yea
#help-28
hot herald
(you can get a numerical value for theta from that)
hot herald
although inefficient, what you've done was alright
you could get
2r = 3rθ
from subtracting rθ from both sides of the equation
north beacon
you could get
> 2r = 3rθ
from subtracting rθ from both sides of the equation
yeah true, idk why I didn't just canceling out the r messed with brain
thank you so much for your help guys!
.close
blissful vine
full forumBOT
blissful vine
Guys
i need help with thos
i dont get how cistheta1 x cistheta2 = cis(theta1+theta2)
can someone help
hot herald
are you familiar with the euler formula / exponential form of a complex number?
hot herald
yes
blissful vine
ok
hot herald
but you made a typo
blissful vine
but you made a typo
i realized, 1 = i
hot herald
that's a special case
hot herald
$e^{ix} = \text{cis}(x)$
glossy valveBOT
ℝamonov
hot herald
no
hot herald
pi isn't the same as theta
blissful vine
where did i make that typo
oh
nvm
thx
that explains everthing
bye!
thx for the help
.close
uneven quail
What does the change of sign represent in this question?
full forumBOT
What does the change of sign represent in this question?
uneven quail
For 1d)
The markscheme says to evaluate 1.2905 and 1.2915, for example but I don’t understand how it verifies what alpha is
uneven quail
I evaluated 1.2, got 0.4916, and 1.3, got -0.0486
uneven quail
Yeah, I see it
simple totem
goes from positive to negative
they want you to find an interval that approximates the solution to 3 decimal places
uneven quail
So I have to find an interval that includes 1.291 ?
simple totem
and whose end points are small enough
to guarantee the solution is approximately 1.291, to 3 decimal points
uneven quail
How does it show what alpha is? I still don’t get it 😭😭
simple totem
if we know there’s a zero between -.00001 and .00001
then clearly the root of the function must be very very close to x = 0
but if we know the root is between -1000 and 1000, the root could be much further from 0, e.g. x = 900
you’re not showing alpha is exactly 1.291, you’re finding an approximation for it
uneven quail
Okay I think I get it now
So just the approximation of what it could be
Why can’t I just put alpha = 1.291 in?
simple totem
because that doesn’t give you an approximation for the root
it could be that f(alpha) is near 0
but f doesn’t have to have a root near alpha necessarily
using the intervals would guarantee it though
i feel the need to stress again
1.291 isn’t a root
it’s an approximation of one
torn jolt
How do you solve this without using the trick?
$$ \int sec(x) dx$$
glossy valveBOT
Brandon H
gritty rose
**Brandon H**
You don't. That's the point of the trick
atomic blade
Mfw sin(x)/sin(x)
torn jolt
You don't. That's the point of the trick
I do. and I will
lone flint
Menacing
atomic blade
You multiply by (sec + tan)/(sec + tan) or something like that
Mental mathing is hard sometimes
lone flint
I thought that's the only way to solve it hmm
gritty rose
I thought that's the only way to solve it hmm
It is at this level. That's the point of the trick
lone flint
I do. and I will
Well then, keep us updated :D
gritty rose
This is a waste of time
lone flint
This is a waste of time
Maths without a calculator
torn jolt
True
I found a way to solve it
but idk if it should also be considered a trick or not
It's way easier to remember than the other method tho
It is not a trick
$$ \int sec(x) dx $$
$$ \int \frac{cos(x)}{cos^2(x)} dx $$
$$\int \frac{cos(x)}{1-sin^2(x)} dx $$
$$\int \frac{1}{1-u^2} dx $$
glossy valveBOT
Brandon H
gilded quarry
Ok, then what are you going to do next?
simple totem
we could try a u sub, u = sin(x)
torn jolt
$$\int \frac{1}{1-x^2} dx = arctanh(x) + C $$
glossy valveBOT
Brandon H
torn jolt
$$ \int sec(x) dx = arctanh(sin(x))+C$$
glossy valveBOT
Brandon H
gritty rose
Inverse hyperbolic trig functions are absolutely not taught before trig functions
torn jolt
I learnt so much today
gilded quarry
**Brandon H**
This has the r/techincallytrue feeling to it.
mental bolt
Is it true that when doing math, you really need someone to explain you?
worn lion
wdym ?
dusty scaffold
Is it true that when doing math, you really need someone to explain you?
...could you elaborate on what this means
mental bolt
If you’re learning math or/and solving it. Is it true that you need someone to help you & actively be with you in order to explain it to you
dusty scaffold
it depends
i've learnt maths off the internet before by just reading stuff that's about maths and trying to understand it
but some people aren't necessarily able to do that
you need to be able to fill in whatever details the thing you're reading skips over
and also be able to like, understand stuff at an intuitive level
look at a pile of symbols and eventually have a clear idea of what it means and why each part of it is there, so that you can think about it in the future without it being really hard to work with
both of those are skills that depend on the person, the maths being learned, and the source they're learning it from
of course it is also possible to "learn" maths from nothing by generating it yourself (i.e. just thinking about maths), but that also depends on how good you are at that, and also on the maths you're discovering, and also on what maths you start from
the tl;dr oversimplified version of the answer is, it depends on how good you are at maths
learning maths from a static document or from nothing is a lot harder than learning with someone who already knows available to help you and answer questions
especially if you're trying to learn from something that assumes knowledge or skill that you don't have
full forumBOT
@mental bolt Has your question been resolved?
mental bolt
Ok
rocky orbit
(n+1)!/(n-1)!=210
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(n+1)!/(n-1)!=210
rocky orbit
i have to find the solution set
worn lion
expand the factorials and ull see that they cancel
rocky orbit
ye but at the end i find n2+n=210 and the answer of the question is S={14}
worn lion
ye but at the end i find n2+n=210 and the answer of the question is S={14}
solve it like a quadratic
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rocky orbit
.reopen
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rocky orbit
if i solve like a quadratic the x'=0 and the x"=-2
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@rocky orbit Has your question been resolved?
viral jasper
if i solve like a quadratic the x'=0 and the x"=-2
what
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torn jolt
what would be the fastest way to solve this?
$$\int \frac{1}{x^{100}+1} dx$$
glossy valveBOT
Brandon H
worn lion
this looks awful to do
kind jay
nobody can do it
kind jay
just approximate it by taking the first two maclaurin terms rofl
gritty rose
what would be the fastest way to solve this?
$$\int \frac{1}{x^{100}+1} dx$$
,w int (1+x^100)^(-1)
glossy valveBOT
Requested by riemann#2529
glossy valveBOT
glossy valveBOT
worn lion
pain
kind jay
remembering that formula is the fastest way
torn jolt
kind jay
$x_2 F_2 \left( 1, \frac{1}{100}; 1 + \frac{1}{100}; -x^{100} \right) + c$ is your answer
torn jolt
I have no idea what that means
kind jay
me neither
glossy valveBOT
Fucktalogist
gritty rose
I have no idea what that means
kind jay
coz they dodnt have games or tiktok to spend time
worn lion
hahahaha
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median cedar
$20 × 12 × 10^{-2} × 5 × 10^{-1} = 6 × 10^{-1}$
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$20 × 12 × 10^{-2} × 5 × 10^{-1} = 6 × 10^{-1}$
glossy valveBOT
Amer
glossy valveBOT
Amer
median cedar
then
hot herald
can you show the original
glossy valveBOT
bee [it/its]
dusty scaffold
$20 \times 10^{-2} \times 5 = 1$ so you can just take out those three terms
glossy valveBOT
bee [it/its]
hot herald
so noone was actually saying this?
median cedar
this is false, the lhs is actually $12 \times 10^{-1}$
the calculator yields 0.6
dusty scaffold
you probably inputted it wrong, try it again
median cedar
it's also 0.6 on my book
hot herald
can you show the original
show the original thing in your book
and what you're entering into the calculator
median cedar
oh you're right
it's a mistake on the book itself
imagine an entire country studies such book
thx
.close
tribal gust
Why does a random variable map from a sample space to a sample space if its the sigma algebra's we define probabilities on?
tribal gust
What happens if the element the random variable maps is not in the sigma algebra? Then it couldnt be assigned a probability through the random variable no?
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tribal gust
<@&286206848099549185>
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@tribal gust Has your question been resolved?
tribal gust
😕
untold pewter
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untold pewter
its possible ?
atomic blade
Part of me thinks it wants you to use polar coordinates
Just by looking at the integrand
If you squash dimensions, your y bounds are 0 to sqrt(1+x^2) and your x bounds are -inf to inf im pretty sure
Try converting to polar
torn jolt
sin(270 + x) = - cos(x)
question: does x have to be less than 90deg and greater than 0, or it can be any angle?
hallow walrus
it can be any angle, but if u choose an angle outside of the 1st quadrant you will have to make sure that u put the right value with the right sign for cos(x)
torn jolt
but if I have to look for the sign (+ or -) then it is not the same formula.
also, what do you mean by "the right value"?
hallow walrus
well it's still the same formula
but it's only really useful when x is an acute angle
torn jolt
Here's the concrete problem I have.
Things that are given:
- Angle alpha is between pi/2 and pi (2nd quadrant)
- cos(alpha - 90) = 1/5
What I need to find
tan(2alpha)
The tricky part is cos(alpha-90). I want to write it in terms of alpha (without 90). I know how to do it when alpha ds acute angle. But if it's not... then am not sure.
I tried to use new made up angle beta, and then... i have no idea what then.. does it even make sense using beta?
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@torn jolt Has your question been resolved?
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@torn jolt Has your question been resolved?
analog fiber
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@torn jolt Has your question been resolved?
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@torn jolt Has your question been resolved?
stoic portal
You could do the second derivative test, where if P”(t)>0, its a minima and vice versa
Or you could check the first derivative at t= 5 and 7 (or any other 2 numbers slightly greater and slightly lesser than 6)
If the first derivative changes from + to -, its a maxima
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@sharp nebula Has your question been resolved?
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@sharp nebula Has your question been resolved?
onyx glen
did you mean to post the exact same question a second time?
or was that a mistake on your part
(if it was then for the love of god DON'T delete your first msg)
oh, good, now we have no question at all, do we?
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onyx glen
task failed successfully.
junior wadi
yay
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severe basin
You are tasked with dividing the city of Knowhere into 11 districts which has a total population of 99,502. You have to ensure that no district has a population that is 20% greater than that of any other district. What is the least population that a district can have?
severe basin
Let x be the population of any district
Then the population of any other district can lie between x and 1.2x
I will set one district to have x population and all others to have 1.2x population so as to find the least population a district can have
x+10*(1.2)x=99502
,calc 99502/13
glossy valveBOT
Result:
7654
this looks sus
how do we know this minimizes the minimum population across all districts
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@severe basin Has your question been resolved?
onyx glen
How should I do minimize then?
good question
alright hang on
is 7654 STATED to be incorrect or do you KNOW it to be incorrect with reasoning
wait no i do see why it is incorrect
the other districts would each have to have a population of no more than 9184.8
but 9184 leaves 8 people homeless
severe basin
Hmm 
onyx glen
lemme try to think how to do this more rigorously
ok so let x be the population of the smallest district
then the others lie between x and 1.2x
whence we get 11x ≤ 99502 ≤ 13x
so 99502/13 ≤ x ≤ 99502/11
so x ≥ 99502/13 for certain
x ≥ 7654
x = 7654 won't work because the rest of the districts can house at most 9184 people so we are 8 short
i believe x = 7655 should work
severe basin
,calc 7655 + 1.2*7655
glossy valveBOT
Result:
16841
,calc 7655 + 10 * 1.2 * 7655
glossy valveBOT
Result:
99515
we got room to spare
glossy valveBOT
Result:
13
There is an extra of 13 people
onyx glen
and?
at no point do we require any district to have a population EXACTLY 1.2 times the smallest.
severe basin
We have 11 districts, and maybe we can manually subtract a total of 13 people form any of the district
Ok
.close
slow spade
What does this notation mean? Does it mean there are D no. of y vectors and each y vector has D scalar values?
onyx glen
can you show a bit more context
brave blaze
What does this notation mean? Does it mean there are D no. of y vectors and each...
D must be a set
and R^D is the set of applications with input from D and output in R
onyx glen
preemptive sully
brave blaze
?
slow spade
brave blaze
it doesnt define D tho
slow spade
Consider D a size
fallen bone
What does this notation mean? Does it mean there are D no. of y vectors and each...
D is a fixed positive integer. (You can try stats with 0 or less values I suppose - wouldn't get very far in applied spaces)
fallen bone
What does this notation mean? Does it mean there are D no. of y vectors and each...
I don't know why there's a superscript but Y^D is a subset then
onyx glen
i think this just means \mathcal Y is a subset of R^D and the D on top just signifies it is a set of dimension D...?
fallen bone
You could also just ask Murphy
slow spade
You could also just ask Murphy
Best advice ever
slow spade
I don't know why there's a superscript but Y^D is a subset then
So what would y^D mean?
brave blaze
your document doesnt define D ?
slow spade
No
fallen bone
your document doesnt define D ?
Earlier in the book
brave blaze
slow spade
Earlier in the book
Earlier it defined D as size of vector but notations in the book is jumpy
slow spade
Ok I will ask there. But what would be your best guess about the notation?
mint grove
full forumBOT
What step are you on?
1. I don't know where to begin
2. I have begun but got stuck midway
3. I got an answer but I'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above
1
willow sedge
what is the perimeter of rectangle in terms of x?
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@mint grove Has your question been resolved?
torn jolt
How do I do the second part
torn jolt
Bruh if I have multiple problems
wet dock
torn jolt
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normal thicket
Whenever i attempt a question, i feel a little bit nervous. But after a few minutes of trying to feel comfortable and when i upload the question here i feel very happy and questions seem easy
halcyon iron
vast fossil
And what's your question?
normal thicket
A mentally question
How to attempt a question with confidence?
So that we can save the time
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@normal thicket Has your question been resolved?
torn jolt
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placid oar
common denominator
placid oar
also please don't ping helpers until it's been 15 minutes as stated in this message
versed silo
placid oar
people in this server aren't here to do the work for you
we just help you understand how to do it
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void rain
hello could anyone help me with this last part on B)
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hello could anyone help me with this last part on B)
onyx glen
do you know what "doubling time" means
onyx glen
how did you get your answer?
void rain
how did you get your answer?
I guess the first time and got it correct
onyx glen
okay so it is just a guess is what you're saying
i think it would be better to look at it this way
void rain
yes
onyx glen
the doubling time is the solution to e^(0.044t) = 2
makes it clear (hopefully) why what you're solving for is really the doubling time
torn jolt
I need to prove that the squared of an odd number is always odd
torn jolt
x = 2k+1
onyx glen
ok keep going
torn jolt
Well
I'm not sure what to do
I did square 2k+1: (2k+1)^2
But I don't know anymore after
ivory cairn
expand it
torn jolt
(2k+1)(2k+1) ?
ivory cairn
yeah
torn jolt
Now what
noble horizon
expand the bracket
ivory cairn
what do you usually do in that situation? Try something. See what you get.
torn jolt
Uh I could plug in odd numbers
ivory cairn
what is (2k+1)(2k+1)?
ivory cairn
can you show that that is odd?
ivory cairn
prove that that is odd, rewrite it so that it meets the definition of an odd number.
ivory cairn
2 * (an integer) + 1, yes
torn jolt
I don't really know but here:
4k^2 + 4k + 1 = 2k + 1
4k^2 + 2k = 0
2k(2k + 1) = 0
2k+1 = 0/2k
2k + 1 = 0 -- ?
Well it's set to equal to zero though
:/
ivory cairn
it's not the same k
4k^2 + 4k + 1 = 2( ) + 1
what should go in the parenthesis to make those equal?
torn jolt
x^2?
ivory cairn
is 4k^2 + 4k even or odd?
torn jolt
Even
ivory cairn
why
torn jolt
It was 2k+1 which is already odd but then it was squared so it's even now
ivory cairn
??
First, that's the opposite of the statement you're trying to prove.
and i'm just talking about 4k^2 + 4k
not with the +1.
is 4k^2 + 4k even or odd? and why?
torn jolt
When you plugin natural numbers you get even numbers
Well, 4k^2 is just 2k * 2k and 4k is just 2k + 2k
and 2k is a definition of even number
The sum of two even numbers is always even
same goes for product?
ivory cairn
right, when it's an even number it's a multiple of 2.
meaning you can factor a 2 out of it.
torn jolt
4k^2 + 4k + 1 = 2( ) + 1
Well what you said matches this
then it would be 2(2k^2 + 2k) + 1
ivory cairn
then it would be 2(2k^2 + 2k) + 1
which is the definition of an odd number
ivory cairn
yes
torn jolt
2(2k^2 + 2k) + 1 is the answer
Ah
Welp I'm very new to math proofs. I just asked chatgpt to give me basic problems for fun
Not very solid in algebra though
Thanks for taking your time c:
.stop
dark shore
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turbid scarab
how do I do l
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how do I do l
turbid scarab
This is what I have
viral jasper
<@&268886789983436800>
shy tinsel
that was weird lol
turbid scarab
what happened?
viral jasper
Just move on
viral jasper
This is what I have
Why do you only have one y solution?
turbid scarab
what do I do after that
turbid scarab
So what's the issue?
how do I find theta using my calculator
viral jasper
arccot?
turbid scarab
I don't think I've learned that
viral jasper
Or do 1/ and then do arctan
viral jasper
I don't think I've learned that
Interesting
turbid scarab
1/ what?
viral jasper
1/y. But nvm
lean dew
do you know cot^-1 (cot inverse)?
turbid scarab
Yes that's how u find theta but how do I do that on calculator
There isnt a cot^-1 button
turbid scarab
recirpocal?
turbid scarab
Do this then
I don't have arctan on calculator
lean dew
do you have tan inverse?
turbid scarab
turbid scarab
wait do I type tan iverse first or 1/y first
lean dew
so does the answer match with the book?
turbid scarab
wiat lemme check
It doesn't 😭?
u know circle quadrant?
idk what its called but like u have to use that
it's late I'll do this tomorrow
thank you sm for helping
🙏
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rotund chasm
Can somebody please help me with finishing this problem? I am kind of stuck here
gritty rose
Can somebody please help me with finishing this problem? I am kind of stuck here
Show the original question and explain all relevant context
rotund chasm
Show the original question and explain all relevant context
Oh this might be unhelpful, its not for a math class but, I just needed my math checked. Its fine if thats not allowed Imma remove my q
gritty rose
Well since you refuse to provide context, can't help too much. What's the justification for swapping the second derivative with the exponential here
rotund chasm
Well since you refuse to provide context, can't help too much. What's the justif...
Okay so this is asking me about the average kinetic energy for the ground state. So, when I try to find the avg kinetic e at ground state we use integral of (psi*)(K)(psi) where psi represents the wave function at ground state
Context: Total energy= KE + Potential Energy
KE= -h(bar)^2/(2m) d^2Psi/dx^2
Is this of any help? @gritty rose
gritty rose
Yes
gritty rose
KE= -h(bar)^2/(2m) d^2Psi/dx^2
Looks like K is an operator so you shouldn't have psi here
rotund chasm
Yup
gritty rose
Can somebody please help me with finishing this problem? I am kind of stuck here
At least from your second line
gritty rose
Well since you refuse to provide context, can't help too much. What's the justif...
Still don't know what you're doing here
rotund chasm
Still don't know what you're doing here
You got to the part where I did second derivative right?
gritty rose
What do you mean "did second derivative"
rotund chasm
Well since you refuse to provide context, can't help too much. What's the justif...
How I completed this part
I can walk u thru the entire process thru where I stopped and then lmk if you can help, how does that sound?
gritty rose
Just answer that question specifically first
Ignoring constants, you have d^2 /dx^2 (exp(-x^2)) in the first line
Then you have them swapped on the next line exp(-x^2) d^2 /dx^2
rotund chasm
Ahh I wish there was a way to vc and talk about this
rotund chasm
Then you have them swapped on the next line exp(-x^2) d^2 /dx^2
I didn’t swap, I basically simplified to this
gritty rose
Using what math
The derivative is an operator
It acts on functions
When you don't have anything following d^2 / dx^2, that's confusing
The stuff you circled is after my screenshot anyway
rotund chasm
Yeah
gritty rose
So if I can't follow from line 2 to 3, I won't get line 3 to 4
normal tree
looks like some weird quantum mechanics notation
rotund chasm
It is quantum mechanics
rotund chasm
The stuff you circled is after my screenshot anyway
its kind of confusing to me, its cool, you tried and I rlly appreciate your help
normal tree
Is <K> supposed to be some operator norm?
rotund chasm
Is <K> supposed to be some operator norm?
Yeah basically
normal tree
I guess then Kphi_0 is not K times phi_0, but K applied to phi_0, no?
usually operator norms look something like <v, Av> or something like that
gritty rose
The physics is explained well
normal tree
I don't know jack about quantum unfortunately 😩
gritty rose
Okay so this is asking me about the average kinetic energy for the ground state....
She explained everything here
gritty rose
I guess then Kphi_0 is not K times phi_0, but K applied to phi_0, no?
This is right
rotund chasm
I guess then Kphi_0 is not K times phi_0, but K applied to phi_0, no?
We are multiplying. So basically what’s happening here is that I am getting rid of the /2 from e^alphax/2
gritty rose
We are multiplying. So basically what’s happening here is that I am getting rid ...
Nope. This is incorrect
gritty rose
The derivative is an operator
.
rotund chasm
Idk how it fully works I just saw my prof doing things like this so I memorized it
I just simplified what I had and did it
gritty rose
That works for multiplication by x
Which is proportional to the position operator
But NOT the momentum operator
So you copied the concept correctly, but it doesn't apply to derivatives
rotund chasm
I see, can you please show me how you would have done it?
Or help me understand which parts I messed up big time
So that we can change what we have
gritty rose
Well since you refuse to provide context, can't help too much. What's the justif...
Exactly what I asked about
rotund chasm
One second
gritty rose
Take the second derivative
gritty rose
There is none
None?!
rotund chasm
I mean the reason for double deriv is because of KE
But I did sth erong
Wrong*
Lemme fix it real quick
gritty rose
Ignoring constants, you have d^2 /dx^2 (exp(-x^2)) in the first line
Surely an upper division physics student can differentiate a gaussian two times
normal tree
so can you at least calculate Kphi_0?
rotund chasm
Surely an upper division physics student can differentiate a gaussian two times
the thing is, I AM NOT A PHYSICS STUDENT LOL
is this better?
I redid it on paper
hopefully this is a bit cleaner to look at
gritty rose
You don't need to redo the whole thing
gritty rose
Ignoring constants, you have d^2 /dx^2 (exp(-x^2)) in the first line
You just need to do this derivative correctly first
rotund chasm
no, I just needed to make sure I was on the right track cuz my brain is so fried from stress im so sorry bro
gritty rose
,w diff exp(-x^2/2)
glossy valveBOT
gritty rose
That's the first derivative
rotund chasm
You just need to do this derivative correctly first
is it still incorrect?
gritty rose
You need to differentiate it again
rotund chasm
,w diff exp(-x^2*x/2)
glossy valveBOT
rotund chasm
I pulled out -hbar^2/2m bc its a constant
so one of those functions w the exp would have to be double differentiated
gritty rose
Yes it does. But your notation in the bottom line doesn't indicate that at all
rotund chasm
Yes it does. But your notation in the bottom line doesn't indicate that at all
how so?
gritty rose
That's just how math works
d/dx f(x) means the derivative of f(x)
f(x) d/dx means f(x) times the derivative operator
gritty rose
d/dx f(x) means the derivative of f(x)
This is f'(x)
gritty rose
f(x) d/dx means f(x) times the derivative operator
This acts on functions
rotund chasm
f(x) d/dx means f(x) times the derivative operator
I see, so how should I alter this?
rotund chasm
this
gritty rose
That's completely different from this
rotund chasm
,w diff xexp(-x^2/2)
glossy valveBOT
rotund chasm
this
gritty rose
Yes. Now you replace d^2/dx^2 (exp(-x^2 /2)) with that. Your integrand shouldn't have any derivatives now
rotund chasm
right
rotund chasm
my third line?
rotund chasm
Yes. Now you replace d^2/dx^2 (exp(-x^2 /2)) with that. Your integrand shouldn't...
yeah I got to this part w that
hold on
gdi
no nvm it was fine
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fast marlin
how do you do f?
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how do you do f?
scenic wren
there are different options here
what i would do is find a vector from the point to the line
we know that the shortest possible vector would be perpendicular to the line
in other words, if this vector is called v and the line has a direction vector r, then
v*r=0
or you could say that this vector v is parallel to the normal
fast marlin
can you show me how i would do that
i think i understand i just need to see how you would do it
glossy valveBOT
~Martin
scenic wren
then we can make a line out of this with starting point R
next up, we need to find the point at which this line crosses the line from P to Q
so we need the equation for that one as well
if we set them equal, then we will get a solution for s
we dont care for t
that will give us the two points on the line or v
fast marlin
oh so we're making the point 20,30 be on a parallel line?
scenic wren
i got a normal of (7,8)
if we consider a line m:
m(t)=m0+t*n
then we can move this line by setting m0 such that it goes through the point R
that way we get v
so we want the difference between:
-point R
-point where v and the line P->Q meet
for some reason my result is not the prettiest though
fast marlin
yeah its not supposed to be
would it be possible to do something with the cross product btw?
scenic wren
hmm i dont know
we could maybe also use law of cosines
if we look at the triangle PQR and we treat PQ as the base
then the height of this triangle is what we want
fast marlin
what if you split it into the parametric equation and set the x and y's equal to each other?
fast marlin
v(s) and l(t)
scenic wren
yes that works
fast marlin
my answer is not very pretty either
scenic wren
hmm
i get two different solutions
with the first option i got something like 9
with cosines i got something over 30
the second sounds better
fast marlin
this is supposed to be the answer-
scenic wren
yeah that is what i got when using cosines
fast marlin
can you show me your work for that?
scenic wren
notation here is not clear i know haha
that is the start
getting the points
then vectors between them
then the length
that gives the side lengths of the triangle
then i check if the two smaller sides together are larger than the hypothenuse
which i think is needed for law of cosines
if we get the angle RPQ
then we can use:
sin(alpha)=h/PR
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fast marlin
@scenic wren i'm getting the second part you did wrong i think
nvm i got it i think
idk what im doing wrong
i feel like its so close
@visual hatch
oops
<@&286206848099549185>
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stark geyser
$$\int_{a}^{b}\frac{lnx}{x^2+ab}dx $$
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$$\int_{a}^{b}\frac{lnx}{x^2+ab}dx $$
glossy valveBOT
『Marius』
stark geyser
where 0<a<b
I tried by parts with a x' there
I skiped some steps along the way but it should be good in the end
and I definetly skipped to put some dxs there
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torn jolt
sqrt(x+7) = x+1
condition x+7 >= 0 is unnecessary.
condition x+ 1 >= 0 is mandatory.
My question is, why is the first condition (unnecessary one) always a subset of the second one. In other words, if the second one (mandatory condition) is satisfied, then the first one will always be satisfied.
I get why the first one is unnecessary, but it just don't feel intuitive that, for example, x+7 or whatever is under root will be greater than zero if the right expression is greater than zero.
vast fossil
Hence if x + 1 is nonnegative, then so is x + 7
hot herald
if you understood that if x+1 > 0
then x+7 > 0
is also satisfied, that's pretty much it
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torn jolt
the more I think about this, the less it make sense..
what if we have this equation:
sqrt(x+1) = x+7
is x+1 >=0 unnecessary condition?
and we can only check x+7 >= 0, x >= - 7
But if i put, x = - 5
then sqrt(-5+1) = sqrt(-4)
In this example, it seems that x+1>=0 is necessary condition.
But, if so, what is the difference between these two examples?
sqrt(x+7) = x+1
sqrt(x+1) = x+7
oh.. if i put -5 then it's not an equation..
the solutions to this equation sqrt(x+1) = x+7 are not real numbers.
Then I have complex numbers..
how do I check complex numbers with these conditions?
Do I even need conditions if the solutions are not real... 👀
lots of questions
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twin raptor
the more I think about this, the less it make sense..
what if we have this equa...
wHAT
oh my
torn jolt
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worn lion
Im stuck, pls help
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Im stuck, pls help
twin raptor
oh my
pearl thicket
Whats even the question you're trying to do?
pearl thicket
Did you try an inspection?
worn lion
wdym ?
pearl thicket
Guessing a function but less hazy than that
worn lion
uhhh
pearl thicket
for instance is it in the form f ' (x)/f(x)
worn lion
i dont think so
worn lion
i did in the 2nd page
but ill try with the base one
forgot the 1/2 here but it simplifies anyways
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wind needle
What level of maths are you operating at
Because I have an idea but don’t know if you can excite it
*execute
wind needle
Have you done any complex analysis?
worn lion
not rly
ive looked at some solutions of this integral and it used complex
but i dont think thats the fastest way
or it might be idk
wind needle
I’ll have another look
worn lion
the result should be
heres the result, the integral is right above
wind needle
Ah I see
One idea, I haven’t tried it mind you, factor the denominator and do decomposition
Will make it easier to integrate
worn lion
One idea, I haven’t tried it mind you, factor the denominator and do decompositi...
partial fractions ?
wind needle
Yes
wind needle
What’s the idea?
glossy valveBOT
Zouni
worn lion
yeah this looked rly bad
worn lion
so let x=i ?
wind needle
Do partial fractions how you typically would to find A and B
wind needle
That’s fine, substitute it for now
You’ll notice that some of it cancels nicely, try it
worn lion
so here, A+B=1 or A+B=ln(x+1) ?
wind needle
Actually before any of that, does your solution have to be analytical?
worn lion
well it gives a real value
wind needle
Oh
Ok well that’s fine
Substitute x=i into the step before the one you’ve just showed me
And you’ll notice that the 2i cancels out
worn lion
but im lookin at the integral calculator website and it gives me something different ?
wind needle
With complex values, the answers can look wildly different
wind needle
Should be pretty chill from here
Also if you’re interested in this field, you can learn series related content and lamberts w function may be helpful in questions similar to this
You should definitely learn about series and sequences if you haven’t as it can make certain integrals much easier
worn lion
i learned abt series and sequences
worn lion
Also if you’re interested in this field, you can learn series related content an...
ok ok ill look into that
idk how to reduce further
i have no idea how they got
wind needle
substitute the bounds
worn lion
wdym ?
worn lion
ok, is there any other reduction possible ?
wind needle
and you can split all of the ln(1+i) into ln(2)/2 + ipi/4
yes you just need to know complex numbers
worn lion
ok i see
wind needle
alot of it cancels
worn lion
but do the absolute values apply ?
wind needle
i've told you what ln(1+i) is
if you're doing this out of interest, try and work it out
might be interesting
i must go, but goodluck, you've done most of it already
worn lion
<@&286206848099549185>
spiral vigil
the arithmetic looks right to me
worn lion
im missing some factors tho
spiral vigil
looking for some paper but i'm starting to think your PFD might be wrong
worn lion
wdym ?
spiral vigil
wdym ?
partial fraction decomposition only works for rational functions so what you really want is like:
$\frac{x\log(x+1)}{(x+i)(x-i)}$
$\log(x+1) \left( \frac{A}{x+i} + \frac{B}{x-i} \right)$
and then you find A = -1/2 and B = 1/2, which matches the integral calculator when you re-distribute that log term
glossy valveBOT
kitten.in.a.teacup
spiral vigil
tricked me too
worn lion
wait isnt A=1/2 tho?
worn lion
im blocked there now
maybe i need to find a way to separate those lns
problem is i have no idea how
spiral vigil
yeah i'm not sure either, I don't think this has an analytic answer using elementary functions
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worn lion
<@&286206848099549185>
it looks like it makes sense up to now i think
but idk how to evaluate dilogarithms with complex
i can maybe separate them and use a similar idea as complex logs ?
ohh or i can use some identities
worn lion
yeah doesnt seem to work
worn lion
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tacit haven
what is the remainer when (345,606)^2 is divided by 20
frosty geyser
what is the remainer when (345,606)^2 is divided by 20
what do you have so far
tacit haven
So I put it as divided by 5*4
frosty geyser
why?
tacit haven
and (345606)^2 is divided by 4
pearl thicket
Try factorising 345606
pearl thicket
😦
frosty geyser
remind yourself what happens when you divide by 2 and by 10
tacit haven
I really don't know
pearl thicket
Darn
tacit haven
yeah it's not even
pearl thicket
So the remainder is 1
tacit haven
ok
tacit haven
if divided by 10 it should have 0
frosty geyser
wait
pearl thicket
That's not necessary
frosty geyser
you divide by 20
tacit haven
oh I get it
pearl thicket
If you take any large number the 3rd digit onwards will be dividable by 20
pearl thicket
Yeah
tacit haven
so remainder is 16
real relic
what is the remainer when (345,606)^2 is divided by 20
an alternative trick is to write this in form (20k+6)^2 which gives the same result
tacit haven
In the xy-plane, line k has a positive x-intercept and a slope of 2. Which of the following equation represents a line that intersects line k at a point above the x-axis?
torn jolt
Consider their slopes
tacit haven
I chose y=2x+1 as the answer
torn jolt
I chose y=2x+1 as the answer
Show your reasoning
tacit haven
because they both have the same slopes meaning they intersect?
torn jolt
Why would 2 lines having the same slope mean they intersect?
torn jolt
Yes, if you picture it in your head 2 lines with the same slope are definitely parallel
,w graph y=2x and y=2x+1
glossy valveBOT
torn jolt
,w graph y=0 and y=1
glossy valveBOT
tacit haven
okay then the answer is y=x+1
torn jolt
Indeed
tacit haven
why though
torn jolt
Well the other answer couldn't be it so by elimination it would be this one
However if you really want to check we can verify that it does fulfill the requirements
torn jolt
You didn't need to graph it to know that 2 lines with the same slope are parallel
Personally I just pictured it in my head
tacit haven
no I'm talking about the other equation
torn jolt
Oh, you mean to check that it meets the requirements?
tacit haven
yes
torn jolt
I also just pictured that in my head but yeah I suppose there's gotta be a simple way to check that
One would be to construct a general equation for a line with slope 2 with a positive x-intercept and show that it must intersect y=x+1 above the x-axis, meaning at a point with a positive y-coordinate
If there is a quicker way I haven't thought of it yet
torn jolt
Actually I guess I can think of a quicker way but it's a bit convoluted
The point of intersection would be at x+1=2x+b, hence at -x+1-b=0, which has a constant slope of -1
Then something something the y-intercept of 2x+b must be below 0 and the y-intercept of x+1 is 1 so -x+1-b is positive at x=0