So the component of A along P is Q, which is in the oposite direction of P?
#help-26
unique tendon
neon iron
The component along P is some negative number. From the drawing I would say something between -0.3 and -0.5
sorry, -0.3 and -0.5
unique tendon
The component along P is some negative number. From the drawing I would say some...
Which length is it represented by? Q?
neon iron
In your drawing, you compare the length of Q to the length of P. The question is: How many times the length of P is the length of Q? half? one third? something like that. If it is half, then the component is -0.5
unique tendon
In your drawing, you compare the length of Q to the length of P. The question is...
Ah i see
unique tendon
In your drawing, you compare the length of Q to the length of P. The question is...
But what does this actually mean? I mean like if i say A and P are forces applied on an object placed at the origin then what does Q mean?
neon iron
This is physics, not math. But the situation is more like this: Sometimes in physics, it is necessary to know the component of one force in a particular direction. For example with Lorentz force, you need to know the velocity component perpendicular to the magnetic field. So you take the velocity vector, and then take one component of it wrt. the vector perpendicular to the magnetic field.
Or you have a slope, and gravitation force, so the slope makes sure the object can only go along the slope. Then you look only at the component of the gravitation force along the slope, because the force perpendicular to the slope is swallowed by the slope.
So there are many situations in physics where this is needed
If you have multiple forces acting on an object and nothing else, you don't need components. You just have a total force which is the sum of all single forces
topaz sinewBOT
@unique tendon Has your question been resolved?
unique tendon
i shouldn't have brought in physics here.
But to sum up:
To find the component of a vector A along another vector B , all i need to do is find the length of the projection of A on B?
I think i have underatood what i needed to.
Thank you for your help.
.close
vapid hollow
Hello , i need your help for Generalized integrals. i want to using comparison theorems determine the nature of
topaz sinewBOT
Hello , i need your help for Generalized integrals. i want to using comparison t...
vapid hollow
I have already done the part for ]0,1] it converge but i don't find for [1,+inf[
I have to compare with 1/t^2 so i try to reduce but i don't find
sweet shard
log(1 + u^2) > log(u^2)
neon iron
This estimation makes the integral smaller, which is useful if we want to show divergence. But I'm pretty sure this integral converges. (and be computed with some trig substitution)
thorny flameBOT
sweet shard
neat
snow nimbus
This estimation makes the integral smaller, which is useful if we want to show d...
IPB
neon iron
it might be, haven't actually computed it
vapid hollow
I don't understand why pi/2 + log(2). were it come from ?
sweet shard
oh log(1+x^2) < log(x^2 + x^2) for x > 1
sweet shard
I don't understand why pi/2 + log(2). were it come from ?
you don't need to
neon iron
my heuristic is that log(1+x^2) is basically 2log(x), and log(x) is slower than x, which means the quotient goes to 0 faster than 1/x, hence the integral exists.
sweet shard
oh log(1+x^2) < log(x^2 + x^2) for x > 1
you could just use this and then some log rules to show convergence on [1,inf)
neon iron
oh log(1+x^2) < log(x^2 + x^2) for x > 1
exactly
vapid hollow
or can i say ln(1+x^2) ~+inf ln(x^2) ?
=> ln(x^2)/x^2 ~+inf 1/x^2
=> so i prove with Riemann a > 1
silent siren
Have I misunderstood your question?
vapid hollow
Have I misunderstood your question?
I just need to find the nature of the integral using comparison theorems
vapid hollow
or can i say ln(1+x^2) ~+inf ln(x^2) ?
Do you think this is enough for proving the convergence for [1,+inf[ ? @silent siren
silent siren
not sure. probably, but I can't justify it.
maybe you need to show f(x) is continuous everywhere before you can integrate.
no, thinking about it, even if f(x) wasn't continuous, you could integrate it piecewise
vapid hollow
in my school we can say it obious that f(x) on [1,+inf[ is continous and it is positive because devide two positive number (V x e [1,+inf[, ln(x) >= 0 )
Thank you for your help.
.close
topaz sinewBOT
topaz sinewBOT
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thorn anvil
So, I am working on part A and I was wondering where I should go from here?
topaz sinewBOT
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oak turret
i need help
fallow igloo
i need help
How can we help you?
oak turret
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What is the significance of the Higgs boson in particle physics, and how does its discovery contribute to our understanding of the fundamental forces in the universe?
The Higgs boson is crucial in particle physics as it imparts mass to other particles through the Higgs field. Its discovery in 2012 at the Large Hadron Collider confirmed the existence of this field, supporting the Standard Model of particle physics. Understanding the Higgs mechanism enhances our comprehension of how particles acquire mass, providing insights into the fundamental forces shaping the universe's structure and behavior.
how can i simplify
topaz sinewBOT
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inner wren
if ABCD is a square of unit lenght
perimeter of FEC = 2
Then find the relation between EC and FC
topaz sinewBOT
if ABCD is a square of unit lenght
perimeter of FEC = 2
Then find the relation b...
oak turret
if ABCD is a square of unit lenght
perimeter of FEC = 2
Then find the relation b...
isn't fec just a triangle?
merry hill
[redacted] obviously
oak turret
[redacted] obviously
language my friend...
merry hill
neon iron
by relation, you mean ratio?
inner wren
ig just an eqauation containibng both the variables.
no not the ratio if you look at it
an eqauation
inner wren
right
merry hill
and is there any relation that you know for right angled triangle
inner wren
pythagoras
inner wren
FE^2 = EC ^ 2 + FC^2
topaz sinewBOT
inner wren
.reopen
topaz sinewBOT
✅
neon iron
I don't know if you have enough information, is AF = AE?
inner wren
I don't know if you have enough information, is AF = AE?
no
merry hill
then how are u supposed to find the angle
inner wren
I don't know if you have enough information, is AF = AE?
not given
merry hill
so substitute the pythagoras ^ here
maybe solve the equation u got from here
that will give you all line segments
then you find AF and AE (using pythagoras)
now u have all sides of triangle FAE
inner wren
i tirired that
merry hill
then use cosine law
inner wren
ok
neon iron
it might have something to do with the square instead of being a rectangle
inner wren
it might have something to do with the square instead of being a rectangle
maybe]
my teacher tloled me toi think outisde iof the box, literally
wait cosin law ain't allowed or something
containing trig
chapter - triangles
plain depot
if ABCD is a square of unit lenght
perimeter of FEC = 2
Then find the relation b...
Without any further information, E is not a point but a locus
So is F
So the ratio is also literally a variable
You can see it wouldn't actually give any result
The angle is the cotangent. And without any information. It is just the cotangent
It's like saying x+y = z, find x,y,z
Unless I missed something in your original question
Does your question give any more information?
plain depot
if ABCD is a square of unit lenght
perimeter of FEC = 2
Then find the relation b...
What does "square of unit length" mean
trim ermine
it means each side is of 1 unit
spiral canyon
<@&268886789983436800> can you remove the helper role from me ??please .......
why
restive inlet
<@&268886789983436800> can you remove the helper role from me ??please .......
Go the id:customize and deselect "I'd like to help others"
spiral canyon
Go the <id:customize> and deselect "I'd like to help others"
Ok .thanks
spiral canyon
Go the <id:customize> and deselect "I'd like to help others"
I can't find it. there are only channels
restive inlet
click on the link
topaz sinewBOT
@inner wren Has your question been resolved?
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@inner wren Has your question been resolved?
pliant briar
Hi i have this equation and i know the differential are equal so i got to the 2nd equation (2nd image).
Now am supposed to express dy with only dx, x and n (without using any arcsin fnction) so idk what to do with the cos(y). I think there is a trig replacement i can t see idk
pliant briar
sudden temple
do you have any restrictions on y?
could possibly use pythagorean identity for sine and cosine
topaz sinewBOT
@pliant briar Has your question been resolved?
sudden sorrel
How do I do this?
topaz sinewBOT
How do I do this?
onyx pasture
How do I do this?
look at the third property
dense lily
How do I do this?
Have you learned properties of limits
sudden sorrel
I’m confused by this specific one maybe I’m overthinking it
Do I plug the 5 to the x’s
Ohhh
6 times 8??
sweet shard
yes
dense lily
6 times 8??
Yep
sudden sorrel
And what do I look for the open or shaded circle?
How do I add graphs even
You add the y values??
They all have the open circle tho so it’s different y values
So DNE?
Halp🫠
<@&286206848099549185>
onyx pasture
Do I look at what happens at x=1 on both graphs??
yes and add them for the first question
sudden sorrel
yes and add them for the first question
But which value is it?
There’s an open circle and a shaded circle
I don’t know what that means
I know the open one is like the limit like the discontinuity in the graph
And that the shaded circle is like a make up for it but I don’t know how it comes to be or which value is the real value
onyx pasture
But which value is it?
the open one
sudden sorrel
And at 0? How do I know what the value of the red graph is??
It can be 0.6,0.7,0.8 or anything
outer delta
A limit only exists if the graph is approaching the same number from the left and the right. at x = 1, does the limit from the left = the limit from the right on the red graph?
sudden sorrel
A limit only exists if the graph is approaching the same number from the left an...
No it’s two different values, so it’s DNE?
outer delta
yep
outer delta
You can calculate the slope of the blue one and once you figure out the limit as the blue graph approaches 0, you should see that you don't have to calculate the red graphs limit at x = 0
outer delta
of the blue one?
sudden sorrel
Lim x—>0 means the limit is 0 right?
outer delta
of the blue one yes
outer delta
lim f(x) * lim g(x)
sudden sorrel
So what am I finding I don’t get it
outer delta
What is the limit of f(x)?
sudden sorrel
They never gave me the function so I have no idea
They never gave me any numbers in this question
outer delta
f(x) is the blue one
sudden sorrel
The limit is the open circle?
outer delta
What is the limit of the blue one as x approaches 0?
sudden sorrel
0 I think
Why is there even a limit there if the graph functions normally
There is no hole there
Doesn’t limit means that it never reaches that number
outer delta
Limits come in use later and its better to understand how they work now
outer delta
Doesn’t limit means that it never reaches that number
Limit means what does the graph approach at some point x
sudden sorrel
Even if it can touch that point?
outer delta
yes
sudden sorrel
So it’s 0 then right?
outer delta
yes
sudden sorrel
So that means the answer is 0 cuz whatever the other answer is it’ll be multiplied by 0?
outer delta
in all except one case yes, (if the red graph was DNE the answer would be DNE)
sudden sorrel
Oo that’s a good point
Thank you! That applies to num4 then and I’ll see if I know how to do 5
What’s the shaded in circle?
outer delta
The "solution" y to the graph at that point x
sudden sorrel
So how can an open circle have a solution at that point if it’s DNE?
outer delta
Thank you! That applies to num4 then and I’ll see if I know how to do 5
Yes, but for (f(x)/g(x)) lim g(x) = 0, then the answer is DNE
Open circle means that is not the solution y at that point x
sudden sorrel
Yes, but for (f(x)/g(x)) lim g(x) = 0, then the answer is DNE
I think you are confusing the two here, the f is 0
So it’s like 0 divided by something
outer delta
I know, but I just wanted to point that out
sudden sorrel
Oh yes like if it was the opposite
sudden sorrel
Open circle means that is not the solution y at that point x
I don’t get this still
outer delta
yes
sudden sorrel
That’s like a hole in the graph
Meaning it doesn’t even exist cuz it’s impossible
So how can we still find a solution anyway for that point
Like in here why is it just a random dot at -1
outer delta
The solution would be the circle/dot if one is given, if it just an open circle there. there is no solution at that point
that dot is the solution at that point
sudden sorrel
The solution would be the circle/dot if one is given, if it just an open circle ...
Yes but I’m trying to understand how to come up with that solution
Why is that the solution at that point?
outer delta
Piecewise function
sudden sorrel
Why doesn’t the graph just go down to that point
What🥲😅
Why do we care about the non solution then and not the real solution
Shouldn’t we use the solution to calculate stuff
outer delta
To show that limit x -> -1 f(x) =/= f(-1)
sudden sorrel
Not equal??
Aren’t we supposed to plug it there
Ohh cuz that’s where it wouldn’t be possible to solve
outer delta
A limit is just what the graph approaches, it may not approach the solution at that point
sudden sorrel
So how do we calculate the shaded dot? And how is it possible for it to exist?
outer delta
It is typically given
piecewise functions are basically a bunch of functions that are smashed into one over given intervals for each "sub-function"
They look like this
sudden sorrel
That looks like a tengant function is that the same thing?
outer delta
Like tan(x)?
sudden sorrel
Yes like the tan function
outer delta
no
sudden sorrel
Damn it
sudden sorrel
Hahaha
outer delta
tan(x) = sin(x)/cos(x)
sudden sorrel
Ohh it’s like literally sin divided by cos that’s pretty cool
Like visually too
But isn’t it also pieces combined together
Since it breaks
sudden sorrel
Ohhh that makes sense actually
outer delta
as sin grows bigger (up to 1), cos grows smaller (approaching 0), so larger/smaller = much larger
sudden sorrel
Thank you🥲 I understand this whole topic better now
outer delta
No problem
ripe aurora
I want to know what I am doing wrong with this problem. My answer is incorrect and I would like to know what process would improve my thinking.
topaz sinewBOT
I want to know what I am doing wrong with this problem. My answer is incorrect a...
safe burrow
what precisely did you do
ripe aurora
I just put 1/x^4
And the same thing with y after moving them to opposite sides of the equation
safe burrow
so y^-2 *y' = -5x^-4?
safe burrow
hmm
golden blade
I want to know what I am doing wrong with this problem. My answer is incorrect a...
My 2nd step would have been $$x^4 : \dd y = -5y^2 : \dd x$$ from here you would just cross divide $-5y^2$ and $x^4$.
thorny flameBOT
bacc
safe burrow
wait
safe burrow
so y^-2 *y' = -5x^-4?
is the point not to integrate by x once we get here?
safe burrow
like this?
ripe aurora
Yes thats what I have
But the answer was still wrong
Maybe I made an error while integrating
safe burrow
sorry for the quality im having to do this in ms paint with a mouse
golden blade
Maybe I made an error while integrating
You should also first integrate both sides when you separated nicely
ripe aurora
Oops
golden blade
**bacc**
After this you should have $$\int \frac{1}{y^2} : \dd y = -5 \int \frac{1}{x^4} : \dd x$$
thorny flameBOT
bacc
ripe aurora
safe burrow
,rotate
ripe aurora
Is there a command for flip
thorny flameBOT
safe burrow
that should be x^3
ripe aurora
Okay
safe burrow
since ur integrating x^-4
ripe aurora
So now that I wrote that in I need to isolate y
safe burrow
tru, but ur basically already there
ripe aurora
Now its just flipping the equation
golden blade
Now its just flipping the equation
yes but be cautious
safe burrow
well it's a little more inticate bc of the constant
safe burrow
put the C into the equation and then flip
golden blade
,, -y = \frac{1}{\frac{5}{3x^3}+C}
safe burrow
so (-5 -3x^3*C)/(-3x^3)
safe burrow
,, -y = \frac{1}{\frac{5}{3x^3}+C}
hold on, where'd u get that from
golden blade
applying (...)^-1 on both sides
safe burrow
but you've turned x^3 into x
golden blade
oh I literally took it from here
safe burrow
doh
halcyon willow
aint no way blud is doing integrals before linear equations 😭
ripe aurora
Okay so how to I remove that 5/3 x^3 from the denominator
thorny flameBOT
bacc
halcyon willow
same logic
halcyon willow
just take the lcm here
dr becomes 5+ 3x^3C
and nr becomes 3x^3
then cross multiply and solve
safe burrow
This might make more sense:
halcyon willow
heiroglyphics ahhh numbers 😭
halcyon willow
lmao jk
halcyon willow
bro
safe burrow
but that's so slow
golden blade
ripe aurora
Okay now I have this
safe burrow
and if u want to actually improve, making a machine do all of ur thinking for u is kinda antithetical to that
safe burrow
it's like sitting at one of those self-playing pianos and calling it practice
safe burrow
i'd leave it like that
halcyon willow
and the nr is having dr as 1
safe burrow
I think that's the most elegant form
halcyon willow
yup thats right
btw
why are u simplyinf that integration ans
C can be whatever tbf
just write +c to whatever u got
safe burrow
it does?
halcyon willow
hmm then i guess u are okay
safe burrow
in that case absorb the 3 into it
ripe aurora
But the professor never went into what can and cannot be absorbed into it
halcyon willow
But the professor never went into what can and cannot be absorbed into it
anything constant afaik
ripe aurora
Okay
safe burrow
which doesn't inlcude the 5 here
halcyon willow
u know what the +c represents right?
safe burrow
bc the C is next to an x^3
ripe aurora
Yes it represents the integration constant
halcyon willow
Yes it represents the integration constant
and what is that
ripe aurora
So that 3 can be absorbed while the x^3 has to stay
safe burrow
ye
ripe aurora
Still says its wrong
Maybe its a formatting error
Btw this is an achieve homework
safe burrow
wdtm?
ripe aurora
That my teacher assigned last minute
safe burrow
maybe he wants the top three absorbed in too and it to be 5/3?
ripe aurora
Oh shit I forgot the negative
ripe aurora
Aight aslong as its the right answer I do not care
okay it was jus the negative I was forgetting
So anything thats directly multiplied or dividing the + C is absorbed?
Thats my last question before I go
safe burrow
it's kinda complicated
if it's within a fraction yes
but if you get e.g. f(x) + C + 5, you could absorb the 5 into C
ripe aurora
Hmmm
So
What if I get like a situation like -(x+C)
Would the negative also be absorbed
safe burrow
ye that'd just become C - x
since you could've had -C earlier for ur integration const
topaz sinewBOT
@ripe aurora Has your question been resolved?
severe lichen
i dont get c
topaz sinewBOT
i dont get c
severe lichen
i chose a test value between 0 and sqrt10 - 1
when x = 2
f'>0
but when x = 1
f'<0
both are in the same interval
so im confused
which is which
mint crescent
,w differentiate (9-x^2) e^x
thorny flameBOT
severe lichen
ohhh
btw im redoing
its because there's a - in front of the e^x
i didnt see that 😭
I didnt realize i pulled out a -e^x
ty
.close
dawn abyss
super quick question if i do 420+3. is that 400 in sig figs?
topaz sinewBOT
super quick question if i do 420+3. is that 400 in sig figs?
@dawn abyss Has your question been resolved?
winter turtle
topaz sinewBOT
finite moth
!status
topaz sinewBOT
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 was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
topaz sinewBOT
@winter turtle Has your question been resolved?
winter turtle
no
topaz sinewBOT
@winter turtle Has your question been resolved?
gloomy fulcrum
Can someone help me on four questions on my math homework? I want to understand the concept of domain and logs I have a test soon :(
gloomy fulcrum
Number 14,15,59,&61
frosty iron
for 14, the domain of an inverse function is the range of the original function
gloomy fulcrum
oh ok
i dont understand the steps to get that
i get confused when trying to find domain..
frosty iron
?
gloomy fulcrum
like so far I have root 1-x has to be greater than 0
frosty iron
yes
gloomy fulcrum
and x cant = 1
frosty iron
x can be 1
gloomy fulcrum
oh
frosty iron
sqrt(0) is 0
frosty iron
x cant be greater than 1 (x <= 1)
gloomy fulcrum
oh ok
so then i make a number line
testing the numbers between 0 and 1
wait
would it be [0,1]
?
gloomy fulcrum
oh so is [0,1] the domain?
im lost on how to ffind range..
do i plug in x and find y?
frosty iron
oh so is [0,1] the domain?
still no, try to think of when x is decreasing
frosty iron
do i plug in x and find y?
yes
gloomy fulcrum
still no, try to think of when x is decreasing
oh umm
frosty iron
you got x <= 1, x has to be atleast 1 for y to be minimal
so you find y for x = 1
and see how y changes as x decreases (if x increases 1 - x would be negative)
gloomy fulcrum
how do i find y for x,=1? plug in -1 into the equatio?
frosty iron
plug x = 1
frosty iron
yep
gloomy fulcrum
-inf,2
frosty iron
correct
frosty iron
yeah
frosty iron
g(f(x)) is g(5x^3 - x) , so g(4) implies 5x^3 - x = 4, solving that you would get g(f(x)) since it = x
but, for example if g(f(x)) = 4x, you'd have to solve for x and multiply that value by 4
gloomy fulcrum
so since g(fx) = x, in the f(x) equation, the x counts as the g(fx)=x?
litele confused
frosty iron
yeah, its the same variable
frosty iron
yeah
gloomy fulcrum
o wow
frosty iron
i might get some things wrong though im in g8
frosty iron
yeah
gloomy fulcrum
middleschoool
frosty iron
for 59 youd use that alogx = logx^a
frosty iron
middleschoool
yes
gloomy fulcrum
for 59 youd use that alogx = logx^a
for which part
frosty iron
both terms
frosty iron
yep
frosty iron
then id do the other term instead
frosty iron
so its js lnx-ln(x^2+3x+2)
what about the second term?
there would be ln(sqrt(x)) as well
gloomy fulcrum
uh the term on the inside of ()?
frosty iron
and also notice how x^2 + 3x + 2 is factorizable
frosty iron
uh the term on the inside of ()?
yes
frosty iron
sorry what do i do with that term the alogx=x^a?
youd get ln x^1/2 which is ln(sqrt(x))
gloomy fulcrum
OH I SEe Tthat now
so far i hace : ln(x+2)+lnx-ln sqrt(x+1)(x+2)
can i get rid of the sqrt?
frosty iron
*the sqrt should be on the second term
so ln(x + 2) + ln(sqrt(x)) - ln(x + 1)(x + 2)
and by using logab = log a + log b
the last term becomes -(ln(x + 1) + ln(x + 2))
ln(x + 2) cancel
gloomy fulcrum
uh waiiit
frosty iron
?
gloomy fulcrum
so ln(x + 2) + ln(sqrt(x)) - ln(x + 1)(x + 2)
i thought this supposed to be ln(x+2)+lnxsqrt? -ln(x+1)(x+2)
isnt there an x
frosty iron
no since the last term is squared
frosty iron
i thought this supposed to be ln(x+2)+lnxsqrt? -ln(x+1)(x+2)
wdym?
gloomy fulcrum
oh so the x is jus sqrted?
frosty iron
yes
frosty iron
and do you see where to go from now?
gloomy fulcrum
is it js lnsqrtx + ln (x+1)
I SE
I SEE
I GOT IT
i accidently crossed out the minus sign lol
thank uu
i will attempy 61 on my own lol
gloomy fulcrum
fuck
frosty iron
im stuck at e^1/2lnp
you can make it (e^lnp)^1/2
gloomy fulcrum
so its p^1/2
frosty iron
yep
frosty iron
lnp would be cancelled already no?
gloomy fulcrum
so ln(p) is ln base e P and it turns into lnp/lne
frosty iron
no i meant this term
frosty iron
yep
gloomy fulcrum
so now its js log e
cuz it canceled out with the other ln p...
wait noo
did i do it wrong
frosty iron
ln e is 1
gloomy fulcrum
waiit i was right
ook
tyyyyyy
u actually stayed for all four problems LMAO thanks sm
round kernel
idk if its separable or not but i tried to separateto no avail so
worthy storm
possibly useful, the left hand side of the equation is d/dt of y^2
worthy storm
what u substitution do you mean?
vernal matrix
u = y^2 + t might make your life a bit easier, I'd think 
silent siren
it's fairly horrible without that transformation XD
topaz sinewBOT
@round kernel Has your question been resolved?
round kernel
My brains having. Minor lapse in judgement and I’m sure this is some bs but I can’t figure out what’s wrong
topaz sinewBOT
My brains having. Minor lapse in judgement and I’m sure this is some bs but I ca...
thorny flameBOT
jan Niku
vernal vale
whats the thought here
round kernel
I could not tell you
Like I know it’s wrong but I’m forgetting how to do substitutions 💀
vernal vale
hmm 
first guess u(y) = y^2
but not quite
well, maybe
$u(y) = y^2 \to \dd u = 2yy'$
thorny flameBOT
jan Niku
thorny flameBOT
jan Niku
thorny flameBOT
jan Niku
vernal vale
do we get cancellation
theres some weird domain restriction maybe
$y = \sqrt{v-t}$ so $\dv{y}{t} = \frac 12 \frac{v'-1}{\sqrt{v-t}}$
thorny flameBOT
jan Niku
vernal vale
which gives the original ODE is
$2 \qty( \sqrt{v-t} )\qty( \frac 12 \frac{v'-1}{\sqrt{v-t}} ) = v^2$
thorny flameBOT
jan Niku
vernal vale
which is $v'-1 = v^2$ 
thorny flameBOT
jan Niku
vernal vale
Yea
i think this works
this is separable
youll need to back out y on the other end though
kind of a pain
round kernel
Ty😭😭
peak edge
Is my equation setup correctly?
peak edge
<@&286206848099549185>
whole geode
@peak edge there's at least one problem that I can see
The intersection looks like this
The integral you set up only covers between 1/4 and 1
You need to handle also the part where only x=y^2 is bounding the region
sacred iris
Can you solve it as the volume of parabola - the volume of conical cavity that forms?
And by the way how can you find the volume of parabola type graphs when they are rotated?
whole geode
The green highlighted region are your bounds on x. I am too inept at desmos tonight to do the total bounds of the integral including the function, but you do the region between the two functions as you wanted.
But you leave out the area to the left of the highlighted region, which is still in the bounded area
peak edge
But you leave out the area to the left of the highlighted region, which is still...
So I need to find two volumes then
whole geode
Yes, and add them together
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fiery aurora
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@fiery aurora Has your question been resolved?
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halcyon willow
.reopen
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coarse tusk
think about substituting t = 2x so that you can invoke the identity you sent
coarse tusk
so what did you get
coarse tusk
yes
halcyon willow
which will differentiate to sec^2t/2
and 1/2
shit i need to sleep rn
😂
thnx
.close
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long fog
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obsidian spade
for a) is it actually possible to know whether it's a positive or negative value? or do we just know it is ± 15?
topaz sinewBOT
for a) is it actually possible to know whether it's a positive or negative value...
neon iron
ok, it is just a convention, and you probably use the right-hand rule like everyone else.
obsidian spade
so with the right hand rule, v cross w would point down so it'd be negative?
neon iron
yes
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cinder oxide
why is p → q such an absolute mess to teach and to understand?
p = true
q = false
therefor if p then q is false
p = false
q = true
therefor if p then q is true
p = false
q = false
therefor if p then q is true
vale juniper
what is p and q used for again
vale juniper
can you give me an example, i remember using p and q for something, just dont remember what for
cinder oxide
a proposition it can either be true or false, but not both at the same time
gleaming thunder
why is p → q such an absolute mess to teach and to understand?
p = true
q = fal...
"if p then q"
If not p then you can't say it's wrong
Refusing this is refusing that things are false if and only if there's a counterexample to them
cinder oxide
"if p then q"
If not p then you can't say it's wrong
Refusing this is refusing t...
a stranger walks up to you and says "if pigs can fly, then ..." but you cut them off and say "hey there buddy, stop talking. i have already heard enough. what you are saying is true. discrete math says so.". you look up in the sky to see pigs flying. 🐷 🪽
gleaming thunder
then you start listening again
cinder oxide
basically if you start with bullshit, it doesn't matter what you say after that, it's gonna be true no matter what
gleaming thunder
the thing that often confuses people is that implication is not logical implication
cinder oxide
if true, then true = true
if true, then false = false
if false, then true = true
if false, then false = true
gleaming thunder
"A => B" is not "B can be proven by assuming A"
Surprisingly, it's not the definition
though there's a whole branch of logic about making it be
cinder oxide
why is p → q such an absolute mess to teach and to understand?
p = true
q = fal...
again, I go back to this question/statement
is it just that our brains are wired for conversational logic? talking like this in conversation doesn't make any sense.
we have to "unwire" our brains to truly understand discrete math logic?
gleaming thunder
"A => B" is not "B can be proven by assuming A"
the difference being that if pigs grew wings, it probably wouldn't change much about the world so you can't deduce much from it
But the implication is true because you don't have a counterexample
gleaming thunder
again, I go back to this question/statement
is it just that our brains are wired...
because we see implication as "provable from"
when in fact it's "there's no counterexample"
so you can have implications about things that are uncorrelated, and they happen to hold, although vacuously
But in general vacuous truths in math are kinda boring and a thing you just accept for the real stuff to work nice
cinder oxide
so it's like hypothesis -> conclusion
if pigs can fly, then I will be an alien ... true
if pigs can fly, then I will be human ... true
frail reef
well yeah
cinder oxide
saying those are false is like making an assumption i guess?
frail reef
since the assumption isnt true to begin with
gleaming thunder
because we see implication as "provable from"
the thing is classical logic includes "False implies anything"
Even "False implies anything and its contrary"
Because False being true implying a contradiction is not gonna break anything that wasn't already broken from False being True
cinder oxide
saying they are true is making no assumption, it's just agreeing, this is bullshit to begin with
frail reef
well you are saying if P is true, then Q is true, but if P isnt true then you cant make any claim about the conclusion Q, and this condition cant even be considered
cinder oxide
i gotta smoke some weed to truly be open minded enough for this philosophical conversation
gleaming thunder
again, I go back to this question/statement
is it just that our brains are wired...
however it's proven that under classical logic, implication implies provability
But classical logic has the axiom (i.e. valid proof) "False implies anything" so deal with that
shrewd horizon
here's a way of thinking about it
suppose that false is true, we want to prove that P is true
well, P is either true or false
if it's true then we're done
if it's false, well, we know that false is true, so P is true
gleaming thunder
well, P is either true or false
(another axiom of classical logic, not not A is A)
cinder oxide
T→T = T you make a promise, you follow through with it. you win
T→F = F you make a promise, you don't follow through with it. you fail
F→T = T you don't make a promise, you follow through with it. you win
F→F = T you don't make a promise, you don't follow through with it. YOU WIN!
moral of the story: don't make promises, and don't follow through with them. you win.
gleaming thunder
no one can hold you accountable
No counterexample of you being untrustworthy
See, makes sense
cinder oxide
no one can hold you accountable
No counterexample of you being untrustworthy
See...
discrete math is gonna turn me into such an asshole with this logic
gleaming thunder
making no promises of being helpful also means you aren't worth much as a friend
cinder oxide
making no promises of being helpful also means you aren't worth much as a friend
not according to discrete math:
p = being helpful
q = being a good friend
if p → q
F → F = T
keen venture
I wouldn't try to attach a real-life connotation for →. Doing so would be missing the point that → is defined to work this way, and there's no need to justify it
cinder oxide
I wouldn't try to attach a real-life connotation for →. Doing so would be missin...
"use less empathy"
keen venture
I don't understand who might be hurt by this definition
gleaming thunder
the difference is math works with the concept of provability, which prevents quite the bullshit in practice
keen venture
Vacuous truth might be weird, until you realize the opposite would be vacuous falsity, and that's also weird
cinder oxide
Oh gawd, there's more stuff to learn on top of this already confusing concept?
`Trivial Proof: If we know q is true then p → q is true regardless of the truth value of p.
Vacuous Proof: If p is a conjunction of other hypotheses and we know one or more of these hypotheses is false, then p is false and so p → q is vacuously true regardless of the truth value of q.`
vacuous just means having or showing a lack of thought or intelligence; mindless.
trivial just means of little value or importance.
keen venture
That's the line everybody finds weird. That F -> X = T
cinder oxide
so we have proof that is of little value or importance.
and proof that is showing a lack of thought or intelligence; mindless
keen venture
"If the moon is made of cheese, then I am made of cheese" is a true sentence
cinder oxide
yeah
cinder oxide
"If the moon is made of cheese, then I am made of cheese" is a true sentence
but so is this:
"if the moon is made of cheese, then i am currently talking to Kaynex on Discord"
or this:
"if the moon is made of cheese, then 1+1 = 2"
gleaming thunder
"if p then q"
If not p then you can't say it's wrong
Refusing this is refusing t...
it's what lets "prove me wrong" be a counter-argument
If something is wrong there is a counterexample
and that's just fundamentally important as a property of our logical system
(while things having proofs is actually more complicated, because some dude named Gödel did too much shit with his brain)
cinder oxide
but if you heard someone say this in conversation, I think there would definately be some red flags or confusion going on in the brain for the point they are trying to make
if the moon is made of cheese, then 1+1 = 2
they would just sound crazy
gleaming thunder
yes, but you don't
Because people put possible things after the if
if things may happen, then it must be that the conclusion may be true
keen venture
but so is this:
"if the moon is made of cheese, then i am currently talking to K...
Indeed, you're right. This throws a lot of people off.
You've got to choose either "true" or "false". You can't choose "doesn't make sense" since that's not an object in our logic.
Note that we already have a symbol where we choose false. It's ∧
gleaming thunder
more precisely
P(A) => Q(A) is one thing
P(A) => Q, where Q is not a function of A, means Q is true if there is at least one A for which P(A) is true
While with Q(A), you have flexibility
gleaming thunder
if things may happen, then it must be that the conclusion may be true
this is also a important fact to notice
cinder oxide
LOL.. i don't know if I will ever agree with the logic, on a human/conversational level if the moon is made of cheese, then 1+1 = 2
but in discrete math I just have to memorize this: p -> q is only false when p is true and q is false. otherwise, it's always true.
maybe memorizing will help me a lot more here than "understanding"
gleaming thunder
yes it's the provability thing
cinder oxide
hopefully the more I use it the more I will better appreciate it
gleaming thunder
yes it's the provability thing
it's that it's not
implication is weaker than provability
but we use the two interchangeably in math, unless you're explicitly working in logic
keen venture
Tbh I don't often think about vacuous truth
cinder oxide
i do find it interesting.. humans are so complex, interpretation of a statement can be WILDLY different from person to person
yet with computers it's just following very basic logic gates, and it's always always always the same result.. it is not open to interpretation on the weather of the day or environment it was raised in. at least not yet.
gleaming thunder
Tbh I don't often think about vacuous truth
its most useful case is starting induction one step earlier than you should, and having a simpler case for it
(or weird edge cases in what you consider)
keen venture
Most of the time we ever construct a p -> q we care about the case where p is true
cinder oxide
Most of the time we ever construct a p -> q we care about the case where p is tr...
yeah, it's easier when it's false, we just brush it off as bullshit
with true we must investigate further
if p is true, then q must also be true
to satisfy truth, or whatever it's called
"have equivilance" to truth?
gleaming thunder
Be true
dusky relic
I need help understanding this question. I can get up to where it says "multiply E3 to (-1/4), but when I get to the add E2 to E3 part, I get stuck, why does E2 get to remain as is when added, while E3 is changed?
topaz sinewBOT
I need help understanding this question. I can get up to where it says "multiply...
@dusky relic Has your question been resolved?
dusky relic
<@&286206848099549185>
limber igloo
reporting for duty
dusky relic
hello!
patent pelican
how old are you?
dusky relic
I'm 21
dusky relic
Yeah i can get up to the last part. Just confused why, when E2 added to E3, E2 still remained
I just need to know the reason for it being left unchanged
patent pelican
all equations isn't changed and just other equations are appeared!
do you understand?
dusky relic
all equations isn't changed and just other equations are appeared!
Yeah, but when E2 and E3 were added together, shouldn't there be just one equation?
patent pelican
another equation is -4z=4
dusky relic
another equation is -4z=4
yes, there is a new equation after adding them together, but in the image, E2 still remained, alongside the new equation
My question was why, after adding them together, there still exists E2
patent pelican
when there are 3 variables, 3 equations must exist.
dusky relic
yes I undestand that concept
patent pelican
so E2 is still existed
dusky relic
so we just pick which one?
patent pelican
it is your choice
dusky relic
ok yeah this makes sense now thanks!
patent pelican
not at all!
dusky relic
.close
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patent pelican
🥹
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solid axle
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plain depot
i dont understand the absolute values in integrals
Split into two integrands
solid axle
yeah but idk how to do the bounds
shrewd horizon
yeah basically you just want to get rid of the absolute value
so you want to split it into the section where the stuff inside is positive, and the section where the stuff inside is negative
slate radish
shrewd horizon
so you want to split it into the section where the stuff inside is positive, and...
(well, sections, in general there could be any number of them, but in this case it happens you only have to split it into two pieces)
so, for which x is x^2 - 1 > 0, and for which x is x^2 - 1 < 0?
narrow badge
Find y intercept. Then split the modulus into its 2 corresponding piecewise functions. And integrate between the y intercept and the extreme values either side
solid axle
so, for which x is x^2 - 1 > 0, and for which x is x^2 - 1 < 0?
when x is -1 it equals 0 though
so what do i do there
narrow badge
Find y intercept. Then split the modulus into its 2 corresponding piecewise func...
^
craggy haven
Find y intercept. Then split the modulus into its 2 corresponding piecewise func...
x-intercept(s)*
solid axle
ok
narrow badge
x-intercept(s)*
Ty long day ahah
solid axle
so one of them is -2,-1 and the other is -1,0
craggy haven
$\int_{-2}^{-1}|x²-1|dx + \int_{-1}^0|x²-1|dx$
thorny flameBOT
solid axle
ok but its still absolute value
craggy haven
ye
solid axle
so
craggy haven
but on the region (-2, -1) we know that |x²-1| is equal to x²-1
narrow badge
ok but its still absolute value
Now u gotta split the modulus into 2 different piecewise functions. One being negated the other just standard
solid axle
but on the region (-2, -1) we know that |x²-1| is equal to x²-1
how do we know that
shrewd horizon
well x^2 > 1, so x^2 - 1 > 0
and the absolute value of a positive number is just the same number, it's already positive so nothing changes
narrow badge
how do we know that
Easier way to think of it. Each time the modulus graph should cross the x axis. Instead the function gets negated and continues being positive. So it bounces and negates off of the x axis
solid axle
Easier way to think of it. Each time the modulus graph should cross the x axis. ...
im gonna be honest idk what ur saying
narrow badge
im gonna be honest idk what ur saying
Well im down and out. Im too cooked to explain stuff lmao
solid axle
well x^2 > 1, so x^2 - 1 > 0
oh so its when it equals greater than 0
for the bounds -1,0. -(x^2-1) is greater than 0
and for -2,-1 (x^2-1) greater than 0 or equal
right?
narrow badge
and for -2,-1 (x^2-1) greater than 0 or equal
If ur curious i was trying to explain what u and the others were discussing but visually. So by thinking of the graph of the function. But yeah. That was it really
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fierce gale
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thorny flameBOT
vale juniper
is 100% 1 or 100 on a graph
fierce gale
1
vale juniper
so from (0,1) to (4,0) on a parabola would look like what
fierce gale
sorry I don't get what it would look like
vale juniper
is a horizontal stretch or compression
fierce gale
horizontal stretch
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solar moth
Prove using induction that for every integer n >= 1 and real number x >= -1, that (1+x)^n >= 1 + nx
I am doing:
P(n, x): (1+x)^n >= 1 + nx
Basis step: P(1, -1): (1 - 1)^1 >= 1 + (1)(-1), thus 0 >= 0, base step is true
Inductive step, assume P(k, j) is true, then the inductive hypothesis (1+j)^k >= 1 + kj is true, then P(k + 1, j + 1) is true, that is to show (1 + (x+1))^(n+1) >= 1 + (n+1)(x+1)
then I am stuck, but i do see some people online to show to prove P(k+1, j) and P(k, j+1) separately. Or am i misreading the question that i in fact only need to prove k+1, and there is no need to show x >= -1 stands?
topaz sinewBOT
Prove using induction that for every integer n >= 1 and real number x >= -1, tha...
sleek haven
sorry
sleek haven
I was typing as you were typing
solar moth
Prove using induction that for every integer n >= 1 and real number x >= -1, tha...
apologies for the ugly formatting, idk latex
neon iron
Hey, I think it’s wrong that you are doing induction over n AND x. Because x is a real number, not a natural number
keen venture
You can't use induction over all real numbers. You must be expected to induct over all n, but leave x to be free
neon iron
In your notation:
P(n): For all x>= -1, it holds that (1+x)^n >= 1 + nx
solar moth
oh damn, i am new to the topic
is it because real numbers are continuous? thats why we cant induct over it
neon iron
Yea exactly. because even if you wrote down a correct proof and did induction over x, then you have proved the statement only for integer values of x, and not all the values of x in between
solar moth
ah i see, that clear things up haha. i will leave this open while i try to write a proof
tysm
sleek haven
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sleek haven
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sleek haven
^^
verbal light
hybrid cars wont be driven any more in 8 years
sleek haven
hybrid cars wont be driven any more in 8 years
that's right everyone will be driving monster trucks
vestal sigil
I worked out b^2-4ac and it = 0
the power line must start from the power station right?
sleek haven
yes
vestal sigil
okay
sleek haven
I think smth like this
vestal sigil
and since it is a line, its equation must be y = ax + b
sleek haven
yes
I know it by y=mx+c tho
vestal sigil
we do!
sleek haven
do we?
vestal sigil
we know c