The numerator is a square, which is always positive. The denominator, sin^2 x - cos^2 x = -cos(2x). Studying its sign is therefore very easy.
#help-0
marsh rapids
marsh rapids
Knowing your basic trig identities is useful yes
You can also just use the Pythagorean identity and have -1 + 2sin^2 x and study it's sign that way. It's just less obvious than cos(2x)
still lynx
Ok ty for the advices
marsh rapids
Fun fact: advice ne prend jamais de s. Tous mes profs d'anglais ont toujours dis "c'est a piece of advice, comme a piece of information ou a piece of luggage (oui, wtf)". Mais généralement on abrège par "an advice" plutôt que "a piece of advice".
Mais pas "advices", on préfère "advice" au singulier à sens général
still lynx
Hmm ok intéressant je pensais qu’on pouvait le mettre au pluriels avec un s
Et si on veut dire some advice
marsh rapids
Hmm ok intéressant je pensais qu’on pouvait le mettre au pluriels avec un s
Et non. Les anglais sont bizarres
still lynx
Haha ok je saurais maintenant
( que advice ne prends pas de s et que les anglais sont bizarres )
marsh rapids
Fun fact: advice ne prend jamais de s. Tous mes profs d'anglais ont toujours dis...
Après évidemment ça c'est la théorie. En pratique advice est assez utilisé comme nom directement, mais on évite quand même d'y mettre un s
Exemple "thank you for your (valuable) advice"
still lynx
.end
marsh rapids
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granite pecan
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delicate ember
@still lynx its .close!!!
still lynx
.close
lone heartBOT
lone heartBOT
Available help channel!
runic hedge
(.0675(x+11.80))+ (x+11.80)=16
Idk where to start with this
Solve for x
runic hedge
.0675x + .7965 +x +11.80=16
.7965+11.80= 12.5965
.0675x+x +12.5965= 16
1.0675x +12.5965=16
Then
Subtract the 12.5965 from 16?
alpine sable
yea
alpine sable
lone heartBOT
alpine sable
Hey everyone. Working in my grade 12 advanced functions practice book over my break
Stuck on question 2. I wrote the answer I got but don't understand how the book got its answer
chrome plank
The first one is wrong, y = f(x + 4) translates the function for units to the left
In the second one looks like you forgot to translate it to the right?
And you translated two units up instead of down
Remember how translation works?
every y becomes (y - [units upwards])
every x becomes (x - [units to right])
Notice the minus sign
alpine sable
Ok could we work on the first one first then?
chrome plank
Yes sure
alpine sable
I've never done grade 12 work. I'm only starting to look at it in a prep book from finishing grade 11
chrome plank
Remember how translation works?
every y becomes (y - [units upwards])
every x be...
Try applying this!
f(x) is a function, so to translate it 4 (units) to the right ...
what would you do?
alpine sable
I'm thinking haha
chrome plank
ok ahah
alpine sable
So minus rather than plus 4?
alpine sable
I see. But how come it's written that way?
Like seemingly opposite than what I had originally thought
Ohhh
Wait
I just read a page back
Is that just because it's just the function itself?
Its JUST that way
chrome plank
What do you mean?
alpine sable
I'm not sure what I'm trying to say
Don't worry about it.
I understand for the first one now.
As for the second question
chrome plank
I see. But how come it's written that way?
It's counter-intuitive, what you did works perfectly fine with points, but not with equations.
Think of a function like something where you give it a number (x) and it gives you another number (y).
Let's use our function as an example, f(x).
Translating it 4 units to the right means that at the y value corresponding to x = 0 of the translated function, you should have the y value at
x = -4 of the original function
So f(x - 4) means that at x = 0, you will have what originally was at x = -4
Let me know if what I wrote makes sense, otherwise I can make a few drawings that will help
ocean sealBOT
Biolysics
alpine sable
I partially corrected myself from my first attempt with the newly found knowledge
But now I'm not sure why it is not -2
alpine sable
Ok yes
chrome plank
||Looks like you forgot the original +1 on the right side||
chrome plank
Like this is unclear
y + 2 = |x - 1| + 1
Move the 2 on the other side
y = |x - 1| + 1 - 2
y = |x - 1| - 1
if this is what you mean yes
alpine sable
Yes. Exactly what I mean
Thank you for the assistance. I'm slowly getting a stronger understanding. Thank you everyone
chrome plank
lone heartBOT
@alpine sable Has your question been resolved?
worn fox
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grizzled plank
hey please someone help me out on this
lone heartBOT
hey please someone help me out on this
grizzled plank
Express Sn in terms of n
by playing around with the first terms i got to this conjecture which im pretty sure is true since it hold for the first couple dozen terms
but i can't find a way to prove it
ocean sealBOT
AimaneSN
mortal trellis
it would probably help to write down how the sum actually looks for the first few values of n
grizzled plank
hmm alright i'll try to look into that
mortal trellis
you are just adding a bunch of the same integer for some time and then switch to the next integer
only need to figure out how often you add each integer
and then I think the RHS just pops out
grizzled plank
that's how i conjectured expression on RHS
idk how to prove it though
since its just observation from the first values of n
mortal trellis
probably induction
grizzled plank
there are 3 values for which it equals 1, 5 for 2, 7 for 3 and so on
grizzled plank
probably induction
yeah thats the part im trying to work out
mortal trellis
and then split up the sum $\sum_{k=1}^{n^2} = \sum_{k=1}^{(n-1)^2} + \sum_{k=(n-1)^2+1}^{n^2}$
ocean sealBOT
Denascite
mortal trellis
plus minus 1 on the bounds
and then induction on the first sum and argue how the second sum looks
runic hull
It's late so I might be wrong but:
You're essentially asking when does \sqrt(k) switch from a whole number to another
which is asking what's the difference between the (n+1)th and nth square number
Which should be easy to compute
mortal trellis
yeah pretty much
alpine sable
visual proof lol, you can add n+(n-1) until you hit the next thing
grizzled plank
It's late so I might be wrong but:
You're essentially asking when does \sqrt(k...
hmmm alright that makes sense let me dig a bit deeper into that
grizzled plank
visual proof lol, you can add n+(n-1) until you hit the next thing
oooo so each time you're adding an odd number of squares to complete the larger square
runic hull
exactly, which fits your conjecture
grizzled plank
yep it does, but to prove it algebraically on the other hand...
let me show you how i did my induction and where im stuck
mortal trellis
if k goes from m^2+1 to (m+1)^2, then what is floor(sqrt(k)) ?
and how many k are there?
slow hound
hint: the expansion of $\sum_{m^2 + 1}^{(m+1)^2} \floor{\sqrt{k}}$ all have the same value apart from the last term
ocean sealBOT
grizzled plank
if k goes from m^2+1 to (m+1)^2, then what is floor(sqrt(k)) ?
it’s all m’s until the last which is m+1
mortal trellis
so $2m\cdot m + m+ 1 = (2m+1)m + 1$
ocean sealBOT
Denascite
mortal trellis
does that term seem familiar?
grizzled plank
ohh yes the conjectured sum
that makes a lot of sense, thanks guys! appreciate you all! @mortal trellis @slow hound @runic hull @alpine sable @worn fox
.close
lone heartBOT
lone heartBOT
Available help channel!
stone dock
- x = 1 - y
- x = .5
- .5 = 1 - y (1,2 transitivity of = )
- .5 + y = 1 - y + y (3 Addition property of = )
- .5 + y = 1 (4 [insert rule])
what's the exact rule that belongs on line 5? is it smthn abt addition?
worn fox
It'll most likely be something about additive inverses
Depends what kind of language your course uses
stone dock
what's an example
worn fox
A number added to its additive inverse is 0
But then you'd have to add the rule that adding zero doesn't do anything
Maybe just say -y and y cancel
stone dock
is there a specific rule for saying adding 0 doesn't do anything?
worn fox
Additive identity yeah
The list you're looking for really is the axioms of the real numbers
Basically everything you need will be in there, depends how specific you wantnto be
stone dock
had some more trouble,
- x = 1 - y
- x = .5
- .5 = 1 - y (1,2 transitivity of = )
- .5 + y = 1 - y + y (3 Addition property of = )
- 1 - y + y = 1 + 0 (additive inverses)
- 1 + 0 = 1 (additive identity)
- 1 - y + y = 1 (5,6 transitivity of = )
- .5 + y = 1 (4,7 transitivity of = )
- y + .5 = .5 + y (commutative property of addition)
- y + .5 = 1 (8,9 transitivity of = )
- y + .5 - .5 = 1 - .5 (10 subtraction property of = )
- 1 - .5 = .5 ( [insert rule])
on line 12 is there an exact rule to cite here, or is it smthn like "addition" or "simplification"
worn fox
I mean this is pretty subjective
Its both how addition/subtraction works
And it is also a simplification
stone dock
can u show me?
worn fox
Show you what?
stone dock
what u'd cite using addition/subtraction & simplification
worn fox
Well like I said it's subjective, it's not clear exactly what you'd need to write
alpine sable
you have to take something for granted eventually. you are basically saying, whats the "rule" for "1+1=2"
you could probably put a few more steps if you really wanted to, which can probably be labelled
2=1+1
2/2=(1+1)/2
2/2=1/2+1/2
1=.5+.5
1-.5=.5
tacit arch
,w random choice ["subtraction", "addition", "simplification"]
ocean sealBOT
stone dock
yeah, more generally im wondering from what rules & assumptions u could deduce "1-.5=.5" or equations of the like
alpine sable
what axioms do you assume?
tacit arch
those 3 rules are all possible
worn fox
Boy have I got a book for you
stone dock
unsure, but looks like potential for needing axioms
worn fox
It takes about 360 pages of pure logic to prove 1+1=2 from the ground up
Cut your losses and just pick one of those 3 options riemann wrote
worn fox
Yeah
lone heartBOT
worn fox
Again?
lone heartBOT
Available help channel!
modern maple
What haven’t I identified yet and why
lone heartBOT
What haven’t I identified yet and why
remote heron
you missed 2
try reducing each fraction
it may make it easier to say like
okay 5/20 is equivalent to 1/4
so just see if each fraction is equivalent to 1/4
instead of 5/20
does that make sense?
modern maple
Oh wait I can multiply a fraction ive already reduced to get another equivalent?
Thanks I got it now
.close
crimson solstice
lone heartBOT
crimson solstice
related rates problem
im not really sure what to do past this point
the 1m deep thing is throwing me off
lone heartBOT
@crimson solstice Has your question been resolved?
waxen flame
@crimson solstice In the question, what are the two values that are "changing"?
lone heartBOT
crimson solstice
<@689942806302556211> In the question, what are the two values that are "changin...
the height of the water
and the volume of the shape
lone heartBOT
Available help channel!
waxen flame
Those are the values that you will be differentiating in your volume equation.
waxen flame
Type .reopen.
crimson solstice
.reopen
reopen
i think it reopened under ur name
let me try differentiating
there are 4 derivatives or rates of change
v b h l
i think im looking for h but and i know V but how do i get b' l'
so that i can solve for h '
waxen flame
And you will need to determine their values in terms of the differentiating value.
If the height increases, how much does the width increase? You can solve that in terms of the height.
The length doesn't change so it will just be a constant value.
crimson solstice
If the height increases, how much does the width increase? You can solve that i...
hmm
i see what u mean but not sure how i would solve for that
or get that rate of change
waxen flame
That's your trough from the side.
I'm sure there are many ways to do this, but mine would be to make an equation for one of the sides.
crimson solstice
hmm
proper tusk
Could someone help me with finding the unit rate, aswell as telling whether an equation has ine solution, no solution, or is an identity?
wait oops
was i not suppose to talk in here? i’m new
waxen flame
You will need to open your own channel. Go to Math Help (Available) and pick one of the open channels to ask your question.
proper tusk
ok thanks!
waxen flame
yw
crimson solstice
ok so im solving for h right
would i manipulate the area formula
and solve for h ? with the values that i do know?
waxen flame
Yes.
waxen flame
One moment, I'm making a graph to show you.
If you look at that graph, you will see how you can calculate the width in terms of the height.
By determining an equation for the side, of which you have the two points necessary to do so, you can solve for the width, x.
lone heartBOT
i need help
lone heartBOT
@frail pivot Has your question been resolved?
somber wedge
keen sinew
Hello,
keen sinew
I wanted to know how in the last part
Why is it that when we do
y= alnx - xln B
Why isn't it this when we derive
y' = a/x - b/x
shouldn't xlnb be derived somehow?
dawn hamlet
it is x multiplied by a constant so the derivative is the constant
keen sinew
what about a; isn't a also a constant
jagged raptor
you mean differentiated the word derived has a mostly unrelated meaning
We are taking the derivative wrt x so we just get the constant ln(B) back. it appears you were doing some mix of differentiating wrt x and b that doesnt work
keen sinew
right im differentiating
3.6 Stewart Differentiation Rules
im confused how
alnx b differentiated turns in a/x
but not on the other side
jagged raptor
what about a; isn't a also a constant
it is
because in xln(B) we dont have an ln(x)
we just have x (times a constant)
keen sinew
but not in a(lnx)?
ill start looking back and see if i can do some more excercises around this
i always hated logarithms lol
.close
balmy epoch
i'm trying to figure out question 13. i know this is a simple algebra problem but we learned point slope form and i feel like it's screwing me up - can anyone help me? thx
balmy epoch
this is my work if you want to see it. i know the answer is 5.8 but i keep getting 10?
<@&286206848099549185>
drowsy lagoon
y-6 and x-5 is wrong
think again what do y and x represent
are they representing t or d
balmy epoch
sorry i got the letters wrong
t is supposed to be y
and d is supposed to be the x
nvm ty
.close
covert juniper
How do i simplify $$(\frac{2}{e})^{\frac{ln(x)}{ln{(2/e)}}}$$ I plugged it into wolfram alpha and it gave back x. How do i show this?
ocean sealBOT
ALBERTO BALSAM
somber wedge
Have you tried breaking this down via the power rules?
covert juniper
The bottom of the fraction with e is simple enough
last ether
$$\left(\frac{2}{e}\right)^{\frac{\ln{(x)}}{ln{\left(\frac{2}{e}\right)}}}$$
ocean sealBOT
Umbraleviathan
covert juniper
yep
last ether
Oh yeah use log fraction rules
covert juniper
$$2^{\frac{\ln{(x)}}{ln{\left(\frac{2}{e}\right)}}}/(xe/2)$$
ocean sealBOT
ALBERTO BALSAM
ocean sealBOT
Umbraleviathan
covert juniper
ah ok so just chnage to base 2?
somber wedge
ah ok so just chnage to base 2?
well if you look at the denominator in the exponent, you can make the base (2/e)
covert juniper
very nice
last ether
$$\begin{align*}
\left(\frac{2}{e}\right)^{\frac{\ln{(x)}}{\ln{\left(\frac{2}{e}\right)}}} &= \left(\frac{2}{e}\right)^\log_{\frac{2}{e}}{(x)} \
&= x
$\end{align*}$$
ocean sealBOT
Umbraleviathan
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
somber wedge
$$\begin{align*}
\left(\frac{2}{e}\right)^{\frac{\ln{(x)}}{\ln{\left(\frac{2}{e}...
via a^(log a (x)) = x
last ether
I will wanna preface though, this only equals to $x$ for $x>0$
ocean sealBOT
Umbraleviathan
covert juniper
Ah yep
last ether
As far as complex numbers go, not too sure if it extends to that
covert juniper
thanks guys
last ether
Np
vital crystal
x²-(5-5i)x-13i=0
lone heartBOT
x²-(5-5i)x-13i=0
rose sigil
last ether
$$\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$
ocean sealBOT
Umbraleviathan
last ether
Uh
It might work with complex numbers
It might get difficult though
Try using it and plugging it back in
vital crystal
Okkay
last ether
I'm getting two answers using a diff method, lemme check on Symbolab
Yeah they did the same thing
They set x = a+bi
vital crystal
last ether
Yeah
Okay I guess quadratic formula works then
I just turned each unknown variable into a+bi and 0 into 0+0i
And did some rearranging
I guess you can use quadratic formula on complex numbers
vast bobcat
lone heartBOT
vast bobcat
how do I find the exact number with linearity of expectation?
last bullet point
<@&286206848099549185>
iron mulch
lone heartBOT
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.
last ether
!15m
lone heartBOT
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 bobcat
i've already posted this question before
vast bobcat
sorry about that ):
lone heartBOT
@vast bobcat Has your question been resolved?
vast bobcat
<@&286206848099549185>
real gazelle
well
if X is emitted with probability p
then around what percentage of the particles will be X
if the number of particles is very large
vast bobcat
yea
real gazelle
but what if it's probability p for X to get emitted
vast bobcat
p - x?
real gazelle
what is x
vast bobcat
possibility of x
lone heartBOT
@vast bobcat Has your question been resolved?
lone heartBOT
lone heartBOT
Available help channel!
brittle hedge
Can someone walk me through an empty set proof ? I struggle with these problems
placid zinc
Do you have an example by chance?
brittle hedge
Yes sorry it’s sending
My internet is taking its time
Hmm I can’t seem to send pictures oh well I’ll write it out
For all sets A, B and C, if (AUB) is a subset of C then (AnB)-C=empty set
I see it is true but struggle with the proof part
neat sierra
how would u start proving set equalities
brittle hedge
You would prove the empty set is a subset of of anb-c and that anb-c is a subset of the empty set?
placid zinc
There's a few ways to prove. We can show that they are each a subset of the other with a logical argument
Or we can algebraically manipulate one into the other
brittle hedge
My professor says that when you see empty set a light should go off in your head to prove by contradiction
placid zinc
We can go that way. Assume there's an element in (A U B) - C. What follows?
brittle hedge
In (anb)- c rather?
I mean when I first saw this I wrote this but i don’t know if it’s right
,rotate
ocean sealBOT
brittle hedge
And if a is not in c and b is not in c
And we stated that a or b are both in c
Then it must be empty
Is my reasoning at least it may be a bad proof though
,rotate
ocean sealBOT
neat sierra
Ok, well look at it this way
There is another assumption in the question that was not used yet
That AuB is a subset of C
brittle hedge
And we stated that a or b are both in c
I used it here
neat sierra
ok, I have to jsut get this out of the way... Do you know the difference between an element of versus a subset of?
Like, it doesnt make sense to say a set is in another set here
brittle hedge
But if a is a subset of b for example all a are elements of b
neat sierra
right..
usually we write sets in capital letters to avoid confusing
like, A and B would be sets and a can be an element of A and b can be an element of B
neat sierra
So, in your question, you showed that x is in A and x is in B. So x is in A n B
But A n B is a subset of A u B
But A u B is a subset of C given in question. Then you have x is from a set contained in C, subtract the set C.
So, what can x be?
brittle hedge
It is empty
neat sierra
right
brittle hedge
So is that the proof
neat sierra
that proves one way
brittle hedge
And then you prove other way
neat sierra
yep
which is basically the reverse of the same argument
Assume x in empty set, ...
brittle hedge
I actually don’t know how to do the other way
Is that why my prof proves by contradiction for this?
neat sierra
Yep
Let x is in empty set, and also assume x is not in the left set
There should be some contradiction. And hence, your assumption must be wrong, x must be in the left set
brittle hedge
What is the left set?
neat sierra
Oh , it’s just (A u B) - C
I just moved to my phone ,so I got lazy mb
Give it a try
neat sierra
X is not in (A u B) - C
So x is in the complement of (A u B) - C
What does that mean?
brittle hedge
We haven’t done anything with complements in this class but I have in a stats class I guess so it’s everything not in AuB -C
brittle hedge
No I do not
neat sierra
Hmm
Ok, so there is a property, (A-B)’ = A’ u B
We can apppply this law to what we have for the complement
But the proof of this requires demorgans. I’m not sure if u r allowed
neat sierra
We haven’t done anything with complements in this class but I have in a stats cl...
But either ways, now you know this. Try to see what happens if u apply this to the above
brittle hedge
Do you want me to send the proof my professor posted and we can go over it maybe then?
brittle hedge
So for contradiction proofs you essentially assume the negation?
And then reach a contradiction with it?
Is that the idea
neat sierra
Yep
Basically
If you want to show a true statement p implies q.
Assume q is false
And work from there
brittle hedge
Ahh okay okay
neat sierra
Since we know this is a true statement, you WILL come across a contradiction. Once you find said contradiction, you are done.
neat sierra
Yeah. It takes some time convincing yourself proof by contradiction works
I didn’t understand it until I took a philosophy course on logic
brittle hedge
Yeah it seems sort of strange
brittle hedge
One this for the third line
Thing*
Did they say AnB in particular AUB because AnB is in AUB?
woven ether
Hello i am sheldon
brittle hedge
And that also connects back with the fact that AUB is a subset of C better?
neat sierra
Did they say AnB in particular AUB because AnB is in AUB?
Yes. Try visualizing the ven diagram.
brittle hedge
Okay okay I see I seeee
neat sierra
And that also connects back with the fact that AUB is a subset of C better?
Yep. That’s the contradiction
neat sierra
Np
lone heartBOT
where should i put the brackets?
old sentinel
around the 42+21 i think
lone heartBOT
Available help channel!
oak wasp
So, i'm having a bit of a problem involving 3d coordinates and angles in a program that I'm writing.
Currently, I have an object that's moving in a straight line in some direction in spherical coordinates. it's in a box, and when it hits an edge of the box, it is supposed to 'scatter', to a new random direction in spherical coordinates.
so every step of the program I have this:
//move the object in the direction specified by theta and phi a constant total distance
xx += dsin(phi) * dcos(theta);
yy += dsin(phi) * dsin(theta);
zz += dcos(phi);
//generate random values for theta and phi.
var randomTheta = random_range(0,180);
var randomPhiNonNormalized = random_range(0,1);
var randomPhi = darccos(1 - (2 * randomPhiNonNormalized));
//check to see if it hits the bounds of the cube, if so, set its angle to the random one
if (xx <= 250){
theta = randomTheta + 270;
phi = randomPhi;
}
if (xx >= 750){
theta = randomTheta + 90;
phi = randomPhi;
}
if (yy <= 250){
theta = randomTheta;
phi = randomPhi;
}
if (yy >= 750){
theta = randomTheta + 180;
phi = randomPhi;
}
if (zz <= 250){
theta = randomTheta;
phi = randomPhi + 270;
}
if (zz >= 750){
theta = randomTheta;
phi = randomPhi + 90;
}
some programming quirks: the dcos and dsin etc. functions give results in degrees. Also, the Y axis is flipped with respect to what we normally have in math (since the screen draws from left to right, top to bottom): some other non-obvious things to understand about this:
the reason that theta and phi have to be rotated in the 6 if blocks is so that the new random distribution of angles is being picked from the correct hemispherical shell based on which face of the box that it's inside it hits.
the deal with these two lines here:
var randomPhiNonNormalized = random_range(0,1);
var randomPhi = darccos(1 - (2 * randomPhiNonNormalized));
is that you can't just pick a random theta and random phi and get a uniform distribution of directions, for the same reason that greenland looks huge on an unwrapped map; the circumference at phi near the poles is smaller, so those values of phi have to be less likely, so phi needs to be drawn from a non-uniform distribution based on some trig. If you don't do it the shell looks like this (simulated by scatering 10000 objects):
if you do, you get the correct distribution which looks like this:
Now, for the problem:
I get the correct distribution (second picture) if the objects are set to collide with the walls of the box (xx and yy), but if it collides with the floor or ceiling (zz, the following lines of code):
if (zz <= 250){
theta = randomTheta;
phi = randomPhi + 270;
}
if (zz >= 750){
theta = randomTheta;
phi = randomPhi + 90;
}```
the directions will give the wrong distribution (first picture). Does anyone know what I'm doing wrong?
winter rune
what language is this?
oak wasp
gml (gamemaker studio 2)
winter rune
are you meant to be adding a value to randomPhi rather than randomTheta in this case?
that seems to be what the difference is between this and other cases as you have typed it
oak wasp
I think it might have something to do with that, since it is the difference. I've tried to do it with theta, but I haven't found any values that work yet
I think I might have to rotate both phi and theta??
winter rune
so phi and theta are angles on a sphere?
oak wasp
yeah. phi is the vertical angle in the z dimension and theta is the angle in the xy plane
winter rune
what I can think of is perhaps it gets confused if phi is not between say, 90 and -90
I don't know enough about the system you are using but I could see it having issues with that
actually you know what, I think adding to randomphi throws off the normalization, like it makes the normalization applied incorrect
oak wasp
ohhhh yeah you're right! damn I have to figure out how to rotate it... I think I have to rotate it first and then normalize it??
winter rune
likely yes
oak wasp
yeah I'll try that and see how it goes
hmm is that really true? why would rotating an angle change the probability distribution?
winter rune
Because the area on the sphere is far smaller around Z = +90 or -90
Thinking how to describe this without visuals
oak wasp
hmm, but why would it matter if you rotate after the fact?
winter rune
oak wasp
oh I think i see. phi and theta are flipped here. theta is the angle controlling the angle in the Z axis and phi is in the XY plane, where it's supposed to be the other way around? i think i rotated to the correct orientation but because of the normalization I need to rotate via a different set of axes??
winter rune
Because the amount of area on the sphere around a certain position is dependent on the angle Phi but not on the angle Theta, so when you rotate around Phi it moves an area near say, the equator, which should have a lot of points, to a much smaller area near the pole, and an area near the pole, which should have few points, to a much larger area near the equator
Rotating Phi is not equivalent to rotating the sphere as though it is a solid object
Think of taking the sphere and flattening it out like a map
oak wasp
ok
winter rune
When you add to Phi, you arent rotating the sphere, youre cutting the map in half horizontally and swapping the two halves
Imagine taking the map of earth and moving antarctica to the equator in the map and not on the globe
It would make antarctica much larger, and distort its shape heavily
So your even distribution becomes an uneven distribution
oak wasp
ohhh yeah I think I understand.
hm I have to think about what to do that will actually rotate it correctly.
i get the sense that it's not so simple, but maybe it is simple and I'm missing it
winter rune
Are you only concerned with rotating by 90 and 270 degrees?
oak wasp
rotating the average new angle by 90 and 270, yes
because when it hits the top and bottom, the new angles need to be distributed in the hemisphere normal to the face
winter rune
Is the angle 0 at one of the poles or at the equator?
oak wasp
phi = 0 , 180 are the poles
winter rune
Try adding 0.25 or 0.75 to PhiNonNormalized, have a little if statement like if PhiNonNormalized > 1 {PhiNonNormalized -= 1 and then do the normalization with that new value
oak wasp
I see, let me try that.
hmm I think I have to do the wrapping around the bounds a bit differently but it might be promising
hmm I don't think this is quite right; shifting around phiNonNormalized will only be shifting the uniform distribution, which won't really do anything
lone heartBOT
@oak wasp Has your question been resolved?
oak wasp
very hacky solution but i think it works
var flipper = irandom(1);
if (flipper == 0){
theta = randomTheta;
}else{
theta = randomTheta + 180;
}
var randomPhiNonNormalizedZZ = random_range(0,1);
var randomPhiZZ = darcsin(1 - (1 * randomPhiNonNormalizedZZ));
phi = randomPhiZZ + 90;
mellow sphinx
lone heartBOT
mellow sphinx
I need the 3 questions above answered especially 2 and 4
<@&286206848099549185> 15 mins
Please help within the next 5 hours
lone heartBOT
@mellow sphinx Has your question been resolved?
mellow sphinx
<@&286206848099549185> please come online and help me, explanations are greatly appreciated and if possible the answers.
lone heartBOT
@mellow sphinx Has your question been resolved?
mellow sphinx
<@&286206848099549185> can one of you guys please assist me
lone heartBOT
@mellow sphinx Has your question been resolved?
tidal zealot
lone heartBOT
alpine sable
its a notational thing, should be fine
to be completely safe (and to get desmos etc to recognise it), (sec theta)^6 is ok too
high granite
I was investigating some ways to derive dx^x/dx and saw you could write x^x as (x*x^(x-1)). Then, you can use the product rule to produce:
(x*x^(x-1))'= x^(x-1) + x*(x^(x-1))'
(x^x)'= x^(x-1) + x*(x^(x-2) + x*(x^(x-2))')
...
(x^x)'=x^(x-1) + x^(x-1) + ... + x^(x-1) = x(x^(x-1))= x^x
This is obviously incorrect. Where did I go wrong?
last ether
$x^x$ is the same as $e^{x\ln{(x)}$; you can use chain rule
ocean sealBOT
Umbraleviathan
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)
last ether
The method you're doing is kinda convoluted
alpine sable
its the classic 'proof' of 0=1: $$1=\frac{d}{dx}x=\frac{d}{dx}(1+1+\cdots+1)\stackrel{?}{=}0+\cdots+0=0$$ The error is assuming that the derivative acts linearly there, as the length of the sum changes
ocean sealBOT
Toby
last ether
I mean
last ether
**Umbraleviathan**
Compile Error! Click the <:errors:504266556054962196> reactio...
This
@high granite
alpine sable
yeah, but they also wanted to know why what they are doing doesnt work
last ether
Convoluted method, brain gets lost that's why
high granite
<@558291236956405760>
i actually solved it by implicitly differentiating it on both sides with $ln(y)=ln(x^x)$
ocean sealBOT
fleur
high granite
its the classic 'proof' of 0=1: $$1=\frac{d}{dx}x=\frac{d}{dx}(1+1+\cdots+1)\sta...
but yeah i was curious as to why it went wrong
last ether
Ig that can work yeah
high granite
.close
flat saddle
Can someone explain to me how they did this
lone heartBOT
Can someone explain to me how they did this
violet bear
are you trying to figure out why p(smoke|fire) is .9?
last ether
Smoke is common even due to BBQs is concerning
violet bear
lol true
flat saddle
are you trying to figure out why p(smoke|fire) is .9?
yes
there is a 1% chance that there is a dangerous fire and then there is a 10% chance that that fire makes smoke?
flat saddle
yeah, I just watched a lecture explaining it but didn't understand anything
A n B means that A and B both happen right
violet bear
yeah
violet bear
because the probability of B is the given condition
flat saddle
why tf don't I understand anything lol
wary stream
Not quite, P(A|B) means the probability of A given B, so in that example, it's the probability of smoke given there is a fire
violet bear
yeah
flat saddle
so, if there is smoke and fire what is the probability that it is a fire
wary stream
That's why you would divide by the probability of fire, because logically, when there is a fire, there is smoke
violet bear
so, if there is smoke and fire what is the probability that it is a fire
if there is fire, what are the chances of smoke
flat saddle
I understand the word version but not the equation
formula
there is a 90% chance of smoke if there is a fire
violet bear
mhm
flat saddle
and there is a 10% chance there is smoke
and a 1% chance that there is a dangerous fire
quick comet
pl scan someone help nme with this
wary stream
pl scan someone help nme with this
flat saddle
how tf am I going to pass probability and statistics when I don't understand this lmao
flat saddle
wary stream
@quick comet Please read #❓how-to-get-help
flat saddle
wary stream
how tf am I going to pass probability and statistics when I don't understand thi...
https://youtube.com/watch?v=ibINrxJLvlM
Maybe this will help?
Also, that equation is defined like that
flat saddle
ok I will watch this
wary stream
violet bear
ok matas
violet bear
i'll be real
the given statements just seems incorrect
i created this two way table
now it's asking for p(smoke|fire)
if there is a fire, then what out of those chances will there be smoke
.001
so if i divide .001 by .01 i get .1
but if it were asking for P(no smoke|fire) it would be .9 because .009 / .01 is .9
flat saddle
okay so in P(A|B) we are dividing by P(B) because we know it is B
I guess I understood this example
I'll try make sense of it, thank you
.close
lament basalt
Is what I did correct? It's about trigonometric identities
lone heartBOT
Is what I did correct? It's about trigonometric identities
lament basalt
I gotta prove that sin x times sec x = tan x
marble coral
Is what I did correct? It's about trigonometric identities
You did
You could do too:
sin × sec = sin × 1/cos = sin/cos = tan
lament basalt
Also
worn fox
What's all this x and y business
Seems over complicated
You have 1/tan(x) * tan(x) :)
marble coral
It's better not to use x/y
marble coral
Is what I did again correct?
1/tan × (1/cos) × sin =( 1/tan×cos) × sin = sin/(tan×cos) = sin/((sin/cos)×cos) = sin/sin = 1
marble coral
You have 1/tan(x) * tan(x) :)
Better way
1/tan × 1/cos × sin = 1/tan × sin/cos = 1/tan × tan = tan/tan = 1
gray isle
many things you were doing are frowned upon
not consistently writing the argument for your trig function x
mixing sides of the equation without the appropriate notation
lone heartBOT
@lament basalt Has your question been resolved?
pulsar hearth
Is what I did again correct?
tanx = sinx/cosx => 1/tanx = cosx/sinx .
Cosx/sinx* 1/cosx * sinx => cosx /cosx * sinx /sinx=> 1*1 =1 . The way you solved it using variables looks pretty confusing . ..
violet bear
Again, trigonometric identities
pulsar hearth
So i can't tell if you are right or wrong .
violet bear
what you should always do is simply the expressions to sine and cosine
marble coral
1/tan × 1/cos × sin = 1/tan × sin/cos = 1/tan × tan = tan/tan = 1
This way is the fastest
gray isle
use $\overset{?}{=}$ or do a LHS, RHS proof
ocean sealBOT
ℝamonov
lament basalt
tanx = sinx/cosx => 1/tanx = cosx/sinx .
Cosx/sinx* 1/cosx * sinx => cosx /cosx...
Ohhhkay, I see thanks for your answers yall guys I appreciate it
Also, one more thing
I'm pretty scared about my upcoming school year
Bcs of my fears of what if I become unproductive again, lazy, and such
I'm trying to finish trigonometry and logarithms before last school year before college and trying to set up schedules of me studying and coding
Any tips, bros? Thanks
Sry for long message
violet bear
you doing precalculus?
lament basalt
you doing precalculus?
I have finished precal
violet bear
ah
lament basalt
But ngl I didn't even understand it only some bcs it was myself being unproductive and lazy and pretty rushed up as well
pulsar hearth
If you feel like your basics aren't good , what i would do is pick up a trigonometry book , any book recommended by your teachers i don't know your curriculum so i can't recommend one . But if i feel like i am not good enough in a topic i will understand it's base and then move on to higher level. But depends on how much backlog you have and how much time you have . Don't stress too much it is better when you enjoy what you are studying. Easier said than done though.
lament basalt
If you feel like your basics aren't good , what i would do is pick up a trigonom...
Ohkay, thanks
lament basalt
If you feel like your basics aren't good , what i would do is pick up a trigonom...
My basis for learning trigonometry is the book that I have last school year and watching yt videos
pulsar hearth
So you feel confident in your trig base ?
If so then solve questions given in the book or given by the school faculty , if you can't solve them and get confused it's alright that just means that your concept weren't clear enough and it is good that you can clear your confusion now rather than getting confused during examinations . I Don't think you ever stop getting confused while you learn ,at least that has been my experience. But you can always ask doubts to your teachers (at least i hope that is facilitated) . Sorry for the lack of commas btw
pulsar hearth
I'm trying to finish trigonometry and logarithms before last school year before ...
As for the schedule it depends upon what you prioritise most and need to spend time the most on . Keep them on the top don't expect yourself to perfectly follow it , stuff doesn't go as planned always but if you can't follow it , make up for it tomorrow , but don't keep this rolling . Just a backup. Things always don't go as planned but that doesn't mean they always don't. Don't study just to clear exams but to understand what you are studying that will make it more fun , at least it works for me . I am not too good at studying but i have fun when i study and understand it so i do it . When i don't it is really frustrating but i eventually do , so it evens out .
Good luck with your studies :)
lament basalt
So you feel confident in your trig base ?
Not pretty yet but I'm learning
alpine sable
What does f(2x-f(x)) having 3 roots mean here
lone heartBOT
What does f(2x-f(x)) having 3 roots mean here
alpine sable
copper stone
What does f(2x-f(x)) having 3 roots mean here
if r is a root then f(2r - f(r)) = 0
alpine sable
i know
im not asking for that
so i know that f(x)=a(x-b)²(x-c)
since b,c are the roots
i think 2x-f(x) has 1 tangent and 1 other one
on y=b and y=c
but even if its the case
i dont know what to do
marsh rapids
so i know that f(x)=a(x-b)²(x-c)
3 equalities for 3 variables maybe ?
alpine sable
theres only 2
marsh rapids
Oh
alpine sable
<@&286206848099549185>
nocturne dove
What does f(2x-f(x)) having 3 roots mean here
fx=2x for 3 values of x
alpine sable
??
lone heartBOT
@alpine sable Has your question been resolved?
lone heartBOT
@alpine sable Has your question been resolved?
lone heartBOT
lone heartBOT
Available help channel!
thin urchin
lone heartBOT
thin urchin
because there are 0-9 for each bit
actually, I have no idea how to even start this one
urban pine
a 64-bit integer has 2^64 possible values
"bit" means "binary digit". only holding a value of 0 or 1.
as opposed to a decimal digit having values zero through nine
quiet delta
How would I solve this limit? I know of the standard limit sin(x)/x, however, this seems like the inerse of it
tacit arch
multiply and divide by 3
tacit arch
right
quiet delta
how does that help though? Do we not need to rewrite the limit so it resembles sin(x)/x?
tacit arch
you can substitute limit variables
t = 3x
so limit as x goes to 0 of 3x is 0, that means limit t goes to 0 as well
quiet delta
Yes, I tried that as well, however, it just results in t/3sin(t)
I gave up on it since I do not konw how to rewrite t/3sin(t) as sin(t)/t
ocean sealBOT
riemann
quiet delta
Well yes, but that still leaves t/sin(t) != sin(t)/t
I feel like I'm missing something important here
violet bear
there is a property that ax/sin(bx) = a/b
quiet delta
there is a property that ax/sin(bx) = a/b
The quotient rule for liits, right+
tacit arch
the function $\frac{1}{x}$ is continuous so if $\lim f(x) = C$, then
$\lim \frac{1}{f(x)} = \frac{1}{\lim f(x)}$
alpine sable
Can i construct a proposition from another proposition without logical connectives?
lone heartBOT
@alpine sable Has your question been resolved?
tacit arch
uhhh, try #proofs-and-logic
alpine sable
.close
alpine sable
Is my working out correct?
lone heartBOT
Is my working out correct?
ocean sealBOT
alpine sable
no, you forgot chain rule
oh wait that's just a scribble
yeah that's correct
Thanks :D
Would it be better as a convention to make the square root into a fractional exponent?
Just to keep the format consistent?
@alpine sable
alpine sable
Thanks :D
Would it be better as a convention to make the square root into a frac...
not necessary, but it depends on the problem
you might want it in a different form
depending on whether you're trying to compute the max/min or not
I see. Thanks!
.close
rotund laurel
what is the difference between
dy/dx (sinx) and dy/dx + sinx
hard thorn
what is the difference between
dy/dx (sinx) and dy/dx + sinx
first one y is the function sin(x)
and second one y is an undefined function + sin(x)
hard thorn
what is the difference between
dy/dx (sinx) and dy/dx + sinx
first is (sin(x))' and second is a function f'(x)+sin(x)
rotund laurel
ooooo so y in dy/dx defines a function.
like x^2 = f(x)=y
it can also writen like
d(f(x))/dx
molten pivot
That first one is not notated properly
The derivative of sin(x) is not written dy/dx sin(x)
You would write
$\frac{d\sin(x)}{dx} = \frac{d}{dx} \sin(x)$
ocean sealBOT
Mr. Gamer
molten pivot
That just means the derivative of some unknown function y multiplied by sin(x)
marsh rapids
y should be a function defined somewhere else
rotund laurel
it can also writen like
d(f(x))/dx
correct?
molten pivot
Yes
rotund laurel
y should be a function defined somewhere else
you meany or why?
so what will happen if we diferentiate dy/dx ?
molten pivot
then you get the second derivative of y
hard thorn
it can also writen like
d(f(x))/dx
yep
rotund laurel
$\frac{dy}{dx}+(sinx) = \0$
ocean sealBOT
yuvan
$\frac{dy}{dx}+(sinx) = \0$
```Compilation error:```! Undefined control sequence.
l.57 $\frac{dy}{dx}+(sinx) = \0
$
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
Preview: Tightpage -1310720 -1310720 1310720 1310720
[1{/usr/local/texlive/2020/texmf-var/fonts/map/pdftex/updmap/pdftex.map}]```
rotund laurel
$\frac{d}{dx}sin(x)+a = {0}$
ocean sealBOT
yuvan
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)
rotund laurel
in first thing what should i do
i second it is obvoius i need to differntiate sinx
heady pollen
first one y is the function sin(x)
i would interprete it as
rotund laurel
**yuvan**
```latex
$\frac{dy}{dx}+(sinx) = \0$
```Compilation error:```! Undefin...
this
its just dy/dx+sinx
should i do anything or its just a first order derivative nothing more than that
and how would you interpret d/dx+(sinx)
.close
dawn hawk
lone heartBOT
dawn hawk
heady pollen
what is your task here?
candid dagger
what is your task here?
I think you need to substitute the value of p, q, r that they give you
dawn hawk
.close
echo crescent
lone heartBOT
echo crescent
so this is an unnecessarily complex problem so stay with me
i’m making a bridge out of 5 meter pieces, here’s a sketch of the 2 towers and the distance between them in multiple units, I need to figure out how many 5 meter pieces ill need for the red line (the rope) and how many increments (in degrees) I need to change each 5 meter piece (if that makes sense)
the pieces are straight lines
like rods
echo crescent
is the rope a circular arc, or catenary?
catenary
urban pine
how's your hyperbolic trig and calc?
echo crescent
how's your hyperbolic trig and calc?
terrible unfortunately
urban pine
https://en.wikipedia.org/wiki/Catenary#Determining_parameters seems like the place to start
echo crescent
lemme check it out
cable length should be ~568 m so ~114 5 meter pieces
just need to figure out the ° difference between the pieces to make the arc
updated sketch
need to know the angle
found this perfect tool
.solved
.close
livid current
Assume you know the values N, B, and R, with some unknown value X. Is it possible to instantaneously prove that for some value of X, (NX) mod B = R will be true without needing to check each value of X from 1 to B?
alpine sable
So you want NX-Bq=R for some q. If gcd(N,B) divides R, then you can use the bezout's identity
alpine sable
whatever satisfies bezout's identity
livid current
Alright, but wouldn't that then mean I'm essentially looking for TWO unknowns in the equation now, rather than just the X like before?
alpine sable
well you get q for free when looking for X using the euclidean algorithm
livid current
I see. Ideally I'm trying not to "look for X" at all, I was mostly just asking if I could avoid that altogether. I'm guessing not though
alpine sable
oh, you want to prove that such an X exists?
alpine sable
well just state "by bezout's"
you don't actually need to calculate anything in the proof, since the heavylifting is already done by bezout's identity
livid current
I'm not sure I understand. Are you saying that if I proved "gcd(N,B) divides R" then I already prove some X exists?
livid current
Okay.. For my case I already know that B will always be a primorial, and N will be some value less than it of course. I should be able to figure out what I'm doing with that if I can just find a VERY fast gcd() function for python
alpine sable
math.gcd lol
livid current
Wel yeah, I just didn't know if there were better ones than the prepackaged solutions lol
Either way, that should speed up my program by several times, thanks!
alpine sable
yw :)
if you want to do it from scratch, euclidean algorithm is pretty fast already
livid current
No need to do it from scratch as long as math.gcd is several times faster than checking the values of X manually for all values 1 to B lol
lone heartBOT
lone heartBOT
Available help channel!
alpine sable
Does 2/0 equal 0 or undefined
lone heartBOT
Does 2/0 equal 0 or undefined
alpine sable
.close
lone heartBOT
chrome plank
(it's undefined)
alpine sable
,calc 2/0
ocean sealBOT
Result:
Infinity
restive hearth
Can someone explain to me why the following problem results in -sin^2
restive hearth
cos^2 theta -1
alpine sable
so you want to prove $\cos^2\theta+\sin^2\theta=1$?
ocean sealBOT
Toby
restive hearth
Yes.
alpine sable
the standard way is to use pythagoras
restive hearth
I don't believe I have learned that.
ocean sealBOT
Toby
restive hearth
oh nvm, I have learned that as Pythagorean Theorem.
alpine sable
yeah
restive hearth
So how would I apply this to my question above?
alpine sable
divide both sides of pythagoras by c^2, then apply the definitions of cos and sin
chrome plank
Another way to do it:
The unit circumference has equation x² + y² = 1
Any point on the unit circumference has the form P(cos(θ), sin(θ))
Big spoilers ahead, ||you can substitute the x and y values from the point P to get cos²(θ) + sin²(θ) = 1||
alpine sable
oh thats neat
chrome plank
oh thats neat
I prefer that one because it's shorter, but the proof with pythagoras theorm is good too!
alpine sable
yup, short is always better :)
restive hearth
I don't believe I have gotten to the proof section of trig identities.
last ether
A squared plus b squared equals to brrrrrrrrr
lone heartBOT
@restive hearth Has your question been resolved?
restive hearth
ah okay, that makes much more sense!
Suppose you have cos^2 theta -1
How would you simplify that using the Pythagorean theorem?
warped stratus
can someone help me please?
lone heartBOT
can someone help me please?
warped stratus
Like- right now, though. Its kind of urgent
violet bear
just ask
abstract fractal
An ambulance is on its way. Hope you have insurance or live in Europe or sumn
violet bear
the question
warped stratus
how do I make this a regular fraction?
warped stratus
if you must know, its due in the morning
but I cant look up these questions because the internet doesnt have them
and the assignment is confusing
violet bear
264*25 and then add it to the top of the fraction
violet bear
can you send a picture
alpine sable
but I cant look up these questions because the internet doesnt have them
type ,w 264 14/25 mixed fraction to improper
warped stratus
warped stratus
type `,w 264 14/25 mixed fraction to improper`
ty
But this 62 1 2 thing
what is that supposed to mean??
abstract fractal
I think it's a typo for 62 1/2
graceful rose
Hi i just wanna ask how do u read this
lone heartBOT
Hi i just wanna ask how do u read this
graceful rose
{(5,1), (2,3), (6,4)}
real gazelle
you can read it whatever way makes sense
one way is this: "the set of ordered pairs 5, 1; 2, 3; and 6, 4"
lone heartBOT
lone heartBOT
Available help channel!