c)
#help-23
pliant inlet
gotcha
foggy salmon
the div n mult is closed under
d i think no
cuz some numbers r not covered
which shld b kinda obv
from the x^3
or u can write out a few terms ig
foggy salmon
um
ye i think that cld work
just show the fn is monotonic
n cant ever hit some number ig
pliant inlet
N \ {x}?
pliant inlet
ill just find the values
foggy salmon
ye
i guess
so then it maps to not all of N
so the set it maps to
cant be equal to N
pliant inlet
right
foggy salmon
ay
safe radishBOT
@pliant inlet Has your question been resolved?
earnest bough
sb help me with this
safe radishBOT
@earnest bough Has your question been resolved?
safe radishBOT
@earnest bough Has your question been resolved?
wet heath
How would I go about proving: Let 𝑎 and 𝑏 be integers. Prove that if 𝑎𝑏 and 𝑎+𝑏 are relatively prime, then 𝑎 and 𝑏 are relatively prime.
Using the theorem: Integers 𝑎 and 𝑏 are relatively prime if and only if 𝑎𝑥 + 𝑏𝑦 = 1 for some 𝑥, 𝑦 ∈ Z.
I know how to do the math as seen below but I do not how to go further than that or describe it in words to form a proof:
(ab)x + (a+b)y = 1
abx + ay + by. = 1
a(bx + y) + by = 1
ak + by = 1 for k = bx+y belonging to the set of integers.
safe radishBOT
How would I go about proving: Let 𝑎 and 𝑏 be integers. Prove that if 𝑎𝑏 and ...
@wet heath Has your question been resolved?
pseudo siren
What you did is correct, you just have to provide the context for it. You can do it the following way:
"We aim to prove that a and b are relatively prime. Meaning, we'd like to find integers x and y such that:
ax + by = 1. We know that ab and a + b are relatively prime."
From there, you can continue as you did with showing that ab and a + b being relatively prime means that a and b are prime in algebraic terms.
"Using the theorem, since ab and a + b are relatively prime, then there exist integers X and Y such that:
(ab)X + (a + b)Y = 1"
As you have shown, this can be rewritten as:
a(bX + Y) + bY = 1
Notice that this is the form that we are looking for. (We were looking for an equality of the form ax + by = 1 where x and y are integers, so we finish the proof by showing how this is what we aimed to achieve.)
"Choosing x = bX + Y and y = Y, we get that ax + by = 1, where, since b, x and Y are integers, then bX + Y and Y are integers as well."
mental patrol
safe radishBOT
mental patrol
I know there is an ss between/ss total formula but not sure what exactly to do to start?
mental patrol
No
This is for exam practice
I can use a calculator but only for simple things
Ss=sum of squares I think
safe radishBOT
@mental patrol Has your question been resolved?
safe radishBOT
Available help channel!
dire fjord
Hii everyone
safe radishBOT
Hii everyone
dire fjord
I have 2 equations $x^2+y^2=2 $ and $x^2/(2-y) +y^2/(2-x)=2$
flat frigateBOT
calculus is fun
dire fjord
I need all real couples (x;y) that satisfy both equations
The couple I reached is (1;1) is there any other one
plucky elk
I need all real couples (x;y) that satisfy both equations
you can solve for one variable and plug it into the other equation
dire fjord
What i did is that I equated LHS of both equations then did common denominator
After that I had an expression with forms of x identical to that of y
plucky elk
it'd be easier if you just showed your work
dire fjord
I got $x^2(2-x)(1-y)+y^2(2-y)(1-x)=0$
flat frigateBOT
calculus is fun
dire fjord
Wait I'll write everything I did and send a pic
So that's my work
Is there something wrong with it
Or something missing ??
plucky elk
you might have multiplied by 0 to get here
what if (2-y) = (2-x) = 0 ?
oh i guess it's undefined since that's in the denominator of the second eqn
dire fjord
Yea it is undefined
I forgot to write the condition
So is it missing anything or this is the correct and complete answer ??
dire fjord
you might have multiplied by 0 to get here
What could I have multiplied by 0 to get here ??
dire fjord
Yea because y can't be different than x If this expression is equal to 0
plucky elk
why?
dire fjord
you might have multiplied by 0 to get here
You can confirm this if you divide this by (1-y)(1-x)
If you do this and then simplify you will get the following
plucky elk
you should include that work then
flat frigateBOT
dire fjord
In this case the expression of x is identical to that of y
So the equation doesn't hold unless x=-y
Wait
x=-y not y
Is this correct now ??
??
safe radishBOT
@dire fjord Has your question been resolved?
sick ice
I can find the derivative of the function, but I'm struggling with plugging in one. Wouldn't that still leave some y's in the answer? The problem wants me to type in a "number" for my answer. I don't think that includes y variables
rigid inlet
safe radishBOT
@dire fjord Has your question been resolved?
dire fjord
.close
granite pike
I’m stuck making 2 equations for this. Can someone help making the equations, you don’t need to solve. Thanks!
safe radishBOT
I’m stuck making 2 equations for this. Can someone help making the equations, yo...
rigid inlet
plucky elk
this channel's occupied. read #❓how-to-get-help
granite pike
<@&286206848099549185>
safe radishBOT
@granite pike Has your question been resolved?
safe radishBOT
@granite pike Has your question been resolved?
safe radishBOT
granite pike
.reopen
safe radishBOT
✅
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@granite pike Has your question been resolved?
safe radishBOT
@granite pike Has your question been resolved?
granite pike
<@&286206848099549185>
idle hearth
I’m stuck making 2 equations for this. Can someone help making the equations, yo...
Chris = x
Anna = y
ur 1st eq is correct which is x=3y
"by his next birthday" that just means plus 1 right, so, x+1=2y (2y bcs he will double her age which is anna's age)
safe radishBOT
@granite pike Has your question been resolved?
marsh granite
why is my answer wrong?
safe radishBOT
why is my answer wrong?
quasi bison
6+3 isn't 12
and 4+3 isn't 10
@marsh granite the question did not say that the numbers on the spinner had to be 1 through 6
you were wrong to assume that
robust sigil
help me for d?
safe radishBOT
help me for d?
robust sigil
how to solve this?
spiral crescent
help me for d?
fg can be continuos only if f and g are both continuous themselves
oh my bad lol i thought i read if fg continous
robust sigil
so the answer is -2 ?
spiral crescent
so the answer is -2 ?
which one?
for d not -2
for d factor the expression and when some expression times some other expression are both in limit, and one of the expression when you put in the limit is non zero or not infinity, then you can evaluate it seperately and take it outside the limit
AH D IS -2
i was reading b
robust sigil
ohh thank u
can u explain why it was -2?
i am not sure about my answr even
.close
quasi bison
need a sanity check over here
is it true that the ring $\bZ/P_n\bZ$, where $P_n$ is the product of the first $n$ primes, has exactly $n$ maximal ideals (the quotients modulo which are $\bZ/(2), \bZ/(3), \dots, \bZ/(p_n)$?
safe radishBOT
need a sanity check over here
is it true that the ring $\bZ/P_n\bZ$, where $P_n$...
flat frigateBOT
Ann
safe radishBOT
@quasi bison Has your question been resolved?
safe radishBOT
@quasi bison Has your question been resolved?
lime stratus
How would I go about calculating y3 here ?
I've tried so many things but so far none of them worked...
peak estuary
unless you have more information you are not giving us, this is impossible to solve
peak estuary
well where are you taking that information from
y3 could also be 703132
for all we know
can you state the whole question?
lime stratus
Okay
X3 and Y3 are in-between X1,Y1 and X2, Y2
Let's just say I know it
What I'm asking is, based off of that information, is there a way to calculate Y3 ?
peak estuary
no
y3 could be anything
if you know that for example y depends linearly on x, then you can do something
that is, y=mx+b
peak estuary
so you were not giving us all information
lime stratus
Okay
peak estuary
do you know that the slope is given as rise over run?
$m = \frac{y_2-y_1}{x_2-x_1}$
flat frigateBOT
Denascite
lime stratus
Ohhhh right
So then m = -70/38500
Which means m = -0.00181818181
So then..
x3 = 1800m+b
peak estuary
well first we now have to find b
lime stratus
But first I need to figure out b
Yes
So
90 = 1500m + b
B = 90-1500m
B=90-1500*-0.00181818181
B = 92.727272715
peak estuary
+92.etc, not *92.etc
violet compass
$\int_{0}^{π/2} \frac{dx} {5cosx + 3sinx}$
safe radishBOT
$\int_{0}^{π/2} \frac{dx} {5cosx + 3sinx}$
flat frigateBOT
Derek ♡
peak estuary
that looks nasty. weierstrass sub maybe?
violet compass
what sub?
peak estuary
often called weierstrass sub. which according to the article is a misattribution. TIL
edgy rover
how do i find cos2(x), i got sin2(x), but when i subbed what i knew into the formula i messed up.
safe radishBOT
how do i find cos2(x), i got sin2(x), but when i subbed what i knew into the for...
edgy rover
i got, (4xroot7)/11 = sin2(x)
the missing side is, root7
when i tried to get cos^2(x) i thought to do, (root7/2)^2 ?
maiden jetty
$\cos^2 x = \left( \frac{\sqrt{7}}{\sqrt{11}}\right)^2=\frac{7}{11}$
flat frigateBOT
秋水
safe radishBOT
@edgy rover Has your question been resolved?
ruby phoenix
f is a function defined on interval [a;b]
f-1 is its reciprocal function
if i have f-1(f(x))
what is that equal to? is it equal to just x ?
obtuse jackal
If f is invertible yes
toxic stratus
If f is invertible yes
then this
toxic stratus
if the inverse exists
obtuse jackal
To have an inverse
ruby phoenix
if the inverse exists
oh okay
ruby phoenix
to prove that the inverse exists should i prove that f is continuous and either strictly decreasing or increasing?
toxic stratus
thats not a necessary condition
obtuse jackal
Strictly monotone is enough
obtuse jackal
Continuous just guarantees f^-1 will be continuous as well
ruby phoenix
ummmm☠☠ PERIODDDD t.
ruby phoenix
Continuous just guarantees f^-1 will be continuous as well
uhhhh okay!
😂👍
thanks yall
still
toxic stratus
its sufficient but not necessary
ruby phoenix
arctan(tan(x)) = x
only if x is in ]-pi/2 ; pi/2[
right?
toxic stratus
obtuse jackal
Yes, because otherwise x is out of arctan's range, so it can't be equal
flat frigateBOT
toxic stratus
you get a chainsaw
ruby phoenix
ummm yeah so wait
obtuse jackal
Otherwise you have 2 different, parallel lines
toxic stratus
i see what you mean now
ruby phoenix
suppose f goes from [a;b] to IR
f-1 goes from IR to [a;b]
then f-1 (f(x)) = x
if and only if x is in [a;b]
is this correct?
toxic stratus
outside [a, b] f(x) isnt defined
toxic stratus
it is
obtuse jackal
suppose f goes from [a;b] to IR
f-1 goes from IR to [a;b]
then f-1 (f(x)) = x
...
f^-1 doesn't have R as its domain, only f([a, b])
obtuse jackal
I mean... x -> x^2 is a function from R to R
That's the entire reason why the concept of surjectivity exists
toxic stratus
no like
"suppose f : [a, b] -> R and f^-1 : R -> [a, b]"
then it would be part of the assumption that f^-1 has domain R
obtuse jackal
Since he skipped 2 lines, to me it didn't feel like a part of the assumption anymore
toxic stratus
if its not then f^-1 doesnt necessarily have domain R
but like
the then occurs after the line saying f^-1 goes from R to [a, b]
so supposedly everything before then is part of the assumptions
safe radishBOT
Available help channel!
solar cloud
split ether
You can use Pythagorean theorem to express h in terms of r and l first
Then you can express l in terms of r using the fact that S = 100
solar cloud
Then you can express l in terms of r using the fact that S = 100
could u help me w somethin
also so i got 100 = pi * r * l
and h^2 + r^2 = l^2
split ether
also so i got 100 = pi * r * l
Yes, now isolate l
solar cloud
alright, 100/(pi*r) = l?
solar cloud
And just take sqrt
how 😐
split ether
h = sqrt(10000/pi^2r^2 - r^2)
solar cloud
100/
split ether
100 got squared, so now it's 10000
flat frigateBOT
A Lonely Bean
split ether
Yeah you can't get rid of it
split ether
**A Lonely Bean**
Just plug this into that
split ether
What did you get for V?
flat frigateBOT
A Lonely Bean
split ether
Yeah, now you let r^2 into the sqrt
solar cloud
Yeah, now you let r^2 into the sqrt
huh
split ether
So you get $V = \frac{1}{3}\pi \sqrt{\frac{10000r^2}{\pi^2}-r^4}$
flat frigateBOT
A Lonely Bean
split ether
So now you need to evaluate the maximum of 10000r^2/pi^2 - r^4
solar cloud
mhm
wait
@split ether what did u get for it?
i got 530.516
that correct?
also what would the domain be
solar cloud
**A Lonely Bean**
also can u explain why the r^2 went above the line? or was that just mistake
split ether
You got that for the sqrt or the entirety of V?
split ether
also can u explain why the r^2 went above the line? or was that just mistake
r^4/r^2 = r^2
solar cloud
oh
solar cloud
You got that for the sqrt or the entirety of V?
entirety
split ether
It also looks like you need to specify the r value for that maximum though
The domain is irrelevant
split ether
Ah
Then just solve 10000r^2/pi^2 - r^4 >= 0
Which you can simplify into 10000/pi^2 >= r^2
split ether
I think there's /rep or smth
solar cloud
/rep
+rep @split ether
rep @split ether
/rep @split ether
hm
alright cya my man, thanks for the help
.close
lament forge
anyone good with like inverse fourier transforms
safe radishBOT
anyone good with like inverse fourier transforms
peak estuary
just ask a question
safe radishBOT
@lament forge Has your question been resolved?
signal granite
The following exercise:
An even function f is right continuous in x = 0. Show that the function is continuous in x = 0
i know intuitively why its right but I'm not sure how to write it down/prove it rigorously.
What i have so far: Since f(x) is an even function we know that for every x in its domain -x is also in its domain and that f(x) = f(-x). lim x-->0+ (fx) = 0, meaning approaching x=0 from [0,x] = f(0). Approaching x=0 from [-x,0] also gives f(0) since for every -x, f(-x) = f(x). Therefore a left limit exists and is equal to the right limit in point x = 0 with lim x-->0 f(x) = 0 which makes the point continuous
safe radishBOT
The following exercise:
An even function f is right continuous in x = 0. Show ...
@signal granite Has your question been resolved?
bronze nymph
The following exercise:
An even function f is right continuous in x = 0. Show ...
so we need to proof that for every epsilon...
let epsilon > 0, f is right continuous on x=0 then exits delta > 0 s.t for every x thats in (0, delta) f(x)-f(0) < epsilon
because f is even, then f(-x) = f(x) < epsilon.
so, for every x in (-delta, 0) and (0, delta) epsilon is greater than f(x)-f(0), hence f is continuous
safe radishBOT
safe radishBOT
Available help channel!
merry atlas
safe radishBOT
merry atlas
I don't understand question c. how would I show this?
isnt the answer already there?
safe radishBOT
@merry atlas Has your question been resolved?
fickle trail
an example of a linear factor is just (x-1) right?
safe radishBOT
an example of a linear factor is just (x-1) right?
fickle trail
power 1 being the highest degree term
what about (x-1)^2? also a linear factor, just twice?
worthy hemlock
what about (x-1)^2? also a linear factor, just twice?
Not
worthy hemlock
Because (x - 1)^2 = (x - 1)(x - 1) results as a quadratic
fickle trail
so (x-1)(x-1) is not a linear factor
worthy hemlock
No
fickle trail
so a linear factor is just one factor on it's own
x=1
therefor the factor is (x-1)
and that's it
in order to be a linear factor it can't be next to another factor?
Wait
but wouldn't (x-1)(x-2) result in a quadratic?
fickle trail
so (x-1)(x-1) is not a linear factor
I think this is linear
worthy hemlock
I think you just worded it weirdly
worthy hemlock
This is correct
fickle trail
so (x-1)(x-1) is not a linear factor
but this is not linear?
an example of linear factorization, rather
i guess i should be more clear
not a single linear factor but two linear factors?
worthy hemlock
That's actually fine, it's linear factors
fickle trail
OK
I think I understand now (finally)
just anything that is power 1
and (x-1)^2 is not in linear form but (x-1)(x-1) is in linear form
thin bridge
linear factorisation is a product of linear factors (and constant)
thin bridge
yeh
fickle trail
sum is addition
difference subtraction
i need to remember this.. so often i see these terms thrown around
thin bridge
(x-1)^2 is the product of (x-1) and (x-1),
it's perfectly fine to have (x-1)^2 in your linear factorisation
fickle trail
wait so it is still considered linear to have (x-1)^2?
fickle trail
Because (x - 1)^2 = (x - 1)(x - 1) results as a quadratic
^
fickle trail
OK
thin bridge
as long as the factorisation is representative of a product of linear factors and constants
thin bridge
by definition of exponentiation, yes
thin bridge
,w expand (x-1)^30(x-7)^8
flat frigateBOT
thin bridge
eg if asked for the linear factorisation of that monster
fickle trail
just a quick and easy factorization -_-
thin bridge
you would not have to explicitly write out
(x-1) 30times
and (x-7) 8 times
thin bridge
the factorisation is linear
thin bridge
no
fickle trail
but the "factorization" is linear
hmmm, so what do we call this? if it's not linear
(x-1)^30(x-7)^8
thin bridge
product of linear factors
fickle trail
interesting..
thin bridge
the expression itself isn't linear as it's a degree 38 polynomial
fickle trail
and obv you can't have any variable with a power inside of the factor itself either, correct? to be considered a linear factor
(x^2 - 1) for example
fickle trail
ty, typo will fix that
(2^2 - 1) would not be a factor at all, it would just be a constant 3
thin bridge
a constant factor is still a factor
fickle trail
to be a factor it must include an x term, right? the whole point is to factorize the polynomial
fickle trail
a constant factor is still a factor
oh...
thin bridge
2x-2 = 2(x-1)
fickle trail
i thought constant is just indicating the dynamic of the graph
and even or odd
but still still the same end behavior depending on the leading term power
fickle trail
2x-2 = 2(x-1)
but if there are multiple factors the 2 can be moved next to any factor, based on the commutative property
fickle trail
I thought the constant is more about stretch, compression, or if it's flipped vertically
so in this case 2x-2 = 2(x-1), (x-1) is the linear factor? 2 is just a constant
but (2x-2) is also a linear factor, afaik?
thin bridge
yeh
fickle trail
weird lol..
thin bridge
though 2(x-1) would be the simplified factorise form
fickle trail
So (2x-2) secretly has vertical stretch in there too
even though it's not as clear as 2(x-1)
and that 2 for vertical stretch is applied to the entire polynomial
but (2x-2) is also considered linear.. so I just gotta be careful. I always wanna factor. factor factor factor. Before anything else, factor.
thin bridge
you can describe stuff relative to other things
2x-2 =2(x-1) is a vertical stretch of (x-1)
though you can also just consider 2x-2 as it is
tranquil fog
Where did I mess up
safe radishBOT
Where did I mess up
thin bridge
you messed up the product of
x^(1/2) and 3x^(1/2)
safe radishBOT
@tranquil fog Has your question been resolved?
safe radishBOT
safe radishBOT
Available help channel!
tranquil fog
safe radishBOT
tranquil fog
Where did I go wrong this time
safe radishBOT
@tranquil fog Has your question been resolved?
grizzled saffron
Suppose triangle ABC has BC = √6, ∠BAC = 60°, ∠ABC = 45°. The length of AC is:
A. AC = 4 B. AC = √2 C. AC = 2, D. AC = 1
grizzled saffron
.close
lean otter
need help understanding how to convert recurrence relations to expressions
lean otter
usually I guess the expression and it works
this time theres a few unknowns
so idrk
what to do
$$4u_{n+2} = 4bu_{n+1} - b^2u_n$$
flat frigateBOT
idecanymore
obsidian oracle
ah, the infamous double order recurrence
There is only one way to do it that I know of
let's say that you have un+2 = aun+1 + bun
the first step is to find all r such that r^2 = ar + b
We call this equation the "characteristic equation of the series"
obsidian oracle
the point is you have to find r that verifies r^2 = ar + b
halcyon carbon
Similar process to differential equation
obsidian oracle
in order to find the correct values of r, you're gonna ofc use the quadratic formula.
Then, you have to distinguish between the cases "2 real roots" ; "1 real root" and "2 complex roots"
obsidian oracle
**idecanymore**
and we're trying to find out if there exists any geometrical series that verify this
lean otter
i think it comes from the assumption that there is some r such that u_n = c*r^n, and then if you plug that into the equation you get:
c*r^(n+2) = a*c*r^(n+1) + b*c*r^n
and dividing it all by c and r^n we get r² = ar + b
obsidian oracle
yep
lean otter
Similar process to differential equation
yeah someone told me about this
it didnt make sense to me why
tbh
then there will be 2 solutions (in most cases) and in the end our equation will look something like this:
u_n = c1 * (r1)^n + c2 * (r2)^n
even youtube'd it
it still didnt make sense
obsidian oracle
**idecanymore**
Point is : there always will be geometrical series that verify this, whether real or complex.
We also use the fact that the set of sequences that are solutions to this form a vector space
obsidian oracle
yes, to find which values of r work
lean otter
$$4r^{n+2} = br^{n+1} - b^2r^n$$
$$4r^2 - br -b^2 = 0$$
flat frigateBOT
idecanymore
obsidian oracle
yep
obsidian oracle
basically
lean otter
$u_n = r_1^n + r_2^n$
lean otter
then there will be 2 solutions (in most cases) and in the end our equation will ...
this
?
and then plug in n=1 and n=2
flat frigateBOT
idecanymore
lean otter
$u_n = c_1 r_1^n + c_2 r_2^2$
flat frigateBOT
rbit ✨
obsidian oracle
Then, three distinct cases depending on the possible values of r :
Case 1 : r can be r1 or r2 distinct real numbers ; then un = A(r1)^n + B(r2)^n
Case 2 : r can only be a single real value ; then un = (A + nB) x r^n
obsidian oracle
Boy here we go
lean otter
then its complex
hypothetically
like
if i get a relation
where r
is complex
what do i do
obsidian oracle
if it's complex, we write them r1 = Ce^(ix) and r2 = Ce^(-ix)
lean otter
does it matter?
lean otter
if it's complex, we write them r1 = Ce^(ix) and r2 = Ce^(-ix)
ik how to write complex numbers thats not my point
lean otter
because what the term will spit out will still be real at the end
obsidian oracle
un = Ar1^n + B r2^n
lean otter
you wrote that its c1r1^n + c2r2^n only for real distinct roots
thats why im asking
no other reason
obsidian oracle
if you're looking for real solutions to un, then the complex roots case is not the same
obsidian oracle
un = Ar1^n + B r2^n
sure, this is also true for complex roots, but you have to make it so un is real
obsidian oracle
so un = A|r|^n cos(nx) + B|r|^n sin(nx)
obsidian oracle
so un = A|r|^n cos(nx) + B|r|^n sin(nx)
@lean otter they're not exactly the "same" because you want un to always be real
obsidian oracle
I mean if you want to keep it to the case where un is complex sure
I was assuming you wanted un to stay real
obsidian oracle
If you extend to un complex, then you only have two cases : distinct roots or unique root
obsidian oracle
np
lean otter
.close
stark robin
How do I solve this? I usually would find the test statistic for the sample but it wouldn’t include the mean or SD, only the sample size and successes.
safe radishBOT
How do I solve this? I usually would find the test statistic for the sample but ...
@stark robin Has your question been resolved?
safe radishBOT
@stark robin Has your question been resolved?
stark robin
So we’d do 40.7-41.6 first?
safe radishBOT
@stark robin Has your question been resolved?
sudden phoenix
1b
foggy salmon
,rccw
flat frigateBOT
sudden phoenix
The answer from the textbook
safe radishBOT
@sudden phoenix Has your question been resolved?
sudden phoenix
<@&286206848099549185>
sudden phoenix
This is qs
sudden phoenix
The answer from the textbook
This is ans
marsh walrus
This is ans
you are unable to get the derivative?
sudden phoenix
1b
sudden phoenix
Got a but idk how to do b
marsh walrus
so did you get to the derivative okay?
sudden phoenix
Yea
marsh walrus
here lemme share uhh
sorry not sure how to share this directly
you need to set the derivative to 0
if you have derivative 0 at some point, then no change
sudden phoenix
Ok
marsh walrus
with one equation, you can do this on a line
sudden phoenix
I think I'll just send u my working
marsh walrus
okay
marsh walrus
,rotate -90
flat frigateBOT
flat frigateBOT
jan Niku
flat frigateBOT
sudden phoenix
use a unit circle
What's unit circle
sudden phoenix
But how do I use this from sinx=cosx
marsh walrus
first coordinate is cosine
second is sine
and the number with pi along the line to each pair of numbers is the angle
sudden phoenix
first coordinate is cosine
Wdym by this
marsh walrus
Wdym by this
look at the pairs of numbers in the image
one is cosine, the other is sine, of that corresponding angle measurement
sudden phoenix
theres another one
Like this?
marsh walrus
yea
so you have the x values where the derivative is 0 right
well, you have conditions on x which make the derivative 0
you got 2 values of of that condition
now you have a formula for getting y values from x values
@sudden phoenix
the problem is also asking you to find their stability, i think
sudden phoenix
now you have a formula for getting y values from x values
How am I supposed to get the y values
marsh walrus
Like this?
if you express these angles as pi/4 and -3pi/4
you should recover the answers in the book, im guessing
sudden phoenix
if you express these angles as pi/4 and -3pi/4
How
marsh walrus
How
its just trigonometry
im using the unitcircle, mostly
do you mean how to get those angle measures?
or how to get the answers or
sudden phoenix
if you express these angles as pi/4 and -3pi/4
Yea how do I get these
marsh walrus
Yea how do I get these
use the unit circle
find 45
what is the radian measure associated with it
sudden phoenix
π/4?
sudden phoenix
The answer from the textbook
The ans doesn't have π/4 tho
marsh walrus
there should be two answers
marsh walrus
5π/4
right, is this between -pi and pi?
the problem tells you it has to be between those numbers
sudden phoenix
Oh
sudden phoenix
The answer from the textbook
Then how am I supposed to get this if the value is out of the range
marsh walrus
Then how am I supposed to get this if the value is out of the range
you can represent it another way
5pi/4 is going forward from 0
maybe you can start at 0 and move backwards?
sudden phoenix
maybe you can start at 0 and move backwards?
135?
But why mmove backwards
I thought moving backwards only for negative
marsh walrus
I thought moving backwards only for negative
because they problem gives you a range
they say you need to find x values that are at most pi rotation away from 0
so pi forward, or pi backward
marsh walrus
once you get those angle measures you are basically there
sudden phoenix
can you vc?
Not at the moment sry
marsh walrus
i just wanna say that
sudden phoenix
This?
sudden phoenix
Oh
sudden phoenix
45 and -135
marsh walrus
what are they in radians
marsh walrus
,rotate -90
flat frigateBOT
sudden phoenix
The textbook ans says π/4 is wrong but -3π/4 is correct
sudden phoenix
The answer from the textbook
This is the answer provided
marsh walrus
im not sure how they say that but okay
maybe they only want the stable one
,w plot e^(-x)(sinx-cosx)
flat frigateBOT
marsh walrus
This is the answer provided
what else is required for the problem
sudden phoenix
So the answer provided is wrong or something?
marsh walrus
So the answer provided is wrong or something?
there are 2
but only one of them is stable
so maybe thats what they want
sudden phoenix
So is π/4 correct?
marsh walrus
it is but its unstable
sudden phoenix
flat frigateBOT
jan Niku
marsh walrus
you have an x
sudden phoenix
**jan Niku**
How did u get this
sudden phoenix
Oh
safe radishBOT
marsh walrus
Thx so much
if youre curious
red is curve, blue is derivative
you can maybe kinda see
anyways
safe radishBOT
Available help channel!
gentle valley
In a density experiment a student measures sides of square object and
measures the mass. Density is mass / volume.
I have height, width, depth and density given. How do I calculate the volume so I can then get density
safe radishBOT
In a density experiment a student measures sides of square object and
measures t...
gentle valley
but is it a cuboid?
gentle valley
In a density experiment a student measures sides of square object and
measures t...
that's all I'm given with no shapes attached
that's the question copy pasted
alright thanks!
.close
safe radishBOT
safe radishBOT
Available help channel!
patent vault
safe radishBOT
patent vault
i made the equation sqrt(x)+sqrt(y)=a/b where a, b Z and gcd(a, b) = 1
not sure what do next
can i get a hint?
toxic stratus
x and y should be irrational
patent vault
x and y should be irrational
ye
toxic stratus
hint: a - a = 0
patent vault
why would that not be true?
toxic stratus
?
patent vault
not sure how to use it
patent vault
whats "it"
flat frigateBOT
patent vault
what would i possibly add
flat frigateBOT
toxic stratus
if you would like
toxic stratus
thats just an arbitrary choice
you want the irrational parts cancelling
leaving behind the rational parts
patent vault
could it be the other way around
toxic stratus
i mean
you dont have to keep the a, b, c around
just
replace them with any random integer
patent vault
?
toxic stratus
no like
we really dont need so much generality
just pick like
your favourite couple of integers
like
okay
a = 1, b = 2, c = 2
so
patent vault
your favourite couple of integers
is there a reason why we can do this
toxic stratus
why not
its just a random example
just
pick any integers you want
provided that they work ofc
i mean you couldve just started with saying like
flat frigateBOT
toxic stratus
so lets set the first thing = sqrt(x) and the second thing = sqrt(y)
then there isnt any picking of integers involved is there
its just a true equation
patent vault
how do you know those values are irrational
bc sqrt(2) is irrational?
thanks for ur help btw
.close
safe radishBOT
safe radishBOT
Available help channel!
misty coral
safe radishBOT
misty coral
what is a "motive"?
woven urchin
a motive is like a reason to be motivated.
"His motive for killer her was because she stole all his sandwiches"
woven urchin
i guess, Ive heard of the other terms related to complex but not that one
misty coral
well I know of all the other ones and know a complex number has them but I have no clue what a motive is
in relation to the
m
woven urchin
modulus is absolute scalar value (or distance from 0), conjugate is when the imaginary part of number is negative (I think), argument is the imaginary component of complex number
never heard of it, so its probably that
misty coral
so if i = sqrt(-1) then I think c and d are both true right
wait
nvm
ty for help
.close
lean otter
can someone help me out with this infinite series. please also tell me what i need to know in order to solve it
safe radishBOT
can someone help me out with this infinite series. please also tell me what i ne...
lean otter
i think it's some kind of algebraic-geometric series, but i'm not sure
please ping
$\sum_{i=0}^{\infty} x^i = \frac{1}{1-x}$
take a look at this sum and take the derivative on both sides
(for |x| < 1)
lean otter
$\sum_{i=0}^{\infty} x^i = \frac{1}{1-x}$
where does that come from?
geometric sum
geometric series
oh wait
could you link some proof please
flat frigateBOT
rbit ✨
lean otter
index should start at 0
i just want to understand it intuitively
$\sum_{i=0}^{n} q^i = \frac{1-q^{n+1}}{1-q}$
flat frigateBOT
rbit ✨
lean otter
prove this per induction, and then assume |q|<1 and take the limit as n goes to infinity
although thats not very intuitive
lean otter
huh?
lean otter
look at what ive written before
toxic stratus
imblind
lean otter
prove this per induction, and then assume |q|<1 and take the limit as n goes to ...
yeah... not very intuitive
i mean we can write it out:
im sure theres some youtube video with some nice visualizations, but if you want to prove it rigurously then induction is a good way
wait i think there was another way
lean otter
im sure theres some youtube video with some nice visualizations, but if you want...
okay, but please just let me know what i need to know in order to solve this
lean otter
https://youtu.be/TGUWt5xcB8M
are you sure
1+q+q²+... = S
q+q²+q³+... = qS
1+q+q²+... = 1+qS
S = 1+qS
S(1-q) = 1
S = 1/(1-q)
works because its absolutely convergent
toxic stratus
assuming convergence tho
lean otter
i don't know man, i've asked this question to many people, a lot of them got it wrong, and the rest just gave me the solution right away
without giving proper explanation, which wasn't very useful you know
soo, what parts are you still struggling with? do you get the derivation i gave or do i need to explain it more?
safe radishBOT
@lean otter Has your question been resolved?
lean otter
soo, what parts are you still struggling with? do you get the derivation i gave ...
give me a sec, let me write it out
btw we need to show first its convergent, do you know any convergence tests for infinite series, in particular the root test?
$1+2\cdot\left(\frac{1}{1,998}\right)+3\cdot\left(\frac{1}{1,998}\right)^2+4\cdot\left(\frac{1}{1,998}\right)^3+...+n*\left(\frac{1}{1,998}\right)^{n-1}+...$
flat frigateBOT
Windy
lean otter
right?
wait we first gotta derive the geometric series
that "*" at the end is just multipilcation
$1+x+x²+x³+... = \frac{1}{1-x}$
flat frigateBOT
rbit ✨
lean otter
is it?
if we then take the derivative of that, we get what you want:
$1+2x+3x²+4x³+... = \frac{1}{(1-x)²}$
flat frigateBOT
rbit ✨
safe radishBOT
@lean otter Has your question been resolved?
safe radishBOT
lean otter
.reopen