it’s also defined in your picture
#help-17
lyric fossil
vast shale
oooh
I see now
they put t_stop into t
Thank you!
My brain is so fried
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vast shale
Would you happen to know how they did this?
vocal sleetBOT
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fossil socket
In general how do you solve questions like this: Is $$ 625^{n} \cong 8 mod 99 $$ possible? If yes, state what n is, if not state why.
twin meteorBOT
sleemo
fossil socket
I myself start doing these by saying it is possible and trying to make a contradiction.
mild flower
if $a$ and $b$ are coprime to $N$ then $a^n \equiv b \pmod{N}$ is possible, but finding $n$ is the discrete logarithm problem and afaik there's not much of a better way than just iterating until you find it
twin meteorBOT
Hayley
fossil socket
So this question would only make sense in a context where either you have to show it isn't possible or the n you have to find is rather small and easy to calculate?
mild flower
i think so yeah
note that it can be possible if a isn't coprime to N
like $8^n \equiv 2 \pmod{10}$
twin meteorBOT
Hayley
vocal sleetBOT
@fossil socket Has your question been resolved?
fossil socket
like $8^n \equiv 2 \pmod{10}$
So when does it not work?
mild flower
wait hang on 8^n = 2 isn't possible
mild flower
yeah
right
okay
i must be really tired
glossy maple
fossil socket
For example n=3
and how would you do that in general? Like when you're not able to calculate it so easily?
is there even a way of saying it works or it works not instantly?
vast shale
There are some tricks, like Fermats little theorem or the euler-fermat with the phi-function
But I dont know a general approach
nimble elm
i dont think either of those helps
mild flower
oh ok i think i remember
from 8^n = 2 (mod 10)
we have that 8^n = 2 (mod 2) and 8^n = 2 (mod 5)
8^n = 2 (mod 2) is somewhat obvious
nimble elm
you can make some reductions with the chinese remainder theorem
then there are some group theoretic results to rule out some things if you compute the orders of those elements
but you will always have to put in work
they key observation is that ||if you compute a^k mod n for k = 1, 2, 3, ... the values will start repeating at some point||
mild flower
if $a$ and $b$ are coprime to $N$ then $a^n \equiv b \pmod{N}$ is possible, but ...
actually this isn't right either because f.e. 4^n = 3 (mod 5) isn't possible
nimble elm
its possible if a is a primitive root hayley
mild flower
ty that's the word i was looking for
nimble elm
in general its not easy to tell whether a value is a primitive root
if you are given one, you can find the others easily but finding one in the first place takes work
nimble elm
thats part of it, yes
because those are the only possible orders
but you still have to check
also discrete log can be solved more easily than just trying
just trying takes O(n), but getting to O(sqrt(n)) is rather easy
(there are even better ways in special cases, but this works universally)
fossil socket
its possible if a is a primitive root hayley
does this mean <a> generates the whole prime residue group of the modulo?
mild flower
fossil socket
I think in english it's called group of units modulo x but I am german haha
mild flower
yhhhh that's the name i know
nimble elm
yeah
the word "primitive root" is older than the concept of groups
thats why its not just called a generator or something
fossil socket
also discrete log can be solved more easily than just trying
but this is something I don't understand really^^
nimble elm
the word root might be misleading
ah actually you can ofc apply this here too, but honestly if you are asked such a question in an exam setting (and with such small numbers), the correct approach is probably to just start calculating
and use what you know about the modulo relation to keep the numbers small
fossil socket
I also had a question going like this: 1003^n is congruent to 6 mod 288. State n if possible or explain why it isn't possible.
Is there a trick/something I can't think of, on doing this with odd numbers and dividing with even?
fossil socket
and use what you know about the modulo relation to keep the numbers small
that sounds pretty good
mild flower
well first of all you can reduce 1003 mod 288
hard atlas
there is something better to do
is 1003 coprime?
what about 6
(with a computer you can check that the order of 1003 is 24, so that's too much to just check by hand)
fossil socket
I tried doing it with gcd:
Since 1003 and 288 are relatively prime their gcd is 1.
Therefore 1=1003 * x + 288 * y | 6
6=1003 * 6x + 2886y | mod 288
6=1003 *6x + 0
Now we need an invers of 6 to somehow find x.
But since 288 and 6 are not coprime, there is no inverse.
hard atlas
ok, so 1003 and 288 are coprime, while 6 and 288 are not coprime
what about powers of 1003 and 288
fossil socket
powers of 1003 are also coprime since if there is no 2 in the prime factorization the number will never be even?
hard atlas
well 2 isnt the only prime factor of 288 but yes
fossil socket
I mean there could be other primes in both of them that are equal, but 288=144 * 2 = 72*4
yeah i wanted to write that right now^^
hard atlas
or maybe in other words, the set of numbers coprime to 288 is closed under multiplication
so now we can ask again, is there an n with 1003^n=6 mod 288 ?
fossil socket
I am not sure if I get the main idea here -.-
fossil socket
therefore 6 and 1003^n can't leave the same remainder?
hard atlas
yes
fossil socket
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vast shale
Suppose that $\set{\vb{p_1},\vb{p_2},\vb{p_3}}$ be an affinely dependent set of points $\R^n$ and let $f: \R^n \longrightarrow \R^m$ be a linear transformation. Show that $\set{\map f{\vb{p_1}},\map f{\vb{p_2}},\map f{\vb{p_3}}}$ is affinely dependent in $\R^m$
vocal sleetBOT
Suppose that $\set{\vb{p_1},\vb{p_2},\vb{p_3}}$ be an affinely dependent set of ...
vast shale
Okay just to make sure because this seems way too simple
twin meteorBOT
paper depot
something something affine dependencies carry over
something something reduce to linear dependence of differences
vast shale
we can use the definition of a linear transformation and affine dependency: [
0 = \map f 0 = \map f{c_1p_1 + c_2 p_2 + c_3 p_3} = c_1 \map f{p_1} + c_2\map f{p_2} + c_3 \map f{p_3} ]
twin meteorBOT
vast shale
for some nonzero scalars c_1,c_2, c_3
vast shale
something something reduce to linear dependence of differences
wait what
hard atlas
thats not the full def of affine dependency
mild flower
was I talking to you about x + p the other day @vast shale or was that someone else
vast shale
like what i was going for is like
\vs{3 mm}
"There exists some nonzero scalars $c_1, c_2, c_3$ such that $c_1p_1 + c_2p_2 +c_3p_3 = 0$ and $c_1 + c_2 + c_3 =0$"
twin meteorBOT
paper depot
waeugh
young blaze
like what i was going for is like
\vs{3 mm}
"There exists some nonzero scalars ...
Indeed true
And because $c_1 + c_2 + c_3 = 0$ (independent of $f$), it follows that the set ${f(\mathbf{p_1}), f(\mathbf{p_2}), f(\mathbf{p_3})}$ is also affinely dependent in $\mathbb{R}^m$. Hence, you have shown that if ${\mathbf{p_1}, \mathbf{p_2}, \mathbf{p_3}}$ is affinely dependent in $\mathbb{R}^n$, then its image under any linear transformation $f$ is affinely dependent in $\mathbb{R}^m$.
twin meteorBOT
adzetto
vocal sleetBOT
cobalt crypt
Suppose that $\set{\vb{p_1},\vb{p_2},\vb{p_3}}$ be an affinely dependent set of ...
\vb{p_1} 
vast shale
what else 
cobalt crypt
$\vb p_1$ or $\vb{p}_1$
twin meteorBOT
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thin root
how do I do this
vocal sleetBOT
how do I do this
thin root
sorry bro I fell asleep
thin root
how do I do this
isn't this when the directional derivative is 0
so gradient dot unit vector is zero
?
young blaze
so gradient dot unit vector is zero
yes
thin root
umm ok
thin root
thus unit vector is perpendicular to the gradient
i know that though but i still don't know how to proceed
young blaze
i know that though but i still don't know how to proceed
Have you calculated the gradient at the given point?
thin root
yeah
young blaze
it must something (1,2)
thin root
nah it's $\langle 1,2 \rangle$
twin meteorBOT
casiofx991exz
young blaze
so (1,2).(a,b)=0 implies a+2b =0 and b=-a/2
so unit vector is (a, -a/2)
normalize it
young blaze
yeap
thin root
instead of ( )
thin root
np
thin root
so unit vector is (a, -a/2)
but I don't know what a is
young blaze
I mean, when only an unknown coefficient a remains, you can simplify a when you normalise it because you know it is a non-zero real number
thin root
I mean, when only an unknown coefficient a remains, you can simplify a when you ...
I got $\langle \frac{2}{\sqrt{5}}, \frac{-1}{\sqrt{5}} \rangle$
casiofx991exz
young blaze
$\left{\frac{2}{\sqrt{5}},-\frac{1}{\sqrt{5}}\right}$ is correct
twin meteorBOT
adzetto
young blaze
,w Normalize[{1, -1/2}]
twin meteorBOT
young blaze
here a is only relevant to the ~~vote ~~ length of the vector, so I ignored it.
twin meteorBOT
casiofx991exz
thin root
here a is only relevant to the ~~vote ~~ length of the vector, so I ignored it.
what do you mean
sorry I am noob
to calc 3
okay I still don't know what that means
so a is like
a scaling factor?
young blaze
yeap
thin root
you can scale the vector
ohh ok
nice
that's pro
btw
I cancelled the $\frac{a}{|a|}$
young blaze
sorry I am noob
sorry for my terminology english is not my first language ^^
twin meteorBOT
casiofx991exz
thin root
I made it 1
young blaze
sure
thin root
without explaining anyhting
young blaze
without explaining anyhting
you can and also you can just consider <1,-1/2> instead of <a, -a/2>
young blaze
okay thanks
np
vocal sleetBOT
@thin root Has your question been resolved?
swift owl
Hi how I can complete the last step of this question? I have underlined it.
vocal sleetBOT
Hi how I can complete the last step of this question? I have underlined it.
@swift owl Has your question been resolved?
velvet bridge
It's a transcendental equation, you can only get approximate numerical solutions, not analytical solutions.
swift owl
So the answer would just be 2?
velvet bridge
solve 0.06=0.3te^{-1.1*t}
velvet bridge
that's right
you can't solve it, so you can't get exact analycital value of it
let's use wolfram to have a try
swift owl
Ok
swift owl
Hmm but on my answer sheet it has t=1.89
velvet bridge
yep
swift owl
Would that just be wrong considering x is greater than 0.06 at 2 in my graph
velvet bridge
because 2.16556-0.268811 gives you 1.896749
swift owl
Nvm I didn’t read question
velvet bridge
Duration of efficacy greater than or equal to 0.06
swift owl
Yep
swift owl
Ok, and there’s no way for me to get values of t by hand?
velvet bridge
That's right.
swift owl
Ok then
velvet bridge
Unless the question tells you the approximate solution.
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thin root
,w critical points of 1/y - 27/x +xy
vocal sleetBOT
,w critical points of 1/y - 27/x +xy
twin meteorBOT
thin root
why y = -1/3
I keep getting 1/3
not negative
so I got x = 9 right
so I plug that in to $f_{y} = \frac{-1}{y^2} + x = 0$\\
$9 = \frac{1}{y^2}$\\
$y^2 = 1/9$\\
$y = 1/3$
casiofx991exz
thin root
What am I doing wrong??
flat whale
I keep getting 1/3
1/3 for what
thin root
1/3 for what
for y
I keep getting (9, 1/3) instead of (9, -1/3) for the critical point
flat whale
There are other solutions you're missing
thin root
There are other solutions you're missing
like?
flat whale
.
thin root
.
how did they even get those
ill type how I got mine
\begin{align*}
f_{x} &= \frac{27}{x^2} + y\\
f_{y} &= \frac{-1}{y^2} + x
\end{align*}
casiofx991exz
twin meteorBOT
thin root
ok so we got the same partials
then I did
\begin{align*}
y &= \frac{-27}{x^2}\\
f_y &= \frac{-1}{\frac{27^2}{x^4}} + x = 0
\end{align*}
casiofx991exz
thin root
,w -1/(27^2/x^4) +x = 0 solve for x
twin meteorBOT
twin meteorBOT
thin root
wait what
thin root
.
why not 1/3
why did they use -1/3
can I use 1/3?
<@&286206848099549185>
,w 27/(9)^2 + (1/3)
twin meteorBOT
thin root
,w 27/(9)^2 + (-1/3)
twin meteorBOT
twin meteorBOT
casiofx991exz
vocal sleetBOT
@thin root Has your question been resolved?
vocal sleetBOT
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sleek flame
I thought I was onto something 😩
vocal sleetBOT
I thought I was onto something 😩
sly sierra
you actually have enough info to compute A
you can write AS = SD, where D is the diagonal matrix with 3 and -1 on the diagonals, and S is the matrix formed by putting the basis for E3 in the first two columns and the basis for E-1 in the last column
then A = SDS^-1
sleek flame
ohhh
sly sierra
to clarify for D, it should be the diagonal matrix with 3, 3, -1 on the diagonal
since 3 is a repeated eigenvalue
sly sierra
ah yes you did
vast shale
I need. help
flat whale
Post your question
vast shale
@flat whale number 23
<@&286206848099549185>
My sirr is arriving rnn
I need help immediately
flat whale
What's a sirr
vast shale
You should try doing it first
I can't I tried
tacit hill
Think of an expression you can use to write out the difference in volume between the two cubes
vast shale
Think of an expression you can use to write out the difference in volume between...
How
My head's not working
dreamy viper
try to find the volume in terms of x
vast shale
What is the volume of the cube before shrinkage?
90
vast shale
try to find the volume in terms of x
What's the equation
tacit hill
Uh where did you get 90 from?
vast shale
90-1
vocal sleetBOT
<@&286206848099549185>
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.
vast shale
Each cube 91
vast shale
Please only use the <@&286206848099549185> ping **once** if your question has no...
Ok
Please help
tacit hill
Just imagine you have a big cube of side length x, what is the volume of that cube in terms of x
vast shale
91
tacit hill
it should be an expression in terms of x
vast shale
Length into breadth
tacit hill
Okay, so what is the length, breadth, etc?
vast shale
Length X Breadth y
vocal sleetBOT
I can't I tried
Show your work, and if possible, explain where you are stuck.
dreamy viper
Length X Breadth y
its a cube
vast shale
Show your work, and if possible, explain where you are stuck.
I am stuck at the full math
tacit hill
You tell us
dreamy viper
whats the volume of a cube
vast shale
91
tacit hill
That is the difference in volumes of our big and small cubes
vast shale
Ohh
tacit hill
What is the volume of the larger cube, if its side length is x
vast shale
Now what should I do to do it 😭
dreamy viper
what is the formula for the volume of a cube
vast shale
What is the volume of the larger cube, if its side length is x
It's volume is X +91?
vast shale
what is the formula for the volume of a cube
L Into B into. height
dreamy viper
in terms of x
tacit hill
Don’t concern the 91 for now
tacit hill
So volume of a rectangular prism is l*w*h, right?
vast shale
Yes
tacit hill
well what is l, w, and h when we are finding the volume of a cube
tacit hill
What is y?
vast shale
Variable
dreamy viper
y would equal x
cause its a cube
also, when a variable is lowercase x, dont refer to it with an uppercase X
tacit hill
So volume of a rectangular prism is l\*w\*h, right?
okay so we know the length width and height are all equal to some number x. Or in other words, l = w = h = x, so what is the volume of our cube
vocal sleetBOT
@vast shale Has your question been resolved?
vast shale
hi all, trying to solve second partial derivative
vocal sleetBOT
hi all, trying to solve second partial derivative
vast shale
I was trying to solve, where I am getting 8 x e^xy x y
but my book solution states that
second partial derivative will be
8 x e^xy x y^2
why is the y squared?
where did this come from
vocal sleetBOT
@vast shale Has your question been resolved?
vocal sleetBOT
@vast shale Has your question been resolved?
waxen harness
hmm dont write f'....f' means df/dx or df/dy or smth like that but this is multi calculus
waxen harness
why is the y squared?
the second partial derivative of f(x) with respect to x is 8y^2*e^(-2xy)
vast shale
the second partial derivative of f(x) with respect to x is 8y^2*e^(-2xy)
Why 8y? I thought when we have e only coefficient (2) will come down
waxen harness
the derivative of this is 8e^(-2xy)y^2
waxen harness
Why 8y? I thought when we have e only coefficient (2) will come down
i m not sure what you are asking but yes (e^(kx))'=ke^(kx) but right now our k is k=-2y
all this with respect to x
vocal sleetBOT
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keen bramble
Just wondering, when we say a derivative is "positive or negative" it is with respect to the positive change in the variable it is differentiated?
vocal sleetBOT
Just wondering, when we say a derivative is "positive or negative" it is with re...
keen bramble
Say f prime x is "negative"
That would mean f(x) decreases for an increase in x?
Just a quick clarification as to the exact meaning of derivatives and what we mean when we say "this derivative is less than 1"
wet tendon
It means that in the domain where f'(x) is negative, f(x) decreases
keen bramble
as x increases?
wet tendon
Yes, when it's in the same domain
keen bramble
Wait why are you specifying so much about the domain
wet tendon
When you say increases/decreases it's always when x is increasing
keen bramble
When you say increases/decreases it's always when x is increasing
Ok that is good to know
wet tendon
Wait why are you specifying so much about the domain
Because the derivative can be positive/zero in other values of x
wet tendon
If you know the derivative is negative when 5<x<10 it shows nothing about the increasment of f(x) when x=20
ruby lichen
vocal sleetBOT
https://artofproblemsolving.com/wiki/index.php/2022_AMC_10B_Problems/Problem_5#V...
ruby lichen
i dont get the solution
why won't the difference of squares be under the square root
why did they seperate it into 2 different square roots
flat whale
why won't the difference of squares be under the square root
where exactly
merry python
why did they seperate it into 2 different square roots
so as to simplify (It would result in some terms which are present in numerator as well so they would cancel)
vocal sleetBOT
@ruby lichen Has your question been resolved?
ruby lichen
so as to simplify (It would result in some terms which are present in numerator ...
nah i mean the right side of the qual sign
nova fulcrum
Because it's the same
They decided to do that way, you can chose not doing that way.
There is not a special reason.
ruby lichen
Because it's the same
well clearly it is
but im wondering how
i don't get why its the same
nova fulcrum
no
nova fulcrum
no
ruby lichen
OHHHH
nova fulcrum
let me show with a paint
nova fulcrum
the exponents add when you multiply same base
ruby lichen
ic ic
nova fulcrum
for example 2^3 * 2^3 = 2^6
here you have another way to see
with the definition of a^x
ruby lichen
o
so in the case where u multiply two different bases but same exponent, then the answer is the product to the power of the common exponent right?
ruby lichen
ty
nova fulcrum
using commutative properties you get to the solution
vocal sleetBOT
@ruby lichen Has your question been resolved?
viral galleon
Ok so
viral galleon
Apparently my x coord is correct
But y is incorrect
It is supposed to be -1
But im not sure where i went wrong
kind ginkgo
hmm?
nova plover
how can we help you
viral galleon
^
nova plover
could you save us trouble and send the question itself?
viral galleon
Do you want ss of the problem online?
Because it wasnt a word problem it was written like this
nova plover
Unless thats what they ask
viral galleon
The only tjing i left out
nova plover
Just send it yeah
viral galleon
Is the answer should be min and max
Ok one sec lemme pull it up again
i used an online calculator
and it said my coord should be 8,-1
however
i got 8,-1/16
...
not sure where i messed up
lime gorge
hey man sorry i had to go somewhere else for a bit
viral galleon
its fine
lime gorge
lets start from here
viral galleon
ok
lime gorge
nono, solve for lambda, not for x and y
i may be using a different method than you, but this is the way i have done it
viral galleon
twin meteorBOT
northsteve
lime gorge
solve for one variable
lime gorge
i solved for x, but either works
viral galleon
ill do for x as well just to follow along
lime gorge
👍
viral galleon
-8y for x
lime gorge
lets start from here
now plug that into the 3rd eqn here and solve
lime gorge
yea,
lime gorge
yea
lime gorge
once u get the (x,y) u just gotta plug into the 4x - y to get the max and the min
lime gorge
nice job
viral galleon
thank you
short coral
Can someone explain how the secant method was used here? Unsure what the two initial points being used are?
vocal sleetBOT
Can someone explain how the secant method was used here? Unsure what the two ini...
flat whale
Can someone explain how the secant method was used here? Unsure what the two ini...
show section 7.6
short coral
it's just a guide to the secant method
flat whale
alpha_0 is not universal notation
short coral
= [[f'(x^k)x^(k+1)]-[f'(x^(k+1))x^k] ]/ (f'(x^k - f'(x^(k+1))
? it refers to the step size
flat whale
some books call that delta x
short coral
the problem is through steepest descent im just confused how theyre calculting step size via secant method
well i mean they layed out what alpha refers to here
flat whale
did you calculate this
short coral
yes, its the line beneath it
flat whale
and did you calculate this
short coral
thats my whole question...what is alpha
flat whale
Can someone explain how the secant method was used here? Unsure what the two ini...
no, you asked about initial points, not step size
short coral
? i aske dhow secant method was used
and that refers to how did they calculate alpha _ 0 which is step size
flat whale
secant method is more than just step size
initial values are also part of secant method
short coral
bro im not sure what youre arguing the semnatics of the question for ?i know initial points matter. my question specified that but is also generally asking how secnat methos is used to solve this
flat whale
telling you to be more specific
so i know what your actual question is
clarifying your question is part of helping
short coral
uhhh not sure what part wasnt specific but ok
short coral
i want a guide through of the sceant method calculation
which is what i asked
the end result is alpha _ not, if im asking how secant method is used, im asking how to get there
flat whale
it's just a guide to the secant method
but you said you don't want to show it
short coral
??
flat whale
show section 7.6
.
short coral
you want me to send a you whole section of thet etxbook that just explains the method
?
im asking how it is applied
flat whale
if you don't understand it and you want to, then yes you should show it
short coral
to this speicfic problem
i just want a general step by step of what the textbook did since they omitted the process
flat whale
this is just the 1d version. do you know multivariable calculus?
short coral
yes
this is the section you have asked for
and thats why im explaining its just a basic intro.
flat whale
Can someone explain how the secant method was used here? Unsure what the two ini...
oh nevermind, they're applying the 1d secant method to the argmin function
let g(alpha) = (2+2alpha - 3)^2 + 4(-1-1024alpha+5)^4
apply secant method to g(alpha)
short coral
yes
flat whale
,w argmin (2+2x - 3)^2 + 4(-1-1024x+5)^4
twin meteorBOT
flat whale
flat whale
let g(alpha) = (2+2alpha - 3)^2 + 4(-1-1024alpha+5)^4
calculate g'(alpha), g''(alpha)
short coral
you want me to do that?
flat whale
right. and plug that into here
you'll have to make up some initial value, so x0 = 0 or x0=1
short coral
why?
flat whale
because you're not given one
since secant method requires one, you try different ones
short coral
i know, but won't that affect alpha_0 vs alpha_1,2,3,...
as i do iterations for steepest descent
and secant method requires two initial values correct?
bc the picture is newtons
flat whale
let g(alpha) = (2+2alpha - 3)^2 + 4(-1-1024alpha+5)^4
in this case you have a polynomial of degree 4. there's at most 3 local min/max
flat whale
and secant method requires two initial values correct?
that's IF the second derivative isn't available
but polynomials are infinitely differentiable
flat whale
right. and plug that into here
i guess you also need to pick a starting point where the second derivative isn't 0 so this is defined
short coral
but its not asking for newtons, its asking for secant
flat whale
i know, but won't that affect alpha_0 vs alpha_1,2,3,...
so to combat this, usual thing to do is to try different initial values and just take the min of the at most 3 results you get
flat whale
let g(alpha) = (2+2alpha - 3)^2 + 4(-1-1024alpha+5)^4
in principle you can set the derivative = 0 to get 3 roots using cubic formula and see which is the minimum
short coral
yeah im not consufsed on how to find a min for it im just lost on the actual showcase of how secant's method is used here. Like i need a walkthrough of the problem step by step just as the textbook is doing for steepest descent in the example
flat whale
well pick two points and plug it in
flat whale
let g(alpha) = (2+2alpha - 3)^2 + 4(-1-1024alpha+5)^4
g'(alpha) = ?
short coral
yes but if it requires two initial points, so far all the examples in the textbook have indicated they need to be specified. How can I pick two arbitrary points that ensures in few iterations i'd arrive at the correct alpha_0
flat whale
try and find out
short coral
not really helpful
flat whale
it's beyond the scope of your course
you're either given a proof of convergence and cases when that happens or you're just supposed to trust it
from this conversation it doesn't sound like you were given a proof
short coral
convergence of gradient methos is fthe following section
flat whale
then go read that
short coral
its after this one, so why would it be explain what im asking now? obviously if the book can skip to the step of solving arg min there is a simple solution to it
its not like im asking you to solve a homework problem for me, its a given exmaple in the book. don't know why youre being so round about with it
flat whale
you're not asking about the problem anymore
How can I pick two arbitrary points that ensures in few iterations i'd arrive at the correct alpha_0
that's beyond the scope of the problem
short coral
its directly related to using secant method
the book wouldn't make me do multiple iterations to solve the problem
flat whale
you're either given a proof of convergence and cases when that happens or you're...
you're just supposed to trust it
short coral
trust what?
flat whale
that two points chosen will converge
flat whale
so yes you do need 2 points
in here
flat whale
the book wouldn't make me do multiple iterations to solve the problem
the book isn't making you do anything
if it wanted you to calculate the secant method steps, it'd give you the initial values
flat whale
Can someone explain how the secant method was used here? Unsure what the two ini...
the book skips that so it's not telling you to do it yourself
short coral
that seems incorrect. if its directly refernecing a method recently taught idk why it would give an exmaple out of its scope?
flat whale
the example isn't telling YOU to do the secant method
you're just making up the problem to do for yourself
but in order to do that you need two initial values
but you can't make up numbers so you're stuck
short coral
mm
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ruby lichen
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https://artofproblemsolving.com/wiki/index.php/2022_AMC_10B_Problems/Problem_5#V...
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floral pike
(product of powers law of exponents)
oh...the signs.
this would be easier to explain if you hadn't cut off the numerator 👀
but what they did was factor the numerator into sqrt(numerator)*sqrt(numerator)
so one of those factors canceled in the top and bottom
I'll try to copy that for you
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floral pike
let's try one piece at a time:
$\frac{\sqrt{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}{1}$
twin meteorBOT
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floral pike
$\sqrt{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}$
twin meteorBOT
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floral pike
ok I think I know what I did wrong. Forgot some curlies
$\frac{\sqrt{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}\sqrt{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}}{\sqrt{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}\sqrt{\left(1-\frac{1}{3}\right)\left(1-\frac{1}{5}\right)\left(1-\frac{1}{7}\right)}}$
twin meteorBOT
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floral pike
There you go. So they factored the top as such
$\frac{\cancel{\sqrt{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}}\sqrt{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}}{\cancel{\sqrt{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{7}\right)}}\sqrt{\left(1-\frac{1}{3}\right)\left(1-\frac{1}{5}\right)\left(1-\frac{1}{7}\right)}}$
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floral pike
@ruby lichen
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ruby lichen
<@776955706431504386>
OHHH ic
ruby lichen
tysm
but then how does he simplify it here
do u just multiply and divide
or is there a fast way to go from step a to step b
nova fulcrum
Use properties of power
sqrt(a/b) = sqrt(a) / sqrt(b)
in your exercise you have sqrt((4 * 6 * 8)/(3 * 5 * 7))
and that's (sqrt(4 * 6 * 8))/sqrt((3 * 5 * 7))
if you do this to both, numerator and denominator
you get ((sqrt(4 * 5 * 6))/sqrt((3 * 5 * 7)))/((sqrt(2 * 4 * 6))/sqrt((3 * 5 * 7)))
so you simplify there
because you have sqrt((3 * 5 * 7)) in both, numerator and denominator
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nova fulcrum
That is right, but that's not how they did
this is how they did
but your approach is also correct
ruby lichen
@nova fulcrum thank you
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brittle storm
excuse me
sonic zinc
Find the equation of the line
And then think wut happens to the x coordinate when a line meets the y axis
silk hollow
,rccw
twin meteorBOT
zinc wren
can i get help with this aswell ?
Find the points of what? It's not visible
vast shale
Okay so you be gay and than get fuck and be gay that all and ypu ba done
brittle storm
Okay so you be gay and than get fuck and be gay that all and ypu ba done
;-;
vast shale
So do that
leaden tusk
can i get help with this aswell ?
dude do u know to form an equation of line
brittle storm
i forgor how to
leaden tusk
well u need to find slope and then take any point in the equation
slope inercept form etc
if u forgot go through a quick revision
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hello, i need help making a vector function of an elipse where the major axis and ending passes trough Point C as seen in the image.
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hello, i need help making a vector function of an elipse where the major axis an...
little marten
the traditional elipse function only allows me to point the major axis vertically or horizontally.
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glacial meadow
tangent ratiios
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tangent ratiios
glacial meadow
im trying to find FKG, BKC, KDF, and BCK
every explanation i can find is like this
is there a formula idk ab 
<@&286206848099549185>
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sacred terrace
im trying to find FKG, BKC, KDF, and BCK
every explanation i can find is like th...
tan (theta) = opposite side / adjacent side
glacial meadow
what is theta?
sacred terrace
and tan theta = sin theta / cos theta
sacred terrace
what is theta?
the angle
glacial meadow
ohh
glacial meadow
and sin theta cos theta?
sacred terrace
wait you have been taught trigonometry right
sacred terrace
and sin theta cos theta?
sine and cosine ratios
glacial meadow
so tan FKG= opposite/adjacent = 45/45?
sacred terrace
so tan FKG= opposite/adjacent = 45/45?
tan (angleFKG) = the side of the triangle opposite to angle FKG/Side of the triangle next to angle FKG(not the hypotenuse of the triangle)
glacial meadow
based on this
glacial meadow
whats tan(60)
sacred terrace
you do know what sin 60 and cos 60 is?
glacial meadow
like soh cah toa
sacred terrace
yeahh
sacred terrace
so t in the toa stands for tan
glacial meadow
like opposite/hypot 60?
glacial meadow
soh- side opposite/adjacent cah cosine etc
sacred terrace
then sin (60) = opposite/ hypotenuse
sacred terrace
soh- side opposite/adjacent cah cosine etc
yes
so this sin(60) is a constant value
glacial meadow
ook
sacred terrace
and will always be equal to opposite/hypotenuse ratio in a triangle
glacial meadow
ok
sacred terrace
and tan (60) = opposite/adjacent side
glacial meadow
so for like
BKC= opposite/adjacent= tan45 = sin45/cos45
thats just like the template
idk if
sacred terrace
and tan, sin and cos have a relation
tan(angle) = sin(angle)/cos(angle)
you can check this by substituting in the triangle side ratios
glacial meadow
oh??
formulas my love
ok im gonna go try the rest, im probably gonna ask someone to check my work do i close this channel or leave it open for that
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weary surge
Can someone help me understand reference triangles? I can't seem to find out where the side lengths come from
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Can someone help me understand reference triangles? I can't seem to find out whe...
weary surge
I also understand that this is a 30/60/90 triangle and that those seem to have side lengths of 1, 2, and radical 3, but I don't know why and I've also seen radical 3 over two
pallid zenith
I also understand that this is a 30/60/90 triangle and that those seem to have s...
the side lengths come from the sine and cosine of the reference angle
weary surge
Would the reference angle be the 150 degrees or 30 degrees?
I mean I could check but just want to double check
Wouldn't that be -sqrt(3)/2 and 1/2 actually? How come the 2 is dropped
pallid zenith
Would the reference angle be the 150 degrees or 30 degrees?
its 30, here
pallid zenith
Wouldn't that be -sqrt(3)/2 and 1/2 actually? How come the 2 is dropped
the 2 is dropped because your radius is 2
all side lengths get scaled up, by similarity
weary surge
Would it be safe for me to just remember to drop the 2 on all of them just for the exam? I assume it won't be that easy with different cases
but this exam review seems to not focus on those
pallid zenith
Do you understand why its going away?
weary surge
Not completely, that's why I ask in case it's something I can't remember. I do want to understand though.
pallid zenith
the end of your terminal side is not on the unit circle, here
like, the hypotenuse is not length 1
youre probably used to seeing triangles where the end of the terminal side is on the unit circle
but here its not, its 2 away from the origin, not 1
do you get what i mean?
I'm using lingo not because I'm expecting you know it just because uhh
weary surge
I think so
pallid zenith
i guess i shouldnt be using this much lingo
pallid zenith
yea, the side thats (probably) not horizontal or vertical
your reference angle here is 30 degrees
so that gives you side lengths of sqrt3/2 and 1/2
but thats for a triangle with terminal side of length 1
because the unit (literally "one") circle
weary surge
OOOOOH gotchu, so the radius of the unit circle is just 1
pallid zenith
yea
but you can just use similarity, which maybe you saw in geometry
for shapes like triangles and stuff
if you scale one side by a certain amount, say you double a side length
the rest just double, too
weary surge
similarity I actually have not learned, I have a really weird and scattered scope of knowledge. Like, I just learned the Unit Circle today, and I can draw it from "memory" (just following the patterns really), but I also don't remember similarity
I've been out of high school for a long time and I didn't really pay attention in class
does similar have to do with the ratio of each side of two triangles?
like a triangle with terminal side 2 and one with terminal side 4, assuming the other lines are similar in that way, that would be similar?
pallid zenith
similarity is like,
similarity is about proportions of sides to each other
you can make a triangle bigger or smaller
as long as you leave the angles the same, then the triangles will be similar
meaning that if you take any two side lengths, and divide them, youll get the same number
which you can check, here
the triangle on the unit circle has hypotenuse 1 and another side of sqrt3/2
your scaled up triangle has hypotenuse 2 and other side sqrt3
so (sqrt3/2)/1 = sqrt3/2
you can try it with the other sides to check, too
pallid zenith
does similar have to do with the ratio of each side of two triangles?
ratio of sides within a single triangle, more so
pallid zenith
like a triangle with terminal side 2 and one with terminal side 4, assuming the ...
assuming you dont change the angles, you can make the hypotenuse however long you want
the relationship between the sides will stay the same
weary surge
okay this helps a lot, but how do we know that this triangle is twice the size of the unit from the polar coordinates 4, 150?
pallid zenith
okay this helps a lot, but how do we know that this triangle is twice the size o...
i am not sure i understand this question
oh i think i see
you know the reference angle
the reference angle gives you a unique collection of angles, if that makes sense
150 gives you the reference angle
you know another angle in the triangle will be 90 degrees
so there is no degree of freedom left for the last angle
then the radius gives you how scaled up or down you are from the reference triangle for that reference angle
weary surge
to clear up my confusing, is 4 considered the radius?
or 2 in this case
I think I'm confusing numbers here
sorry for my misunderstanding, you're being really helpful
here let me draw
I'm not understanding whether 2 or 4 is the radius, and how the 4 becomes a length of 2. Perhaps I shouldn't be studying this late lol
pallid zenith
yea i think i'm confused where you are getting 4 here
polar coordinates are angle and radius
lemme see
weary surge
4, 150 from the polar coordinates
pallid zenith
so, i guess the question is if you got your point from the triangle or your triangle from the point
weary surge
Triangle from the point I believe
weary surge
this is from the video they provided :o
pallid zenith
pallid zenith
hmm you know i think i see why youre confused!
i dont know why they chose to draw that triangle
weary surge
haha! did he get it wrong in the video?
pallid zenith
i think so
i mean you can count out when he says 2 lol he places it 4 away
maybe just absent-mindedness
weary surge
interesting, okay so it should be radius 4? Then shouldn't the initial side be length of sqrt(3)/2 * 4
pallid zenith
😮 am i mistaken wait
weary surge
😮
pallid zenith
ugh its hard to find a good resource
pallid zenith
interesting, okay so it should be radius 4? Then shouldn't the initial side be l...
it should be radius 4, yea
but it should be clear what happens with the rest of the side length
we changed 2 to 4
but its a similar triangle
what happens to the side of length 1, and the side of length sqrt3
weary surge
I think I'm starting to get it
I'll ask again if I run into a hiccup, but I think I might have it now
thank you so much!!!
I really appreciate your patience and help!
pallid zenith
no problem, good luck
supple fractal
Can someone help me on this problem.I don't know where to start.
x and y are real numbers such that 2/x -1/y=1/(2x+y) (obviously x , y and 2x+y are not equal to 0
paper depot
what's the question
supple fractal
Calculate x²/y² +y²/x² (forgive my ignorance, I totally forgot it)
supple fractal
Nothing
pallid forge
what have you tried?
supple fractal
Nothing worth mentioning
paper depot
may i suggest as a first step that you multiply both sides by xy(2x+y)
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pallid forge
nice
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brave ruin
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brave ruin
If Y can take on the value 0, would that be the number of paths being like zero?
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wise hinge
when Y=2, all three valves must be open, therefore P(Y=2) = 0.8x0.8x0.8
that should give you an idea for how to solve for P(Y=1) and P(Y=0)
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river sierra
