axis but yea that’s right

[quick pasture]

i dont follow how this would be 3 though if i were to take a guess based on how the answers come out

oh no this is just one turn hold on

[spring tiger]

Yea

[fading monolith]

Yeah, but at least your polynomial have 2 roots and we dont know about the complex ones

[quick pasture]

im guessing 2 but thas wrong

[spring tiger]

How many turns does it have?

[quick pasture]

3 turns

[spring tiger]

So?

[quick pasture]

the answer shown here says 4

[spring tiger]

Yea

[quick pasture]

not 3

[spring tiger]

Degree 4

[quick pasture]

i dont get how its 4 when theres only 3 turns

[spring tiger]

Remember the parabola?

One turn, degree 2

[quick pasture]

is it always just one more degree per turn?

[spring tiger]

Because degree one has 0 turns, since it’s a straight line

[quick pasture]

is that the rule im missing?

[spring tiger]

If that helps you understand

[quick pasture]

Yes that does help a lot more

[spring tiger]

Pretty much

For now, yes

[quick pasture]

I got really confused by all the other explanations

when it was that simple

[spring tiger]

It’s not always quite that easy but it doesn’t get much harder

[quick pasture]

ok thank you for that tip, i was going crazy

[spring tiger]

Yea that’s a good way to understand it

[quick pasture]

long winded explanations lose me quick

[spring tiger]

Same

[quick pasture]

Is f the x and n the y axis its referring to?

[spring tiger]

F just means function

So F(n) just means you do the stuff on the right hand side to n

Imma be honest I got no idea what an f intercept is

Apparently it’s not taught where I live

[quick pasture]

so when im graphing it i look at the intercepts on the right side of the graph?

its ok no worries

[spring tiger]

You graph it as normal

So it intercepts the x axis at -3, 5 and 4

I’m assuming the f intercept is the y intercept

And the n intercepts are x intercepts

So the n intercepts are -3, 4,5

[quick pasture]

This is how desmos graphed it

[spring tiger]

And the f intercept is what you get when you sub n=0 into the equation

[spring tiger]

Yea notice how the line cuts the x axis at the coordinates in the function? Just like a parabola

[quick pasture]

yea

[spring tiger]

That’s bc it’s the same thing

Just with three instead of two

[quick pasture]

so how do i get the f intercept?

[spring tiger]

It’s 60 right?

[quick pasture]

yea when i scroll out its basically 60

[spring tiger]

Ok

[quick pasture]

im just trying to understand equation wise how i would get 60

[spring tiger]

Well the y intercept (or f intercept in this case) is just where x =0

[quick pasture]

dont wanna rely on desmos entirely

[spring tiger]

So sub x=0 into the equation

You get y=-5x3x-4

Which is 60

[quick pasture]

thats -57?

oh wait *

[spring tiger]

Sorry the asterisk made it italic and disappeared lol

[quick pasture]

sorry didnt see your edit

oh ok got it

i was mixing up my pemdas a little

overthinking it

[spring tiger]

Pretty simple actually

[quick pasture]

i dont know why letter variables throw me off so hard

i cant stop seeing them

[spring tiger]

Just substitute a value for x

The y intercept is always at x=0

[quick pasture]

right

[spring tiger]

So when calculating it you just do that

[quick pasture]

gotcha

thank you

very helpful again

[spring tiger]

no problem

[quick pasture]

i feel confident with power and polynomial functions now

[spring tiger]

Great

Now do some ig

[quick pasture]

ig?

[spring tiger]

I guess

[quick pasture]

oh sorry

acronyms

[spring tiger]

Yep

Even I don’t know most of them and I’m a teen

[quick pasture]

alright off to bed, got work tomorrow morn but will be off next two days after so will prob be back with other sections i was backtracking on. thanks for the help again Ango.

[spring tiger]

No worries

I’m going to bed too

What country are you in? We are on same time

[quick pasture]

US west coast

[spring tiger]

Damn how late is it bro

[quick pasture]

145am

[spring tiger]

Wow ok

We are not on the same time then

[quick pasture]

lol

[spring tiger]

It’s 8:45 pm here

[quick pasture]

lucky

well gn

[spring tiger]

Gn

[crisp bluff]

Is there any value in learning how to use the definition of a derivative to find the derivative of a function outside just passing an exam? It's kind of a long 4 step process, at least the way my teacher taught it. Can you know the definition of a derivative without knowing that method? Is it useful?

it is when you set the function with lim h->0 and then use some equation with f(x-h) and it seems needlessly complicated and annoying to remember

tag me if you respond

[fading monolith]

Of course there exist rules for derivatives of elementary functions but its always necessary know the formal definiton with the limit. Its useful for lateral limits, compute some limits, during proofs where you dont know the form of the function, etc

[fading monolith]

Its very easy to remember

[crisp bluff]

So for more complex functions later on I will need to know this?

[fading monolith]

Not for more complex functions but necessary to check derivability and differentiability

Its always good to remember the definiton, trust me

[crisp bluff]

ok thank you I will learn it

[summer ruin]

it is very useful in more complicated cases, but there are no such cases in calculus 1, 2 or 3

(aside from synthetic examples that use some poorly behaved function to illustrate something about derivatives)

I'm talking about being useful for well behaved functions

[modest oasis]

why is this A i said it is C

what i did was factor out pi/2 and i got -pi than since d is opposite i said it was pi to the right which is 0.5 rad

[summer ruin]

the question is poorly formulated

it's not clear what is meant by $\cos\frac{\pi}{2}x - 0.5,$ is it $\cos \left( \frac{\pi}{2}x\right) - 0.5$ or $\cos \left( \frac{\pi}{2}x - 0.5 \right)$?

[obsidian monolith]

Transparent Elemental

[modest oasis]

this question makes pi/2 look like k

and to get the phase shift u need to factor out k

so the last option is what i think is meant by the question