help-23

[lean otter]

Sin =5x square

[true crescent]

?

[lean otter]

Sec =x+1/4x

[true crescent]

What is the question asking?

[lean otter]

Sec theta + tan theta is

[true crescent]

(1+x+1/(4x))/(sqrt(1-(x+1/(4x))²))

4|x|(1+x+1/(4x))/(sqrt(-16x⁴ -8x²)))

2(1+x+1/(4x))/(sqrt(-4x² -2)))

(2+2x+1/(2x))/(sqrt(-4x² -2)))

Oops I forgot (4x)²

4|x|(1+x+1/(4x))/(sqrt(4x²-16x⁴ -8x²-1)))

4|x|(1+x+1/(4x))/(sqrt(16x²-16x⁴ -8x²-1)))

4|x|(1+x+1/(4x))/(sqrt(8x²-16x⁴ -1)))

(4|x|+4|x|x+|x|/x)/(sqrt(8x²-16x⁴ -1)))

That last one should be correct

[lean otter]

Final answer?

[true crescent]

I think! maybe there's more to simplify but I have no clue how to get rid of the square root and the absolute values

[lean otter]

Because here answers are 2/x
X/2
2x

?

[true crescent]

I'm not sure

[true crescent]

Is this a test? Because that's forbidden in this discord channel

[lean otter]

No

Exercise

[true crescent]

ok then.

[lean otter]

Is there an error

Cause sometime the answers to the questions are not there

[true crescent]

maybe that's your techer's problem

[lean otter]

Yh I know

Then just leave it I'm going to ask him

To check it

[true crescent]

ok

[lean otter]

What about this

[true crescent]

simplify the term on the right to get 1/cos²x

or which is the same sec²x

now you can use your identities

[lean otter]

1/cos²x=1-e²

[true crescent]

no 1/cos² x = sec² x = 1+ tan²x=2-e²

[lean otter]

(2-e²)½?

[true crescent]

why?

sec x + tan³x cosec x = 1/cos²x=sec²x=1+tan²x=2-e²

[lean otter]

Oh I see now but here the at ½ or 3/2 at the end

?

[true crescent]

what do you mean? it's just 2-e²

[true crescent]

Ok I'm wrong

sec x + tan³x cosec x = 1/cos³x

=(1/cos²x)^(3/2)

=(2-e²)^(3/2)

which 2

[lean otter]

3/2

[true crescent]

the 2 in 1/cos²x cancels with the 3/2 to get back to 1/cos³

[lean otter]

OK clear

A big thanks Sir

[true crescent]

You're welcome

[true crescent]

If you don't have any more questions type .close to let others use this channel

[lean otter]

OK sir

Thanks again

.close

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[urban dock]

Guys simple question. What is a derivative in terms of limit?

[urban dock]

Every time I look it up it gives me the formula but how does the formula work

[cosmic grove]

$f'(a) = \operatorname{lim}_{x\rightarrow a} \frac{f(x) - f(a)}{x-a}$

[flat frigate]

Ｈｅｒｅｌｓ

[cedar rover]

$f'(x) = \lim_{h \to 0} \frac{f(x) - f(x + h)}{h}$

[flat frigate]

Nonna

[cedar rover]

Sometimes you see it like this too

It's just Δy /Δx over very small distances

[mossy wigeon]

someone help me in my mathh homeworkk

[cosmic grove]

no

[mossy wigeon]

oh

[urban dock]

But basically you take the average distance of two points in a graph?

[mossy wigeon]

uhm its kinda hard

[cosmic grove]

infinitesimal distance

[urban dock]

What do you mean by infinitely distance?

[cosmic grove]

infinitely small rate of change

[main lynx]

Geometrically the tangent line, say at a point (a,f(a)) seems to be the limit of the secant line as h approaches zero

[urban dock]

Oh I see I think I am starting to understand.

Basically I wanted to know this because I wanted to explain how just the idea of a limit makes up calculus

[main lynx]

I mean we can not talk about the limits of the lines, but can talk about the slope of this line so the slope of the tangent line at (a,f(a)) is $\displaystyle \lim_{h\to 0} \frac {f(a+h)-f(a)}{h}$

[cosmic grove]

geometrically thats the main idea

[flat frigate]

kartheek

[urban dock]

Okey guys thank you

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[unique valley]

How do I show this, do I do so via induction? However when applying L’Hosiptals rule for n=1 it doesn’t work

[split ether]

Perhaps start with n = 0 then?

[thick sparrow]

Apply squeeze theorem with $$\frac{e^{-1/x^2}}{x^{3n}} \leq \frac{1}{x^{3n}}$$.

[flat frigate]

jimmy1234

[true crescent]

$\lim_{x\to 0}\frac{e^{-x^2}}{x^{3n}}=\lim_{y\to \infty}y^{3n}e^{-y^2}$

[flat frigate]

Mohammad

[true crescent]

$\lim_{y\to \infty}\frac{y^{3n}}{e^{y^2}}=0$

$\lim_{y\to \infty}\frac{y^{3n}}{e^{y^2}}=\lim_{z\to \infty}\frac{z^{\alpha}}{e^{z}}=0$

[flat frigate]

Mohammad

[true crescent]

Where $\alpha=3n/2>0$

[flat frigate]

Mohammad

[true crescent]

Here I'm using the theorem in Apostol's calculus which states that for all $\alpha,\beta>0$ the limit $\lim_{x\to \infty}\frac{x^{\alpha}}{e^{\beta x}}=0$

[flat frigate]

Mohammad

[safe radish]

@unique valley Has your question been resolved?

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[iron escarp]

Hey, quick question. I know this may seem simple, but this is Australian Year 8 level mathematics.

[compact wraith]

yeah send

[iron escarp]

So I need to prepare notes for the tests and one of the things I want to add is the formula for the distance between two points on a graph. I was searching through the internet and found this; d=√((x2-x1)²+(y2-y1)²)

is this correct or should I have something else (possibly simpler?)

[stray bluff]

yeah thats right

[iron escarp]

Alright sick

[compact wraith]

so this is just pythagoreas's theorem!

draw the two points on a graph

join the points up with a line

then create a right angle triangle where the base is a line parallel to the x axis and the height is a line parallel to the y axis

[iron escarp]

Mhm

Well that's it for now, thanks.

.close

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[vapid kiln]

Write down the formula for the quadratic polynomial that satisfies a=2 and the roots are x=1 and x=5.5.

[vapid kiln]

The teachers have told me that the result is f(x)=2x^(2)-13x+11

I just need to find out the steps inbetween

[lean otter]

you can use the discriminant for that, i think

or, even easier, use the factored form of the quadratic polynomial

as in, you know that

[vapid kiln]

how

[lean otter]

$(x + m)(x + n) = 0$

replace $x$ with $1$ and $5.5$, and see what $m$ and $n$ can be - then expand the product under the standard form of a quadratic polynomial

[flat frigate]

trololol !

[vapid kiln]

wait what

how does that give me the formula though

[lean otter]

wdym

[vapid kiln]

well

one second

i just added random values, but it adds up to being 0

so how does that give me the formula though

[lean otter]

what do you mean

[vapid kiln]

my result has to be this: f(x)=2x^(2)-13x+11

[lean otter]

where did you add random variables

[vapid kiln]

didnt you say see what s and m can be

n and m*

[lean otter]

well, you have to find out what they are

[vapid kiln]

how

[lean otter]

$ab = 0 \iff a = 0$ or/and $b = 0$

[flat frigate]

trololol !

[lean otter]

do the same for $(x + m)(x + n) = 0$

[flat frigate]

trololol !

[vapid kiln]

but i just did that

and it became 0

[lean otter]

ok, let me show you step by step

[vapid kiln]

the result is the same as long as one of the parenthesis is 0 right?

[vapid kiln]

yes please

[lean otter]

$x = 1 \iff (1 + m)(1 + n) = 0 \iff 1 + m = 0 \iff m = -1$ and $n$ can be anything

$x = 1 \iff (1 + m)(1 + n) = 0 \iff 1 + n = 0 \iff n = -1$ and $m$ can be anything

[flat frigate]

trololol !

[lean otter]

so there you have it, for x = 1, then either m or n have to be -1

BUT

don't forget that in the problem, they state that $a$ (the coefficient of $x^2$) must be 2

[flat frigate]

trololol !

[lean otter]

you can use that to find out the other number too, knowing that the first one is -1

also, don't forget that, due to being a quadratic polynomial, it has two roots

thus, the product must be 0 for both x = 1 and x = 5.5

[vapid kiln]

i still dont get how that leads me to this as a result: f(x)=2x^(2)-13x+11

thats what i dont get

i understand everything you explained

about 1 and 5.5

but how do i get that as a result

@lean otter

[lean otter]

wdym get that as a result

after you found out the values of m and n, do the multiplication $(x + m)(x + n)$

[flat frigate]

trololol !

[lean otter]

which expands to $x^2 + xn + nx + mn$

[flat frigate]

trololol !

[lean otter]

but since they said that the coefficient of $x^2$ is $2$, it means that it is $(x + m)(2x + n)$, (notice that $2$ near the second $x$)

[flat frigate]

trololol !

[vapid kiln]

okay let me ask like this

where does 11 and 13 come from

in this formula $f(x)=2x^(2)-13x+11$

[flat frigate]

KIJ Designs

[vapid kiln]

.close

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[shadow silo]

Hello, i need help with my homework

can you help me by any chance?

[hardy lion]

Multiply by the denominators

[lean otter]

Cross multiply

$\frac{2}{3M} = \frac{1}{M+1} \iff 3M = 2(M+1) \iff 3M = 2M + 2 \iff M = 2$

Oh garlic already got it

[flat frigate]

trololol !

[shadow silo]

tysm

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[spring wasp]

need help with both parts

[safe radish]

@spring wasp Has your question been resolved?

[spring wasp]

nvm i did part b but still dont get part a

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[chilly latch]

Can some one point me in the right direction for filtering time series data? I want to make a multi band gate with minimum phase.

[chilly latch]

<@&286206848099549185>

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@chilly latch Has your question been resolved?

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[waxen estuary]

not sure if this kind of question is allowed but i'm doing some programming where i need to figure out how to store something similar to a permutation matrix (https://en.wikipedia.org/wiki/Permutation_matrix) more efficiently so knowing the properties of what i have would be very useful, but i don't know what the name of that would be.

A definition of the kind of matrix that i need to learn more about would be a matrix with at most one 1 in every column and every row (as opposed to a permutation matrix where it's exactly one), while all other elements are zeros.

also my matrix doesn't have to be square, in most cases it won't be

[waxen estuary]

so what would this be called?

and where would i learn about it more

i'm finding this very hard to google

[unique bison]

Well if a column or row doesn't have any 1's then it's all 0's so the determinant of it is zero

[waxen estuary]

yeah

[unique bison]

Otherwise it's a permutation matrix

[waxen estuary]

there are no other interesting properties?

[unique bison]

Idk this seems quite specific. Not sure if there's a name for this

You can try #linear-algebra

[waxen estuary]

ah, that sucks, thanks

wait that sounded mean

[unique bison]

[frozen forge]

https://tel.archives-ouvertes.fr/tel-02084015/file/2018TOU30069b.pdf

[waxen estuary]

i mean it sucks that this might be too specific to have been researched, but thanks for telling me that

[frozen forge]

You can search some properties in this thesis i found but it's in french

you'll have to translate

But it seems very complete

[waxen estuary]

i think i have a friend who speaks french that could help, this thing is very useful

thanks

[frozen forge]

idk if it's worth your time but you can try, if you have to know something, it's in there

[waxen estuary]

idk, maybe i could just accept that that part of my program is gonna be either slow or take a lot of memory, i'll see

i guess my issue is solved if i have this document? idk

.close

[frozen forge]

I hope

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[waxen estuary]

thanks a lot again!

[safe radish]

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[jolly jolt]

Hi, can someone explain to me how Cos turned into 1-sin*2?

[lean otter]

$sin^2x + cos^2x = 1$

[flat frigate]

♡LexQa♡

[jolly jolt]

where did the 1 come from? or is just a rule?

[lean otter]

It's the most basic rule of trigonometry yes

It describes the unit circle

[jolly jolt]

alr but how did cos turn to sin?

we got the derivative?

[nova creek]

If sin²x + cos²x = 1

Then cos²x = 1 - sin²x by algebra

[jolly jolt]

but it's sin²x = 1 - sin²x

not what u sent

[nova creek]

It's given in the question sin²x = cos²x

And it's an identity that sin²x + cos²x = 1

So cos²x = 1 - sin²x

[lean otter]

Don't take it the wrong way, but I think you need to take a step back from calculus and relearn some of your foundations

Such as trigonometry and algebra

It might really confuse you in the future if you are doubtful of something such as this

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[obtuse heath]

I'm kind of confused on how to do these questions and kind of like describe them. This is Grade 11 Exponential Functions. I'm kind of confused on how to know what k value is which and the fractions and decimals just mess me up.

[obtuse heath]

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[fiery bronze]

i'm trying to do step d, i have no idea how to find the coordinates of
point R

[dense lynx]

Hmm,try measuring gradient

ER = RF

[fiery bronze]

the what

[dense lynx]

the gradient(?) the steepness

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[cosmic epoch]

3(sin2x) = 2(sin4x) solve for x where 0<x<pi/2

[cosmic epoch]

how would I do this question?

[lean otter]

try using double angle formula on sin4x

[cosmic epoch]

so it looks like 3(sin2x) = 2(4sin0cos0)

Is that what happens?

[lean otter]

those zeros on the right should be 2x

move everything to one side then factor out sin2x

[cosmic epoch]

so 3(sin2x)-2(4sin2xcos2x)=0

sin2x(3-8cos2x)=0

[lean otter]

yeah

now either sin2x=0 or (3-8cos2x0)=0

[cosmic epoch]

so would the values be 0, 90 for sin

ad then I'll figure out values for cos later

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[night pasture]

Find F'(x) if F(x) = (integral sign, with x+2 on the top and x on the bottom) (4t+1) dt

[night pasture]

ye

[flat frigate]

ΣAC

[final halo]

You need to use fundamental theorem of calculus

[night pasture]

in what way

[plucky elk]

Have you ever used it before

[night pasture]

yes

so F(x+2) - F(x)

but how do I get the derivative of that

wait what did i do wrong

[lean otter]

@night pasture

So basically

Split the integral into two

At some point c

It can be 0 doesn't matter

Then make it so that they are in the form where FTC can be used

[night pasture]

so one interval is 4t and the other is 1

right?

[lean otter]

Don't actually split the integral

Like

Split the bounds

Icba to latex someone will do it

[night pasture]

Im sorry im confuseed

[lean otter]

Okay so you know how you can make an integral $\int_{a}^{c}f(x)dx = \int_{a}^{b} f(x)dx+ \int_{b}^{c}f(x)dx$ @night pasture

[flat frigate]

♡LexQa♡

[night pasture]

yeah

[lean otter]

Okay so for ur integral make the 'b' be some constanr like 0

[night pasture]

ok

[lean otter]

After fhat

[night pasture]

alr

[lean otter]

Then u can apply the thingy

U gotta do chain rule

[night pasture]

oh

[lean otter]

Well doesn't matter but

[night pasture]

ye that makes sens

ty!

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[heavy glade]

uh i need some help with B rn

[gilded vault]

okay

[heavy glade]

I have this so far

[gilded vault]

oh okay thats good

literally just take A, B, C, plug in their x and y coordinates

[heavy glade]

i dont know if it's clockwise or counterclockwise

oh so i just stick it into the x and y

[gilded vault]

usually assumed that its counterclockwise

[heavy glade]

ohh okay ty

lemme do that

bruh and Part c is just multiply the x by 3 right?

[gilded vault]

yeah

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[heavy glade]

.close

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[vague coral]

Let (a,b,k) $\in \mathbb{N}^{3}$ such that $k=\frac{a^{2}+b^{2}}{1+ab}$ Prove that k is a perfect square.

[lean thorn]

bot is slow/not working tonight 😦

[vague coral]

ah i see

I'll go to chegg then and ask later

[vague coral]

yea

u got it

i saw the problem in the hallway on campus today but idk how to solve it

like where would i start to prove this

.close

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[heavy glade]

do you know how to combined the first two?

Look at the bounds and how they align

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@tribal silo Has your question been resolved?

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[barren pewter]

I am confused on what the question is asking, I know a rational function needs a vertical asymptote; however, the current function already has one so I am just confused on what it wants me to do to the original function? I hope this makes sense sorry if its such a dumb question...

[plucky elk]

Use the definition of cosh(y) for y=log(x)

[barren pewter]

Is this what you mean by defining the cosh expression?

[plucky elk]

The first one is the definition yes

[barren pewter]

Oh okay, thank you! I'll try to figure it out with that help 🙂

Would we have to do the same thing for ln? would we turn it into:

Omg nevermind Im so dumb I figured it out, its like the chain rule where I plug in ln(x) to the defined cosh equation.

Thank you, I figured it out and youre the one who helped me find that cosh definition ! ❤️

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[proper axle]

How did he get from the first line to the second line

[split ether]

Expanded

[proper axle]

Not quite sure

Wym

Do you mind showing it out

[split ether]

$x^2(\frac{300}{x} - \frac{x}{4}) = x^2\cdot{\frac{300}{x}} - x^2\cdot{\frac{x}{4}}$

[flat frigate]

A Lonely Bean

[proper axle]

Where did the x go under 300

[split ether]

$x^2\cdot{\frac{300}{x}} = x\cdot{x}\cdot{\frac{300}{x}} = x\cdot{300}$, the x's got cancelled out

[flat frigate]

A Lonely Bean

[proper axle]

Why is it 1/4 and not x^3/4?

[split ether]

1/4 * x^3 and x^3/4 are the same things, author just preferred the first one

[proper axle]

Tyy

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[honest slate]

how to solve

[safe radish]

@honest slate Has your question been resolved?

[swift palm]

Ok for the numerator

It can be written as

And that can be simplified to 4xy^2

Then cancel out a factor of 2 from the numerator and denominator and you have

$\frac{2xy^2}{3x^5}$

[flat frigate]

Hydro

[swift palm]

And now cancel out a term of x to finish

I'll let you do that though so I don't do everything, let me know if you have trouble though.

Ah actually since you need to write it in the form $ax^by^c$

[flat frigate]

Hydro

[swift palm]

You have a little more to do after cancelling the x

You will be left with

$\frac{2y^2}{3x^4}$

[flat frigate]

Hydro

[swift palm]

And I'm hoping you know how to use negative exponents to make that not a fraction but yeah lmk if you can't finish it off.

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[wispy sentinel]

what type of matrix is (symmetric matrix) * (skew symmetric matrix)

[safe radish]

@wispy sentinel Has your question been resolved?

[wispy sentinel]

<@&286206848099549185>

<@&286206848099549185>

[open wagon]

$\begin{pmatrix}
a & b \ b & a
\end{pmatrix}
\begin{pmatrix}
0 & b \ -b & a
\end{pmatrix} = $

[flat frigate]

xpoes

[open wagon]

$=
\begin{pmatrix}
-b^2 & 2ab \ -ab & b^2+a^2
\end{pmatrix}$

[flat frigate]

xpoes

[open wagon]

I'm not sure what you specifically want, but there's that

$det(A) = -b^4+a^2b^2$

[flat frigate]

xpoes

[open wagon]

if that's interesting

[wispy sentinel]

thats my question btw if u can help

[open wagon]

what have you tried?

[wispy sentinel]

i want to know what happens is we multiply these two types of matrices

but our tutor told only about what happens if we raise some power to it

[open wagon]

well try a case

and see what happens

[wispy sentinel]

@open wagon thanks for your help

.close

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[open wagon]

np

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[mossy remnant]

helllo! so I have a project about fermented products and a way of selling it so my product is manggo jam and the total investment is 140 pesos and the manggo was for 80 peses, sugar 25 pesos, kalamansi 5 pesos, jar 25 pesos, and the label for 5 pesos how much do you think I'll be able to "sell" it and how much will I "profit"

[sudden forum]

you can sell it for literally any price lol

provided the buyer is willing to buy

[mossy remnant]

yeah but how much

[stray socket]

Profit is determined by (sell price) - (production price), neglecting taxes and other governmental bullshit

It's up to you how much you wanna sell it

[mossy remnant]

do you think 200 pesos is a good price?

[stray socket]

Well no

[mossy remnant]

too much?

[stray socket]

The sell price has to be more than the cost of production

[mossy remnant]

then how much

[stray socket]

Well calculate how much it costs to produce each jar

[mossy remnant]

its 140 each jar

so whats a good price for that?

[stray socket]

Oh I see

Well 200 could work then sorry I misread your thing

140 for each jar to produce, 200 could work

Although personally I'd do a 50% mark-up minimum

So 210 pesos would be better imo

[mossy remnant]

ooh okie thank you so much!

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[elder loom]

how do I know the a and b if I don't know the mean and variance

[drowsy moss]

properties of mean and variance

$E(aX-b) = aE(X)-b$\
$Var(aX-b) = a^2Var(X)$

[flat frigate]

Zybikron

[safe radish]

@elder loom Has your question been resolved?

[elder loom]

Yeah ik

Let me send the current step now

So it’s a substitution equation now?

[drowsy moss]

yeah, use the second one to solve for a in terms of sigma
then sub that into the first one to find b

also the first one should be a(mu) - b
not +

[safe radish]

@elder loom Has your question been resolved?

[elder loom]

But I still don’t know both the average and the variance

So there are still two unknowns

[safe radish]

@elder loom Has your question been resolved?

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[pliant smelt]

i have no idea on how to do this please help

the help video linked is very confusing for me too so i would also appreciate an explanation if you prefer

[obtuse plover]

so we have our standard form of

ax^2 + bx + c

[pliant smelt]

yeah

[obtuse plover]

(Part a)

[pliant smelt]

i understand that

[obtuse plover]

Looking at the first 2 terms

x^2 + 8x

We wanna take 1/2 of b then square it. So (b/2)^2. What would that be?

[pliant smelt]

so half of b is 4

since 8/2 is 4

then 4^2 is 16

[obtuse plover]

correct

[ember sphinx]

complete square

[obtuse plover]

So now we’re going to add 16 to our expression, but also subtract 16, so that we are only adding 0

So x^2 + 8x + 16 + 19 - 16

Do u see what just happened

[pliant smelt]

yes we added the result of this and added it to the epression

[obtuse plover]

And then we subtracted it from the expression because we can’t change the expression. By both adding and subtracting 16, the expression doesn’t change. Do u see this?

[pliant smelt]

yeah bc they both cancel each other out

[obtuse plover]

Yea

So now we have

x^2 + 8x + 16 + 19 - 16

Now that we’ve found (b/2)^2, we’ve completed the square and can now make the binomial squared term

So can u factor x^2 + 8x + 16

[pliant smelt]

yeah

hold on

(x+8)(x+2)

[obtuse plover]

Not quite

[pliant smelt]

oh wait

io see

(x+4)(x+4)

[obtuse plover]

Yes

[pliant smelt]

or (x+4)^2

lmao

[obtuse plover]

Exactly

[pliant smelt]

sorry my brain wasnt braining

[obtuse plover]

But we cannot forget about the +19 - 16

So now we have

(x+4)^2 + 19 - 16

Can u simplify?

[pliant smelt]

is it (x+4)^2 +3

or am i wrong

[obtuse plover]

Yes

That’s correct

[pliant smelt]

oh

[obtuse plover]

Do u understand how we got here

[pliant smelt]

because +19 and -16 are like terms?

so basically 19-16

[obtuse plover]

Ye

[pliant smelt]

yeah everythign else i understand

[obtuse plover]

Ok cool

[pliant smelt]

are you still here?

[pliant smelt]

oh wait @obtuse plover i meant like until we got to here

i still dont understand after this 😭

[obtuse plover]

Oh part b?

[pliant smelt]

yes

[obtuse plover]

do u know vertex form

[pliant smelt]

and on part a, the answer is (x+4)^2+3 or no?

[pliant smelt]

no i dont

[obtuse plover]

Well take a look, does it match that form?

[pliant smelt]

hold on i'll look it up

[obtuse plover]

Vertex form?

[pliant smelt]

y=a(x-h)^2+k

[obtuse plover]

It’s

Yes

^2

[pliant smelt]

so i have to sub it in now

[obtuse plover]

We alrdy have it in vertex form

[pliant smelt]

oh so thats my answer to part a then

[obtuse plover]

Well they alrdy give u a form in part A, so just see if it matches, that part doesn’t necessarily concern vertex form

[pliant smelt]

oh ok

so c is the y-intercept

but i dont understand what is d

[obtuse plover]

No, c is not y intercept

U get y intercept from standard form quadratic

ax^2 + bx + c

Not from this

From vertex form, u get vertex

Or turning point

[pliant smelt]

oh

so what is c and d in this case

is d=3?

[obtuse plover]

Well what’s the form given in the problem part a?

[pliant smelt]

(x+c)^2 + d

OH

c is 4 and d is 3

omg im so daft

[obtuse plover]

Lol

[pliant smelt]

jesus

so how would you do part b?

[obtuse plover]

So this is vertex form: y=a(x-h)^2+k

With (h,k) being the vertex

[pliant smelt]

so (h,k) is what we want for this question

?

[obtuse plover]

Yes

So if -h = +4, what’s h

[pliant smelt]

-4

[obtuse plover]

There u go

And what’s k

[pliant smelt]

so thats what we got so far

[obtuse plover]

Our vertex is (h,k)

[pliant smelt]

oh thats h

the -4

[obtuse plover]

Ye

[pliant smelt]

so is k +3

[obtuse plover]

Yea

[pliant smelt]

omg finally tysm

i spent a good a hour on this quesiton

[obtuse plover]

Np

[pliant smelt]

how do you close this

!close

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[arctic ingot]

I don't understand an algorithm, to find perpendicular and orthogonal vector parts.
I have a formula
G(a1...ak)*[X1...Xk] = [dot(a, a1) dot(a, a2)... dot(a, ak)]
where G is Gram matrix, a1...ak is base of linear space W, and dot(a, b) is dot product with vectors a, and b.

What is vector a in this case on the right side of equation?

[arctic ingot]

it's be easier with formula's screenshot

[gilded vault]

oh alpha is the vector made up of X_1, X_2, X_3,..., X_k

im pretty sure

let me double check though

actually another helper should do this sorry

i dont know

[arctic ingot]

so alpha is any vector which is in linear space made of a1...ak, so i can get the perpendicular and orthogonal part of alpha?

[safe radish]

@arctic ingot Has your question been resolved?

[arctic ingot]

I have an example linear space:
W = lin([1 2 0], [1 0 1])
i have an alpha = [3 2 2]
i compute a gram matrix for my space, and i get

[5 1][x] =  [7]
[1 2][y]    [5]

i solve the equations, and i get
[x y] = 7/8 * [1 3]
i use them in this equation, and i get perpendicular part equal
7/8 [4 2 3]
Is it correct? <@&286206848099549185>

also, the perpendicular part of vector alpha is longer than vector alpha

[arctic ingot]

.close

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[iron glacier]

can e^-x ever be equal to 0?

[thorn chasm]

Only in the limit as x goes to infinity

Not for any actual x values

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[iron glacier]

how do we go from lim x->inf x/x+1 to 1/(1+1/x)

[thick sparrow]

Multiplying $\frac{1/x}{1/x}$ to the original limit can give you what you want.

[flat frigate]

jimmy1234

[iron glacier]

I multiply 1/x to the top and bottom of x/x+1 ?

[dusty ermine]

yes

[iron glacier]

ty

.close

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[opal storm]

?help

[opal storm]

hi

14.16

from N3

kinda

idk

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@opal storm Has your question been resolved?

[opal storm]

no

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[reef radish]

.close

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[dense hamlet]

Working with Anti derivatives

or indefinite integrals if you are so inclined to call them that

Were just learning these so I dont know how to undo the quotient rule

[solid shell]

Have you worked with u-substitution?

[dense hamlet]

Nope

all we have used is the anti derivative "power rule"

[solid shell]

Okay that could work too but we need to rearrange the integrand

[dense hamlet]

Wait

This is an inverse trig

is this inverse tan?

[solid shell]

,w derivative of arctangent

[flat frigate]

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Requested by lexitorius#4749

[solid shell]

Yup, it is

[dense hamlet]

okay

[solid shell]

You'll want to factor out the 16 first but yup

[dense hamlet]

And then to find the C value I just plug in 1 for x

or in this case, t

,w derivative of arccosine

[flat frigate]

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[dense hamlet]

,w derivative of arcsin

[flat frigate]

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[dense hamlet]

.close

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@potent needle Has your question been resolved?

[buoyant shadow]

the width is 8

and you made 4 measurements

i'd say it's 2 × ...

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@potent needle Has your question been resolved?

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[normal barn]

hmm

[normal barn]

i need help

how do i prove that the product of Primary numbers from 1 to n is <= 4 to the power of n

like 2.3.5.7 <= 4° 7 or 4°8 or 4 °9

[plucky elk]

is n the nth prime number or is it the number of prime numbers?

[shut inlet]

I guess it's not required to be a prime number

[plucky elk]

so what is n in this case? 4 or 7?

[shut inlet]

for exemple primary numbers between 1 and 9

n = 9 in this situation

[normal barn]

right

[plucky elk]

i see

[shut inlet]

if we prove that is true for <=4^p when p is the last prime

it will be true for the rest

[normal barn]

yeah u could look at it that way

[plucky elk]

did you try induction?

[normal barn]

lemme google what induction is

yeah

but found nothing to put in beetwenn

i tried with n! didn't work

all sorts of fonctions that have n!

[plucky elk]

do you know bertrand's postulate?

[normal barn]

no

[plucky elk]

maybe that's not useful in this case

[normal barn]

well i thought of trying to quantify primary number

with the PI (x) fonction

but that didn't lead me anywhere

[plucky elk]

if you have the patience, it's lemma 3.2 here
https://web.williams.edu/Mathematics/lg5/Chebyshev.pdf

they give a hint to prove the intermediate result in (b) first

[normal barn]

you just saved my life

ty

hmm im still kinda lost

[plucky elk]

better hint: strong induction

[normal barn]

nah i get it

i mean

obviously 4 ° m > 4 ° m-1

what i don't understand is the Theorem

i don't know why our Analysis 1 teacher gave this task

[plucky elk]

a lot of analysis is bounding random things

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[wicked tapir]

If I have a funnel with a big circle opening that's 4 inches in diameter and a small circle opening that's 2.5 inches in diameter. It smoothly transitions the whole length of the funnel. The length of the funnel is 8 inches. How do I find the diameter of the opening at any particular given length along the funnel?

[stray socket]

Is it gonna be a frustum

[safe radish]

@wicked tapir Has your question been resolved?

[wicked tapir]

What is a frustum?

[safe radish]

@wicked tapir Has your question been resolved?

[queen parcel]

A frustum I don’t remember, but you can definitely look it up

Try drawing a picture of the problem

That may help

[gentle lintel]

Find the radius of both circles

[wicked tapir]

Radius is 2 and 1.25

[gentle lintel]

Nevermind, the answer is the line that pases through (0,2.5) and (8,4)

[safe radish]

@wicked tapir Has your question been resolved?

[wicked tapir]

For example if I was looking for an opening diameter of 3, how do I find that length?

[safe radish]

@wicked tapir Has your question been resolved?

[fathom wadi]

I used some trigonometry to find the diameter

At a distance x from the smaller circle

@wicked tapir

[wicked tapir]

What if we find the slope of the line for half the funnel

Then input coordinates for the desired diameter

[fathom wadi]

That will be bit tedious using co-ordinate geometry

[wicked tapir]

I'll try and sketch something up to give the idea

Cuz then you can make a triangle

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@wicked tapir Has your question been resolved?

[safe radish]

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[empty gyro]

Cool problem mitch

What have you tried

[dapper shale]

Well

All angles can be found

And I used sin rule

However I’ve done this repeatedly and my answer was incorrect

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@dapper shale Has your question been resolved?

[empty gyro]

Well hard to say where you went wrong if you don't show you work

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[next flare]

Am I allowed to switch the order of these summation signs? mu_n is in [0,infinity]

[next flare]

I’m pretty sure I can, but I don’t know how I can convince myself

.close

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[tired bison]

which steps should i take?

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[elfin sandal]

<@&286206848099549185>

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Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.

[elfin sandal]

sorry lad

[halcyon carbon]

Have you tried expanding and simplifying

[elfin sandal]

i dont even know what this it

is

[halcyon carbon]

like (n-2)(n-2)?

foil?

[elfin sandal]

so do that for n-3 and n+3?

[halcyon carbon]

where did that come from

[elfin sandal]

oops i had i different question on the side sorry

so when i do that

what is it equal to

or do i just aim to get an answer that is factorised

like n(2+2) for example

[final halo]

You need to expand and see what cancels before you think about factorising again

[elfin sandal]

okay give me a few mins

i got 2(2n-3)

and i got it right

[halcyon carbon]

yes

[elfin sandal]

is this the exact same

[final halo]

Yes do the same

[elfin sandal]

okay

wait i got zero

hold on what

n^2 +6n + 9 - n^2-6n-9

what have i done

<@&286206848099549185>

[junior knot]

12n

(n-3) squared - (n-3) squared

@elfin sandal

expand the brackets

n squared + 6n+9-(n squared -6n+9)

n squared +6n+9-n squared +6n-9

=

12n

[elfin sandal]

oh minus mius

[elfin sandal]

it says its wrong

[junior knot]

whoops my bad

try

-(n-3)

if not im confused

[elfin sandal]

wrong and i cant put any more answers anymore

[elfin sandal]

can u help with this

[junior knot]

sry man have no clue

<@&286206848099549185>

[elfin sandal]

okay

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@elfin sandal Has your question been resolved?

[elfin sandal]

.close

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[round oriole]

please help ive been stuck on it for hours

[queen parcel]

h(x) is just a slight twist on f(x)

Instead of x being the input to f(x), g(x) is now the input

f(x) → f(g(x))

[round oriole]

thanks

.close

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[iron sundial]

How do you make a two column proof to solve 15 = 3x – 18.

[gentle lintel]

3x=15+18=33, x=33/3=11

[iron sundial]

thanks

.close

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[sonic sapphire]

I have an antiderivative...

[sonic sapphire]

$F=32\theta+16\sin(2\theta)$

[flat frigate]

Dr. Leonard McCoy

[sonic sapphire]

and now I'm trying to write it in terms of x

[queen parcel]

You need θ in terms of x

Do you have the original problem?

[sonic sapphire]

I'm like 5 stages deep

when I set up the problem, I got the integral of...

$\sqrt{a^2-x^2}$

[flat frigate]

Dr. Leonard McCoy

[sonic sapphire]

so I figured this would be a trig sub problem

it uses equation of a circle it looks like

this problem

with r=4

I think I can re-write...

[queen parcel]

Do you have work on a piece of paper?

[sonic sapphire]

$F=32\theta+16\sin(2\theta)=32\theta+16*2\sin(\theta)\cos(\theta)$

[flat frigate]

Dr. Leonard McCoy

[sonic sapphire]

ignore the part above the squiggly line

[queen parcel]

I’d say your area function should be s^2

s = 2r

[sonic sapphire]

sorry I don't get what you mean

oh

uh

[queen parcel]

Your area function should be the area of a square

[sonic sapphire]

shouldn't it be squared?

[queen parcel]

Closer, yes

[sonic sapphire]

oh fuk

[queen parcel]

The area of a square is s^2

The side length of your square is 2 times the x or y coordinate

[sonic sapphire]

right, that's what 2f(x) means

but it looks like I didn't include it

ooh

I know what my reasoning was

I was only going to integrate a quarter circle

and multiply by 4

integrate [0, 4]

I figured that might be simpler

yeah, I was going to do the integral for just the quarter of the circle

and then 4x the result

ok fuck this problem lmao

[queen parcel]

Well from x: 0 → 4, you’ll want to integrate (2 times y)^2

You know y in terms of x

[safe radish]

@sonic sapphire Has your question been resolved?

[sonic sapphire]

so it sounds like it's actually easier than I set it up to be

I totally made it harder didn't I

rofl, I did

[queen parcel]

You may have lol

It’s so easy to overcomplicate things

[sonic sapphire]

I've attempted this problem several times now... makes me wanna cry tbh

[queen parcel]

Close, keep in mind the 2 is in the parentheses

[sonic sapphire]

uh, sorry I'm not understandin ya

doesn't the outer square undo the inner square?

[queen parcel]

Well, in the parentheses you should have (2 √stuff)^2

[sonic sapphire]

I put the 2 outside the integral

[queen parcel]

So you should net a 4 outside

[sonic sapphire]

I thought it would come out the same

but alright

[queen parcel]

It’s almost the same, but not quite

2x^2 ≠ (2x)^2 = 4x^2

[sonic sapphire]

huh, ok

I thought I would just integrate half a circle

and then double it

that was my reason anyway

[queen parcel]

That’s one option, although you’d miss some of the solid

Because the squares would be much smaller, doubling wouldn’t compensate enough

[sonic sapphire]

ooooh, dammit

XD

I see what you're saying

sweet jesus, I got it right

thx for ur help

[safe radish]

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[clever gust]

I don't know what to do next

I don't know the next step

[mortal sandal]

You want to show a pair of sides is congruent, right

[clever gust]

yes

So i put KG = JH

or no

[mortal sandal]

yeah

see how steps 5 and 6 help

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[iron sundial]

anyone able to help with this?

[gentle lintel]

Two column proof?

[iron sundial]

yeah so like

statements and reasons

[gentle lintel]

Squaring both sides you get 4x-6=15^2

[lean otter]

Tf you mean squaring both sides

[gentle lintel]

Use the fact that a=b implies a²=b²

[lean otter]

It's 1/2 not sqrt

[foggy salmon]

wait this is not even a proof

solving is not a proof what

[lean otter]

Just say "distributive property" or whatever when you change things on each side

Yea it isn't technically a proof i think it's just meant to teach the concept but

[nova creek]

Maybe it means prove x = whatever is the solution

Effectively by giving each step

[foggy salmon]

lol

prove by solving

[nova creek]

Hey, if it ain't broke, don't fix it ¯\_(ツ)_/¯

[iron sundial]

hm

alright what about this one

[gentle lintel]

Are the formulas supposed to be the angles?

[iron sundial]

yes

[gentle lintel]

So 2x+19=4x+7, and 2x+19+y=180

[iron sundial]

alright

.close

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[fervent terrace]

hello i want to ask how to what is the multiplicity of a matrix ( eigen)?

[nova creek]

What are you confused about specifically?

[fervent terrace]

hi again, I dont know what is the multiplicity of the eigen

the question like thid

but I have no idea what the multiplicity is about

[nova creek]

In polynomials, the multiplicity of a root is the power of the factored polynomial

i.e., the root 2 has multiplicity 2 in (x - 2)²

[fervent terrace]

ok

[nova creek]

For eigenvalues, the multiplicity is also related to the dimension of the eigenspace

[fervent terrace]

so is the number of free variable?

[nova creek]

You can think of it like that, yes

[fervent terrace]

i see thanks again

.close

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[tall fern]

can someone tell me how to do (4+3)^2

[snow ice]

1. do 4 + 3

2. do ^2

[hardy lion]

We just went over this

[foggy salmon]

i think thrs some abbrev thing

pemdas

brackets go first

then that exponential like plaisk said