precalculus

[summer ruin]

piecewise doesn't imply the limit doesn't exist

absolute value is the simplest interesting piecewise function

[viscid thistle]

good point

[summer ruin]

but also f(x) = e^(-1/x) for x>0 and 0 for x<=0 has a limit at 0 which is zero

in fact it's even infinitely differentiable at 0

[viscid thistle]

Wait could you explain this

thats kind of interesting

[summer ruin]

well you could show that k-th derivative is zero at 0 using the definition of derivative and l'hopital

it's true for any k so it's infinitely differentiable

the left limit is always zero, so might as well only consider x->0+

[viscid thistle]

This graph right?

im failing to see it

[summer ruin]

you didn't define it piecewise

you defined e^(-1/x) for all x, not for x>0

[viscid thistle]

ohh

[summer ruin]

it's an interesting function because even though it's infinitely differentiable at 0, it's taylor series is an infinite series of zeros so it's zero, meaning it doesn't converge to the true function at all

[viscid thistle]

oh wow thats interesting

[wanton robin]

I've tried finding a good method to solve this in, but I can't seem to.. without a calculator. How would I solve this?

[summer ruin]

what is x+x+x equal to?

[wanton robin]

3x

[summer ruin]

that should answer the question

[wanton robin]

9^15?

[summer ruin]

that's not how multiplication works out in this case

[wanton robin]

Well

My only other option is getting 9^45

[summer ruin]

what does 3^(15) even mean

[wanton robin]

Without calculator, I just know it's a very big number

But it's 3 times 3, 15 times

[summer ruin]

what does the operation ^(15) mean

yes

[summer ruin]

now if you multiply that by 3

[wanton robin]

Oh

[spice yoke]

it means do 15x 3*3

so:

[wanton robin]

I get 45

So 3^45

[summer ruin]

think about it a bit more

[wanton robin]

:S

I mean, if it's 3.. 15 times, happening 3 times, wouldn't it be 45 then?

3 times 3 happens 45 times

[summer ruin]

no

[wanton robin]

I am not sure I follow then haha

[summer ruin]

3^(15) = 3 x 3 x 3 x 3 x ... x 3

multiply that by 3 again

[wanton robin]

Once? Then we get 3^16, right?

[summer ruin]

yes

[wanton robin]

I think I understand, it's multiplication and I am dealing with addition in this case

But I don't see how I solve it

[summer ruin]

you already solved it

you can just start reading this convo from the beginning and you write the answer and the process yourself

[wanton robin]

Took me a bit, but I finally understand..

3^16

Literally, it was just one more 🥴

Thank you, Elemental!

[woeful grove]

yo someone wanna teach me pre calc

[slim steppe]

You have resources online to help you, Khan Academy, Albert.io, Organic Chemistry, etc...

[azure dove]

could i have help with domain and range interval notation?

[opal tree]

Post a specific question

[north remnant]

Finding the limits…

Question 11:
Find the limit.. X approaches (7/2) from the right Int(2x-1).

Looking at question 11 in the image, how are you suppose to know to let x=(7/2)+h where h is in (0,(1/2)?

[north remnant]

Forgot to mention, without graphing

[primal quail]

What exactly is int here?

[north remnant]

The natural integer

As X approaches (7/2) from the right. The Natural integer will be 5.

I’m just not understanding why the answer key, the visual of question 11, is allowing x=(7/2)+h where h is in (0,(1/2)).

[brittle mauve]

https://math.stackexchange.com/q/4651552/1156125
im sorry for the ping no one is answering me from stackexchange if you have answer please ping me

[tardy thicket]

how would I graph this

what I got so far

[uncut mulch]

there isn't a unique solution
there are multiple things that you can draw
try make a full attempt

[tardy thicket]

what I got so far

the one I underline does it mean it doesn’t exist?

why they both not equal each other

I need 7 approaching both sides but they don’t equal to each other

[north remnant]

Is this some form of quiz/exam?

[tardy thicket]

Noo

He

Hw

what I got so far

[viscid thistle]

Because the angle from eye level is given as 34 degrees, then the angle behind him (where i drew) should also be 34 degrees right?

[willow bear]

yes

[quiet spear]

Yes, you can see this by noticing that the two triangles share the upper left angle so that one is the same, and the bottom left angle is a right angle for both so they are also the same. Because the sum of the angles in a triangle is 180°, the other angles must also be the same.

[wanton ocean]

Taco

[slim owl]

What are the subjects cutoffs for where trig ends, precalc starts/ends, where algebra ends (before linear)

[viscid thistle]

Thank you!!

[north remnant]

Question 44.

How would you be able to distinguish when to move -(1/3)pi(a)^2H in front of lim as h approaches 0?

Also where did the “-“ go?

[hushed sphinx]

A general strategy is to move all common factors that don't depend on the variable out of the limit.

[north remnant]

Ah I see, that makes much more sense. Thank you very much!

[lilac current]

@wind apex twizzy
#1939

[half geyser]

does anyone has the exersises for difference quotient pls i learn it online and dont have any kind of textbook

it helps a lot if anyone can send

pls ping me if you send

[thorny geyser]

1/x

[barren tulip]

x^2

[viscid marsh]

Try typing difference quotient mcq

on google or your web browser.

[slender ice]

So am i crazy or is this incorrect?

It says the inverse of h(x)=4-x is h(x)^2=4-x

my head hurts quite a bit atm so I'm hoping someone can sanity check me

[proven siren]

they're correct

so y = 4 - x

the inverse means that you reflect on y = x

therefeore

x = 4 - y

y = 4 - x

[slender ice]

yeah I was just loosing my mind for a moment thanks

[fringe sable]

determine the quadratic equation given the following characteristics:
minimum value of -16 that has x intercepts of -8 and 4

answer is y = 4/9*(x+2)^2-16

how do i get that answer?

[primal quail]

With roots -8 and 4, you have the quadratic equation C(x+8)(x-4) for some C constant, then expanding you get, Cx^2+4Cx-32C , the vertex of an quadratic is just f(-b/2a) , then subbing in -4C/2(C)= -2, you have C(-2+8)(-2-4)= -36C which is -16
Thus -36C=-16 => C= 4/9 => the function is precisely (4/9)(x+8)(x-4)

[keen nexus]

hey guys is this legal
$-2 < \frac{1}{x} < -1.99 \Leftrightarrow 1.99 < \frac{1}{x} < 2$ ?

[obsidian monolith]

fluffy snail

[solar olive]

u need to change the <'s to >

actually

yeah

thats right

[keen nexus]

okay nice!
I don't have to change $\frac{1}{x}$ to $-\frac{1}{x}$ ?

[obsidian monolith]

fluffy snail

[solar olive]

actually yeah u do

[keen nexus]

thank you, was really unsure

[chilly hazel]

if ln(x)' = 1/x
then why not ln(ln(x))' = x?

you are taking reciprocal of the reciprocal right?

[willow bear]

the derivative of a composition is not the composition of the derivatives

look up chain rule

[hot reef]

Hi!
Can anyone provide me the simplest description of reflexive, symmetric and transitive relations?

[pearl agate]

is everyone perfectly happy with the notation for logarithms? or do you find the notation overly confusing and would prefer to see them changed to be less confusing? from what I have observed, confusion seems to be a common issue for those new to learning logarithms

[wide ibex]

what’s confusing? Only that “log x” without specific base can mean many things, but if you specify the base everything is easy.

[floral ore]

Doesn't log x mean log to the base 10 if no base is specified?

[wide ibex]

It depends on the convention used normally it’s 10, e or 2

[floral ore]

2

[wide ibex]

computer science

[floral ore]

lol

I've mostly seen log used with base 10, ln with base e

[wide ibex]

I would just specify the base

[floral ore]

yeah, probably convenient that way

[thorn creek]

log base 2 is normally written as "lg"

[floral ore]

well there we go

life is simple again

[pearl agate]

I don’t know why my professor says this, like “log” without base can mean anything for base and you want to check with the author. Its just confusing for new students. Log without base always means log base 10. ln always means base e. And lg always means base 2.

Imagine a new student learning about logarithms for the first time, being told from one person log without base means base 10. Then the next day someone else says log without base means base 2… How are you supposed to know without emailing the author of a textbook to find out from them directly?

But it’s not just this it’s John Napier’s notation.. one of my professors stated some in the math community are not a fan of it

[floral ore]

fair

[thorn creek]

mathematicians don't make much use of or care much about log base 10, logs base 10 are the province of those filthy engineers that nobody really caers about

[pearl agate]

Math = ln
Engineering =log
Computer science=lg

[thorn creek]

log base e is all over math, and log base 2 shows up a lot in discrete math, computer science, and information theory

[floral ore]

log base 10 is there to give you an ego boost in log problems I assume?

[pearl agate]

But it sounds like computer science textbooks can sometimes writes lg as log, which I think is incorrect.

[thorn creek]

i've lterally never seen that

[pearl agate]

It’s always lg?

[thorn creek]

every textbook i've encountered uses lg or log_2 when a base isrelevant

[pearl agate]

Well that’s what one of my professors said I don’t think she’s basing that on anything

[thorn creek]

and in O-notations, lg(x) and ln(x) are the same

[pearl agate]

That’s good to know

[thorn creek]

since they literally only differ by a constant, and constants don't affect asymptotic equivalence

[pearl agate]

So it’s just confusing to hear this and it’s an incorrect assumption being made from teachers to students

[thorn creek]

sadly, teachers are often wrong

the confusion is between mathematicians on one side, and engineers (and especially chemists) on the other

engineers and chemists use log base 10 by default

computer science isn't a problem here, computer scientists are firmly in the natural log by default camp in my experience

enginers use log base 10 because it plays better with their slide rules and such not

humans are much better at dividing numbers expressed in decimal notation by 10 than they are at dividing by e

from what i've seen most mathematicians simply write ln for natural log, lg for log 2 (when needed) and log_b for any other base. the people who are the problem are the bloody engineers, who insist that they can write whatever the fuck they want and expect everyone else to just understand, and who thus use log with no indicated base for base 10.

of course, since 95% of students in intro math classes will not go on to be mathemticians, teachers of intro math classes often feel compelled to teach the engineering practice (which far more of their students will go on to become) as if it made any sense at all

in other words: don't blame the mathematicians

[hushed sphinx]

I don't think I've actually encountered "lg" in a computer science context. My experience it that is is generally one of

• "log_2" in serious publications.
• "log", and who cares about the base anyway for asymptotics purposes.
• we're among friends here, so "log".

[thorn creek]

i'm sure i have

but then again my formal study of CS was in the late 1980s and early 1990s

[hushed sphinx]

I think the asymptotic growth of log*(n) actually depends on which base, but it is so slow anyway that nobody cares a lot.

[thorn creek]

oh iterated log? maybe

[hushed sphinx]

Yes.

[thorn creek]

wikipedia says that log* is base e and lg* is base 2

take what you will from thta

lg* grows "faster" than log*, but, as you note, the difference is so small that nobody cares

since lg* < 5 for any practical n, and so log* would also be < 5

[violet bobcat]

x + 2cosx = 0, how do i solve for x

[hushed sphinx]

Numerically.

[thorn creek]

yeah, there's no exact analytic solution to that

although if you introduce a constant for the fixed point of cosine (that is, the unique number ζ such that cos ζ = ζ), you can express the solution in terms of elementary operations along with that constant, using double-angle identities

[hushed sphinx]

Interesting. Is that a special property of the multiplier 2?

[solar olive]

theres actually a name for that constant, but i forgor

[hollow elbow]

out of curiosity, would it be possible to solve for soley x in the function y = xsin(x^2) + 1. I couldnt really find an answer for it, but I was wondering if anyone else might be able to, or at least lead me in the direction?

I hate to keep asking the same question across different channels, but the only answer I've been provided other than it not necessarily being easily possible was x = sqrt(eW[(1-y)/e]), however I can't seem to find how that would be correct?

[proven siren]

where W is the lambert function?

you cant do it without making a new type of function like W

[hollow elbow]

yea, it was the lambert W function they used

[hollow elbow]

The issue is I just have no idea which function would necessarily be the best function to utilize in this case

[proven siren]

what grade are you in

W is like graduate level math

its not like you can just use y^2

[hollow elbow]

The question has nothing to do with what grade I'm in, however I'm taking calc 1. it was just some personal practice I was trying to do then I ended up giving myself a poor function to work with , then my curiosity got the best of me when I was trying to figure out how one would work it out

[proven siren]

the reality is that when you do math problems on your own, you often cant solve them

math problems in class are rigged to be solvable

without using W you cant

[hollow elbow]

yeah thats generally how it is unfortunately. Well, if anyone wants to try it for themselves and they happen to find an answer, I'd love to know. Also, why necessarily "W"? Couldn't there possibly be a better function to use?

[proven siren]

i dont even know what W is

you could approximate x = sin(x) at small values of x

oh actually

i found something

use the taylor series

which is calc 2

and you get a series for sin(x^2)

so you basically have y as a sum of infinite terms

[hollow elbow]

Ok, I'll try to see how that works out

I'm familiar with the taylor series

enough to understand them

[thorn creek]

no, you can do it for any rational multiple

x=sin(x) has exactly one solution, which can be found by inspection

[dark niche]

@surreal berry
(lg(f))^2 = (lg(g))^2
try to change it to solve

i wonder what you will get

$\lg^2{f}=\lg^2{g}$

[obsidian monolith]

kisek

[dark niche]

where f and g - some functions

maybe even...

[surreal berry]

lg?

[dark niche]

$\lg^2{(x-5)(x+5)}=\lg^2{(x+2)5}$

[obsidian monolith]

kisek

[dark niche]

$\lg = log_10$

[surreal berry]

what is lg?

[dark niche]

oh shit dont look

$\lg = log_{10}$

[surreal berry]

just write log_{10} then

[obsidian monolith]

kisek

[surreal berry]

at least write log

[dark niche]

idk lg is common thing

[surreal berry]

lg is confusing

[surreal berry]

i dotn think so

[dark niche]

google thinks that lg is log_2
my math textbook says it is log_10

[surreal berry]

f = g

[dark niche]

[surreal berry]

different science and math fields have different conventions

in CS log_2 is more common, in math log_e is more common

there's no univeral log notation

that's why we have log_b

[dark niche]

[surreal berry]

also ksp 2

[dark niche]

is that your complete answer?

i just found two ways to solve it

im curious, which one you choose

$\lg^2{(x-5)(x+5)}=\lg^2{(x+2)5}$

[obsidian monolith]

kisek

[dark niche]

[thorn creek]

$\lg^2$ is ambiguous notation

[obsidian monolith]

rome of oxtrot

[opal tree]

Not really

The hell is lg

$\lg$

[obsidian monolith]

Umbraleviathan

[opal tree]

Lmao what kind of base is lg

[proven siren]

it just means log

ig cause laziness

and for some reason ^2 means base 2 lol

thats such a bad way though oml

oh no nvm im completely wrong

lg is base 10

wtf

[opal tree]

Why is the base a superscript

[proven siren]

its not

that was meant to be squared

lg^2 = (log base 10(x))^2

[hushed sphinx]

There are people/communities that use "lg" for base-10 logarithms, and other people/communities that use "lg" for base-2 logarithms.
As convenient as it would be if the notation had a single True meaning, that simply doesn't check out with actual usage.

[To make the confusion complete, Wolfram MathWorld cites a number theory paper from 2003 which claims that $\log_k(x)$ "as usual" means $\underbrace {\mathrm{ln(ln(\cdots ln}}_{k\text{ times}} (x)\cdots))$.]

[obsidian monolith]

Troposphere

[floral edge]

Hey anyone is able to help me by any chance?

i cannot post the question here not sure why but just dm me its regarding limits!

[proven siren]

You should be able to

[viscid thistle]

How would I do/approach something like this?

Show that the polynomial x^3 - x^2 - 1 cannot be factored using only integers.

I was thinking perhaps rational roots theorem?

[summer ruin]

by using continuity and intermediate value theorem

and monotonicity

[viscid thistle]

Isn't there a simpler way?

[summer ruin]

no

if you attempt to apply rational roots theorem you will fail

and it's pretty easy to realize given that neither 1 nor -1 are roots

[viscid thistle]

i am not even familar with intermediate value theorem lol

its to late here, off to bed

[opal tree]

That's what I thought lol

[carmine geyser]

is there a channel for conics

[jaunty fern]

Hi I’ve been having a bit of Trouble with this question and was wondering if anyone could help explain what to do

[carmine geyser]

Hi I’m wondering why b2-4ac identifies which conic is which

I saw this explanation on the internet and learned about some projectile geometry and the line of infinity points, but why in the end it became an equation of (x/y)

Does that have anything to deal with homogenous form

[summer ruin]

ac-(b/2)^2 is determinant of the matrix ((a,b/2), (b/2, c)) which defines the quadratic form, it's easy to figure out what shape it describes when it's diagonal and when it's not diagonal it's proven that it can be diagonalized

depending on the positive/negative definiteness of the form you get ellipse/hyperbola and in semi-definite cases you get things like parabola or a point or a couple of straight lines, etc

[fringe sable]

convert to vertex form: y = x^2 + 8x

i have gotten to
y = (x^2 + 8x + 16) -16

but i dont know how to get the answer y = (x+4)^2 - 16

can someone show me the all the steps to get from where i am to the answer please?

[proven siren]

This is called completing the square

Do you know what that is

Let me ask you this

Why did you add and subtract 16

That just makes it more complicated

[analog nova]

that's how you show your work

[hardy idol]

depending on the school of thought the add and subtract might be standard

but it can be confusing at the completing the square level

[analog nova]

the formula is just $(a+b)^2=a^2+2ab+b^2$

[obsidian monolith]

redoftwored

[analog nova]

$x$ is $a$ and $4$ is $b$

[obsidian monolith]

redoftwored

[analog nova]

thats why the middle term is 8x because 2 * 4 * x is 8x and you add 16 because 4^2

[proven siren]

I was asking for them to understand..

I know how it works

[analog nova]

lollll

[proven siren]

It’s necessary when you’re first learning it

[analog nova]

can any of yall answer a question regarding polar coordinates

[proven siren]

Ya

[analog nova]

how would you graph the y axis in polar form

im trying to do something in matlab but i cant just write x=0 cuz my system is already in polar

[hardy idol]

"theta = pi/2 or theta = 3pi/2"

still figuring out LaTex

[analog nova]

ohh LMAO

i forgot you can do that

[hardy idol]

yes and you would need both to be clear

[analog nova]

i was trying to do $x=rcos(\theta)$ for that

[obsidian monolith]

redoftwored

theREALyumdub

[viscid thistle]

I can use rational roots, it will show me possible roots are -1 and 1, then i let x = -1 or 1 and substitute, f(x) = -3 or -1, which is not zero. Therefore, it can't be factored with integer.

[jolly sandal]

Where did I make a mistake I dont get it...

[crisp raft]

I feel like it's correct...

[jolly sandal]

Ok. Because when you do u=sinx it gives different values but that doesnt matter I guess? It follows the same path as when you do u=sinx. But if the question was like "when does the integral cut the x-intercept" both answers would be correct?

Or are there any scenarios where that would matter is what I'm asking I guess

[crisp raft]

Yes if you do u=sinx it finally was a function with term sinx, but I guess that doesn't matter what the term is

Maybe function with term sinx is just more simple I guess

We could try to make the function we got with term cosx into a function with term sinx by using sinx2+cosx2=1 to see if they are the same I guess

[uncut mulch]

antiderivatives differ by a constant,

don't drop the + C
at the end

But if the question was like "when does the integral cut the x-intercept"
it wouldn't make sense for a question to ask that

also this belongs in #calculus or a claimed channel

[jolly sandal]

oh true im dumb

[steel stratus]

its like the discriminant of a quadratic

in the equation

becuz i think that decides if the roots are real or imaginary

and this decided te type of conic

[carmine geyser]

Yeah but

If so

[steel stratus]

and you complete the square on that equation to get standard form for some conic

[carmine geyser]

Then why would you identify a conic by deciding the number of its roots( intersections with x axis)

If it’s imaginary or real

[steel stratus]

no its based on the completuon of square

[viscid thistle]

God I hate conics

[steel stratus]

lol im learning them rn in shcool

[carmine geyser]

Chapter 10 Precalculus

They just have some theories that don’t give us proof and i really wanna know why

[steel stratus]

its math 2 for me🤣

u could try completing square on general equaiton

ive only done it when applicabel

[carmine geyser]

I saw an understandable explanation using homogenous coordinates

Does anyone know when this is gonna be taught, in college?

Homogenous coordinates

[turbid phoenix]

sm1 pls help

[viscid thistle]

@turbid phoenix

[turbid phoenix]

?

[proven siren]

!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'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above

[proven siren]

also you showed a picture of like 6 problems

pick ONE

[mortal bane]

i need help with this problem

[azure thorn]

First convert everything to the same base then you can use this log rule

wait I'm dumb

wrong rule

gimme a sec

You use this one

[mortal bane]

ok thanks

[viscid thistle]

How does the mans question even convert?

[azure thorn]

Like this I think

[short saddle]

Let $x\neq \frac{\pi}{2}+2k \pi ,k \in {\bf Z} $ .Find $A$ and $B$ ?
$$A=\sum_{k=0}^{n} \frac{\cos kx}{\cos^k x}$$$$B=\sum_{k=1}^{n} \frac{\sin kx}{\cos^k x}$$

[obsidian monolith]

aSome1gussy

[short saddle]

how would we use demoivre formula

[open badger]

2+2=4

[mortal bane]

answer is 3

solved it

[lavish dagger]

It's so easy

You can prove it easily by using mathematics induction.Ok?

[short saddle]

show me how

[lavish dagger]

um

moment

[short saddle]

thanks in advance

[lavish dagger]

okay?

[viscid thistle]

How’d you do it? The rule didn’t look like it applied to it

[mortal bane]

it did

ill send a photo

[maiden raptor]

guys I have a question

I just started calculus

and I want to know how $da/dx = x^2$

[obsidian monolith]

Sanaltsalt

[maiden raptor]

cause if I input A = 3.001 and A = 3

divided by dx which is 0.001

I get:

$3.001 - 3/0.001 = x^2$

[obsidian monolith]

Sanaltsalt

[maiden raptor]

x^2 here is 3^2

or this:

[summer ruin]

you need to better formulate your question and be consistent in your explanation and examples because right now it's not clear what you're asking and why the examples don't match what you say you want to show

[maiden raptor]

here he says that if we subtract one point in the graph A from another point and divide it by dx we get x^2

but that doesn't exactly work

I'm really new to calculus and basically just started so sorry if I kind of act stupid

[obsidian monolith]

Transparent_Elemental

[maiden raptor]

its a 3blue1brown video

[summer ruin]

the second formula in the upper right corner is correct though

[maiden raptor]

ic

can you maybe just go through that part of the video and explain it to me please

cause it's 5 years old

and nobody has commented about it

[summer ruin]

he wants to make an intuition, which is why he simplifies things, but that's not what the notation dA/dx means in reality

[maiden raptor]

ic

[maiden raptor]

but isn't dA the change in the area of the rectangle

and 3.001 - 3 is the change in the area

therefore dA/dx = x^2 applies here right?

ok sorry

if that's not what it means then can you please explain this expression

[summer ruin]

I don't even understand what he's trying to say in this video given that you aren't supposed to know integrals at this point, it just seems like a backwards way to look at derivative through the lens of an integral

[maiden raptor]

alright

[summer ruin]

maybe you should just watch the derivative video instead, given that there's a far better interpretation on the difference between derivative and differential which aren't the same thing and it would justify the approximate equality sign better

[maiden raptor]

ok so should I just forget about what he said there?

and da/dx = x^2 becomes a better and better approx as it approaches infinity though right?

[summer ruin]

I guess so, because even later in the video the formula dA/dx approx f(x) hides the fact that this kind of thing would be possible only if A was the integral of f(x) and not in other cases

moreover even writing dA/dx approx f(x) makes no sense in context of calculus course because we operate with derivatives as precisely known functions and not approximations

[maiden raptor]

and f(x) x dx = dA

which here is basically the length x breadth of the rectangle is the area

f(X) being the length and dx being the breadth

dA = area

[summer ruin]

none of this is true if A(x) is not an integral of f(x)

[maiden raptor]

but isn't it only true as x -> 0

[maiden raptor]

hmm

[maiden raptor]

@summer ruin

is this correct?

it is right?

[summer ruin]

maybe, maybe not

you can't say until you know what s(t) is

[obsidian monolith]

kappa

[shadow vector]

would complex numbers be discussed here?

[proven siren]

yes

the basic level though

[shadow vector]

Aight, cause i understand the idea of complex numbers, but no clue why they are the way they are

Like how does raising a number to i rotate it? How is that even something you can do?

Like integer and decimal exponents are simple enough since a decimal exponent is just a series of roots (took me way too long to figure that out), but how are you supposed to multiply a number by itself ‘i’ times?

[proven siren]

great question

[shadow vector]

I feel like people teach math too much by explaining that x does y, but not HOW

[proven siren]

I can explain in 10 min, busy rn

[shadow vector]

kk

Also is there a way to get an arbitrary root without having to guess?

like x^0.001

[proven siren]

okay so

we define the imaginary plane to have real numbers on the x axis and imaginary on the y axis

the reason multiplying by i rotates

is because when you multiply i by itself you get -1

and this negates the real component

so it ends up rotating by 90 degrees

if you multiply by i again (so i^3) you get -i

so that negates the imaginary component

so it rotates 90 degrees again

and so on..

[shadow vector]

ah

[silent smelt]

Sure, and that’s just frustrating

[shadow vector]

i’ve taught myself most of it which has helped me learn the why

[granite tinsel]

Not sure how to do part b

[opal tree]

First find m • n

Just the numerical answer

[granite tinsel]

So m is 9

I think i solved n to be 9+3 root 3

Using a formula for subtracting vectors that looks like the cosine rule

[opal tree]

Well wait I meant find m • (m-n)

But yeah I guess you could find n

Well

(m-n) • (m-n) = 36

So 9 - 2(n • m) + |n|^2 = 36

Oh it's actually not bad

[opal tree]

Sorry I meant find (m-n) • (m-n)

Or use that

It should work out fine

[opal tree]

From here

Because you can find |n|

Because $m \cdot (m-n) = 18\cos(\theta)$

[obsidian monolith]

Umbraleviathan

[opal tree]

Hm maybe you don't need to do the (m-n) • (m-n) route

[granite tinsel]

I’m not sure

It seems like there is an easier way but I can’t figure it out

[opal tree]

|m|^2 - (n • m) = 18 cos(t)

And then |m|^2 - (|n| • |m| cos(30)) = 18cos(t)

But we don't know |n|

[granite tinsel]

Idk if we can find it either

This guy sent me this formula to find n but idk if it’s right and I’ve never seen it before

[opal tree]

You can't square a vector

[granite tinsel]

Oh yeah

Idk then

[opal tree]

Well maybe you can find |n|

You should know that a vector dotted with itself is its magnitude squared.

So

(m-n) • (m-n) = 36

That also implies that:

|m|^2 + |n|^2 - 2(m•n) = 36

[granite tinsel]

It’s like the other formula then

[opal tree]

So:
9 + |n|^2 - 6(|n|cos(30)) = 36

Oh you can definitely find |n|

[granite tinsel]

Looks like algebra from here then

[opal tree]

Yeah

Well it's all algebra

[granite tinsel]

Cos30 is just 1/2 I think

[opal tree]

Yeah

[granite tinsel]

U would probably get a quadratic then?

And the positive would be the value we use

[opal tree]

Yeah

It should work

[granite tinsel]

So I think it’s

(3+3root13)2

[opal tree]

Yeah

Well I haven't done the work myself

[granite tinsel]

Ye im on my laptop so trying to see it in my head

[opal tree]

Try that for |n|

[granite tinsel]

K

Wait so I plug this into the scalar product formula

[opal tree]

Up here

It should work

[granite tinsel]

Don’t think so

I got 86.4 degrees unless I’m doing it wrong

The answer is 135.5

This questions actually unfair

It was only the last one on an AS paper

[viscid thistle]

I haven't learnt mathematical induction, and I don't know what the "simpler" way is that is referred to in the notes. Can someone help point me in the right direction.

[granite tinsel]

Are u just proving the sum of r=1/2n(n+1)

[uncut mulch]

gauss technique for arithmetic series

[solar olive]

norm(m - n) = 6
36 = norm(m)^2 - 2m.n + norm(n)^2
36 = 9 - 2(3 norm(n) cos(30)) + norm(n)^2
solve for norm(n) and then when u get norm(n),

cos(t) = m.(m - n) / (norm(m) norm(m - n))
cos(t) = (norm(m)^2 - m.n) /(3 * 6)

t should be the angle between m and m-n

in the end u should get cos(t) = (1 - 3sqrt(5))/8

since its not a special angle u can use ur calculator to do arccos((1 - 3sqrt(5))/8)

135.5 degree

u choose the positive root btw cuz norm cant be negative

[viscid thistle]

I don’t know?

[short saddle]

https://artofproblemsolving.com/community/q1h2277546p17803223

Let $(P_n(x))_{n \in \mathbb{N}}$ be an array of polynomials defined as:

$P_n(x) = x^n sin(a) - x sin(na) + sin((n - 1)a)$ where $a \in (0, \frac{\pi}{2})$. Prove thath there is unique monic (first coefficient is $1$) $Q$ such that $Q(x) | P_n(x)$ for $n > 2$ and degree of $Q$ is $2$.

[obsidian monolith]

aSome1gussy

[proven siren]

This is precalc??

[viscid thistle]

no

[thorn creek]

lol

[brittle monolith]

YOOOOO, can someone help me study for my test please, im kinda screwed thanks

give me a call

[viscid thistle]

cant do call but whats the test on?

[grave flower]

How does #17 even make sense?

[willow bear]

,rccw

[willow bear]

@grave flower do you still need help with this

[grave flower]

Yes.

[willow bear]

@grave flower what is it that you find nonsensical

i've drawn here the asymptotes and focus according to the equations and coordinates given in the problem

[grave flower]

Usually asymptotes have the same constant added to x, no?

But one of them here's + 1 and the others - 9.

[willow bear]

asymptotes are just lines.

your hyperbola isn't centered.

that's why your asymptotes don't have the same y intercept.

[grave flower]

Alright, nevermind.

Thank you for your patience.

[short saddle]

i don't understand this one bit. what are the prerequisites?
https://math.stackexchange.com/a/3003581/654289

[proven siren]

this is precalc?

[short saddle]

yes

[proven siren]

Well I finished calc and I don’t get it either

[viscid thistle]

Does parameterization belong in pre calc?

[willow bear]

sure why not

if you post in the wrong channel you will get redirected anyway

[proven siren]

Depends on what kind but at least precalc yeah

[viscid thistle]

This is a tutorial question. Never been taught this at all. I know derivatives and integrals but not this stuff. I can't find a decent introduction video anywhere that explains what paramataztion is and used for. Can someone please help.

[viscid thistle]

<@&286206848099549185>

[severe pond]

If f(x)>_a then f’(x)>_0 ?

[willow bear]

is >_ supposed to stand in for ≥?

so your question is: if f(x) ≥ a for all x, is it necessarily true that f'(x) ≥ 0 for all x? @severe pond

[obsidian monolith]

겨울

[severe pond]

yes

[willow bear]

then the answer is no

sin(x) ≥ -1 yet it is not true that sin'(x) ≥ 0

[viscid thistle]

Did i do this right?

[solar olive]

for question a u can find two points and thhen use the point formula for line

for that choose two random t's and plug it in

t = 0
(3, 2)
t= 1
(4, 4)

2 = 3m+b
4 = 4m+b
m = 2
2 = 6 + b
b = 4

y = 2x + 4

[viscid thistle]

Thanks yeah I solve A and also half of B, then got stuck on the last part of B but only just figured out how to solve it.

[solar olive]

ah okay then

[viscid thistle]

Thank you

[tepid narwhal]

how do i study for trigonometry

my teacher is not the best

i don’t have much material other than my homework to study from

where can i go to understand this material

i have a test next monday

[past island]

@tepid narwhal organic chemistry tutor on youtube

can someone explain quadrants to me im having a hard time understanding it

[alpine loom]

Lol

[past island]

trig

[alpine loom]

Trig is in al2

which is 11th grade for most people

[past island]

i am taking in community college

[alpine loom]

What do you not understand about quadrants?

[past island]

im taking it to refresh my memory

@alpine loom

[alpine loom]

Yes

what about that

[past island]

i don’t get it

[alpine loom]

If sin is -, it is in the bottom 2 quadrants, if cos is negative it is in the left two quadrants, if tan is negative, it is in the top left or bottom right

The rest can be infered by what is left behind

[past island]

OHHHHH

[summer ruin]

just use trig circle and forget about memorizng the combination of pluses and minuses

[tepid narwhal]

ok thanks

are there any websites i can use to practice?

[alpine loom]

Khan academy

[tepid narwhal]

good idea

thanks

[long ravine]

Hello , i have been struggling for over an hour in this exercise:

If anyone can help me, please ping me

[willow bear]

@long ravine do you still need help with this

[long ravine]

@willow bear Yes, no luck :/

[willow bear]

so, have you made any progress?

i can see a calculus-y way and a non-calculus-y way to do this.

not sure which one would be less effort.

[long ravine]

@willow bear I have tried setting f(x) <= 4 and f(x) >=-1
i have tried to set y=f(x) and then try to solve some kind of 2nd degree polynomial

im a math school teacher

this is an exercise for last year highschool

[willow bear]

one of your inequality symbols is the wrong way around, presumably due to a typo

[long ravine]

yes yes

[willow bear]

i think that looking at the discriminant of the equation y(x^2+1) = ax + b ought to be fruitful

namely that we want this discriminant to be positive precisely when -1 < y < 4, i think...

and it ought to be a quadratic in y by the looks of it

[long ravine]

yx^2 -ax + b +1 = 0

is quadratic with x

as the variable

are you talking about this

[willow bear]

+1 should be +y

actually, no, you've got another sign error there

[long ravine]

uhh ok sorry sorry im just writing it fast

[willow bear]

$yx^2 - ax + (-b+y) = 0$ is what it should be

[obsidian monolith]

Ann

[long ravine]

i have written it like this on my notebook

but i was just tryin to ask if thats the equation you were talkin about

[willow bear]

so take the discriminant of this, and require that it be factorable as (y-4)(y+1) times a constant...?

should work.

[long ravine]

cause its quadratic with x as a variable

[willow bear]

but yes we are on the same page here.

[long ravine]

the discriminant to be factorable?

the discriminant is $Δ = -4y^2 + 4yb + a^2$
I want Δ>0 , so i took $Δ' = 16(b^2 - a^2)$

[obsidian monolith]

ShaolinMonk

[long ravine]

but really dont understand where i should go with this

[willow bear]

...you say you're a teacher, right?

anyway, you don't "just" want Delta>0, you want the inequality Delta>0 to have a specific solution set

[long ravine]

this is some corny insulting behavior which is not allowed in this server as far as the rule

● Do not insult, attack, troll, gaslight other people.

I have just started teaching my career. Even if i was an old teacher, and I asked a question you have no right to be making those spoiled remarks.

I'll have to ping the <@&268886789983436800> , this behavior

Thanks for the "help"

[willow bear]

weh?

that's some massive projection you've got going on there.

[long ravine]

Sure okay. Whatever , i let this to the mods, i won't reply

[willow bear]

i did not and still do not intend to insult you in any way, for the record.

[steady cypress]

This doesn't really constitute an insult worth pinging moderators for

[silent smelt]

Hey

[opal tree]

???

[silent smelt]

Where can I learn the Texit bot commands?

[opal tree]

Google

[silent smelt]

thanx

[opal tree]

Undergraduate teacher

A TA?

[obsidian monolith]

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

My commands!

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

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

LaTeX Rendering

Render LaTeX code and configure rendering options.
​ ​ ​ ​ ​ ​ ​ ​ ​ ​ tex: Render LaTeX code.
​ ​ ​ ​ ​ ​ ​ ​ ​ ctan: Searches the ctan
​ ​ ​ ​ ​ ​ ​ texdoc: Searches the texdoc
​ ​ ​ ​ ​ ​ autotex: Toggle whether your LaTeX is automatically rendered.
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Guild administration
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​ ​ ​ ​ ​ ​ ​ disable: Disable commands in this guild.
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forgetrolesfor: Forget stored persistent roles for one or all members.

Page 1/4

[coarse marsh]

#bots

[silent smelt]

Didn’t see that, sorry

[viscid thistle]

Has anyone used completing the square for optimisation stuff? Like finding the maximum area for fencing a rectangle field..?

[solar olive]

whats the full problem

[vocal parcel]

Could somebody help me

[viscid thistle]

I’ll help you if you help me lol

[viscid thistle]

@long ravine @willow bear guys take a chill pill. ;))))))))))

[opal tree]

LOL

[stuck lark]

pls dont do that. just move on

[dire oriole]

Can someone explain where this property came from to me ? If i am differentiating with respect to something and inside the bracket theres stuff other than that something why do i multiply by the derivative again ?

[summer ruin]

chain rule

[dire oriole]

Can you elaborate ? In my book the rule is explained as such ,but i cant make out the relation between that and that

[summer ruin]

y^n is a composition of a function f(t) = t^n and y(x), f(y(x)) = y^n

[hazy hollow]

Are those first two correct and can anyone help with the second two?

[summer ruin]

it's not hard to spot that the answer is non-sense

how could possibly sine be equal to -5?

[hazy hollow]

No idea. That’s what my teacher got 💀

[willow bear]

did your teacher give any explanation whatsoever as to how they got it

[hazy hollow]

Nah I asked him and he did it all like in his head I guess

[willow bear]

[hazy hollow]

Exactly

[solar olive]

if tan(t) = -5/4 and cos(t) >0 then its in the third quadrant so sin(t) < 0 so sin(t) = -5/sqrt(41)

[viscid thistle]

If cos(t) > 0 it means cos(t) in first for fourth quad. Not third

[tulip onyx]

Sketch the graph of the equation by translating, reflecting, compressing, and stretching the graph of y = x2, y = √x,
y = 1/x, y = |x|, or y = √3 x appropriately. Then use a graphing utility to confirm that your sketch is correct. y = −2(x + 1)2 − 3
what does this question says

[solar olive]

ooosyeah fourth

0indexing fucked me

[viscid thistle]

But how did you solve the question to know sin = -5/sqrt41? Please explain

[solar olive]

tan(t) = y/x
cos(t) = x/sqrt(x^2+y^2)
sin(t) = y/sqrt(x^2+y^2)

the sign changes depending on the quadrant (x, y) is

[analog meadow]

Just wanna ask, if a limit is going to positive infinity but cannot leave pass zero, will it still be positive infinity?

Like this problem

[opal tree]

The 1 doesn't do anything against negative infinity

What would 4/(-infinity) be @analog meadow

[analog meadow]

Thanks, though i would like to apologize since I've written the equation wrong

[opal tree]

:0

What's the actual question then

[opal tree]

It'll still becomes 4/(-infinity)

But you should know what 4/(-infinity) is

Or rather -4/infinity

Doesn't really matter

Basically simplify -4/infinity

[analog meadow]

I asked on help and somebody said the anwser would be zero, though i dont really understand how it happened

[opal tree]

Yeah it would be zero. We can first use end behavior, that is, keep leading terms of the numerator and denominator, since we're evaluating a limit to some infinity.

The leading term of the numerator is the only term there: 4. The leading term for the denominator is x because adding 1 has literally no impact on infinity or negative infinity.

So the limit is equivalent to $\limit{\frac 4x}{x}{-\infty}$

[obsidian monolith]

Umbraleviathan

[opal tree]

@analog meadow does this part at least make sense

[analog meadow]

Thank You!

[opal tree]

And do you understand how 4/-infinity = 0?

[analog meadow]

Is it because the value is increasing but the value would never be a positive number?

[opal tree]

Eh no

Basically imagine -4/8

That's -1/2 right?

[analog meadow]

Yes

[opal tree]

If you increase the denominator, -4/16, it becomes -1/4

And if we increase that

-4/28282839939229

It starts to get closer and closer to 0

Eventually the denominator just becomes infinity

So -4/infinity = 0

In fact, any real number, a, divided by infinity is 0

[analog meadow]

Ohhh

Thanks! I think i understand it now

[long ravine]

Its like you have 4 pizzas and you share them with infinitely many people , basically y'all wont eat anything

[long ravine]

That's just fancy notation, I suggest you understand the chain rule first by doing some examples, and then head back to this theorem, i believe you will understand it better. If not, ping me i will help

[long ravine]

what's the problem here, do you know how to do matrix operation ? Just let matrix
X = ( x1 x2 )
( x3 x4 )

then do ur operations

[long ravine]

Can you send what you have done until now? I need to understand in which place you are stuck

[long ravine]

Well. I'll just ignore this person.

Maybe instead , instead of the drama, someone could actually give me some push on this exercise? Ill re write my answer and send it

[viscid thistle]

Im lost on how you came up with those steps lol.

[viscid thistle]

Ok ill send it later, ill ping you or dm you.

[viscid thistle]

that was basically just constructing triangles

if tan theta is negative, and cos is positive, sin must be negative

if tan = O/A

O = -5

A = 4

Get hypotenuse using pythag theorem = root 41

sin = O/H

so -5/root 41

[pulsar pagoda]

I think my question fits here but if it's more advanced math let me know.
Does the power of a power property, (x^n)^m = x^(nm) have any relationship to logarithms?

[viscid thistle]

i mean logx (n) raised to the power m = either logx (mn) or m logx (n)

[vapid plaza]

if you take log on both sides it is equivalent to m (n log x) = mn (log x), so i guess you could say it follows from the property log (a^b) = b log a

[olive scarab]

How to solve this?

and this

[viscid thistle]

Tried to dm you but you don’t allow messages haha

[long ravine]

Add me again then

[viscid thistle]

No it’s ok, basically I had trouble creating the initial expression. I knew at some stage I need to use a=lw but didn’t know how to create initial expression. Secondly, I have no idea what completely the square is doing in terms of this problem, like I can’t see the relationship. I’ve solved similar problems using derivatives but completing the square makes no sense to me for this problem, like what is it even doing?

[long ravine]

one side is y the other side x

fencing means... the perimeter

Can you write a formula , which contains, x y and the all the costs

forget the area for now

[opal tree]

Well we don't give out answers but

[opal tree]

!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'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above

[opal tree]

Also if it helps, and x and y is the length of whatever respective side (in meters)

[long ravine]

Its not my question, it's craig's question

and second, i am not giving out answers, i gave directions

[opal tree]

Oh I thought you were the one asking

[opal tree]

Given that sin(x) > 0 and cos(x) = -0.8 < 0, what quadrant does the reference angle lie in

[olive scarab]

2nd

[opal tree]

So tangent must be negative. $\cos(x) = -0.8 \implies$ the adjacent side is $-8 $and the hypotenuse is $10$. You need to find the opposite side.

[obsidian monolith]

Umbraleviathan

[sturdy gale]

heyyy guysss!! i've got my math board exam on tuesday......any time management tips?? 🙂

[thorn creek]

try not to fall asleep during the exam

[viscid thistle]

Yeah so it will be 8y + 16x and we set equal to $800 because we know that is all available for fencing. So 16x + 8y = 800

[viscid thistle]

Do the pomodoro study technique and schedule everything around study times

[long ravine]

So now you need to write the formula of the are as a function , and try to find it's minimum. To write on terms of x ONLY... Use the previous formula u just wrote..

[viscid thistle]

Yeah so I can write it in terms of x, but is the completing the square step used because it finds the vertex (i.e min/max) ?

[proven siren]

Yeah

Or it can be used to find zeros

[long ravine]

yea.. or you can use derivatives.. idk if you have reached derivatives yet

[daring roost]

precalculus
derivatives

[gaunt iris]

calculator says that the answer is 27.5 and rounded up to the nearest degree is 28

why does B equal 29?

i can't see any way im inputting it in wrong

[proven siren]

,w simplify in degrees asin(3*sin(98)/6.2)

[obsidian monolith]

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

Requested by Kefir101Awesome#7665

[proven siren]

@gaunt iris

[gaunt iris]

wait so

to plug it into the calculator

the way ive been doing it is 3sin(98)/sin(6.2)

and thats what gets me to 27.5

[proven siren]

its not sin(6.2)

just 6.2

and then asin() to get the angle

[viscid thistle]

I can use derivatives but they want us to use completing the square lol thanks for you help!

[fervent lily]

I factored out pie.h to give v / p.h = R^2 - r^2, thickness is R-r? But I’m not sure where to factor from here.

[viscid thistle]

yell CAUUUUUUUUUUUU out loud during the exam...

[tepid narwhal]

i have so much trouble verifying trig identities. how do i make it easier

is there anything to do first when solving them or something?

[summer ruin]

use identities you already know

[tepid narwhal]

i feel like i can start the problem off but i dont know how to continue it

[summer ruin]

the goal is to make the equation obviously true

e. g. sin(x) = sin(x)

if it doesn't look simple enough then you probably should simplify it further

[proven siren]

Can you send a picture of all your homework?

I don’t want to just do some of it

[tepid narwhal]

alright

thanks

[viscid thistle]

Hey, what is the name of this book?

[viscid thistle]

Trig identity cheat sheet will help a lot.

[tepid narwhal]

yeah

[pulsar pagoda]

Does the property log(1/a) = -log(a) have a name?

[summer ruin]

that's just the power rule for logs

[pulsar pagoda]

Is it? I thought the power rule was log(a^b) = blog(a)

Wait wtf you're right

[long ravine]

Your comment is completely unnecessary.
Craig might be a person revisiting maths, and trying to solve an exercise. I gave him another way to solve the exercise.

[long ravine]

you dont have to divide with pi h
just factor the expression, maybe see if there are some identities...

[long ravine]

James Stewart, Lothar Redlin, Saleem Watson - Precalculus Mathematics for Calculus

[willow bear]

it's a bit snarky but it is nothing more than a redirection to #calculus.

[pulsar pagoda]

Does the expression 1/(1/a)+(1/b) have an actual operator to relate a to b? It was in a 3blue1brown video where he called the operation "o-plus" using the ⊕ symbol

[willow bear]

well you can always make up a name just like 3b1b did

[fervent lily]

Precalculus by James Stewart

[thorn creek]

that feels familiar but if i used it it would have had to be the first or second eedition because the third edition was published in 1999 and i took precalc in 1985

i haven't been able to find a publication date for the 1st or 2nd editions

[solar olive]

what i would do is convert that parametric equation into its cartesian form and integrate it

prob not the best option tho

the cartesian form is $y=-\frac{g}{2\left(v\cos\left(\theta\right)\right)^{2}}x^{2}+\tan\left(\theta\right)x$

[obsidian monolith]

Impractical

[solar olive]

$\int_0^{\frac{2 v^2 \sin (\theta) \cos (\theta)}{g}} \left(-\frac{g}{2\left(v\cos\left(\theta\right)\right)^{2}} x^2+\tan(\theta) x\right) , dx$

k

m

[obsidian monolith]

Impractical

[solar olive]

that should result in something like $\frac{v^4}{g^2} \frac{2\cos(\theta)\sin(\theta)^3}{3}$

[obsidian monolith]

Impractical

[solar olive]

$c = \frac{2}{3} \cos(\theta) \sin(\theta)^3$

[obsidian monolith]

Impractical

[solar olive]

i think i made a mistake it should be 1/3 not 2/3

actually no its 2/3

[solar olive]

also i think this should go to #calculus

literally area under the curve

[viscid thistle]

Can

• Sinx/cosx
  Be written as
  Sinx/-cosx

?? Pls let me know, cause if it does not then I am gonna lose 4 marks 😥

[uncut mulch]

$-\frac mn = \frac{-m}{n} = \frac{m}{-n}$

[obsidian monolith]

ℝamonov

[uncut mulch]

ideally you'd use the first form

[viscid thistle]

Yeah but it's true right ?

Thank God, sometimes I forgot basic algebra while doing calculus

[viscid thistle]

Can the same be applied for
-cosxtanx/cosecx
As -
Cosxtanx/-cosecx

[spring stirrup]

You can flip The - from bottom to top
But (-cosxtanx)/(-cosecx)=(cosxtanx)/(cosecx)

[viscid thistle]

No but - is in only numerator, that can be transferred to denominators with no issues right ? But is their any condition where it's not true ?

For example 5/x-5 , can bring -5 from here to above ?

[old isle]

so -5/x+5?

that wont be valid

-5/(-x+5)

would be valid

[spring stirrup]

You can do -5/-(x-5)=-5/-x+5

[viscid thistle]

You multiplied whole side x-5 by -1 ?
Which is -x+5 if i i am not wrong

[spring stirrup]

Ye

[viscid thistle]

Ohk got it, but trigno functions are kinda confusing here, any tips on how to improve their ?

[old isle]

remember all the identities

or keep an identity table next to you

for trig

and unit circle!!!

[viscid thistle]

Ok thanks

[split holly]

quick question, if sec(-theta) = sec(theta);
does this mean arcsec(-theta) = arcsec(theta)?

or does it not apply for inverse functions?

[void matrix]

i got the problem in a precalc class...

also @solar olive the prompt is asking for a common fraction for c

[eternal thistle]

think about what arcsecant does

it doesn’t take in the ta

you could plug it in to test some values

but no i don’t think it would work

multiple angles can give the same “length”

but not the pppisite

[sharp marten]

hi

I was wondering if there is anyone who is expert in VECTORS

[solar olive]

no it didnt?

wtf

[chilly hazel]

can someone explain the difference between tangent lines and linearization?

[viscid thistle]

I'm not an expert but what is your question and I may be able to help

[summer ruin]

they're the same thing

[solar olive]

isnt linearization just approximation using first order of taylor series

[summer ruin]

yes that's the third term to describe the same thing

[solar olive]

ah okay

[bitter compass]

can someone tell me how this number came up

[slender river]

Adding to both sides of equal sign is valid move

see how right hand side also has 100

pls ping me if unclear too i move channels frequently

[bitter compass]

does it matter what number is added?

Because If what u said is the case I can just any number that cancels another one ot

[viscid thistle]

Yep

[slender river]

nope as long as its the same number added to both sides

[slender river]

Yes

[bitter compass]

cool thanks guys

[slender river]

The reason they do this is to make it easier to simplify the left hand skde

[bitter compass]

Yea I was just going over the notes since I have an exam on Thursday and saw this which made me wonder where it came from

[viscid thistle]

If you wanna underdtand these types of rules you should try to use real number instead of variables to intuitively understand how these rules work

100x5 = 20x25

100x5+55 = 20x25+55

[bitter compass]

interesting way to look at it

[proven siren]

theres an easy explanation

they just want to complete the square

which requires that (x-a)^2 so they need an a^2 which is 100

they could add +100 - 100 to the left side but thats the same thing

[spice yoke]

thats just simple algebra, this is hardly calculus

[hollow dragon]

hi

[viscid thistle]

Hola, Hallo, Bonjour, Privet, Nǐ hǎo, Namaste!

Spanish, German, French, Russian, Chinese (simplified), Hindi/Sanskrit

[slim steppe]

Privet, kak dela?

[coarse kernel]

Ploho.

[fringe sable]

how do i find the vertex of this quadratic equation?

y = (2x - 1) (3x + 1) + 4

[slim steppe]

Eto ne harasho

[willow lynx]

First apply the distributive property to remove those parentheses. Then reduce terms.

When you get y=ax²bx+c

The "x" of the Vertex is:

X = (-b) / 2a

When you find the x, calculate the image for that x using y=ax²+bx+c and in this way you will get the y.

The Vertex is V(x,y)
X and y are numbers
V(-b/2a,y)

EXAMPLE:

If x = 7, the y of the Vertex is:

y=a(7)²+b(7)+c

(Of course a, b and c are values ​​you already have)

---

Another way is to find the roots of the equation. Then add both equations and divide it by 2. That value is the x of the vertex. To find the y, do the same as the first method. This does not work when the roots are double (y=x², y=(x+7)²,y=-2(x+11)², for example)

[fringe sable]

i got x = 1/12

can you show me how you got x=7 please

[willow lynx]

It was just an example, hehe. I did not calculate the value of x!

[willow lynx]

It's perfect

Now calcule y

[fringe sable]

thanks i got it

[white bear]

You can also complete the square

But i think it's only worth trying for monic quadratics

[marble bramble]

bx would be negative so no, you couldn’t complete the square

Altho yeah in general that is significantly easier, since in the format (a+b)^2 + c, C is the vertex

[viscid pelican]

hello

[opal tree]

This is just false lmao what

A vertex is a point not a value

Unless you're referring to extrema

[white bear]

Just to demonstrate
Again I wouldn't use it for non monics

[old isle]

I always used differentiation to get the vertex 💀

I fr learnt calculus before learning the vertex form of quadratics cuz i slept during online school

[white bear]

Definitely use calculus over a lot of other techniques
But it's nice to know different methods

Especially in proof...

[solar olive]

a nice trick to convert ax^2 + bx + c to a(x - h)^2 + k is to expand a(x - h)^2 + k and compare coefficients and solve a system of equations, it only counts as a trick if u forget how to do it.

ax^2 - 2ahx + k + ah^2

b = -2ah
h = -b/(2a)

c = k+ah^2
k = c - b^2/(4a)

[viscid thistle]

Yes

[vast patrol]

what is the use of F_(n+2) -1 , which is the way to find the sum of fibonacci. like, is there any purpose in finding the sum of an nth term?

[hushed sphinx]

It's a kind of problem that arises fairly often in algorithm design.
It often happens that you need to compute the amount of resources necessary for some task (or, conversely, the size of a task that can be handled within a given resource limit), and you can somehow split it into phases, with a rule for the "sizes" of each phase being some combination of the sizes of the last few phases. That gives a Fibonacci-like sequence of sizes, but as often as not you're really interested in the total size of everything, i.e. the sum of the entries in the sequence.

[bitter bolt]

hoowwwww

[white bear]

It really depends on the question
i'm from Australia btw so the application of calc might be different to that in USA
but once you get an understanding of the fundamentals you find its a much better method than what was previously used

[bitter bolt]

I'm from 🇨🇦 so idfk'

is it better by speed or efficiency? or both???

[white bear]

I would say faster and elegant

[bitter bolt]

could you possibly give me an example?

[white bear]

Say you have a graph of a vehicle's displacement which is some weird curve and you want to find its speed at a certain point
without calculus you would probably have to get out a ruler and approximate the gradient at that point
calc allows you to find the exact speed at that point in time using differentiation

if you were also given a weird curve of a vehicle's velocity, its displacement is the area under that curve
this can be approximated by counting the number of squares within that region under the curve
calculus lets you find the exact displacement using integration

The way these two techiques are proven is interesting and is primarily what makes it so elegant
just the fact that it provides accurate answers based off clever ideas

theres probably one or two cases where calc is used over other techniques but i can't recall any

[jagged sun]

whats wrong with me i cant figure out where the 3t comes from

OHHH

WTF

i figured out lol

[fringe sable]

im given a rectangle with the dimensions: (20-2x) by (x+2)
the equation is given when i multiply the polynomials
the screen shot shows the graph and its equation

what are the answers to these questions?

1. what do the x-intercepts represent in this situation? how do they relate to the dimensions of the rectangle?

2. what does the domain and range represent?

3. what does the maximum value represent?

[viscid thistle]

when u multiply l *b u get area of a rectangle

so (1) would be the values of x for which the area of a rectangle is 0

meaning either the length is 0 or width is 0, or both r 0

(2) domain is just all possible values of x u can input into the lenght and width to increase/decrease it and the range is in regards to those values, how is the area for this function affected

(3) maximum value means for what value of x u can input, that ur area of rectangle reaches the MAX area it can reach

[full relic]

i cant get the domain in the form of n+1/n

[solar olive]

just a nitpick its actually either the length or width is 0,they both cant be zero at the same time because of different factor (ie not in the form of (x-something)^2)

[viscid thistle]

yep

[umbral sandal]

does anybody have advice for trig verifications

[humble drift]

Can someone walk me through this please?

[proven siren]

is that the answer key

[humble drift]

no this is my work

[proven siren]

youre asking someone to walk you through your work?

[humble drift]

not my own work lol the question

[proven siren]

did the help channel you made answer your q

[humble drift]

yeah more or less

[dry locust]

this is just like

solving 50=0.88^n

and check when the RHS is larger than LHS right?

equal sign prob not the most precise

but thats the idea right

[proven siren]

!show

[tender quest]

Show your work, and if possible, explain where you are stuck.

[marsh cave]

am i trippin or how is this wrong

[vast mountain]

Seems right

[opal tree]

You didn't use i j form

[marsh cave]

is that not it?

[opal tree]

No

[opal tree]

i-j form is when you have a vector v = <A, B>, and then you make it Ai + Bj

[marsh cave]

ik im late to respond but ty it worked

[viscid thistle]

From the information I've seen online, I believe D
should be a circulant matrix, but I'm unsure how to best find it (and find it's inverse D−1
). Thanks 🙂

[opal tree]

Probably better in #multivariable-calculus

[viscid thistle]

oh whoops sorry i thought i was in that chanel

thanks

[cyan pumice]

can anyone help me with this please

[summer vigil]

thats ez

[proven siren]

helpful

[wintry coral]

,tex $\int_{0}^{1}\frac{\ln \left | 4(1-2x) \right |}{\sqrt{x(1-x)}}dx$

[obsidian monolith]

D3STROY1NG

[wintry coral]

Idk what to do

I've tried trig substitution x=sin^2 (theta)