how is it II??
#help-0
solar glade
heavy delta
left to right i mean the left side of function (a) to right
not whole actually
just a small part
heavy delta
yeah
solar glade
which part of it is slope zero and und??
heavy delta
wait hold on
solar glade
ok
heavy delta
veritcal line
lone heartBOT
heavy delta
oh lol
broken pivot
-# whoopsie, i accidentally closed discord and got distracted watching the yt video that was behind it
lone heartBOT
Available help channel!
supple raven
good ev
north rover
didn't get the PB = 10 squareroot 3
you know your trig, right?
supple raven
you know your trig, right?
yes
north rover
yes
then consider tan(60).
since tan=opposite/adjacent=30/PB here.
and we want PB.
north rover
i can onlk imagine PB= 30/squareroot3
okay,
do you know how to rationalize the denominator?
supple raven
do you know how to rationalize the denominator?
ahhhhhh, i forgot that...
yes, i do
north rover
better yet, note that $10\sqrt{3}\cdot\sqrt{3}=30$.
ocean sealBOT
;(
supple raven
thank you, now i got it
north rover
so what can we do to that pesky rt(3)?
north rover
thank you, now i got it
👍
supple raven
so what can we do to that pesky rt(3)?
multiply both, up and down and get the result there
thank you
.close
vapid sundial
hi so
vapid sundial
im really confuseed with imagtinary numbers
like how am i supposed to do this
what am i supposed to do
vapid sundial
sqrt of 72 is 36
safe meadow
sqrt of 72 is 36
no
vapid sundial
idk
twin nimbus
Have you learned about prime factorization of numbers in school?
vapid sundial
get tuah the point psl
vapid sundial
Have you learned about prime factorization of numbers in school?
yes
twin nimbus
What are the prime factors of 72?
safe meadow
i dont think 4,6 are prime
vapid sundial
8
winter bay
I mean 4 isn't prime but it is square so that's important to note
vapid sundial
can u visiualize this
winter bay
Start with 2 and keep dividing 72 by it until u can't anymore
vapid sundial
where did the 28 come from
safe meadow
where did the 28 come from
it is only an example
winter bay
why
The amount of times you can divide a number by a prime shows how many times that prime divides the number
vapid sundial
What.,
winter bay
Ok so since 72/2 = 36
That means 2 is a factor of 72
Since 36/2 = 18
That means 2^2 is a factor of 72
And so on
safe meadow
hiido, i dont think mela understand about prime factorization
winter bay
hiido, i dont think mela understand about prime factorization
Yeah I guess not
Hmm
Idk how else to explain it
safe meadow
it would be better if you could explain it first
vapid sundial
can you guys draw it
safe meadow
can you guys draw it
we dont need to draw it
vapid sundial
im a visual learne
winter bay
Sadly being a visual learner can be unfortunate with math
safe meadow
alr let me try to help
winter bay
Idrk if there's a way to visually show this
vapid sundial
draw the equation and the steps
winter bay
I guess let's try to find sqrt(18)
winter bay
Oh yeah that's a cool way
vapid sundial
factor tree?
winter bay
Yes
And the amount of primes u get at the very bottom
Is the prime factorization of the number
So 32 = 2^5
Bc there's 5 twos
winter bay
how do ik where to put the i
U just multiply it at the end
Since $\sqrt{-x} = i\sqrt{x}$
ocean sealBOT
hiidostuff
safe meadow
Useful for college research too
yeahh
winter bay
where can i learn it
I learned by teaching people on here
safe meadow
I learned by teaching people on here
i still used my keyboard like 2^2 or 3*4
i still use ^ and *
winter bay
I mean it's good for short form stuff
But if ur doing more complicated math then texit is the way to go
Anyways @vapid sundial is everything good now?
ocean sealBOT
hiidostuff
winter bay
And 6^2 * 3 isnt 72
vapid sundial
yea
vapid sundial
12
ocean sealBOT
hiidostuff
vapid sundial
2
ocean sealBOT
hiidostuff
winter bay
<@608508888953782307>
well you know that $\sqrt{96} = 4\sqrt{6}$ now
ocean sealBOT
hiidostuff
winter bay
since you factorized it
so just multiply by -i
since you have a negative outside and inside the square root
winter bay
oh
no its not but ig for now it can be
dont forget i tho
you need to multiply ur answer by i bc u have a negative in the square root
vague atlas
can someone help me checking my problem if it’s good or not?
vague atlas
charred jewel
can someone help me checking my problem if it’s good or not?
what were you trying to do here
lone heartBOT
@vague atlas Has your question been resolved?
vague atlas
what were you trying to do here
It's about the lines, the slope and that
charred jewel
and the question was?
vague atlas
if it’s good or not
i mean, i already make the problem but my question was, if the procedure was good
tardy stag
that line does go through those points
lone heartBOT
@vague atlas Has your question been resolved?
lone heartBOT
lone heartBOT
Available help channel!
coral stone
A is (at most) countable if there exists an injection f: A --> N. How do i show that AxB is countable if A and B are countable
mortal trellis
can you find an injection NxN -> N ?
coral stone
i did not find it myself but we were given one
mortal trellis
think about how you can use it here
coral stone
ok
inner fable
Does anyone know how to integrate this one, i tried many times but dont see the way to approach it tho
quasi vector
Doesn't look like the antiderivative would be nice
tight pier
Maybe integration by parts
quasi vector
maybe convert to infinite series and then integrate term by term
Doesn't sound fun for an indefinite integral
inner fable
I did try D I method, but it wasnt any better tho
lone heartBOT
Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
inner fable
hmm
wide cairn
,w integral of x^2/(1 + e^x)
ocean sealBOT
inner fable
ok wow it does not look promising
quasi vector
Are you sure you don't have to do a definite integral
tight pier
not an ordinary one
inner fable
Are you sure you don't have to do a definite integral
yeah
well not really
It's actually a piece of a question, I just divided into smaller one cuz I thought it would be easier tho
lone heartBOT
Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
inner fable
Are you sure you don't have to do a definite integral
This is the original question tho
little drum
Yes this is indeed a definite
inner fable
yeah
junior kraken
It was definite 🥲
wide cairn
😭
little drum
Invert it about y and you get rid of the denominator
junior kraken
Kings rule and add, the deno will vanish
wide cairn
what the fk is king's rule
little drum
Wherever man is, it's king's rule 
wide cairn
😭 oh it's just a substitution
inner fable
But the question is letting that definite integral= api + bpi^2+c*pi^3, So basically i was finding a, b, c (a, b, c belongs to Q)
inner fable
what is king's rule btw
wide cairn
yeah idk
little drum
But the question is letting that definite integral= a*pi + b*pi^2+c*pi^3, So bas...
Lol. Read what I said
little drum
Invert it about y and you get rid of the denominator
Basically let y = -x
junior kraken
wide cairn
it's just a sub from x -> a + b- x where a and b are your integration bounds
inner fable
hmmm
quasi vector
This is the original question tho
This is why !xy exists
inner fable
ok, letme try to do it then and see where it goes
quasi vector
Invert it about y and you get rid of the denominator
You have to add the two integrals
little drum
quasi vector
It's really useful for integrals of the form $$\int_{-a}^a\frac{E(x)}{1+e^{O(x)}}\dd{x}$$ where $E(x)$ is an even function and $O(x)$ is an odd function
ocean sealBOT
kheerii
quasi vector
And that's what you have
inner fable
Um what am i supposed to do next
um still confused
Like now I have to put it back into integral or what?
quasi vector
Simplify it first
inner fable
Like letting y=-x ?, and having nearly the same function?
quasi vector
Like letting y=-x ?, and having nearly the same function?
That's what you did
And yes, the point is that you have a function very similar to the original
inner fable
yeah
quasi vector
Try to see what's different about it
quasi vector
You just undid your work
quasi vector
Simplify the top expression
inner fable
splitting it into two?
quasi vector
Why did the -x in the exponent turn into x?
little drum
Tihsho
inner fable
Why did the -x in the exponent turn into x?
u mean in the sine then yeah?
quasi vector
u mean in the sine then yeah?
No, in the denominator
inner fable
oh yeah
little drum
$2I = \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \left(\frac{x^2 + \sin^2 x}{1+e^x} + \frac{x^2 + \sin^2 x}{1+e^{-x}} \right) \dd x$
ocean sealBOT
Arya
little drum
is what you should have
wdym by that arrow on the left 
at the end you'll forget to halve it and get wrong ans
lone heartBOT
inner fable
.reopen
lone heartBOT
✅
inner fable
wait, how do we get this one, cuz like can we use it for any function?
or is there any specific requirement?
quasi vector
wait, how do we get this one, cuz like can we use it for any function?
You can substitute u=a+b-x
and yes it works for every function
It's a property of the integral operator, not the function
inner fable
ohh ok thkss
lone heartBOT
@inner fable Has your question been resolved?
twilit dew
can you solve this by taking infinite circles using dynamic radius?
tardy stag
yeah
twilit dew
but to do that we have to take account of the slope right using arc length formula?
ripe perch
twilit dew
but i do not understand why we need to take account of the slope when we have infinite circumferences of circles
ocean sealBOT
kronium_
tardy stag
but to do that we have to take account of the slope right using arc length formu...
nah
when the thickness of each cylinder is small enough, the slope of the line is nothing
twilit dew
when the thickness of each cylinder is small enough, the slope of the line is no...
thts exactly my thinking too, but apparently u DO have to take account of slope
twilit dew
twilit dew
yes for SA you need the slope
but why?
tardy stag
consider the surface area of these two curves revolved
twilit dew
dont follow you
tardy stag
they have the same horizontal width (both of them go from x=1 to x=2)
but the green one is clearly bigger
twilit dew
but while integrating from 1 to 2 wont the y axis and x -axis dynamically change?
tardy stag
the axis??
tardy stag
when you integrate you break it up into tiny slices. for the blue curve, a slice of width 1/10 will contribute 2πr = 2π*1/10 to the surface area
for the green curve, a slice of width 1/10 will contribute much more
twilit dew
dont follow you
tardy stag
hmm
can you see that the green surface will have more surface area than the blue surface?
twilit dew
when the thickness of each cylinder is small enough, the slope of the line is no...
this is the point im stuck in
tardy stag
yeah because that's only true for volume, where the rest of it dominates
for surface area the slope is all that's there
so it doesn't vanish
twilit dew
yeah because that's only true for volume, where the rest of it dominates
dominates?
tardy stag
if you've done arc length this is similar to that
tardy stag
dominates?
"is substantially larger than"
twilit dew
if you've done arc length this is similar to that
yea u use that formula too
twilit dew
can you see that the green surface will have more surface area than the blue sur...
yes
wont u split it into infinite circles whose circumferences you can stack to form the SA
tardy stag
wont u split it into infinite circles whose circumferences you can stack to form...
each one of those tiny circles (really squished cylinderish things) has a surface area which is bigger than that of a cylinder
twilit dew
each one of those tiny circles (really squished cylinderish things) has a surfac...
ok...
lone heartBOT
@twilit dew Has your question been resolved?
simple lodge
I need help with math
lone heartBOT
@twilit dew Has your question been resolved?
summer dirge
so I'm trying to prove part a)
lone heartBOT
so I'm trying to prove part a)
summer dirge
I've been able to show that S^n-1 is compact, and that the annulus A is open
then there's a theorem stating that in such a situation, there's a smooth bump function from h: A \to R
that means 0 <= h <= 1 on A, h(x) = 1 on S^n-1, and h is compactly supported in A (i.e. h(x) = 0 outside a compact subset of A)
you can smoothly extend this h to be defined on all of R^n by defining it to equal 0 everywhere outside of A
call that extended function h'
now I want to define a smooth g: R^n \to R that agrees with f on S^n-1, and is 0 outside of A
but I am struggling to do this
I tried
\begin{align*}
g(x) =
\begin{cases}
h'(x)f(x) & x \in S^{n - 1} \
h'(x) & x \in A \setminus S^{n - 1} \
0 & x \in \R^n \setminus A
\end{cases}
\end{align*}
but this has the issue that it's not necessarily continuous, let alone smooth
ocean sealBOT
ourfallenstars
summer dirge
but this has the issue that it's not necessarily continuous, let alone smooth
because the function doesn't quite smoothly transition from S^n-1 to A \ S^n-1
so I'm stuck 
summer dirge
because the function doesn't quite smoothly transition from S^n-1 to A \ S^n-1
that's because h'(x) is at most 1, so g(x) <= 1 on A \ S^n-1
but h'(x)f(x) = f(x) could be much greater than 1 on S^n-1
maybe this is the wrong approach? 
mental python
you need some kind of smoothing function that is 0 on x=|1-e| and 1 on x=1
that is in C^inf
summer dirge
I have such a function: h (or h')
the issue is using it to extend f to all of R^n
I can't seem to find a way to do that
summer dirge
that doesn't fix the issue that that f(x) could be much much larger than 1
in which case the extension is discontinuous
so obviously not smooth
mental python
if you have a point x on the annulus then you can find the nearest point that is on the unit sphere
thats the point you want to smooth with
summer dirge
sure, and I could multiply the h'(x) in the annulus with that point (to get a smooth extension), but how would I formally write that down?
summer dirge
the closest point on the circle to a point p is clearly the intersection point between the circle and the radial line emanating from the origin 
but idk how to turn that into a formula to extract that point
mental python
like this is what you wanna do
mental python
the distance is $||x_1-x_2||_2$ with $x_1$ the point and $x_2$ the point on the circle
ocean sealBOT
Bonk
mental python
it should be, yes
summer dirge
we're gonna be multiplying points a short distance away from each other by different constants
mental python
if you smooth it radially, it should be i think
like, imagine its a pie and you take a slice, that curve on the slice should look kinda gaussian around r=1
mby its easier to work in spherical coordinates?
then r=1, ezpz
summer dirge
like this is my situation rn, yeah?
those are the graphs of f and h', resp.
then like, we have a point x in the annulus, and we take the intersection of the radial line to x with the circle, and consider the value of f there
mental python
whats that intersection point
summer dirge
wdym?
mental python
like, how do you find it
summer dirge
call it a, idk
summer dirge
okay updated graph
mental python
$\frac{x}{||x||}$
ocean sealBOT
Bonk
mental python
😄
summer dirge
okay updated graph
this is my situation now
mental python
that gives you the point on f
mental python
okay updated graph
that not necessarily smooth, but yes i see what you mean
you want like a sin or smth
summer dirge
like this
summer dirge
that not necessarily smooth, but yes i see what you mean
yeah ik
mental python
s.t. for $||x||=1$ sin(x)=1
ocean sealBOT
Bonk
mental python
yup
summer dirge
but idk how to draw so that's what we have
now uhh
okay I just took a section of the graph of f to show what I want g to look like
but is that actually what g will look like? 
I'm worried about horizontal non-smoothness
like say between the two yellow lines
mental python
$\sin(p(x))=\begin{cases}0 \text{ for } ||x-\varepsilon||=1\1 \text{ for } ||x||=1\end{cases}$
ocean sealBOT
Bonk
mental python
you need smth like this
summer dirge
no don't worry about the sin thing
mental python
okok
summer dirge
like I have this already
h is my bump function
we don't need a new one
I just need to extend f
mental python
no need to give a definition?
summer dirge
then there's a theorem stating that in such a situation, there's a smooth bump f...
.
and the message below that one
summer dirge
like say between the two yellow lines
but this remains my concern
with the possibility that the function g isn't smooth as we move sideways
obviously it's smooth in the radial direction
mental python
it depends on your smooth bump function
summer dirge
we don't know anything about it
mental python
but im pretty sure you can define your smooth bump function in a way such that its also smooth when going sideways
mental python
can you give the theorem?
summer dirge
it is guaranteed to be smooth sideways?
cause if so, then I'm done
if not, then I'm stuck
mental python
i think so
summer dirge
start here
mental python
does it not come from C is compact?
summer dirge
what does?
mental python
that its smooth sideways
summer dirge
how does it follow from compactness?
mental python
idk, thats the only way i can think of it being smooth sideways lol
so its probably somewhere in there
mental python
compactness is smth like every cauchy sequence converges?
summer dirge
uhh, that's completeness, no?
compactness of C means every open cover admits a finite subcover
mental python
fk
summer dirge
hmm
limpid turret
lone heartBOT
mental python
what was the solution??
summer dirge
what was the solution??
it's indeed smooth sideways
f(x/|x|) is smooth cause f is smooth
product of smooth is smooth
so g is smooth
lone heartBOT
Available help channel!
summer dirge
so g is smooth
.I had bad intuition and that's what made me fail to see it
whole quiver
|x| does not have a derivative at 0
lone heartBOT
|x| does not have a derivative at 0
north rover
|x| does not have a derivative at 0
yes, since it is defined as $|x|=\begin{cases} x& x\ge 0 \-x& x<0 \end{cases}$
ocean sealBOT
;(
lone heartBOT
summer dirge
|x| does not have a derivative at 0
that's okay, because the question ignores 0
the annulus is defined to not include 0 on purpose
lone heartBOT
Available help channel!
mental python
Im really confused how this step works. Its concerning the simplex algorithm
tight pier
Im really confused how this step works. Its concerning the simplex algorithm
looks like they did -row1 + row2 -> row2 and then x1 became a basis variable
and added row1 + row3 -> row3
lone heartBOT
candid ore
guys
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Available help channel!
lusty root
lone heartBOT
lusty root
It says two sides of the triangle ABC are AB=BC=18dm Bisector AE together with side AB make a 15 degree corner find 1) Bisector BK length 2) length of AC 3) Area of triangle ABC
whole quiver
i would do part 2 first and then part 1 and then part 3
whole quiver
yes
lusty root
and bka =bkc=90 degrees?
whole quiver
yeah because sin of 30 is 1/2
lusty root
yea so BK =9 right?
whole quiver
correct
whole quiver
apply the pythagorean theorem because you know 2 of the three sides of a right triangle yes
whole quiver
yeah that is correct
lusty root
ok, but now there is a problem
3rd question
area of triangle ABC
1/2ah?
so 1/2 * 18 square root of 3 * 9?
and i dont get a normal answer
whole quiver
the big triangle is made of two congruent triangles so if you find the area of those two you have the area of the whole
whole quiver
yeah
lone heartBOT
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Available help channel!
pseudo folio
I’m not sure how to evaluate integral(4,0)(4/pi arccod(x/4)-2+sqrt(x) dx, which is what I think the area is.
lone heartBOT
I’m not sure how to evaluate integral(4,0)(4/pi arccod(x/4)-2+sqrt(x) dx, which ...
@pseudo folio Has your question been resolved?
pastel belfry
lone heartBOT
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Available help channel!
timber condor
lone heartBOT
timber condor
Can someone help me with this I need to find the value of x
heavy delta
log X/ log 10 is not equal to log(X/10)
u are wrong at the second line
log X - log 10 = log(X/10)
timber condor
ok and then i changed it to log4X+log4 (x+1)-log4 10
idk what to do with the log 5/x
north rover
idk what to do with the log 5/x
$\log(\frac{a}{b})=\log(a)-\log(b)$
ocean sealBOT
;(
north rover
$\log(a\cdot b)=\log(a)+\log(b)$
ocean sealBOT
;(
mental python
,tex .log rules
ocean sealBOT
Bonk
quaint thicket
bro im getting f in middle school what i do
mental python
bro im getting f in middle school what i do
!occupied
lone heartBOT
bro im getting f in middle school what i do
Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).
heavy delta
idk what to do with the log 5/x
what is the original que
heavy delta
for the first term use this but reverse
so u get base 10 of log (x^2+x)
hmm wait
this que is weirder than i initially thought
timber condor
i got log 2 x(x+1)^1/2 for the first term
north rover
so, $\frac{\log_4(x^2+x)}{\log_4(10)}+\log(\frac{5}{x})-\log_2(27)=7$? i recommend you convert everything into $\log(x)$ first before you proceed.
ocean sealBOT
;(
north rover
you can.
north rover
i got log 2 x(x+1)^1/2 for the first term
thats not true here.
timber condor
hmm
heavy delta
oh wait
north rover
for the first term use this but reverse
as well as this.
timber condor
ii dont understand
north rover
ii dont understand
i have stated the properties.
heavy delta
log (x^2+x) + log5 - logx - log_2 27 = 7
combine log (x^2+x) with -logx
so is
log (x+1) +log5- log_2 27 = 7
north rover
log (x^2+x) + log5 - logx - log_2 27 = 7
combine log (x^2+x) with -logx
so is
lo...
please let OP figure that out.
north rover
why log (x+1)
because of change of base.
timber condor
kk
ocean sealBOT
;(
north rover
now, can you isolate the log(x+1) term for me?
north rover
7-log5+ log2 27
nice!
ocean sealBOT
;(
timber condor
not necessarily.
Oops my bad you reverse the (x+1) and the 7-log5+log2 27
I’m pretty sure
north rover
Oops my bad you reverse the (x+1) and the 7-log5+log2 27
what do you mean?
timber condor
10^7-log5+log2 27 = x-1
north rover
10^7-log5+log2 27 = x-1
yeah, for that whole constant term.
then, you add one, and you’re basically donez
lone heartBOT
@timber condor Has your question been resolved?
timber condor
Ok thx
lone heartBOT
@timber condor Has your question been resolved?
digital musk
$\int_{0}^{\infty} \frac{1-cos(\pi\omega)}{\omega}*sin(x\omega)d\omega$
ocean sealBOT
TheOnlyCheezIt
digital musk
The question is show that the above integral is equal to pi/2 for 0 < x < pi, and 0 if x > pi
I'm pretty confused, and dont really know what direction to go
If i try to expand it out and do integration by parts i feel like it leads me no where
Pretty sure im supposed to use this equation
but i dont really know what do do with it
tacit arch
Plug in f(v) = (1-cos(pi v))/v
digital musk
I get the B(omega) equals (1-cos(pi omega))/omega
tacit arch
Or plug in B(w) = pi/2 if 0<w<pi and 0 otherwise
digital musk
but how does that become f(v)?
tacit arch
Or plug in B(w) = pi/2 if 0<w<pi and 0 otherwise
This one will show
Much easier to compute
lone heartBOT
tacit jewel
tacit jewel
does this function look close to anything
late turtle
permutation
lone heartBOT
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Available help channel!
kindred obsidian
is this right?
lone heartBOT
is this right?
hushed locust
no
kindred obsidian
how to fix
hushed locust
(ln 5)² ≠ ln(5²)
little drum
,w (ln 5)^2
ocean sealBOT
little drum
You might want to use that to un-ugly the curves you had on you
lone heartBOT
lone heartBOT
Available help channel!
paper trellis
what happened
lone heartBOT
what happened
paper trellis
help me for b and d
exotic canopy
what happened
how did you get 4?
surreal zenith
@exotic canopy why are you fucking ignoring me?
paper trellis
isnt it 8,4
exotic canopy
<@474467399207223307> why are you fucking ignoring me?
what?
exotic canopy
i help those who i can and want to help, idk how to solve your question
paper trellis
hi
paper trellis
you're confusing the x and y axis
isnt it 8 and 4
r 6
pls help me
<@&286206848099549185>
.close
static wagon
A ferry is attempting to cross a river. The current of water is moving downstream at a cosntant rate of 3.0m/s. The ferry can move at 4.5m/s relative to the water in the river, as measured by an observer on the ferry. Propose a way that the ferry can cross the river to reach a point directly across from its starting point.
static wagon
How do I draw the diagram for this question??
little drum
2. draw a downstream current
3. draw a ferry on one side of river, label it A, and a point exactly opposite at B
4. realize that for the ferry to reach point directly across, its velocity component upstream should cancel the velocity for downstream current
5. send the ferry off at an angle theta to shore```@static wagon
static wagon
so like the bottom diagram??
little drum
so like the bottom diagram??
? where's the river ... where's the ferry, points A, B angle theta??
if you don't want to listen, why ask 
ur wlc
raven prism
A ferry is attempting to cross a river. The current of water is moving downstrea...
the "along the river" component of the velocity of the ferry wrt water should be equal and opposite to the velocity of the river.
raven prism
the "along the river" component of the velocity of the ferry wrt water should be...
proceed from there
static wagon
the "along the river" component of the velocity of the ferry wrt water should be...
i dont get this
raven prism
i dont get this
this is the only thing needed to solve this.
If you don't get this, then try again
little drum
could you explain this diagram pls
wdym explain this
static wagon
wdym explain this
like explain the process of drawing the diagram
raven prism
If you don't get this, then try again
You won't learn anything if you are spoonfed
little drum
downstream current is 3
ferry is sent upstream at angle theta from shore
the green line is ferry's ideal motion
v_y is ferry's velocity's vertical component that should negate downstream current
v_x is horizontal component that gets the ferry across river
little drum
like explain the process of drawing the diagram
.
static wagon
downstream current is 3
ferry is sent upstream at angle theta from shore
the gre...
so we're just solving for costheta then?
which is 3/4.5
little drum
yes
static wagon
but would that mean that the ferry has to depart 48 degrees downstream?
so that it would travel along its ideal path
little drum
and it wasn't "just". You couldn't even figure the diagram so this question was significantly tough for you
also, no.
cos theta = 3/4.5 = 2/3 gives an acute angle
so it'd be 48.19 degrees upstream
static wagon
so it'd be 48.19 degrees upstream
oh so the river would push the ferry downstream and allow it to travel along its ideal path right
lone heartBOT
Available help channel!
nimble zealot
Hey guys, I am currently working through the Book of Proof by Richard Hammack (mostly to get back into math).\bigskip
1. $\{ 5n+2b: a,b \in \mathbbm{Z}\}=\mathbbm{Z}$\\
2. $\{ 6n+2b: a,b \in \mathbbm{Z}\}= \{all \; even \; numbers\}$\bigskip
Assume $\{ 5n+2b: a,b \in \mathbbm{Z}\} = A$ and $n \in A$ and represent all integers.
Also assume $5n+2b$ represent all integers.\\
Therefore:\\
for $a=n,b=-2n$\\
$\rightarrow n=5(n)+2(-2n)$\\
$\rightarrow n \in A$ and $A = \mathbbm{Z}$\bigskip
I know this is not pretty and complete but shouldn't I also show that $A \subseteq \mathbbm{Z}$ (what i remembered from Uni) ?
manonmars
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
lone heartBOT
@nimble zealot Has your question been resolved?
nimble zealot
<@&286206848099549185>
lone heartBOT
@nimble zealot Has your question been resolved?
lone heartBOT
@nimble zealot Has your question been resolved?
dire gazelle
seems like you assumed the conclusion you wanted to prove
also, shouldn't it be {5a +2b : a,b in Z} and not n?
the idea of using n and -2n seems right though
nimble zealot
also, shouldn't it be {5*a* +2b : a,b in Z} and not n?
@dire gazelle ty for answering! {5a +2b : a,b in Z} is indeed in Z but i couldn't find where i wrote that it is in n
nimble zealot
ohh right my mistake it meant to be a
nimble zealot
seems like you assumed the conclusion you wanted to prove
care to elaborate how i should be doing it in the right order? TY!
dire gazelle
i'd get rid of this part
you can say "Let A={5n+2b:a,b in Z}"
if you just want to give it a name
nimble zealot
my small brain was thinking, ok we want to prove that there exist an n in A, which can be any element of the integer, therefore yada yada and thus we can put n = 5a+2b
and through that we can prove that A is indeed equal Z
or do i overcomplicate it ?
dire gazelle
For all $n\in\mathbb{Z}$, let\
$a=n$, $b=-2n$\
$n=5(n)+2(-2n)$\
$n\in A$ and $\mathbb{Z}\subseteq A$
ocean sealBOT
Axe
nimble zealot
ohhhhhh
so just to understand the steps:
- we define for all n in Z, there exist an a = n and b = -2n (since a,b are elements of Z)
- now put them into equations and say that n is indeed an element of A which we defined as Let A={5n+2b:a,b in Z}
and also that Z is a real subset of A
dire gazelle
yeah, a exists because n exists. b exists because you can multiply n by -2 no matter what n is
a is an integer because n is an integer
b is an integer because integers are closed under multiplication
oops i fixed a typo
nimble zealot
and we dont have to prove that Z is a real subset of A right? kinda redundant
dire gazelle
yes that implies A=Z
nimble zealot
thank you so far! i will read more on how to prove that these subsets are kinda equal and will come back with more questions!
.close
lone heartBOT
nimble zealot
.reopen
lone heartBOT
✅
nimble zealot
@dire gazelle i looked up the definition of subsets, is this maybe correct?
since we know $n\in\mathbb{Z}$ we can say:
$\mathbb{Z} \subseteq A \Leftrightarrow \forall n : (n \in \mathbb{Z} \Rightarrow n \in A)$
ocean sealBOT
manonmars
nimble zealot
so i just have to do it from the other direction again correct?
dire gazelle
**manonmars**
that's right, and you don't even need to know n is in Z
this is the definition of a subset
dire gazelle
so i just have to do it from the other direction again correct?
yes now you need to show n in A implies n in Z
nimble zealot
we also proved that $n \in A$, $A \subseteq \mathbb{Z} \Leftrightarrow \forall n : (n \in A \Rightarrow n \in \mathbb{Z})$
because of $A \subseteq \mathbb{Z} \wedge \mathbb{Z} \subseteq A \Leftrightarrow A = \mathbb{Z}$
ocean sealBOT
manonmars
nimble zealot
**Axe**
or did you mean i had to do it all over again for n \in A
so that i can come to the conclusion that n is indeed in A ?
because that might be my mistake
dire gazelle
we don't need to prove $A\subseteq\mathbb{Z}\Leftrightarrow \forall n : (n \in A \Rightarrow n \in \mathbb{Z})$ \
we only need to prove $\forall n : (n \in A \Rightarrow n \in \mathbb{Z})$
ocean sealBOT
Axe
dire gazelle
yes now let n be an element of A
nimble zealot
hmm ok, i will need a minute on that, ty for the input!
dire gazelle
and show that n is in Z
nimble zealot
i see i see, it's just reversed
nimble zealot
jup i am typing up the solution 1 sec
dire gazelle
ok
nimble zealot
ok
For all $n\in A$, let\
$a=n, b=-2n$\
$n=5(n)+2(-2n)$\
$n \in \mathbb{Z}$ and $A \subseteq \mathbb{Z}$
since $n \in A$ $\forall n : (n \in A \Rightarrow n \in \mathbb{Z})$
$A \subseteq \mathbb{Z} \wedge \mathbb{Z} \subseteq A \Leftrightarrow A = \mathbb{Z}$
ocean sealBOT
manonmars
dire gazelle
this doesn't seem like the right approach
we should say
For all $n\in A$ there exist $a,b\in\mathbb{Z}$ such that $n=5a+2b$
ocean sealBOT
Axe
dire gazelle
since multiplication and addition are closed over the integers, and since a and b are integers, we can say 5a+2b is an integer and so n is an integer
nimble zealot
i feel so stupid now.... so the first direction we literally proved that n is in Z because of the combination of a,b in Z, now we want to prove that n is in A which is 5a + 2b and it is done?
dire gazelle
the first direction we proved that $n\in\mathbb{Z} \implies n\in A$
ocean sealBOT
Axe
nimble zealot
jup
dire gazelle
and now i just showed that $n\in A \implies n\in \mathbb{Z}$
ocean sealBOT
Axe
nimble zealot
understood
dire gazelle
does everything make sense?
nimble zealot
it does now haha, i am just sitting here thinking that i might be too stupid for proofs lol
dire gazelle
i don't know, practice should help
lone heartBOT
lone heartBOT
Available help channel!
primal cipher
guys if you have 5^4+(-4)^2+(-2)^7 and now betwen 5^4 and (-4)^2 there is a + now what will be done first the + and - to go and be - or first that (-4)^2 which will make 16 in + and + and + gives +
polar ivy
The - sign is inside the bracket so it will get squared first before anything else, so the second thing you said
5^4 + 16
left arrow
i need help
mental python
i need help
!help
lone heartBOT
i need help
To ask for mathematics help on this server, please open your own help channel or help thread. See #❓how-to-get-help for instructions.
@primal cipher Has your question been resolved?
lilac hawk
Im doing definite integrals, area between curves, and need feedback where I went wrong and if im using substitution correctly.
tulip ore
Im doing definite integrals, area between curves, and need feedback where I went...
😭 put that on latex
don't worry
i'm going to
polar ivy
Im doing definite integrals, area between curves, and need feedback where I went...
I noticed an error you did: you forgot to change the integrand bounds when you changed variable
lilac hawk
For the sqrt? It gave me the same answer afterwards if I used x or kept it u
polar ivy
oh, yeah i see now that you switched back to x before using the integrand bounds, so that's fine actually
lilac hawk
Okay, so the substitution is fine. The answer is wrong on webassign for some reason. I plan on talking to my teacher, but I want to make sure my math is good first
polar ivy
Could you send the problem statement?
lilac hawk
Evaluate the integral and interpret it as the area of a region.
polar ivy
A picture perhaps? Your calculations are entirely correct, I also got apprx. 11.58. The only options are if the webassign is incorrect or you've misunderstood the problem statement
lilac hawk
My sketch matched the correct answer on webassign, so I guess its webassign
polar ivy
I think you made a latex error in the top right corner
lilac hawk
The sqrt in the integral from 6 to 9 is sqrt(5x+6)
polar ivy
I think you made an error with the bracket here
lilac hawk
Second bracket its 1/2(9^2-6^2) instead of subtracting the squares from 1/2
polar ivy
yup xd
lol
All the calculations of that integral look correct to me. That's all I can say with the info you've provided
lilac hawk
All right. Thank y'all for the help
lilac hawk
I think i misinterpreted the question. It wanted the equation after evaluating the integral, not the answer
polar ivy
Yo this is why I told you to send the problem statement lmao
lilac hawk
Yea this is what I typed before
polar ivy
I think i misinterpreted the question. It wanted the equation after evaluating t...
I'm confused, did you find the issue?
lilac hawk
This is the answer format it wants
lilac hawk
Yea no calculator. Clearly I did to evaluate everything on this one, so I guess that was my issue
I think I have my own stuff to work on and doing simplifying radicals without a calculator
sharp marsh
why is it multiplying everything by 2
lone heartBOT
why is it multiplying everything by 2
sharp marsh
you add 9/4 to both sides you get 9/8
jagged cobalt
im not sure what you mean
they only added the sec^3 integral to both sides
so on the left you have 2 of those
sharp marsh
no the sec^3 integral has 9/4 in front
jagged cobalt
yeah 2 * 9/4
sharp marsh
right but then you multiply the other side by 8/9
sharp marsh
im stupid why tf did i keep getting 9/8
okay yea it should be 9/2 lmao
wait but shouldnt everything be over 2
jagged cobalt
the left is the integral * 9/2
since you only want the integral =... you multiply both sides by 2/9 to get rid of it
(i imagine)
unless you actually want the result of 9/4 * integral
in which case yeah divide everything by 2
sharp marsh
so we're just manipulating it to get what we want lol
jagged cobalt
thats maths
lone heartBOT
crimson dawn
I think if you haven't already you could try expanding (x+p)(x+q) to get an understanding
If I'm reading this correctly
wind atlas
Wait are you asking how you go from the bottom line to factors of the equation?
crimson dawn
So the constant term c must equal to pq
And the coefficient of the x term must be equal to p+q
wind atlas
and you are stuck at the bottom line?
it’s right
i’m just trying to understand what you are stuck on
wind atlas
so on the bottom you have x(ax+p)+1(qx+c) correct?
wind atlas
i see why you are confused because now i’m looking at this confused lmfao
i see what you are saying, you can’t just do the normal thing since what is in the parentheses is different
wind atlas
This just shows that c*a is not equal to p*q
ah i c
lone heartBOT
@fallow quest Has your question been resolved?
wind atlas
i think what this does is literally proves that it only works for whenever (ax+p) and (qx+c) are equal
@fallow quest do you know what splitting the middle term is?
see if this video helps
can you rephrase?
mhm
it’s just a way to formalize factoring by grouping
or splitting the middle term
yep
same principles
in both of them, you are seeing what adds up to, or multiplies to
try both the x method and factoring by grouping on the same equation
(ax+b)(cx+d)
wind atlas
(ax+b)(cx+d)
how do you factor this out?
sorry
how do you expand it
tell me what you multiply in each step
buoyant gyro
yo
wind atlas
yep
wind atlas
Someone please help me
!help
lone heartBOT
Someone please help me
To ask for mathematics help on this server, please open your own help channel or help thread. See #❓how-to-get-help for instructions.
buoyant gyro
And the teacher’s teaching way too fast
wind atlas
i think the issue might actually be that variables are y used
yeah, because then there are issues with like terms
for instance, in any example, for b, d, a, and c, they would be numbers
wind atlas
for instance, in any example, for b, d, a, and c, they would be numbers
so logically if this is true, the only variable would be x
both only have x therefore they are like terms and can be added together
so it just gets a little weird because they aren’t technically like terms when you make them all variables
x^2+ 6x+8
(x+4)(x+2)
xx + x2+4*x+4times2
wind atlas
x*x + x*2+4*x+4times2
simplifys to x^2+6x+8
so the two terms multipled (first and last, second and first) became the middle term when added. The last term is simply last times last
it’s really hard to conceptualize since you are dealing with like terms, multiplying and adding
but it will definitely make sense later
you can still wait tho if you would like, i’m sure someone will be better at explaining it than me
yeee
lone heartBOT
@fallow quest Has your question been resolved?
lone heartBOT
@fallow quest Has your question been resolved?
lethal galleon
find prime number a, b, c, d, e, f so that:
a^2 + b^2 + c^2 + d^3 + e^2 = f^2
lethal galleon
i use module 3 and able to work out in a, b, c, d, e at least one of them is divisible by 3
north perch
should they be different?
lethal galleon
no
north perch
its d^3 ?
lethal galleon
so case 1 if d is divisible by 3 then d=3 (d is prime) so d = 27
27 + a^2 + b^2 + c^2 + e^2 = f^2
i use module 4 and 2 of them is divisible by 2
so 2 of them is equal to 2
35 + c^2 + e^2 = f^2
then i use mod 5 and there is no solution to this case
case 2: a or b or c or e is divisible by 3
little drum
$a^2 + b^2 + c^2 + d^3 + e^2 = f^2$?
ocean sealBOT
Arya
lethal galleon
yea
clever nimbus
d^3 is interesting to me here
lethal galleon
yes