precalculus

[viscid thistle]

But you had to decide whether to use sqrt(4-3x) or 16/x. How did you decide?

-4.01 <= -4 or -4.01 > -4?

Ok.

So if that's true, which of the formulas do you substitute -4.01 into?

Why?

Right.

So now we know that g(-4.01) = sqrt(4-3*(-4.01))

But (a) isn't actually asking us about g(-4.01), it's asking about something slightly different.

It's asking, as x gets closer and closer to -4 from the left, so -4.01, -4.001, -4.0001, ..., does g(x) approach some value?

g(x) approaches -1 as x approaches -4 from the left?

Why?

Calculate g(-4.01) and what do you get?

What is 4-3*(-4.01)?

Ok then what is the sqrt of it?

Close to 4.

So then what is g(-4.01)?

Right.

Not -1.

Yes, but the reasoning you gave wasn't quite right, I think.

And understand the way it works is more important for part (b) I think.

(a) isn't asking you to find g(-4.01)

But what it is asking you is related to that.

Is this helping? I don't want to confuse you more.

Ok, but you're original (c) was incorrect.

[vagrant ingot]

how do i do this

tanx=-sqrt3 over the interval [0,2pi]

[viscid thistle]

You want to integrate?

[vagrant ingot]

no i have to solve

[viscid thistle]

You want to find a value a, such that tan(a)=-sqrt(3)?

[vagrant ingot]

i want to find radians

so i would move tan over
x=tan^-1(-sqrt3)

idk what to do after that

[viscid thistle]

The way I would do it is, I would try to guess values that make tangent close to -sqrt(3).

Then once I guessed the write value, I can write down the verification.

[vagrant ingot]

no...

i have the answers but idk how to get to them

x=2pi/3, 5pi/3

[viscid thistle]

Well, do you know what sin(2pi/3) is?

[vagrant ingot]

isnt that like .8 something

but its sqrt3/2

[viscid thistle]

What is cos(2pi/3)?

[vagrant ingot]

.5

so 1/2

[viscid thistle]

What is sin(2pi/3)/cos(2pi/3)?

[vagrant ingot]

idk let me do it rq

-sqrt3

[viscid thistle]

Okay.

Which makes sense, because tax(x) = sin(x)/cos(x), right?

[vagrant ingot]

yeah

it does

[viscid thistle]

That's how I would have done it.

[vagrant ingot]

but we didnt do that in class

like nothing like that

[viscid thistle]

You might be able to compute tan^-1(-sqrt 3) directly, but I don't know how to.

[vagrant ingot]

yeah same

[viscid thistle]

Maybe do a Taylor expansion?

But that requires derivatives, so wouldn't make sense in a pre-calculus class.

[vagrant ingot]

idk

[vagrant ingot]

how would i get x=2pi/3, 5pi/3 from this

or 5pi/3

[viscid thistle]

You know that the answer is -sqrt(3) and you also know that tan(x)=sin(x)/cos(x).

So if you think about the unit circle, then you know whatever the answer is, it must be an angle in the second or fourth quadrant.

Do you understand why?

[vagrant ingot]

yeah i understand it

but we did nothing like this in class

[viscid thistle]

I don't think there is an elementary way to just calculate the inverse tangent of something directly.

But I might be wrong.

[vagrant ingot]

ok thanks

ill lyk if my teacher says anything

[viscid thistle]

Yeah, ask your teacher.

[rapid sundial]

<@&286206848099549185> can anyone explain this how 2 came from

[viscid thistle]

@rapid sundial What is (n+2)(n+1)?

[rapid sundial]

factorial and premutation 😕

[full pagoda]

they're asking for a product

[viscid thistle]

What expression is (n+2)(n+1) equal to?

[rapid sundial]

i dont know im kinda confused

[viscid thistle]

What about (3+2)(3+1)?

[rapid sundial]

20?

[viscid thistle]

What about (n+2)(3+1)?

[rapid sundial]

i dont know

[viscid thistle]

Ok. Do you know the distribution rule for multiplication and addition?

[rapid sundial]

yes

[viscid thistle]

Okay, type it in here.

[rapid sundial]

yo i just need simple explanation on how 2 comes after n2 + 3n

[viscid thistle]

That's what I'm explaining.

It comes from the distribution rule being used to compute the product (n+2)(n+1)

[rapid sundial]

explain more please

[viscid thistle]

Okay, so the distribution rule is this: a*(b+c) = a*b + a*c

[rapid sundial]

so whats a b and c in this question

[viscid thistle]

a,b,c can be any number, or expression that computes a number (4+5, n+8, 4/3, etc)

What do you get when you apply the rule to 4*(1 + 8)?

[rapid sundial]

36

[viscid thistle]

4*1 + 4*8 = 36.

Cool.

If you put a backslash before the star, then it will be normal.

Then can you apply the rule to a*(1+8)?

[rapid sundial]

so the distrubition rule is a*(b+c) = ab + ac and how did i get 2 here

[viscid thistle]

Apply the distribution rule to (n+2)*(n+1).

[rapid sundial]

n*n= n2 2=1 = 3 and 2?

[rapid sundial]

ohhh got itttt now thank you

[rapid sundial]

but how do i distribute it

[viscid thistle]

Right, so how would you distribute 3*(n+1)?

[rapid sundial]

3n + 31

times

[viscid thistle]

Yeah.

So then do the same thing with (n+2) instead of 3

What do you get?

[rapid sundial]

so n+2*(n+2)*(n+1) how with this partcular example

[viscid thistle]

So you just told me that: 3*(n+1) = 3*n + 3*1. What do you get when you replace 3 with (n+2)?

Because we want to know (n+2)*(n+1).

Another way to think about it is this: (a+b)*(c+d) = a*c + a*d + b*c + b*d.

But that equation comes from applying the distribution rule 3 times.

[rapid sundial]

🙏 thank you got it now will practice more questions

[viscid thistle]

Okay, but also think about this one: (a+b+c)*(d+e+f)

[rapid sundial]

so what about for this type of questions (n+4)(n+3) (n+2)

what do i do after expanding

[viscid thistle]

How far did you get?

[rapid sundial]

i got (n+4)

is there easy way to factorize

[viscid thistle]

Actually I have to go. But the key to understand most formulas of that form is the distributive rule: a*(b + c) = a*b + a*c

[rapid sundial]

Alright thank u tho

[uncut mulch]

you already factorised n^2+3n-40 to (n+8)(n-5)
what seems to be the issue

[old isle]

quadratic formula

or calculator

i generally use casio calculator

[sour cairn]

Hey guys, I'm doing calc and learning proofs and such and realized my basic skills (polynomial manipulation, factoring, quadratics, nth roots, logs, etc) are all really weak. I would like to improve, so is there a recommended website or book thats just full of precalc problems? How many a day should I do? Is there ever a point where I'll feel "perfect" about my ability or should I just move on after I can intuit why it works and apply the rules to calc

[willow bear]

khanacademy would be decent i think

[floral holly]

if im gonna seperate them as log 10^6 + log x^3 do they both get the 1/3 or does it have nothing to do with the log?

[willow bear]

well you can write $\frac13 (\log(10^6) + \log(x^3))$ if you specifically don't want to expand afterward

[floral holly]

thanks

[karmic frigate]

Is the limit operator distributive over multiplication?

[willow bear]

yes, so long as the limit of each factor exists and it's not a 0*∞ scenario

[pulsar path]

would this be infinity or DNE?

[gleaming owl]

Lim=∞= d.n.e

Infinity isn't any limiting value, and function is going towards infinity at 4

[finite heath]

^ yes, but lim=infty is a more descriptive version of dne

It's to distinguish it from stuff like this (lim as x approaches 0)

So you could put either, but I'd put infty

[gleaming owl]

Is that a valid thing to do?

Just asking

I mean if limit does not exist, it need not to be because of lim x--->c= ∞

[finite heath]

Yeah

In the example I gave, the limit towards 0 doesn't exist

The limit towards $0^+$ and towards $0^-$ exist, but they're different, and limits are unique, so the whole limit dne

[obsidian monolith]

hiiistrex

[gleaming owl]

Yeah

[lean cove]

This isn't related to precalc but do I need a role to see the general channel? I don't see one but it shows there is one in the info channel.

[stuck lark]

@lean cove u have the studying role, do ,iam study in #bots to remove

[lean cove]

thank you!

[short saddle]

Given real numbers $x, y$ that satisfy $0 \leq x, y \leq 2$, $x + y \leq 3$. Find the maximum value of $A = x^2(x^2+1)+y^2(y^2+1)$

[obsidian monolith]

erictheeonicpizhao

[short saddle]

i think the answer is 40, rght?

[viscid thistle]

No

x=2
y=2
2+2<=3?
No

so answer can't be 40

you need to satisfy the inequality too

[limber heath]

can someone tell me why inverse trig function notation is the way it is?

it makes no sense at all

like sin^2(x) is sine(x) * sine(x)

makes sense

but sin^-1 does not equal 1/sin(x)

but x^-1 = 1/x

so what the hell

[finite heath]

It's like function inverse notation

[limber heath]

but the logic is inconsistent

[finite heath]

If you don't like it (which is understandable) just use arcsin etc, it's equally valid

[limber heath]

why write it that way when they also use the exponent spot for other meanings

[finite heath]

sin^1(sin^-1(x))=x
If you don't think about it too hard they look like exponents that cancel out, which is in fact what we want

It's not great notation tho, I agree

[limber heath]

what if you actually did want to say sine to the power of -1?

would you have to do (sinex)^-1?

[finite heath]

Yep

Or csc(x)

[limber heath]

so what about sine^-2

[finite heath]

csc^2

[limber heath]

is it?

[finite heath]

Yeah

[limber heath]

so only ^-1 is inverse

[finite heath]

Cosecant = 1/sine

[limber heath]

every other exponential number

follows exponent logic

[finite heath]

Yep

[limber heath]

the only one that doesn't is -1

😐

kill me

[finite heath]

Idk who made the notation but they didn't think that far ahead

Fortunately you don't usually see exponents there besides 2, occasionally 3 or 4

But that's it

And at least we have an elegant replacement for the reciprocal, even if it's not as consistent as just exponent notation

[limber heath]

yeah

it's not the end of the world

just rubs my ocd the wrong way

on a related note

what does f^2(x) even mean? is it the same as (f(x))^2?

no it wouldn't be would it

[finite heath]

Seeing notation like that is pretty rare

[dusty kite]

depends on context

[finite heath]

^

[dusty kite]

it can either mean composition or square, depending on how the book defines it, but it's usually a composition

[finite heath]

The first thing I think of is f(f(x)) but that's not universal

But for trig it's always squaring, partly because sin(sin(x)) rarely has any conceptual significance

[viscid thistle]

what does find a reasonable domain and range mean??

[stuck lark]

reasonable=makes sense for the problem

[viscid thistle]

so there are no

correct answer?

[fiery saffron]

Think about it. For the domain, does negative time make sense in the problem?

[viscid thistle]

no because negative months wouldn't make sense oh thanks

[tall prism]

Can someone help me figure out how to graph this?

The cos (3/2x)

[low slate]

can someone help im a little confused on why this is wrong

[uncut mulch]

how are you getting 70.86

[low slate]

cos^-1(19.4/58.18) i believe

[uncut mulch]

58.18?

[low slate]

sorry 59.18

[uncut mulch]

the value in the question is 59.15

[low slate]

oh shit

thanks i didnt know i got 8

[toxic cradle]

I cant get past Trigonometry. Please recommend courses/youtubePlaylists/books or any kind of resource for Trigonometry.

[uncut mulch]

khan, prof leonard, organic chem tut

[granite lynx]

hello

Im having diffuvulties proving this ture

true

its with induction

Its c

[granite lynx]

I do it for 2

Dude come on atleast give me a sec

[echo hare]

Can you answer it man??

[granite lynx]

Bruh

Im asking a question let me ask then ill read

[echo hare]

The question is that how to answer.

[granite lynx]

As i was saying

I ma having difuculties proving this

[vapid plaza]

Can you at least do the base case for n=1 first

[granite lynx]

I was continuing to wrote

I was going to show it

Damm my english on the phone sucks

Aight

So you cant do for n=1

Cuz denominator 1-x

So i did for n=2

and i get to

[vapid plaza]

Is just can’t do for x = 1
n = 1 ok

[granite lynx]

Oh

So maybe im right

N 2 gave me this

[vapid plaza]

n=1 is easier

[granite lynx]

No wait still wrong

[vapid plaza]

Try it

[granite lynx]

Aoght

[vapid plaza]

For n=1 the left side is just (1+x)(1+x^2)

[granite lynx]

Ye it gave me that to

[vapid plaza]

Wait bruh can we do n = 0?

[granite lynx]

Nah

[vapid plaza]

n = 0 then left side just 1 term

[granite lynx]

I think we cant do n=0

But you get what i mean now right

[vapid plaza]

Ok cool we’ll do n = 1 then

Have you done ;=1

[granite lynx]

For n=1

It gives me supposedly the “n =2” term

And i think i am supposed to get x +1

Only

[vapid plaza]

Uh what are you writing

[granite lynx]

I did for n=1

[vapid plaza]

Bad presentation

[granite lynx]

Sorry professor

Was quick

[vapid plaza]

Your teacher would not like that

At the end you shouldn’t just put the finished expression as numerator

Open a new line

[vapid plaza]

So the right hand side is equal to left hand side

Good.

[granite lynx]

Nono

Is wrong i think

Thats me just simplifying

[vapid plaza]

(x+1)(x^2+1) matches left side tho

[granite lynx]

I think it should only give 1+x

[vapid plaza]

If n = 1 left side would be (1+x)(1+x^2) tho

[granite lynx]

How do you discern if it n=1 is supposed to be (1+x)(1+x^2) or just (1+x)

Because in sums normally n is just one term

[vapid plaza]

Well if n=1 then 2^n = 2
Right?

So the “sequence” ends in (1+x^2)

[granite lynx]

Now im confusion

I know that

I understnad completly

But i dont get why they would write it like such

On the left side

So i got to pay attention?

So n= 1 are 2 terms

And n=3 are 3 terms

[vapid plaza]

The “…(1+x^2^n)” part indicates that the left side expression “ends” when we reach (1+x^2^n)

[granite lynx]

I was used to n representing only one term

[vapid plaza]

Bruh

[vapid plaza]

Nah n=3 is 4 terms

[granite lynx]

Bruh

I did it manually wut

[vapid plaza]

n=3:
(1+x)(1+x^2)(1+x^4)(1+x^8)

[granite lynx]

Brrrrrrrrrr

Im a do it manually agane hol up

I meant n=2

Mb

Mb

I meant n=2

Gotcha chief

[vapid plaza]

Oh cool

[granite lynx]

Aight tight ty

Ok sorry to bother once agane Mr.Mask

But i was just doing for n+1

[vapid plaza]

Ye

[granite lynx]

And from what i understand you just simplify it and test

So here you would multiply the formula with n+1 correct?

Or would you put n+1 where n is?

[vapid plaza]

Multiply the formula with (1+x^2^(n+1))

[granite lynx]

Ah

There you mean 2^(n+1)+1 correct?

[vapid plaza]

?

We’re given the n case

And we want to prove the n+1 case

[granite lynx]

The formual already is 2^n+1

Ahhhh

Ok

Wait

So you do this

[granite lynx]

.

Ye multiplying with n+1 wouldnt make sense

[vapid plaza]

Multiply both sides of the first equation with (1+x^2^(n+1))

So that the left side matches that of equation 2

And then check that the right sides matches too

[granite lynx]

On it

Ye

The answer it gives is

1-x^2^n+2

/ 1-x

So ye works

Tu

[vapid plaza]

Yeh

Done

[pseudo dew]

help with this please

ive already changed it into exponential form but dont know where to go from there

[sick steppe]

apply log laws

[pseudo dew]

would be helpfgul to know which one and how it applies

[old isle]

for 17(a)
ln ab=ln a+ln b

for 17(b)
ln a/b=ln a-ln b

[pseudo dew]

oh just 17a

[old isle]

oh

okay

[pseudo dew]

i can try the rest after that

[old isle]

👍

[pseudo dew]

wait so i would have ln6+ln3?

ohh wait

i see now

[old isle]

you can distribute ab as 2 and 3

[pseudo dew]

i would have ln(3*2) because of ln 6

yea

thank you

[old isle]

no problem

[rustic forge]

No calculator allowed

[old isle]

I just used a calculator

to find the value of tan 20/√3 tan 10

then i used inverse tan

imma remove it

😀

[nova surge]

how do u even do this

im g9 i wanna learn

higher grade math

[old isle]

im not sure either w/o calculator

im in grade 10

[nova surge]

w calc tho?

[old isle]

havent delved deep into trig other than some periodic relationships

[nova surge]

u need scientific

[old isle]

oh with calc it is easy

[nova surge]

what do u do

[old isle]

you can use inverse tan

to find x

[nova surge]

im just stupid

what tf is tan

[old isle]

oh

the ratio between the opposite line and the adjacent line

[nova surge]

g10 next yr for me requires pre calc

[old isle]

they will teach trig in precalc

[nova surge]

ok so there is opposite and adjacent

[old isle]

tan=Opposite/adjacent

[nova surge]

why not just slove for tan

solve tan

then

divide it

you alr know that the square root of 3 is multiplied by tan

correct or incorrect

[old isle]

correct(?)

elaborate maybe

[nova surge]

tan x 1.73205080757?

[old isle]

yeah

that is correct

[nova surge]

then u have 10 and 20

cant you just divide

to solve

oh its degree

then i have no clue

i thought it would be

[old isle]

I will show you the calculator method

[nova surge]

tan 2

1.73205080757 x tan

[old isle]

hol up

[nova surge]

ok

ε @old isle is the sign beside X ε

[old isle]

epsilon has no significance to the question tho

epsilon is used in limits

[nova surge]

hmmmmm

i see

im rly confused then

(0,90)

like

huh

i might just do trig

its easier

[old isle]

@nova surge

I didn't show the final answer cuz that'd be spoilers but you can just calculate that inverse tan in the calculator

[nova surge]

huhh

okx wait

so u found out that tan is equal to 1.1917...

[old isle]

yeah

[nova surge]

and you flipped the equasion

[old isle]

yup

[nova surge]

which suprisingly works as there is -1

but the -1 exponent

its kinda tricky

[old isle]

-1 is not an exponent there

[nova surge]

then what is it

[old isle]

it is signifying that the tan is inverse

[nova surge]

OHH

nvm

man

pre calc is too confuisng

ill get my mom to explain to me

[viscid thistle]

anyone know what the vertical squiggly line is called

[old isle]

∫?

integral

[viscid thistle]

yes

[atomic pelican]

why is there a need for the lim h->0 ? Since it's division isn't like "common sense" that h cannot be 0 ?

[viscid thistle]

because you want an answer

you want to see the behaviour as h goes to 0

not just no answer/undefinable

it makes a lot of sense when you see the def of a derivative on a graph

[atomic pelican]

I see , thank you

[viscid thistle]

no problem

[finite heath]

In general there's no problem with dividing by a limit towards 0, the problem is only with 0 itself

[old isle]

If the limit cannot be evaluated using the limit laws or algebraic manipulation, it just means that the limit DNE

so there is no problem when dividing by lim h--->o because there are methods to evaluate the limitt

[rustic forge]

not quite.

[rustic forge]

it does not use calc >_<

[nova surge]

Yea

im g9

so im kinda

dumb

mb

[rustic forge]

And is a very usual question when studying trig identities

[nova surge]

ah

[short saddle]

Any ideas?

[short saddle]

Lmao

[finite heath]

Find a common denominator probably

[narrow marsh]

Probably one of the inequalities

cauchy, etc.

[brittle ice]

Hello. I wanted to verify a problem I did.

[obsidian monolith]

Jef Costello

[brittle ice]

I need to show that the slope of a line containing (2, 11) and the point on the graph of q very close to (2, 11) is approximately 21.

so would it suffice if I take a point arbritrarily close to (2, 11) like let's say at around (2.001, 11.021)?

[uncut mulch]

based on the wording yes

but you should be using 11.021012

instead of the rounded value

[brittle ice]

[rose basin]

I have a question about a homework problem that I have. It's telling me if f(x)= 300+15x, find f^-1(30). But I'm not sure how to solve this problem. Do you know how to solve it?

[dusk skiff]

@rose basin You have to find the inverse of the function. The equation is already written as f(x) = 300 + 15x. Now, swap the x’s and y’s and then solve the new equation for y. This will give you the inverse of the function. Then, plug 30 into wherever you see x, and the output value is your answer.

[rose basin]

Alright thanks for the help.

[dusk skiff]

Yeah, no problem.

[willow dagger]

Is there any actual difference between taking a partial and normal derivative? Or are they just written differently in the case of multivariable functions?

[old isle]

this is the precalc chat

[willow dagger]

Fair

[viscid thistle]

Can anyone do my Precalc homework? i’ll give you $3 per question. it’s 5 questions on Interest and Compound interest. Wont take longer than 10 min

[ebon barn]

Instant ban

[willow dagger]

We can help with specific questions

[viscid thistle]

ok sorry

[dim cosmos]

I'm currently in the functions unit that covers inverses, graphing functions, limits, proving discontinuity etc. and I have my Math unit test tomorrow lol

I'm mainly struggling with the graphing portion and no amount of videos seem to help me ingrain the knowledge

Is there a general rule I can use for graphing, sorta like the y intercept is always outside of any groupings etc.

[viscid thistle]

Nope, don’t break rules pleade

[granite lynx]

Yo

I am having difuclties with the proving of this theorem( Its exercise 2)

Its asking me to prove with the pascal identity that the sum of natural numbers with 0 to the power p is equal to what is in the RHS

I used the binomial theoreom so that both of sides look a like

and this is where im at

It then asks me to calculate the sum of natural numbers to the power of 4

Could anybody help plz, any help would be appreciated

I think i can do the second part of t he ex but not the first

This is where i grt to

[mossy vapor]

Can anyone help me solve this I’m so confused and have no idea how to start : 8sin2øcos3ø = cosø - cosøcos(4ø)

[tight compass]

what are you trying to do? Show left hand side is equal to the right hand side?

[mossy vapor]

Yes

[mild swan]

@mossy vapor My guess would be to start using identities for the LHS to change the arguments of the sine and cosine functions

From all that you'll need a factor of cosine to pop out and a 1 - cos 4x

x or phi - the dummy variable doesn't matter a whole lot

[pallid widget]

how does the cosx get above sinx from cos^2x?

[fathom rain]

how do u set up the problem in the formula?

for 19

[steel tulip]

cos^2(x) = cos(x)*cos(x)

[stone field]

what age do you start learning pre calculas

in school

[vivid void]

people usually learn it in in last year of high school or the first year of college i’d think

[mild swan]

Probably depends.

There are people who never take anything past Algebra II.

There are people who take Calculus as a sophomore in high school.

There are probably people who are extremes in terms of this standard

I believe people normally take their first calculus course as a senior in high school or as a freshman in college

[sullen lichen]

what's difference between domain and implied domain

[trim hemlock]

well they are identical in most cases, but in cases where one such function can be rewritten as another function but with a different domain, the difference becomes clear. For example, a function like f(x)=(x+2)^2/(x+2) has the (implied) domain of R\{-2}, but notice that f(x) can also be simplified and rewritten as another function, say, f(x)=g(x)=x+2, here the implied domain of g(x) is R, but the domain of g(x) is still R\{-2} since g(x) is still the original f(x) function

this is seen more regularly in math problems that have to do with real life quantities, for example a polynomial function in x that takes in the population of a country as an input and then output something like a number of people that will land a job in the following year. The implied domain of this polynomial would be R, but however, the domain of this function is x>0

since the population can not attain a negative value

[finite heath]

@empty rain read the problem again. The 33 comes from the 90 degree force and the 105 comes from the x axis force

[empty rain]

i dont understand
how do u know when do add these things

[finite heath]

The 105 is in the x axis so we add it to the x component

The 33 is in the y axis (90 degrees off x) so we at it to the y component

A more general way is to look at the angles (where the first angle is 0) and use the same method as for any other entire

cos(0)=1 so we put 1*105 in the c component, and sin(0)=0 so we put 0*105=0 in the y component

cos(90 deg)=0 and sin(90 deg)=1, so we add the force vector <0*33,1*33>

[empty rain]

ohk

[swift flume]

you know how

[obsidian monolith]

Leon

Leon

Leon

[swift flume]

is there a name for that point or am I just making all this up

cuz I think my teacher requires that point to be drawn when plotting by hand and I kind of want to know its name

so I can look it up and stuff

[willow bear]

...there does not appear to be a special name for that point

[swift flume]

NOOOOOOOOOOOOOO

😭

It's okay I will name it and it shall be known as "bob" from now on

say hi to bob

[past meadow]

unfortunately we don't help with marked tests/quizzes here.

[magic mantle]

<@&268886789983436800>

[stuck lark]

ty

[finite heath]

It's the vertex https://www.cliffsnotes.com/study-guides/algebra/algebra-ii/conic-sections/hyperbola

Well kinda

In a special case it's the vertex

The link I gave gives a very different interpretation of hyperbolas, but with n=1 it's the same shape but rotated

[olive parrot]

Hi, can someone share some resources on how to find equivalents to trig functions that involve identities?

[lean apex]

you mean like a formula sheet

[olive parrot]

Hi, someone helped in another group. Thank you.

This was the question. I needed to use identities repeatedly to simplify

[lean apex]

you can just fgraph that on your calculator and finfd the answer

[olive parrot]

#16?

Sorry 10

[lean apex]

#10 the one u sent

graph that big thing and then graph the options and see which one is the same graph

[olive parrot]

Oh but if I have to do it by hand since I won't get a calculator on the test

[lean apex]

then youll have to use identities i assume

[olive parrot]

Thank you!

[outer briar]

Hello, I am self preparing for ordinary differential equations and partial differential equations because my exams are nearing up. I'm an open University student feeling stuck. Like I'm studying maths after 2 years so I'm recalling the prerequisites which I've learnt in my high-school!

I am surprised that i haven't understood trigonometry well during my high-school so i understood from the first!
I completed the algebra requisites.
Now I'm onto limits and continuity

Sometimes because i have less time i just look for problems with solved solutions because it becomes ease for me to make my concepts grasp fast!

Will anyone guide me?

[viscid thistle]

https://www.khanacademy.org/math/calculus-1/cs1-limits-and-continuity | https://tutorial.math.lamar.edu/classes/calci/continuity.aspx | https://www.sparknotes.com/math/precalc/continuityandlimits/summary/ these helped me quite a bit

[empty harness]

someone help

[glacial dragon]

for fuck sake that's not preclac

[willow bear]

this person seems unwilling to respond to my question of whether they still need help with this so it's probably best to assume they don't

[viscid thistle]

i was the one who directed him here my bad

idk where it goes i learned compound interest in alg 2 which i guess counts as precalc

[willow bear]

this is nothing to do with compound interest

[native urchin]

can someone teach deritatives

[mild swan]

You can look stuff up online for that

[dusty mesa]

yeah just youtube power rule, product rule, quotient rule, chain rule
though i'm not sure why you need it in precal and if you do ig it might involve something basic like power rule

[wheat radish]

Just learned precalculus, feeling good!

[olive parrot]

Hi, yes. Do you have specific questions or need to know all in general?

Can anyone help with this question:

I know cos(-t) = cos(t), sin(-t) = -sin(t) and tan(-t) = -tan(t)

Also cot(t+5pi) =__ cot(t)cot(5pi) - 1__
cot(t) + cot(5pi)

I cannot seem to simplify it all correctly

[vivid void]

over complication

cot(t+5pi) = cot(t+pi) = cos(t+pi)/sin(t+pi) = cos(t)/(-sin(t)) = -cot(t)

[vivid void]

are you asking why c is the answer or why c isn’t the answer?

because i got c i think

[olive parrot]

no, the answer should be B. I figured it out.

so we move the cot(t+5pi) term to the numerator as tan

and tan 5pi is 0 so that crosses a couple things out and then the tan t can be crossed out and you're left with 4 cos(t) sin(t) which can be replaced with an identity. 2sin(2t)

[olive parrot]

Thank you for taking the time to solve it to help me 🙂

[vivid void]

oh shoot the denominator is -1 not 1 lol

[olive parrot]

🙂

here is another question. Finding the period is fine, but how to find the max?

without a graphing calculator

[willow bear]

do you know at what value(s) of x the function sin(x) [raw sine, no modifications] attains its maximum?

no need to list all of them, just one is enough

[olive parrot]

pi/2

[willow bear]

correct

[olive parrot]

figured it out. x = 0.5 and max is 26?

[willow bear]

no need to find the x value itself

it is enough to know that there exists x such that pi*x - 3pi = pi/2

and that sin(pi/2) = 1

[olive parrot]

oh

[willow bear]

but yes

[olive parrot]

thank you!!

[opal garnet]

How to calculate the volume of a function rotated about a particular line?

[finite heart]

that would be calculus

[opal garnet]

ohk fine i'll ask there

but what if they expect me to know too much, which i don't

[finite heart]

do you know integration?

[opal garnet]

yes

[finite heart]

oh okay you should be fine then

[opal garnet]

thanks a lot

[olive parrot]

for this, how do you account for the 47, 38, and 23 in front of the common unit circle values?

[finite heart]

recall that trig functions repeat every 2pi

[mild swan]

@olive parrot ^^

In other words:

csc(x + 2π) = csc(x)

Same with the rest of the trig functions

[native urchin]

im still hella confuse

about like what exactly derivative points are

[full pagoda]

the derivative evaluated at a point is the slope of the original function at that point

[native urchin]

wdym'

how can one point have slope tho?

[full pagoda]

not exactly what i mean

it's the slope of a function at a point

not the slope of a point

[native urchin]

so different points can have different slopes

[full pagoda]

points in a function, yes

given it's not linear

[native urchin]

what about linear equations

o

ok

[full pagoda]

constant slope

[native urchin]

ok

[full pagoda]

take the derivative of any linear equation

you get a number

[native urchin]

but for example if it was like a parabola then it would hva different slopes rtight

[full pagoda]

yes

at different points

[native urchin]

ok

ty

[forest oriole]

Hii, how can I test myself to see my calculus level? I want to know if I'm capable of passing calculus III

[sick steppe]

do problems from a Calc3 resource

[jade belfry]

if your right and left hand are not equal, where will you put the DNE?

[grim zephyr]

idk if this belongs here

but what transformation would happen if u did f(x)^2

[sick steppe]

the y values would be squared

describing further I think is near impossible

example x becomes x^2

[grim zephyr]

would there be a term to describe how the function is stretched?

[sick steppe]

stretched under squaring?

not that im aware of

[grim zephyr]

like how f(x-3) would shift the graph right 3, 2 f(x) would horizontally stretch the function

ok

i was just curious idk if there was anything interesting about doing stuff like that to functions

[slender elk]

Help

[lean hill]

m is a odd number, prove that for a big enough $a$ there isn't any x from the set of real numbers such that:
$(x+a)^m=x^m+a$
$m>1$

[obsidian monolith]

AeroBennu

[sullen lichen]

<@&286206848099549185>

i got

exam tommorow

wsh me luck

[fiery quartz]

good luck

break a leg

[sullen lichen]

@fiery quartz help pls

this is hardest exam ever

;-;

[fiery quartz]

eh it's not that bad

i believe in u

[willow bear]

@fiery quartz hi 👁️👁️

didn't know you were here too

[fiery quartz]

yep 😄

how's it going ann

[willow bear]

fine ig?

been a bit burned out on nethack

but have taken up dwarf fortress instead

[fiery quartz]

hahaha

[olive parrot]

can you show me specifically for csc(47pi/4). What would it equal? just csc(pi/4)?

[mild swan]

How did you get there? Do you have any ideas as to what the process is?

Hint:

csc(π/2) = csc(5π/2) = csc(9π/2) = csc(13π/2) = ...

[olive parrot]

csc(pi/4) = csc(9pi/4) =csc(17pi/4) and on until csc(41pi/4) and csc(49pi/4)

so what about the 47 in between?

[mild swan]

Find some other angle where you can get 47

[olive parrot]

7pi/4?

and I get -sqrt(2). Thank you!!

[sullen lichen]

can someone help me

with my practise exam

fuck me bro, i got no fucking idea what this is

i havent leanrt it...

i swear i havent leant this

<@&286206848099549185>

[sullen lichen]

hi

<@&286206848099549185>

[gleaming owl]

#elementary-number-theory try asking there

[mellow tapir]

You can only ping helpers ONCE, and it must be 15 minutes after nobody has helped you since the beginning of your question

[quasi rivet]

Can any one having a link for a good physics server

[hushed sphinx]

#old-network?

[quasi rivet]

Tell me pls

Any one

[fallow quiver]

#old-network check there.

That is what they meant.

There is one, it's a good one.

[mild swan]

Someone literally just told you where to find one

[viscid thistle]

i need some help for some basic linear functions

im having trouble with knowing if a straight line passes through 3 points

any help is welcomed

[brittle ice]

If you know the equation at hand, you can plug in the co-ordinates of those points in the equation and see if the equality holds

[magic mantle]

If you don't know the function, take 2 points and make the function based on those points. Then plug in the third point and check if it lies on the graph of that function

[finite heath]

And for that function, use point-slope form, because it's the easiest from that setup. You can get the slope from the ratio of the differences of each coordinate (so, rise over run)

[quasi rivet]

Can any one teach me calculas

Sorry i not seen precalculas

[viscid thistle]

I could

but for calculus

I really recommend a video series

starting with the vid "calculus in under 10mins"

[late dirge]

Can anyone please help?

[grim zephyr]

how many times does it cross the x-axis

[covert hawk]

Can someone help me with this please

[hushed sphinx]

The only real problem is that there are so many ways to solve it that it can seem overwhelming to pick one.

But it really is just a matter of picking -- do not attempt to derive what a, b, C have to be, because they can be anything you want.

(Oh, and I suppose another problem would be that this is in no way form, or shape PREcalculus).

[covert hawk]

Wait so what exactly do you do I’m very confused?

[hushed sphinx]

You start by making some arbitrary choices.
E.g. decide that f(x)=x or that |f(x)| must be constant, or that a=0 and b=2, or something else to narrow down the possibility space.

Well, actually you start by building an intuition for how it is even plausible that the conditions can be satisfied.

I.e. what must be the relation between f(x) and |f(x)| such that the integral of one can be twice that of the other ...

[covert hawk]

hmmm thank you

[covert hawk]

are we integrating the absolute value because i havent learnt that

[hushed sphinx]

You haven't learned what abolute value is?

[covert hawk]

ahh i have but we've never integrated it

[hushed sphinx]

Since you are free to choose what f is, you could choose it such that |f(x)|=1 for all x. That is at least straightforward to integrate.

You would then need to split the integral of f(x) itself into intervals where f(x) is -1 or +1, respectively.

[covert hawk]

ah ok

[kind arrow]

use angle addition formula for tan, and that should get you the first factor of the result you're trying to verify

[thick wren]

can someone help me out on this question please

[full pagoda]

with what exactly

[haughty latch]

So what is the slope of the function?

[ornate cargo]

well, basically in my precalculus homework im working with sets, my questions is in 2 points appear that the intersection of B and C is equal to φ but if one of the sets is contained in other why the intersection is equal to φ

im so confused

if I draw the sets it doesnt fit me

[ornate cargo]

is 0 because is constant, horizontal line

[late dirge]

Anyone know this answer?

[full pagoda]

what have you tried

[haughty latch]

Why not both? It is constant zero

[molten garnet]

can someone show their wokring to me

i subbed 1000 as x

I followed this

<@&286206848099549185>

[tepid python]

what is the problem asking you to do, just to find the limit?

and are you familiar with the limit laws/rules? if not, what methods of solving this were you taught?

[molten garnet]

lim
x -> 5

i learnt that u just sub in the value thats closest to 5

like 4.99 for e.g.

[molten garnet]

ye just the limit

@tepid python

[molten garnet]

except for infinity then u just follow the table

[molten garnet]

I applied this but i used 10000 instead

for x

[tepid python]

well if that's how you're supposed to solve it then make a table with an x column and a $\lim_{x\rightarrow \infty} \left( 1 + \frac{7}{3x} \right)^{x-1}$ column (and maybe intermediate columns for $\frac{7}{3x}$ if you need them) and plug the values in

[obsidian monolith]

crossbeam

[molten garnet]

what does e me in limits

@tepid python

[tepid python]

e is Euler's number

$e = \lim_{n \rightarrow \infty} \left( 1 + \frac{1}{n} \right)^n \approx 2.71828$

[obsidian monolith]

crossbeam

[molten garnet]

ohh i see

@tepid python what value did u sub in x?

same as mine?

[tepid python]

if this is the correct answer to the question then plugging in values is the wrong approach

[molten garnet]

ohh

but how do u get 2.71?

is it just the rule n formula?

@tepid python

[ornate cargo]

yes

[hushed sphinx]

The limit in Crossbeam's latest formula is sometimes even used as the definition for e. Its value is well known with lots and lots of digits.

https://en.wikipedia.org/wiki/E_(mathematical_constant)#History

[tepid python]

e is an irrational number. basically you can't write it down completely because it has an infinite amount of non-repeating decimal places.
2.71828 is just an approximation, not the exact answer.
you can get an approximation by plugging in values but you can't get the exact answer. for example

$(1+\frac{1}{999})^{999} = 2.7169225742266474\dots$

[obsidian monolith]

crossbeam

[tepid python]

which is close but not equal to e.

if i take the problem you posted and plug in a large number

$(1+\frac{7}{3\cdot9999})^{9999-1} = 10.3070465953\dots$

[obsidian monolith]

crossbeam

[tepid python]

we get something close to the real solution but not exactly it.

[ebon barn]

How hard is precalc compared to calc and alg 2 I’m probably gonna end up doing an independent study on it over the summer

[tight compass]

precalc is just algebra and trig with extra stuff added. Have a decent foundation in algebra and trig will let you enjoy learning about calculus.

People do bad in calculus because of a weak foundation in algebra and basic trig.

[digital plover]

stuck at Qa

[uncut mulch]

what have you tried?

[tight compass]

Everything looks fine, just finish up zw.

[digital plover]

Literally can’t

[tight compass]

Why?

[digital plover]

Idk

[tight compass]

Just add like term the sum of 2 and 2(-1) is 0.

Like you were doing for the previous ones.

2(-1)= -2.

[tight compass]

Can you be more specific so I can help you?

[digital plover]

So the answer would be 0+5i

[tight compass]

Yes.

Can you figure out how to do z/w?

[digital plover]

Oh great

I’ll try it

[molten garnet]

thx @tepid python

is derivatives hard btw

[proud raven]

is this a question

[outer briar]

@proud raven how should a question be?

Sir/ma'am

Next time I'll ask a question according to your "defination of question".

[proud raven]

i didnt mean it like that

[vapid plaza]

Bruh moment

[proud raven]

but i think you have it

[outer briar]

Even if it's silly it's a question.

[proud raven]

its not silly

[outer briar]

Okay thankyou!

[proud raven]

i think people missed your question since you posted it as an image is all im saying lol

people tend to ignore people who just post an image

because

reasons

historically

[outer briar]

Yes, i use an app on mobile

[proud raven]

but you actually just wrote out your question

[outer briar]

I actually practice on mobile writing app when I'm outside. Okay next time I'll make sure I'll post it in text!

But sir/ma'am, I don't know to type in LaTeX

[proud raven]

das okay dont pay attention to what i said about your image

it was just a comment

[outer briar]

Okay, it's okay sir/ma'am! I didn't get offended!

[finite heath]

Btw just put $'s around the math part and put \'s before names of symbols and if it has an argument (for example square root) that argument goes in {}'s

Beyond that just try calling the symbol by different names until it stops yelling at you, although there's a good chance you'll get it first try

Also _ for subscript

[steel fern]

what is the domain here

[tepid python]

you need to give more context, what exactly is the question asking?

[steel fern]

nvm im just dumb and dont know how to read

[steel fern]

how do i read this on the decreasing

why is it (-4,-3),(-2,infinity) and not (-4,3),(-2,-infinity)

[tepid python]

that's not quite right, take a look at your minuses and the lack thereof

[steel fern]

specifically concerning the infinity

[tepid python]

oh, i missed the question under the image.
(-4, 3) goes from -4 to +3, so from where it starts decreasing to the other side of the y axis and then some

and (-2, -infinity) goes from -2 where it starts decreasing again to -infinity which lies all the way to the left

[steel fern]

right but the computer says the answer is (-4,-3),(-2,infinity)

not -infinity

[tepid python]

yes, because -infinity is going in the wrong direction.
it's decreasing starting at -2 and going to the right, which is where positive infinity is

@steel fern, that make sense?

[steel fern]

no im sorry

[tepid python]

ok let me make a diagram

in red is where the function is decreasing

are you with me on that?

[grizzled veldt]

Your graph is fundamentally incorrect due to ∞ and -∞ not being real numbers

you should have arrows on both sides of the x-axis to show that it's approaching ∞ and -∞ instead of being ∞ and -∞

[steel fern]

i mean you are right

[tepid python]

it's just a quick and dirty sketch, but yes

[steel fern]

but its a nitpick

[tepid python]

so the first interval (-4, -3) should be clear now, right?

[steel fern]

yes

[grizzled veldt]

But then, the answer should be (-2,∞) for the next one, because the function is going right (so to speak)

mathematically it's incorrect to say that a function goes to the right any more than you could say a number is on the right of another on the number line (I got marks off my test in October for that)

but at least, if it's going right, it's approaching ∞, or rather to say, a value greater than -2.

When you are unable to see the whole graph, just pretend it's gonna go WAY all the way to ∞ and -∞ where applicable and where not told by the questions.

[steel fern]

im genuinly confused what the relative min and max values are for maxima

i just plain dont know how to read to find it.

it seems like the lowest points on the graph are (0,1) (2,1)

[tepid python]

do you know what a relative (or local) maximum or minimum is?

[steel fern]

not entirely

[tepid python]

intuitively it's a peak (for maxima) or valley (for minima) in a graph

[steel fern]

right

that makes sense

ok

[tepid python]

now the answer is going to be what value(s) x would have to be for f(x) to be at those minima or maxima

[steel fern]

so im lookin at the y axis

[tepid python]

the answer will be the x values

[steel fern]

oh

[tepid python]

every function passes the vertical line test. that is you can draw a vertical line anywhere you want and it will only intersect the graph of the function (at most) once.

a consequence of that is that every x value uniquely identifies one "place" in the function. so if you answer with an x value it's always clear to where you're "pointing" in the function

if you were to answer with a y value it could be referring to infinitely many "places" in the function

[steel fern]

ok

but why

[steel fern]

so what would be the answer for here be

[tepid python]

in the previous example y=-1.5 intersects the graph 4 times but every x value is guaranteed to identify one and only one point on the graph

[steel fern]

im confused

[tepid python]

you correctly identified two "lowest points", now the answer is the x component of them

[steel fern]

0,2

[tepid python]

yes

[steel fern]

those are the answers ?

[steel fern]

i entered 0,2 and it says its wrong

[tepid python]

oh, it's only asking for maxima, didn't see that my bad

[steel fern]

where did it get this num from

oh

...... its the highest point of 5

[tepid python]

hm, that's weird that it wants the y value

[steel fern]

yea

what is this quesion asking

[tepid python]

i'm not sure what at what numbers is supposed to mean. it can only be the x or y values though

[steel fern]

yeah

its weird

it said -1 is wrong

[simple cobalt]

hello guys

sketch the graph of the function y= f(x)= 2x^3 - 5/2 x^2 - 6x - 3 !

I'm confused

[swift gale]

help

[strong ermine]

https://tenor.com/view/discord-ping-pingus-bingus-cat-gif-18905873

[stoic crater]

You are given 15 points on a piece of paper with 5 points that are colinear. How many quadrilaterals can be formed using these points?
This is a combinations/permutations question I've been stuck on for a while (precalc)

[tardy ridge]

case 1: using 0 of the 5 points

case 2: using 1 of the 5 points

case 3: using 2 of the 5 points

@stoic crater

[narrow marsh]

hmmmmm this question is slightly faulty

imagine an equilateral triangle with a point in the middle

The vertices and that point ---

I think there is more than 1 possible quadrilateral isn't there?

[tardy ridge]

o that's interesting.

[wide yacht]

I am trying to find the number of unique ways to multiply 2 2 167 3 together

so I know i need to find the number of unique subsets of this multiset since multiplication is commutative and we have 2 2's
so [2,2,167,3]

one method I tried

was

doing (4 choose 2) + (4 choose 3) + (4 choose 4)

but this gets me one short

[cinder briar]

why is this mathematically incorrect

oh

its not timsing

holyyyyyyyyy

[delicate dragon]

anyone can help me rq?

just with 1 question

[tardy ridge]

same as the number of ways to order AABC

[tardy ridge]

because (2A+B)/2 is not (A+B)/2

[cloud hinge]

@cinder briar

[visual belfry]

Can some teacher me something

[steady warren]

[neat trail]

all points on this major arc subtend an arbitrary angle with z1 and z2

or

[obsidian monolith]

Staasi

[neat trail]

where z consists of points on this major arc

can someone give me an ideally intuitionistic argument as to why this holds (i guess otherwise i have to algebraically solve this)

for alpha=pi/2 i can prove the semi circle case with simple geometry

[cinder briar]

I did something similar

I made rooted part base 2 for the 4 and 4e^y

[cloud hinge]

You need to divide that rooted part by 2 also

,w -x^2+2x-e^y=0, x=

[obsidian monolith]

Results provided by WolframAlpha

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

Requested by Joe Mama#8435

[cloud hinge]

,w -x^2+2x-e^y=0

[obsidian monolith]

Results provided by WolframAlpha

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

Requested by Joe Mama#8435

[cloud hinge]

I mainly use wolfram alpha to verify ans

[jolly trout]

how do I factor this x^3 - 2x - 2 (by hand)?

[viscid thistle]

am i right in making the assumption (just to check my answer) that the highest volume given this surface area would be a lidless cube?

because if so i think i screwed up in my method

also tag me if you wanna answer cos ima go offline

[cloud hinge]

,w x^3-2x-2

[obsidian monolith]

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Requested by Joe Mama#8435

[cloud hinge]

Its irrational solution

[cloud hinge]

Use depressed cubic formula

[cloud hinge]

can you figure out the tank dimension?

@viscid thistle

[viscid thistle]

Wdym, like calculate the maximum for x?

[cloud hinge]

Nono

wait

which part are you asking?

[viscid thistle]

So I used the method of finding the maximum value for x and then I substituted that into the volume formula

[cloud hinge]

Oh

[viscid thistle]

But I'm wondering if it's true to assume the maximum volume of the tank is when it is cubic

[cloud hinge]

well

it is true

but

If you want to solve it with step

You need to understand that "the maximum or minimum value of a function is reached when its first derivative is 0"

Have you learn this?

[viscid thistle]

Yeah

[cloud hinge]

Yeah

[viscid thistle]

Yeah I get that part I just wanna check my answer against itself if ygm

[cloud hinge]

if you wanna just check

You can use wolfram maybe

[viscid thistle]

Bc when I did a fifth of the surface area (bc lidless) and square rooted that to find the side length of the cube and then cubed that, I got a different volume

[viscid thistle]

Wolfram?

[cloud hinge]

yeah