precalculus

[spring thunder]

$$A=\pi r^2 + \frac{50}{r}$$

[spring thunder]

$$A=\pi r^2 + \frac{50}{r}$$

[spring thunder]

Dem my phone is really a potato

[dusky vale]

damn you're typing this on your phone? lol

[spring thunder]

So the goal is to minimise this

(Yea)

[sullen oracle]

how can i solve lim (tg(3x)/sin(4x), x->0?

[trim yarrow]

Is tg short for tangent?

[sullen oracle]

yes

[dusky vale]

You learn L'Hospital's rule?

@sullen oracle

[hexed ermine]

You can find it analytically

If not

[thick raptor]

🔨

[sleek path]

anyone want to help with my beginner level vectors?

Two vectors, a and b have a common starting point with an angle of 120° between them. The vectors are such that |a| = 3 and |b| = 4

a) Calculate |a+b|
The textbook says its √3 but I keep getting √13

[civic plaza]

@sleek path I get the same answer as you

[viscid thistle]

I don't know why, but I am really struggling to get beyond this. Without a calculator how does 11pi/4 equal pi/4? I can't seem to figure out that calculation.

[civic plaza]

? that's not true @viscid thistle

is there any more information ?

[viscid thistle]

Sorry, working with radians. I couldn't figrue out the following problem so went on mathway to enter the question and work through the solution with the example but stuck on how they got pi/4.

I know the sin(11pi/4) = x and cos(11pi/4)= y. But I keep getting the wrong answers and can't figure out how 11pi/4 equats to pi/4 radians. For some reason I just can't wrap my head around it. I have the answer but not sure how they got it.

Sorry.. i guess my question is how do I figure out sin(11pi/4) without using a calculator?

[civic plaza]

well the period of the sin and cosine function is 2pi so that means the value of sin(theta) and cos(theta) is the same for any theta+2piK, where K is an integer

so you should add/subtract 2piK to get a theta in between 0 to 2pi

11pi/4-2pi is 3pi/4

which you should know the sin and cosine of @viscid thistle

[viscid thistle]

Awesome thank you!

[full palm]

I'm struggling in Trigonometry. Everything else in precalculus has come pretty easy so far and now I'm losing my pace. Can anyone explain some key concepts for a few minutes?

I'm currently learning from Stewart Precalculus with Khan Academy as a supplement.

[wooden ocean]

if u wanna be good at trig, you need to understand the construction of the unit circle, understand the sine/cosine/tan graphs and how they relate to the unit circle, and know the identities of sine,cosine,tan

[full palm]

In the textbook, to find a reference number I can find the solution by subtracting the radians from pi or 2pi. I get it hit and miss, I'm not sure how that relates to what quadrant the terminal point is in. And I'm also having trouble evaluating trig functions exactly.

[wooden ocean]

You need to look at a unit circle

to eval trig functions

for each angle (in rad), there will be a corresponding coordinate: (x,y)

the x coordinate is a solution to the COS(angle)

likewise, the y coordinate is a solution to the SIN(angle)

tan is just sin/cos

[viscid thistle]

how do I identify the intervals on which the function is increasing, decreasing, and constant, for h(x) = 0.5(x+2)^2-1

[bitter nova]

try to sketch it and hope it makes it clear

[rocky bison]

differentials

and the nature of the curve

@viscid thistle

[viscid thistle]

?

I graphed it already

[rocky bison]

o

Then it should be kinda obvious

There will be a single point of which the rate of increase is 0

I'm assuming that's what you meant by 0

Although it's never increasing/decreasing by a constant amount

the rate of change is always instantaneous

[viscid thistle]

alright then

but I really need help understanding this

[hexed ermine]

Differentiate and set the derivative = 0 to find critical point, with a quadratic, you have one critical point, at this point use a table with test values to the left and right of the critical point to show if that interval is decreasing or increasing

[full palm]

@wooden ocean
I finally got it, thanks to you!

[wooden ocean]

Got what

solving trig equations

[stray obsidian]

number 25 im having trouble setting it up

im setting it up wrong because i cant subtract cuz i get a negative number

[echo smelt]

you'd have to multiply the 12000-X by both .045 and .04

So I think it's (.04(12000-X)) + (.045(12000-X)) = 525

I'm like

80% sure

[stray obsidian]

oh ok ty

[hexed ermine]

Yeah Nick has it

[echo smelt]

Sweet

[honest path]

Using synthetic division and the rational zero test, it should be able to solve for x^4 - x^3 - 2x^2 -4

The zeros are -1 and 2

but I cant seem to solve it with synthetic division

unless I am just doing it wrong in the first place

whenever I use 2 or -1 in synthetic division, I end up with a remainder

[hearty plinth]

I need help guys

I don't know how to graph this

[proven marten]

Can't see anything.

[gritty blaze]

Not a perfect picture but it's a high resolution one.

[proven marten]

That still doesn't change the fact I can't see anything.

[gritty blaze]

Zoom in.

[proven marten]

no u

[wooden ocean]

LOL ^

[brave void]

How do I simpifiy this

<@&286206848099549185>

When i try to simp it

I get nasty factions

[wide frost]

first of all, u can cancel out x

[brave void]

yes

So I got h + 4/h+x + 4/x

common denominator?

[wide frost]

ye

or just split the sum apart

[brave void]

which would be

4(x+h)?

wait

wait wtf

I'm getting tripped

[vivid harbor]

maybe give everything a common den of (h + x)

[brave void]

4(h+2x) over x(h+x)

[vivid harbor]

oh wait, x(h + x)

[brave void]

okay

So I got that

But I still got it over h

[vivid harbor]

yep 😄

[wide frost]

=tex \frac{h}{h}+\frac{4}{h^2+hx}-\frac{4}{hx}

[vivid harbor]

scale num and den by 1/h

[brave void]

too many numbers

holy shit

[dire sand]

Letters 😉

[brave void]

I'm stuck

Because if I plug in h

I get 0 here

[wide frost]

i would simplify to this:

=tex \frac{h}{h}+\frac{4}{h^2+hx}-\frac{4}{hx}

[brave void]

then?

[wide frost]

then h/h=1

[brave void]

yes

[wide frost]

and then u need to get a common denominator for the rest

because in its current form, when u plug in h=0, u get inf-inf which is indeterminate

[brave void]

yes

so I got 1

Plus those other two fractions

I'm still stuck thjere

Those fractions suck

@wide frost

[wide frost]

=tex \frac{4hx-4h^2-4hx}{h^3x+h^2x^2}

[brave void]

what the hell

Where'd that come from

[wide frost]

common denominator

[brave void]

I'm so confused

[vivid harbor]

these would be the steps before what he just gave you

though i’m seeing if i can solve w/o expanding

[brave void]

okay

So we're there

[wide frost]

and now u can cancel h^2

[brave void]

yes

h is 0

[wide frost]

and then its not indeterminate anymore

[brave void]

-4 over x^2

[vivid harbor]

i get 1 - 4/(x^2) as the final answer (sorry i can’t latex)

[wide frost]

exactly

[brave void]

ty guys

💗

[dire sand]

If I got a question do I hope over to the MATH HELP channel or should I be asking it here? 👀

[brave void]

ask it here

I didnt' mean to hog the channel XD

[wooden ocean]

question channel

[dire sand]

How do I know which ones are available?

How long does the last message have to be to give me the a okay

Or as long as nobody is currently talking is it fine?

[vivid harbor]

i assume if the last question seems resolved you can just go ahead

[dire sand]

Oke

[vivid harbor]

the third channel seems open

idk what greek symbol that is

[past yoke]

delta

[vivid harbor]

ah, ty

[past yoke]

in cursive delta looks like an eighth note with a half note body

[vivid harbor]

i’m assuming the triangle is uppercase delta then

[past yoke]

yes, a triangle is uppercase delta (unless it's on its tip, then it's nabla)

and if the floor is missing (so it's an upside down V) it's uppercase lambda

[vivid harbor]

ah

[viscid thistle]

how do I find the range of f(x) = (x+4)/(x+2) ?

[vivid harbor]

think of the range as all the possible outputs the function could give

[viscid thistle]

yeah

[sullen shoal]

...

[patent beacon]

For example, f(x) can never be 1, since that's the horizontal asymptote

So that's not part of its range

[viscid thistle]

I see

but looking at the graph I don't get how to do it

[patent beacon]

= pup graph (x + 4)/(x + 2)

[vivid harbor]

n i c e wao

[viscid thistle]

yeah so how do I find the range

[vivid harbor]

(-inf, 1) and (1, inf)

[viscid thistle]

but how

[patent beacon]

Well, you can tell it's a rational function, and numerator/denominator are the same degree, so there must be a horizontal asymptote

The horizontal asymptote is the ratio of the leading coefficients

So since it's (1x + 4)/(1x + 2)
The asymptote is 1/1 = 1

[viscid thistle]

I don't get how you got (1x + 4)/(1x + 2)

[patent beacon]

That's your function

[vivid harbor]

it’s just the function with the 1 coefficient shown

[patent beacon]

Let's say your function was (3x - 1)/(2x + 2)
The asymptote would be 3/2

[viscid thistle]

yeah I see that

[vivid harbor]

the values of f(x) will forever approach but never touch 1

[patent beacon]

Or, if it were (5x² + 7x - 2)/(2x² - 4x - 2)
The asymptote would be 5/2

[vivid harbor]

and extend to neg and pos infinity

which is why it’s split into (-inf, 1) and (1, inf)

[viscid thistle]

but why 1?

I don't get it

the graphs aren't anywhere near 1

[patent beacon]

Because the horizontal asymptote is 1
That is, the graph never touches the line y=1

[spring thunder]

$$\frac{x+4}{x+2} = 1 + \frac{2}{x+2}$$

[vivid harbor]

you need to consider the graph as it extends infinitely

it’s just cut off here

[patent beacon]

So there is no x value that gives y = 1, so 1 isn't in your range

[viscid thistle]

ok then

but so what if the line doesn't touch 1?

[vivid harbor]

soft bracket )

[viscid thistle]

what

[patent beacon]

Then 1 isn't in your range

So your range is all real numbers, except 1

[vivid harbor]

yeah

i shouldn’t have assumed you were taught the bracket notation rip

[viscid thistle]

I was

[patent beacon]

But yeah, bracket notation is a good way to show it.
(-∞, 1) ∪ (1, ∞)

[viscid thistle]

yeah I know that

[dire sand]

Quick question folks

What happens when you square root a radical

Or put a radical in another radical

[past yoke]

what do you mean mean by put into? radicals contain no x, but a variety of coefficients

[dire sand]

Ex:

Do you addition or multiply the roots in order to find the simplified form?

[past yoke]

aaaaah

those are

$$(x^{\frac{1}{3}})^\frac{1}{2}$$

[past yoke]

which you multiply

$$(x^{\frac{1}{3}})^\frac{1}{2}=x^{\frac{1}{3}\cdot\frac{1}{2}}$$

[past yoke]

you can quickly try it out with naturals on paper

just write out what (x²)³ means when you write it out completely and count the xs (to understand what it means for exponents. you then have to apply that roots are just exponents, too)

[dire sand]

When you multiply exponents add to each other correct?

That would just get me two fractions as exponents

[past yoke]

$$a^b\cdot a^c=a^{b\cdot c}$$

[past yoke]

if you mean that

crap

$$a^b\cdot a^c=a^{b+c}$$

[dire sand]

Yeah

[past yoke]

now it's correct

[dire sand]

So that means I have two fractions up as exponents

[past yoke]

messed up the exponent in the first rendering

[dire sand]

Which isnt really simplified

Although it still works, how would you put it under a radical?

[past yoke]

$$\sqrt[3]{\sqrt[2]{x}}=\sqrt[3\cdot 2]{x}=\sqrt[6]{x}$$

[past yoke]

I hope that's what you asked for

[dire sand]

Thats the one

Thanks

@past yoke YO, I helped someone out, in the same channel you helped me. Feels good man 😤👍

[past yoke]

nice

[upbeat comet]

I need some help with factoring 6x^3+125

[slender river]

sum of cubes

wait fuck 6 isnt a cube

[upbeat comet]

lol

I was thinking of that and said the same thing

[bitter nova]

you can factor it it'll just look bad

[upbeat comet]

The last 5 questions on my study guide are these suckers

Can't even do that one

[viscid thistle]

(a^3+b^3) = (a+b)(a-ab+b^2)

(i think)

[upbeat comet]

Yeah we were supposed to memorise those tonight for tomorrow

[viscid thistle]

You shold have heard the acronym SOAP

same opposite always positive

[upbeat comet]

Algebra Identities

I have never been taught SOAP

[vivid harbor]

same

but good to know now

[upbeat comet]

yeah lol

[viscid thistle]

Pay close attention that

a^3 - b^3

SOAP

(a-b)

same

(a-b)(a^2+ab+b^2)

same opp always +

[upbeat comet]

Oh that makes sense

[viscid thistle]

Thats how you factor the sum or difference of cubes

[upbeat comet]

is my a and b 6 and 125?

[viscid thistle]

6x^3 + 125

oof

[slender river]

not easy to make into what we want aha

u gl lee

[viscid thistle]

not ez fren

=pup factor 6x^3 +125

[granite stirrup]

Do more with Wolfram|Alpha pro

Query made by

@languid wind

[viscid thistle]

double check your problem

@upbeat comet

[upbeat comet]

Okay

[viscid thistle]

Make sure is 6x^3

[upbeat comet]

OOOOOOOOOOppss

[viscid thistle]

yee fren

[upbeat comet]

64 not 6

[viscid thistle]

[upbeat comet]

[viscid thistle]

4^3

whats 4 x 4?

x 4

[upbeat comet]

16

oh

[viscid thistle]

yep

so we know it's cubes now

64x^3+125

so if ax^3 = 64

cube root of 64 = ax

so ax is how much?

[upbeat comet]

gimme a sec

[slender river]

oh if it's 64 then its nice and pretty

ez soaping haha

[upbeat comet]

So are you

[slender river]

aw ty ❤

[upbeat comet]

Lemme think though

[upbeat comet]

I need to find a number that when cubed it is 64?

[viscid thistle]

yes

[upbeat comet]

Then i'm on track

[viscid thistle]

what times itself times it self is 64?

[upbeat comet]

8

[slender river]

y0sss

[viscid thistle]

8x8x8 = big numbre

[upbeat comet]

oh wrong number

[slender river]

wait i cant read haha

[upbeat comet]

Lol yeah same

[slender river]

should texed that

haha

[upbeat comet]

I read that too fast

[viscid thistle]

Does 5^3 work?

whats 5 x 5 x 5

@upbeat comet

Is that too much or too little?

[upbeat comet]

Too much

[slender river]

waht

ty

[viscid thistle]

how about 4?

[slender river]

that was bad

[viscid thistle]

whats 4 x 4 x 4

[upbeat comet]

OOOO

[viscid thistle]

yee fren

👏

[upbeat comet]

I was thinking 4 but I doubted myself

[viscid thistle]

4^3(x^3) + 125

what times it self times itself = 125

4 isn't enough

lets try 5?

whats 5 x 5 x 5

[upbeat comet]

125

[viscid thistle]

4^3(x^3) +5^3

amirite?

You agree with this?

[upbeat comet]

Yep

[viscid thistle]

What if I said

(4x)^3 +5^3

is that same?

[upbeat comet]

Yeah that's the same

[viscid thistle]

so what does a =?

[upbeat comet]

4?

[viscid thistle]

Well be careful here

its 4x

because a^3 is (4x)^3

how much is b ?

[upbeat comet]

5

[viscid thistle]

so lets try the soap method

(4x)^3 + 5^3

whats the first term?

( )

[upbeat comet]

Umm

[viscid thistle]

remember the form (a+b) is the first term

where the sign of SOAP

says it is the S, the same

so fill it in

(a+b)

what do you get?

[upbeat comet]

4x+5

[viscid thistle]

(4x+5) ( )

Whats the first term in the trinomial ?

[upbeat comet]

a+b

[viscid thistle]

ok we got that

[upbeat comet]

a

[viscid thistle]

lets go to the second part now

[upbeat comet]

a^2

[viscid thistle]

how much is a

[upbeat comet]

Wouldn't it be 4x^3?

[viscid thistle]

if a = 4x

and we square it

you should get (a)^2

plug a in now

what do you get?

[upbeat comet]

16x^2?

[viscid thistle]

yep

(4x+5)(16x^2 )

whats the sign now?

• or - ?

SOAP

so we're on O now

what did O stand for?

[upbeat comet]

it's minus

Opposite

[viscid thistle]

(4x+5)(16x^2- )

whats the next term ?

[upbeat comet]

ab

[viscid thistle]

how much do you get when you get a * b

[upbeat comet]

20x?

[viscid thistle]

whats the next sign?

AP

[upbeat comet]

Always Positive

[viscid thistle]

(4x+5)(16x^2-20x+ )

whats the last term?

[upbeat comet]

b^2

so it is 25

[viscid thistle]

(4x+5)(16x^2-20x+25)

[upbeat comet]

[viscid thistle]

You know how to foil?

check the first and last term

[upbeat comet]

First Out Inside Last?

[viscid thistle]

4x(16x^2)

and 5(25)

what do you get

[upbeat comet]

64x^3

125

[viscid thistle]

[upbeat comet]

AHHHHHHH

[viscid thistle]

That all you needed for this problem?

[upbeat comet]

Well the others are pretty much similar to this

[viscid thistle]

Use the soap

lol

[upbeat comet]

I will ping if I need help though

[viscid thistle]

Ok

[upbeat comet]

SOAP is useful af

[viscid thistle]

S O AP

[upbeat comet]

FOIL is used to check my answer?

[viscid thistle]

If you want to yes

The soap works

[upbeat comet]

I will attempt the second one by myself then

64x^3+27

[viscid thistle]

how much is a?

[upbeat comet]

(4x+3)(16^2-16x+9)

[viscid thistle]

the middle term is (4x)(3)

the right is right

==4*3

[granite stirrup]

12

[upbeat comet]

OOps

So now, my answer is right?

[viscid thistle]

If you changed the 16 to 12 yes

💯

[upbeat comet]

I will do one more to check with you and then finish the last two without bothering you

[viscid thistle]

you missed the x though as well now that I see it

(4x+3)(16^2-16x+9)

(4x+3)(16x^2-12x+9)

[upbeat comet]

Oh yeah I did lol

8x^3+1

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

@viscid thistle

[viscid thistle]

yo

lemme check

yee fren

=pup expand (2x+1)(4x^2-2x+1)

[granite stirrup]

Do more with Wolfram|Alpha pro

Query made by

@languid wind

[viscid thistle]

@upbeat comet

[upbeat comet]

I have been looking foward to playing all night. Time to speed through this!

@viscid thistle

[granite stirrup]

Do more with Wolfram|Alpha pro

Query made by

@upbeat comet

[upbeat comet]

(4x-3)(16x^2+12x+9)

=pup expand (4x-3)(16x^2+12x+9)

[granite stirrup]

Do more with Wolfram|Alpha pro

Query made by

@upbeat comet

[upbeat comet]

=pup expand (3x-5)(9x^2+15x+25)

[granite stirrup]

Do more with Wolfram|Alpha pro

Query made by

@upbeat comet

[upbeat comet]

=pup expand (10x-1)(100x^2+10x+1)

[granite stirrup]

Do more with Wolfram|Alpha pro

Query made by

@upbeat comet

[upbeat comet]

All because I forgot an x LOL

Thanks again! @viscid thistle

[viscid thistle]

yee

SOAP fren @upbeat comet

yw

[neon vale]

Help with 2?!?!?

[past yoke]

the hint kinda gives it away already

but maybe this helps: the transformations done to the function affect it only in one direction at a time. range only lives in y-direction, domain only in x-direction

[full palm]

What does it mean when a prime is added to a variable or series of variables and not a function?

Like a0', a1', a2', etc

[clever dune]

In calculus, maybe d/dx of a0

Or maybe it’s just a new separate variable.

[full palm]

I don't think it's in the context of a derivative, I've seen it in CLRS to describe a permutation. What would be a clear, formal definition of it's usage instead of my broken understanding?

[past yoke]

sometimes prime, asterisk, tilde, bar, roof are just used to signal that a variable is based on another one, but isn't the same

[sweet niche]

<@&286206848099549185>

[viscid thistle]

alpha + (90 - theta) = 90

is the line from the bottom to the top a straight line?

making 90 degree angle?

[trim coyote]

Can someone give me a visual demonstration of what happens when you compose two functions? I know how to perform function composition, but I imagine a point on a line. You then go over on the axis to the same value of the height (y) then either up or down to find the point whose y value is the output of the second function.

But what does it actually look like in the math universe to compose two functions?

It's easy to imagine a single number as an input to a function, but imagining a function as an input is somewhat difficult if you aren't just thinking of the output of one as the input of the other.

So does anyone have a good example?

[sweet niche]

@viscid thistlethe line from the middle is indeed straight down

@viscid thistle I don't really get how you got that expression

[viscid thistle]

oh

alright look at it like this

you have 3 triangles in that

one on the right, one on the left and a squishy looking one in the middle

right?

[sweet niche]

Yeah

[viscid thistle]

you tell me on the right, one has length 90 - theta

and you tell me the line in the middle goes straight down

and the horizontal line is straight

so we know it forms a 180 degree angle

the right one has length 90 - alpha and makes a right angle triangle

of angle 90

so the 3rd angle of the triangle on the right is alpha

it forms a straight line with the squished triangle

does it not?

then one of the angles of the squished triangle is 180-alpha

and another is theta

all angles in a triangle must sum to 180

so 180 - ((180-alpha) +theta) is your final angle

or alpha - theta

rearrange that

after you plug it into the final angle on the left hand triangle

where 90 -(alpha - theta) or 90 + theta is the angle

knowing all the angles of the left hand triangle must sum to 180

you get (90 + theta) + alpha + (90 - alpha) = 180

rearrange those and the equation before

and you get alpha = 180 - theta

all divided by 2

[timber saffron]

Is there a way to graphically find the complex roots of

=tex f(x)=x^3+1

[thick raptor]

why would u do graphically when this is obviously sum of cubes

[gritty blaze]

No.

[thick raptor]

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

[timber saffron]

Just curious. I know there are easy ways to do it

[clever dune]

@timber saffron Something about roots of unity on a unit circle?

[past yoke]

rearrange it to x^3=-1. see that it has a trivial root at -1. As it's a simple root, all roots are arranged evenly spaced along the circle of radius sqrt(|x|). so you draw your trivial root and then draw the regular polygon corresponding to your degree, which in this case is 3. so you get an equilateral triangle centered at the origin, with one vertex at -1 and the other vertices on the circle at the exact positions of the other roots

luckily the 3gon is constructible, so you can actually find them with a pair of compasses

[viscid thistle]

how do I find the domain of radical(4-x-2)

[rocky bison]

You mean

=tex \sqrt{4-x^{-2}}

[rocky bison]

@viscid thistle

[past yoke]

yea, but mathbot wouldn't render again for some reason

[rocky bison]

GO AWAY EMERIC

🔨

[spring thunder]

[rocky bison]

But the domain basically tells us what numbers we're allowed to input into a function

So what numbers are disallowed there

[past yoke]

anyways, the domain is the set of numbers for which the function does work. the square root works as long as its argument is greater or equal 0. so you have to examine that function and see where it is greater or equal 0.

[rocky bison]

👀

you ignored -2

0 is not allowed

[past yoke]

I mean you ahve to examime 4-x^-2

[rocky bison]

Ah

oke

my mistake

[past yoke]

like, when you enter a big number, you get basically 4-0

[rocky bison]

yea

[past yoke]

which is greater 0

[rocky bison]

well ok

missy

don't need to get sassy

[past yoke]

also, you function is even, as you hopefully saw

so it doesn't matter whether x is positive or negative, you get the same results

basically (since this is precalculus) you could plot your function manually to understand it. then find the critical values (mathematically, not graphically) and then argue with those

in this case you get an upside down, even hyperbola that is shifted 4 upwards, so all negative numbers are around 0 and all positive numbers are away from 0

it's critical (i.e. interesting) at 4-x^-2=0, which gives you x=+/- 0.5

[viscid thistle]

I mean radical4-x-2

with the -2 on the outside of the radical

[past yoke]

yes, that's what I was talking about

[fresh oar]

so I've been asked to find the complex roots z of some quadratic equation, and I found two roots via the quadratic formula: z = c +- sqrt(i), where c is just some constant that doesn't matter here. I'm wondering about the sqrt(i) part. Clearly there are two complex numbers whose square is i, but does sqrt(i) have two values? How do I deal with it inside this question? Can I just assume sqrt(i) to be the solution p to p^2 = i with the lesser angle/argument or do I have to account for both second-roots of i?

[past yoke]

you had a quadratic equation with complex coefficients?

[fresh oar]

yes

[past yoke]

wait, +-/sqrt i is just two solutions

not 4

you get a quarter circle and three quarter circle and if you then try plus minus, you will find they just switch places

[fresh oar]

oh

[past yoke]

but feel free to correct me. I'm just imagining what the solution looks like 😄

no

not quarter, eigth 😦

[fresh oar]

ah I'm beginning to see it now

thanks

I should've plotted them first

[past yoke]

I'm wrong

they are different.

sorry for confusing you

[fresh oar]

no you are actually correct
both roots are equal to the negative of the other root

just negate both the real parts and imaginary parts

[past yoke]

OH, right!

[charred rune]

Im having trouble figuring out how to do these fundamental theorem problems

[tiny totem]

well the second one u could easily factor by grouping

the first one you should do the rational root theorem first and then test the roots using synthetic division

[sleek path]

if X(-2,1,2) and Y(-4,4,8) are two points in R^3, determine the following:
A) XY and |XY|

both XY and |XY| are vectors just dont know how to put the symbol

nevermind I got it

nevermind I didnt

still need help

[proper stirrup]

How does one say what the domain is of a point of a piecewise function? Like 2x {x not equal to 0} and 1 {x=0}. Are the x-coordinates of the points just the domain?

[pure skiff]

in the above functuon domain is all of R

R being real numbers

because the functuon exists for all real numbers

you plug in 0 you get 0 and you plug in any other real number a and you get 2a

the function is discontinuous at point x=0 but it still has x=0 in its domain

@proper stirrup hope that helped

[indigo steppe]

What's the difference between composite and compose?

My teacher explained it weirdly

and google isn't helping much either

[viscid thistle]

@indigo steppe Composite and compose in what context?

[indigo steppe]

functions

[viscid thistle]

a composite function is a function with a function, like f(g(x))

[indigo steppe]

like f(g(x))

[viscid thistle]

ya, thats a composite

[indigo steppe]

ok and a compose is like domain stuff?

[viscid thistle]

im pretty sure they're the same thing

there's nothing that's a "compose"

[indigo steppe]

yeah that makes sense

ty

[viscid thistle]

np man

[proper stirrup]

@pure skiff thanks

[pure skiff]

np

[hearty plinth]

Whos willing to teach me one on one?

[proven marten]

'oceanography'

https://www.embibe.com/exams/real-life-applications-of-trigonometry/

[fallen cloud]

trigonometry, can be used for architecture (combined with stress tensors which i admittedly don't know much about, also calculus too), engineering, and generally building things

[viscid thistle]

@viscid thistle Trig tends to define the base of almost anything it seems
Constructions? Sure
Electricity? Sure
Volume analysis? Sure

[gilded timber]

what was the one theorem that stated that if a graph goes from negative to positive or positive to negative, it must have at least one real root?

[gentle hedge]

descarte's theorem

[viscid thistle]

hi can someone help me with this: "Dikembe has reflected the gunction g(x) = x^3 in the x-axis, vertically compressed it by a factor of 2/3, horizontally translated it 13 units to the right, and vertically translated it 13 units down. Three points on the resulting curve are (11, -23/3) , (13,-13), abd (15,-55/3). Determine the original co-ordinates of these three points on g(x)."
I traced the mapping rule back to (x-13 , (3/2y)+13). My x is right but my y isn't, what did I do wrong?

[viscid thistle]

<@&286206848099549185>

[past yoke]

today I noticed that german highschool books seem to very slowly phase out the definition notation for functions

in lower classes and somehow randomly later, function are written like this:

$$f:x\mapsto x^2+2x+1$$

[past yoke]

only in higher classes f(x)=x²+2x+1 gets used somehow

[spring thunder]

same in france

and this is the more formal definition in fact

[past yoke]

it's pretty weird, as the definition notation is pretty useless for a 6th grader

[thick raptor]

imo:

=tex f=x\mapsto x^2+2x+1

[thick raptor]

bettre notation

[past yoke]

well, that's far to abstract to be of any use there

[thick raptor]

=tex f^1=f\f^{k+1}=f\circ f^k

[past yoke]

I hate the chain symbol

[thick raptor]

=tex I_1=f\mapsto n\mapsto f^n(n)

[thick raptor]

[past yoke]

it's so unintuitive. why not just use brackets?

[thick raptor]

wdym?

[past yoke]

or is that just me being a programmer?

[spring thunder]

brackets for waht?

[past yoke]

$$f\circ f=f(f)$$

[thick raptor]

no

[past yoke]

it looks weird without arguments though

[thick raptor]

bad

Writing it like that leads to problems for higher order functions

For example

[past yoke]

(expecting some weird lambda shit)

[thick raptor]

=tex f=n\mapsto n+1\I_1(f)(3)=~?

[thick raptor]

=tex I_1(f)\ne I_1\circ f

[past yoke]

what is that I1 supposed to mean?

[thick raptor]

=tex I_1=f\mapsto n\mapsto f^n(n)

[thick raptor]

It maps f to the function n -> f^n(n)

where f^k is function iteration previously defined

[past yoke]

is that ^n as in iterated n times or as in power n?

[thick raptor]

=tex f^1=f\f^{k+1}=f\circ f^k

[thick raptor]

I defined that too ^

[past yoke]

iteration, I see

but that actually doesn't hurt my point, as I was talking about highschool

the circle symbol only appears once in highschool and that is in the definition of the chain rule

[thick raptor]

My point is why wouldn't you?

[past yoke]

and only if it is defined using it, which isn't necessary

[granite stirrup]

Rendering failed. Check your code. You can edit your existing message if needed.

[thick raptor]

@past yoke what were u tryin' to write

[past yoke]

the thing I wrote at the bot channel 😉

[thick raptor]

[warped anvil]

Um what

What do you need help with

[midnight veldt]

@warped anvil He wants you to do those tasks and PM it to him.

[copper saddle]

Paying 100 for someone to help with my final i have the pdf i want you to do it then explain (I am only allowed 15 mintues of questions)

[viscid thistle]

Please see #❓how-to-get-help

6. specifically

[copper saddle]

I did

[viscid thistle]

Nah you didn't

[copper saddle]

Its not a "test" its review of last year but its super long

Its last year finals test but we doing it as review for honework

Thats why i have pdf lmao

[viscid thistle]

Willing to pay 100

Sounds based

[copper saddle]

Ya

Based?

[viscid thistle]

The fuck out of here

[copper saddle]

Free works to lmao

[viscid thistle]

Yeah, if it's not test. Attempt questions yourself and ask in the questions channel. Don't expect people to just do your work for you.

[copper saddle]

Questions channel is unauthorized

[viscid thistle]

You really didn't read rules did you

[copper saddle]

What

Anyone know what the sume of this series is (-5 -7...

[viscid thistle]

#help-5 bruh

[random loom]

can i get some help with this?

Question D

[drowsy plaza]

Do you know the formula for slope?

[random loom]

i know but idk how to do it when its in fraction form

can i get a step by step on it?

[drowsy plaza]

okay can you show me?

yeah sure

[random loom]

show you what?

[drowsy plaza]

let's not worry about the fractions

plug the numbers in and show me what it should look like

we'll deal with the fractions after

[random loom]

( x1 - x2 ) , ( y1 - y2 )

isnt that how the formula for slope should look like

[drowsy plaza]

almost

let m be the slope

we have m = (y2 - y1)/(x2 - x1)

[random loom]

ohh ok

[drowsy plaza]

okay so now we have to select a pair of points to be (x1, y1) and (x2, y2)

can you do that?

[random loom]

yeah

[drowsy plaza]

show me

[random loom]

in fractions or no?

[drowsy plaza]

yeah

like I said we'll deal with the fractions after

[random loom]

oh so just give you some numbers?

[drowsy plaza]

using the slope formula I gave you

plug in the numbers

[random loom]

ohh ok

[drowsy plaza]

and let me know what you get

we will work it out step by step

but that's the first step

[random loom]

ill just give out random numbers alright?

[drowsy plaza]

lol no

[random loom]

oh what?

[drowsy plaza]

what do you mean random

[random loom]

i mean you told me to plug in "the numbers" but what numbers?

[drowsy plaza]

well you have (1/3, 3/2) and (2, 7/2)

and the formula is

=tex m = \frac{y_2 - y_1}{x_2 - x_1}

[drowsy plaza]

so you have to decide what you want your

[random loom]

ohhhhhh, i thought you meant to like do the formula with some numbers cause i took your meaning of doing fractions later too seriosuly

[drowsy plaza]

=tex (x_1, y_1)

and

[random loom]

yeah

[drowsy plaza]

=tex (x_2, y_2)

to be

[plain rose]

I don't think this is pre cal

[random loom]

i mean my whole course in grade 10 math is pre calculus

[drowsy plaza]

ok ok focus

[random loom]

alright.

[plain rose]

woah dang

[drowsy plaza]

what do you get?

[plain rose]

Grade 10 for me is alg 2

[drowsy plaza]

for m

can you show me?

[random loom]

ok Question, how do i ignore the fractions? in this problem?

[drowsy plaza]

well we'll deal with it after

okay for example

[plain rose]

convert to decimals

[drowsy plaza]

if I give you two points

(1, 2) and (3, 4)

then m = (4 - 2)/ (3 - 1)

that's what I want to see

the numbers plugged in

[random loom]

ohhhh alrighty

[drowsy plaza]

yeah

we can do the computations after

we're just taking it step by step

[random loom]

m = ( 1/3 , 7,2 ) (3/2 , 2 )

[drowsy plaza]

nope

remember

=tex m = \frac{y_2 - y_1}{x_2 - x_1}

[drowsy plaza]

there's a subtraction and division there but I don't see it in your answer

[random loom]

ohhhh then

m = ( 7/2 - 3/2 ) / ( 2 - 1/3 )

[drowsy plaza]

nice!

okay so now we will deal with the fractions seperately

so on top we have 7/2 - 3/2

do you know how to subtract two fractions with the same denominator?

or how we can rewrite it

[random loom]

unfortunately i forgot, is it numerator - numerator?

and simplify it by the end right?

[drowsy plaza]

exactly

so we have 7/2 - 3/2 = (7-3)/2

which gives us 4/2 = 2

do you follow?

[random loom]

yeah lol just taking notes on it

[drowsy plaza]

cool

so we have

=tex \frac{2}{2 - \frac{1}{3}}

[drowsy plaza]

now we have to deal with the denominator

[random loom]

yeah since 1 and 3 isnt the same

[drowsy plaza]

What do you mean by that?

[random loom]

isnt there a hidden 1 denominator on the 2

[drowsy plaza]

yeah

so now how can we rewrite 2 to have the same denominator as 1/3?

[random loom]

uhm 2/3??

[drowsy plaza]

no

okay so let's say we have 2/1

now what is 3/3?

what number is that equal to

[random loom]

1

[drowsy plaza]

right

now if we multiplied 2 * 1

what does that give us?

[random loom]

2

[drowsy plaza]

so the number didn't change

now what if we did 2 * 3/3

what does that give us?

[random loom]

still 2

[drowsy plaza]

right because we're multiplying by 1

so if we do

=tex \frac{2}{1} \times \frac{3}{3}

[drowsy plaza]

what does that give us?

[random loom]

6
3

[drowsy plaza]

=tex \frac{6}{3}

[drowsy plaza]

^ that?

[random loom]

yep

[drowsy plaza]

cool!

and it's still 2 because we multiplied it by one

but now we have it in a form that's useful to us

[random loom]

yeah

[drowsy plaza]

so now we have

=tex \frac{6}{3} - \frac{1}{3}

[drowsy plaza]

what does this give us?

[random loom]

5

[drowsy plaza]

are you sure?

[random loom]

yeah

oh wait

5
3

[drowsy plaza]

right!

nice

so we have

=tex \frac{2}{\frac{5}{3}}

[drowsy plaza]

now we're almost done

okay so that fraction in the denominator is pretty annoying

what do we get when we have

=tex \frac{5}{3} \times \frac{3}{5}

?

[random loom]

15
9

[drowsy plaza]

are you sure?

[random loom]

yeah

[drowsy plaza]

do you know how to multiply fractions?

[random loom]

that's how i thought you did it, please correct me if im wrong.

[drowsy plaza]

okay so to multiply fractions

we multiply the numerator

and we multiply the denominator

so

=tex \frac{5}{3} \times \frac{3}{5} = \frac{5 \times 3}{3 \times 5} = \frac{15}{15} = 1

[drowsy plaza]

Does that make sense ?

[random loom]

oh alrightt

so just multiply it by the numbers next to it?

[drowsy plaza]

Yeah

[random loom]

oooohhh alrighty

[drowsy plaza]

okay!

so now we have

=tex \frac{2}{\frac{5}{3}}

[drowsy plaza]

also one more thing

notice that

we are multiplying 5/3 by 3/5

3/5 is called the reciprocal of 5/3

anyway so you saw how when we multiplied 5/3 by 3/5 it gave us 1?

[random loom]

yeah

[drowsy plaza]

well that's kinda nice

it gets rid of the denominator!

from

=tex \frac{2}{\frac{5}{3}}

[drowsy plaza]

and remember when we multiplied 2 by 3/3

because that's the same as multiplying by 1?

[random loom]

yeah i remember

[drowsy plaza]

well let's do that again

=tex \frac{2}{\frac{5}{3}} \times \frac{\frac{3}{5}}{\frac{3}{5}}

[drowsy plaza]

=tex \frac{2}{\frac{5}{3}} \times \frac{\frac{3}{5}}{\frac{3}{5}} = \frac{\frac{6}{5}}{1}} = \frac{6}{5}

[granite stirrup]

Rendering failed. Check your code. You can edit your existing message if needed.

[drowsy plaza]

=tex \frac{2}{\frac{5}{3}} \times \frac{\frac{3}{5}}{\frac{3}{5}} = \frac{\frac{6}{5}}{1} = \frac{6}{5}

[drowsy plaza]

does that make sense?

[random loom]

took me a bit to analyze it but yeah!

[drowsy plaza]

COOL!

so we have

=tex m = \frac{6}{5}

[drowsy plaza]

now I encourage you to do exercises on multiplying fractions

adding and subtracting fractions

[random loom]

yeah i have a booklet on that gonna go and work on it after

thank you so much!!

[drowsy plaza]

https://www.khanacademy.org/math/arithmetic/fraction-arithmetic

go through this when you have time

fractions aren't so scary!

[random loom]

yeah over thinking it just gets me sometimes

thank you so much again!

[drowsy plaza]

just keep practicing

and when you're confused ask yourself why

try not to let these "gaps" in your knowledge go unquestioned

they can add up later on

Good luck 😃

[random loom]

thanks dude! have a nice day

[drowsy plaza]

Cya

[void patio]

@random loom i was in your shoes once, and i'm much better now. although, I can't say i feel any different compared to before I got better. if you need any help feel free to ask me. i know exactly where you're coming from.

[viscid thistle]

Hello?

I need someone to teach me Sigma notation

[ocean cove]

@viscid thistle do you still need help?

[viscid thistle]

I’m back

And yeah

So anyone up?

[viscid thistle]

I dunno how to use mathbot

<@&286206848099549185> can someone guide me with how to solve this?

[viscid thistle]

Separate them to start

Then add them

After that you get Sum(2ak) +sum(k)

In this first sum, 2 is a constant so you can multiply the sum of ak by 2

2(Sum(a_k) +sum(k)

@viscid thistle

I tried it

My final answer was 3133

What do you do about the k=4

Sums was long time ago

=pup summation 18 to 4 of n

[granite stirrup]

Do more with Wolfram|Alpha pro

Query made by

@languid wind

[viscid thistle]

%pup summation 4 to 18 of n

=pup summation 4 to 18 of n

[granite stirrup]

Do more with Wolfram|Alpha pro

Query made by

@languid wind

[viscid thistle]

=pup summation 4 to 18 of 2a_k

[granite stirrup]

Wolfram|Alpha didn't send a result back.
Maybe your query was malformed?

[viscid thistle]

I’m kinda screwed

[viscid thistle]

term x compound *

[viscid thistle]

solving for pmt..... it is returning -607 instead of the desired 284.....

pv = 179900 * 0.1

[drifting hollow]

for this question

$$x^2 - 3x - 5 = (x-\alpha)(x-\beta)$$

[drifting hollow]

i'm told that

without finding the roots

i need to find a new quadratic with same roots alpha^2 and beta^2

i've tried this:

but now i don't know what to do

it seems that the answer would have been x^2-29x+100

if i square it to try to get it out of x root y

it just goes to a y version of the x variable quadratic above

i got a different question in a mock exam which was similar but instead of finding a quadratic with roots alpha^2 beta^2, it was just alpha-1, beta-1

for that one i set y = x-1, x = y+1, plugged (y+1) in all of the x positions in the quadratic and simplified

with this one i'm not sure because if i do the opposite it should be square root, but the final form i have doesn't seem like it's a "complete" quadratic

since there's a square root

am I doing something horribly wrong or is the root(y)^2 etc etc correct?

[viscid thistle]

<@&286206848099549185>

[drifting hollow]

feelsbadman

[random loom]

lf someone to correct my set notation on Domain & Range, been worrying about it lately since a unit test is coming up

[hexed ermine]

Okay

[random loom]

im just gonna jot down some random numbers

lets say lowest at X is -6 and highest is 10
and lowest at Y is -3 and highest is 12

tell me if im wrong please and what the right answer would be step by step

{ X | -6 < X < 10, XER}

{ Y | -3 < Y < 12, YER}

i dont know when to use the more than or equal to sign and to use < or > so that would be cool if you explained it too

[viscid thistle]

looks good

except per your language it would be
{ X | -6 <= X <= 10, XER}
{ Y | -3 <= Y <= 12, YER}

its X <= 10 if the domain/range includes 10 and X < 10 otherwise

[random loom]

ohh alrighty thanks

[sleek path]

A mass of 20kg is suspended from a ceiling by two lengths of rope that make angles of 30° and 45° with the ceiling. Determine the tension in each of the ropes