Kk
#help-27
tiny carbon
tiny carbon
$Sin(x) = sec(x)/{cot(x) + tan(x)}$
woven radishBOT
Breeziboi
pseudo basin
$\sin(x) = \frac{\sec(x)}{\cot(x) + \tan(x)}$
woven radishBOT
Ann
tiny carbon
I suck at bot
Thank you ann
This might help in some way
Same for cos and swap sec out with csc
cos(x) = csc(x)/(cot(x) + tan(x))
I dunno how to prove this either
safe grail
i dunno much abt identities
safe grail
but im pretty sure you can prove boths sides being equal to sin(3theta)/sin(theta)sin(4theta)
tiny carbon
but im pretty sure you can prove boths sides being equal to sin(3theta)/sin(thet...
Wait how
safe grail
use some trig identities
tiny carbon
Show me lol
safe grail
like sin(t)cos(s) - cos(t)sin(s) = sin(t-s)
etc
theres like a list of them
wait
this might come in handy
devout snowBOT
@subtle fiber Has your question been resolved?
subtle fiber
Is my ques solved ? 😅
atomic lintel
Trying
Idk how much time but
Uh
I didnt get -cot4θ
I got +cot2θ
Nvm i messed up
viscid crown
I need to create partial fractions however, the denominator is not factorable. Is it still possible to create partial fraction through some other way? The equation is $\frac{x^3+x^2+2x+1}{-x^3-x^2+1}$
devout snowBOT
I need to create partial fractions however, the denominator is not factorable. I...
woven radishBOT
meguuuuu
viscid crown
in particular -x^3-x^2+1 is not factorable.....
viscid crown
ah i see
viscid crown
hm
placid rover
,w -x^3-x^2+1
woven radishBOT
viscid crown
yeah....
no other alternative huh
so if i do long division i get $1 + \frac{2x+2}{-x^3-x^2+1}$ but that denominator is still there :/
woven radishBOT
meguuuuu
placid rover
wht do u want partials for
viscid crown
well this fraction is a generating function for this binary strings i found, and i needed to do partial fractions to find its closed form
placid rover
,w decompose (2x+2)/(-x^3-x^2+1)
viscid crown
the issue is the generatign function is correct, i just checked, but partial fraction doesnt exist, so how tf is there closed form :/ confusing of highest order
viscid crown
i mean i guess we can do decomposition as complex roots right?
viscid crown
i guess thats the only way
placid rover
,w -x^3-x^2+1 =0
placid rover
if you want exact roots you're gonna have to go with cubic formula
placid rover
Post in an open channel
viscid crown
.close
brittle pier
How do I get started on simplifying this?
devout snowBOT
How do I get started on simplifying this?
mortal sapphire
Can i pls have some help
brittle pier
Go to an available help channel!
placid rover
factor.
devout snowBOT
brittle pier
.reopen
devout snowBOT
✅
brittle pier
yeah i'm stuck
brittle pier
How do I get started on simplifying this?
no clue how to get started on this.
supple knot
did you factor?
novel compass
Could you explain, don't know English terms.
take common out of both the denominator and the numerator
supple knot
Could you explain, don't know English terms.
something like
$\frac{a - ab}{c - cd} = \frac{a(1-b)}{c(1-d)}$
woven radishBOT
riemann
trail wing
hello
trail wing
just got a quick question:
this is what it asks:
here is the context
i did a, basiacally proved what it said above ^
im just confused for how i would use that result and log diff to find f'(x)
trail wing
but then left side is derivative of ln f(x)
not deriv of f(x)
thats what im confused on
patent tartan
chain rule
trail wing
oh ok, could you explain that briefly (i know the rule but how to use)
patent tartan
the left side will be 1/f(x) * f'(x)
trail wing
ohh ok
devout snowBOT
Available help channel!
signal holly
I had this exercise in a test.
devout snowBOT
I had this exercise in a test.
slender mirage
last zephyr
I have done the work for a) and I can send that. I'm mainly confused on the bottom bit where I need to find a set of curves which characterize the solution set. I'm hesitant to just put it into desmos as I really don't think that's going to do anything for me but any hel pis appreciated.
devout snowBOT
I have done the work for a) and I can send that. I'm mainly confused on the bott...
devout snowBOT
@last zephyr Has your question been resolved?
sour girder
sour girder
I have the answer, but am looking for someone to talk to me in a vc to show me the step by step solution
sour girder
<@&286206848099549185>
modern glade
I have the answer, but am looking for someone to talk to me in a vc to show me t...
Ypu can use the series formula of e^x
sour girder
.close
gilded sedge
Can someone verify my working and let me know if i did this correct? I’m unsure :/
A person is on a rollercoaster that had a path that looks like y = 7 + 10x - 4x^2, and is looking in the direction of the rollercoaster’s path at all times. When x = 0, is the person looking above or below a bird flying at the point (3,6) ?
devout snowBOT
Can someone verify my working and let me know if i did this correct? I’m unsure ...
lean matrix
Ur part 1 seems correct to me
For next part try tangent of parabola
Ya
Second part seems correct too
full patrol
hi, i have a question regarding vectors
devout snowBOT
hi, i have a question regarding vectors
restive river
Vectors represent more magnitude and direction not really length so dot product is really just a way of combing two vectors
full patrol
then what would be the difference of llall + llbll vs a.b
lilac heath
a.b = |a||b|cos(theta)
full patrol
i dont think i was thought cos(theta)
velvet notch
|a|+|b| = squareroot(ax+ay+az) + squareroot(bx+by+bz). As for dot product you are looking (abx+aby+abz)
full patrol
|a|+|b| = squareroot(ax+ay+az) + squareroot(bx+by+bz). As for dot product you ar...
geometrically speaking what does the dot product represent? i can see that lal + lbl would be the sum of length of both vectors
lilac heath
geometrically speaking what does the dot product represent? i can see that lal +...
if a is a unit vector, a.b is the length of b projected onto a
velvet notch
Ye
lilac heath
lilac heath
here, b/|b| is the unit vector
lilac heath
cauchy ineq comes from definition of dot product, a.b = |a||b|cos(theta)
you dont need vectors to prove the triangle ineq
thats just how triangles work
velvet notch
Right triangle
lilac heath
all triangles
sum of 2 sides is >= remaining side
well
not equal
it wouldn't be a triangle then
dusky trench
how do i solve this?
devout snowBOT
how do i solve this?
dusty violet
devout snowBOT
Available help channel!
zealous oriole
when do i use xer and yer and x > 0 and y>0
devout snowBOT
when do i use xer and yer and x > 0 and y>0
main gull
What is xer and yer?
main gull
I'm asking was xer and yer means
main gull
Because I've never seen those terms before
main gull
That means any real x
zealous oriole
mhm
so my question is when i would use x >0 or xer
acc nvm i think i got it thank you
restive river
hey
restive river
i just wanted clarification proving the special triangles
45 45 90 and 30 60 90
so i believe you first form an equilateral triangle
then you represent all sides as x
I dont see why bd is half the length of the original triangle
!close
.close
devout snowBOT
devout snowBOT
Available help channel!
lament glen
can someone explain why this would be negative infinity? I dont see how this counts as indeterminate if it is undefined - undefined
devout snowBOT
can someone explain why this would be negative infinity? I dont see how this cou...
storm crater
it's not undefined - undefined
lament glen
I thought 1/0 was undefined
lament glen
oh ok I kind of understand that
woven radishBOT
storm crater
better
exotic stump
not how this works
main gull
just answer it rq
No
storm crater
read #❓how-to-get-help
it's occupited by i-can-beat-goku
@lament glen So if you see some thing like 1/x² - 1/x^4
the first term seems to approach infinity
(1/x²)
and the second term does too
so we can't see yet which one is dominant
so we need to rearrange.
lament glen
do we use l'hopital's rule or is there something really simple?
storm crater
simple 🙂
lament glen
ok
storm crater
You could combine it into one fraction and use Hôptial's.
storm crater
yeah but there is an even simpler way
you can just use distributive law
1/x²(1 - 1/x²)
now it's clear that the bracket goes to -infinity
the first factor goes to infinity
and there is no conflict, so it goes to -infinity in total.
storm crater
You are welcome 🙂
lament glen
.close
proven hamlet
Can someone help me integrate this? I've tried u sub but I get my answer as -72 instead of -72/15
devout snowBOT
Can someone help me integrate this? I've tried u sub but I get my answer as -72 ...
proven hamlet
the upper limit is 6 and lower is 0
formal haven
@proven hamlet What are you setting u = to?
proven hamlet
<@!276982812635299840> What are you setting u = to?
i tried it to be that entire equation so its u^4 and i even tried it so that its only sqrt x
but either way it didn't work out
main gull
If you u sub, you have to take the derivative of u
proven hamlet
If you u sub, you have to take the derivative of u
i did
I'm not sure how i end up without a denominator in my final answer
main gull
Post your work
proven hamlet
woven radishBOT
Kurama
fervent jewel
this would get rid of everything inside,
||solve for dx = 2sqrt(x)du, then rearrange that u-sub to get sqrt(x) = u+sqrt(6) so your dx = 2(u+sqrt(6))du||
devout snowBOT
@proven hamlet Has your question been resolved?
proven hamlet
||solve for dx = 2sqrt(x)du, then rearrange that u-sub to get sqrt(x) = u+sqrt(6...
hmm ok i'll try that
tired oak
devout snowBOT
tired oak
can someone pls help me
ive never been so lost doing homework i dont understand any of it
the ones in the box are correct
i cant find the first 4
and i have one try left
i thought they would be -1
<@&286206848099549185> sorry for ping but its timed and i dont have alot of time left
woeful shale
can you see that you approach different y values as you approach x = 1 from the left and the right?
sorry, x=-1
holup that's one
mb reset
the limit is what you are approaching
which can be different from the actual value there
tired oak
can you see that you approach different y values as you approach x = 1 from the ...
im confused do we look at the red thing thats on the left when we go towards the right
woeful shale
so start on the line off to the left, walk right along it until you get close to -1
where do you end up?
tired oak
so start on the line off to the left, walk right along it until you get close to...
we hit -1
-1 no?
woeful shale
looks like you hit 2
woeful shale
yes, as I'm talking about the y value here
woeful shale
so more accurately, the y value is approaching 1 as we walk along the line towards x=-1
from both directions!
that's what we call the limit
at x=-1
woeful shale
note though that the actual value of f at -1 is not 1
that's why there's a hole in the graph
ah I misread the axes, I am blind today
y=2
can you see that?
tired oak
omg lol yes
i completely forgot y axis is above aswell, when we go to the right we hit 2
from the left
2
woeful shale
yes
tired oak
what about when f(X) is at -1
the third box
just to be clear
the first 2 boxes are 2
woeful shale
yes, the first three in fact
bc we have the same value from the left and right, the limit exists and is that value
careful though bc the limit need not be the actual function value there
that solid dot denotes what f(-1) actually is, the hole tells you that f(-1) is not 2
even if that is the limit
tired oak
oh yea i forgot about that
when left and right are the same
when its at -1 its the same as those
woeful shale
right, the limit is
woeful shale
so where's that solid dot at x = -1
woeful shale
yep!
woeful shale
sure
woeful shale
do you know how to graph a piecewise function?
woeful shale
mm
tired oak
i have one that looks more simple
and takes less time hopefully
those are not right
woeful shale
alright so for these purposes, we'll define a function as being differentiable at a point if it is "smooth"
this means no discontinuities
no jumps or anything
and also no sharp corners
see anywhere like that on that graph?
formal haven
Also remember a good way to know if a function is continuous or not is if you can draw it without lifting your pencil
woeful shale
^^ yee
so which x values do you see this happening (where might you have to lift up your pencil?)
woeful shale
right, which x value is that?
tired oak
that separatess the two
0 and 2.5 i think?
are we talking about when it first breaks?
woeful shale
talking about at which x value it breaks at
fervent jewel
look at how f(x) is defined
when x >= some value, its defined in one way. when x < some value, its defined in another
woeful shale
we on a different question rn fam
fervent jewel
oof nvm
tired oak
lmao
tired oak
talking about at which *x* value it breaks at
the left line it breaks/stops 0
at the x axis
formal haven
@woeful shale At what x value do you lift your pencil?
tired oak
its either 0 or -3 lol
im getting confused
i havent dont math in 2 years and this if fuckin me up
woeful shale
start from the left, trace the graph
stop once you have to lift your pencil while tracing
then see where you are on the horizontal axis
woeful shale
ayooo
tired oak
the graph looks like -3
woeful shale
no worries
tired oak
2,1?
formal haven
I think you mixed up the x and y coordinates
tired oak
thats were the other sharp turn occurs
woeful shale
exactly, there's a sharp corner there
in other words, if you were tracing again with the pencil, you'd have to stop moving and change direction
it ain't smooth
woeful shale
it only asks for the x values
tired oak
so just -2,2
woeful shale
yes
woeful shale
yeesh
tired oak
just gonna accept the L
woeful shale
we can try
woeful shale
ah yes
so this function is defined as one thing when x is less than three, and another when x is greater than or equal to 3
tired oak
ok
woeful shale
so its graph is gonna be two things stitched together
in this case
the red part is where x is less than three, the blue where x is greater than three
tired oak
okok
woeful shale
the question is asking for the limit at 3
so approach it from the left and right and see if they're the same, as before
woeful shale
here, this screenshot might be clearer
ah we are looking at what happens as x approaches 3
not y
formal haven
the very point at the top
Which part of the piecewise function has x=3?
woeful shale
bigger text last screenshot
woeful shale
ah, notice how one part has x < 3 and one part has x ≥ 3?
≥ means greater than or equal to, so this is in fact defined at x = 3
woeful shale
lessgoooo
tired oak
thank you winston
woeful shale
anytime
woeful shale
and thanks for the tip @formal haven
tired oak
.close
devout snowBOT
devout snowBOT
Available help channel!
restive river
can anyone help pls with 6c
devout snowBOT
can anyone help pls with 6c
hidden crown
Have you finished b?
restive river
Have you finished b?
yeah
hidden crown
I hope it would be obvious that the max height would be the diameter of the wheel. The total distance covered by one complete revolution is $2 \pi R$ with $R$ being the radius of the wheel
woven radishBOT
[TEB] darthlothins
restive river
I hope it would be obvious that the max height would be the diameter of the whee...
cant i just find the height at T=0 then find the next time that the height is the same?
i tried that but its giving me the wrong answer?
or is that wrong?
hidden crown
cant i just find the height at T=0 then find the next time that the height is th...
this is what you might end up calculating with this method because cos and sin are periodic functions
(excuse the bad writing)
restive river
this is what you might end up calculating with this method because cos and sin a...
so would that be right?
hidden crown
No, say you find the height at t=0 to be H
restive river
oh wait no since its not complete
hidden crown
Exactly
hidden crown
One thing I'm not certain of is if one can assume that two half revolutions take the same time as one full revolution
You can plug in the maximum height into the height equation to find the value of t when the height is maximum, that would be equivalent to one half revolution (Of course now I realise that would be the case when H=0 at t=0)
restive river
~~You can plug in the maximum height into the height equation to find the value ...
could you try that with the minimum value?
hidden crown
I guess you find the height at t=0 ($H_{t=0} = 5$ units) and subtract that value from the maximum height, to get the height when one half revolution has been completed. Find the time taken for this half revolution to be completed and multiplying it by 2 would give you the time for one complete revolution, assuming the time for two half revolutions is equal to the time for one full revolution
woven radishBOT
[TEB] darthlothins
hidden crown
Not sure if there would be a more straightforward way to tackle the problem but I hope you get my logic
restive river
I guess you find the height at t=0 ($H_{t=0} = 5$ units) and subtract that value...
this is what the ms did
but if you can do that with the max height shouldnt it work for the minimum?
hidden crown
I guess that is just easier 😄
Ah well it should, but you took t=0 which is different
But maximum height was asked by a previous part of the question which you end up using here
Which is why the maximum height was used in the solution, the minimum height should yield the same result
restive river
Ah well it should, but you took t=0 which is different
so if we had a pendulum moving back and forth would that work with t=0? or should i always use the max height in situations like these?
hidden crown
well for a pendulum the time for one oscillation doesn't depend on the amplitude (maximum height)
$T = 2 \pi \cdot \sqrt{\frac{l}{g}}$
woven radishBOT
[TEB] darthlothins
restive river
well for a pendulum the time for one oscillation doesn't depend on the amplitude...
ahhh alr thanks
lilac heath
sum the area of each side of the shape
misty grotto
Hello there, I have a question from an non-inverting amplfiier (opam) (electric circuits).
Here the current I is leaving the inverting terminal. The current (I) is negative because is leaving the inverting terminal, right?
Because if I do the KVL without the consideration of it, I (current) is going to give me positive, and it's wrong.
I hope I explained my self enough, thanks in advance
devout snowBOT
Hello there, I have a question from an non-inverting amplfiier (opam) (electric ...
main gull
Hello there, I have a question from an non-inverting amplfiier (opam) (electric ...
Maybe a STEM/Engineering/Physics server would be better to ask
devout snowBOT
devout snowBOT
Available help channel!
left pulsar
Is this the only characteristics of a function?
"A function is a relation in which each possible input value leads to exactly one output value. We say “the output is a function of the input.” The input values make up the domain, and the output values make up the range."
devout snowBOT
Is this the only characteristics of a function?
"A function is a relation in whi...
left pulsar
okay thanks!
btw how about the limits of a function?
Our teacher asked us to explain the limits of a function. He hasn't discussed it yet.
Is it this one?
"The limit of a function at a point a in its domain (if it exists) is the value that the function approaches as its argument approaches a. The concept of a limit is the fundamental concept of calculus and analysis. It is used to define the derivative and the definite integral, and it can also be used to analyze the local behavior of functions near points of interest."
copper mango
a doesn't have to be in the domain
devout snowBOT
Available help channel!
full patrol
hello
full patrol
can someone please help me with part b
i have arrived at the partial step of F(25/s.d.) - F(-25/s.d.) = 0.99
but not sure how to proceed
on finding the s.d.
devout snowBOT
@full patrol Has your question been resolved?
devout snowBOT
@full patrol Has your question been resolved?
full patrol
.close
red ermine
hey guys how can I solve this? I'm trying to solve for beta and I combined the TAB equations but kinda confused. Any help is appreciated.
devout snowBOT
hey guys how can I solve this? I'm trying to solve for beta and I combined the T...
red ermine
devout snowBOT
@red ermine Has your question been resolved?
supple knot
what are you given and what are you solving for
unborn token
If one is asked to prove te following
devout snowBOT
If one is asked to prove te following
unborn token
$\sum\frac{1}{(2n-1)\cdot 2} =\sum\frac{1}{(2n}$
woven radishBOT
Mathematics
solid trail
huh
solid trail
this clearly isn’t true
copper mango
can you post the question verbatim?
unborn token
I can send a picture of it
1 sec
the one with blue arrow
"Show that for all integers n >= 1 that "
unborn token
yes
copper mango
assume true for $n=k$, namely that $\sum_{j=1}^k\frac{1}{2j(2j-1)}=\sum_{j=1}^k\frac{1}{k+j}$
woven radishBOT
Mosh
unborn token
wait, why is the right hand side 1/k+j and not 1/2k?
copper mango
cause each term is of the form 1/(k+j)
unborn token
but doesn't the "1/2n" in the last part mean that it follows 1/2n and not 1/k+j?
what I mean is, what happens with the factor of two that is given in the last part?
copper mango
but doesn't the "1/2n" in the last part mean that it follows 1/2n and not 1/k+j?
no.
when j=k, you get 1/(k+k)
unborn token
ohhhhhhhhhhhhh
from here would one want to manipulate both sides to equal each other
like how you started
ah I think I got it
@copper mango thank you ❤️
devout snowBOT
@unborn token Has your question been resolved?
devout snowBOT
devout snowBOT
Available help channel!
silk loom
Im going over some math from last year and I’ve forgotten why square rooting both sides results in the +- symbol. Could anyone explain why this happens?
devout snowBOT
Im going over some math from last year and I’ve forgotten why square rooting bot...
main gull
Because you have the positive answer and the negative answer
A simpler example would be $$x^2 = 4 \implies x = \pm \sqrt{4} = \pm 2$$
woven radishBOT
dldh06
silk loom
So putting in 5 or -5 for x would result in the same y value? $$y = -4 \pm sqrt{(x+50)/3}$$
woven radishBOT
Coal_CC
silk loom
Wait that’s not right
wanton aspen
I need help with this
devout snowBOT
I need help with this
wicked turtle
With what aspect?
wanton aspen
i understand cycle notation and transpostions/permutations but idk how they got a*b
i got a different answer
pseudo basin
what answer did you get
wicked turtle
$a \cdot b = (23)(12)$. Look where each element is sent: $1 \mapsto 2 \mapsto 3$. $2 \mapsto 1$. $3 \mapsto 2$.
woven radishBOT
OurBelovedBungo
wicked turtle
Recall that these are functions. $a\cdot b(1) = a(b(1))$, i.e. you apply $b$ first and then $a$.
woven radishBOT
OurBelovedBungo
wanton aspen
Sorry give me a sec, library just closed
ok sorry about that, for a*b i got (3 1 2)
which was by applying b first, then a
wicked turtle
So when you apply b first, then a, you're telling me that 1 gets sent to 2?
I suggest checking that again. 😁
wanton aspen
1-> 2-> 3 , no?
wicked turtle
Yep, so $a\cdot b$ sends 1 to 3
woven radishBOT
OurBelovedBungo
wicked turtle
So $a\cdot b$ certainly cannot be $(312)$.
woven radishBOT
OurBelovedBungo
wanton aspen
b is (1 2) right so 1->2 , a is (2 3), so 2->3 so overall , 1->3
ok wait a sec, i think im not understanding something
wicked turtle
No, that's not true.
(312) sends 3->1, and 1->2, and 2->3
Therefore $(312) = \begin{pmatrix}1&2&3\2&3&1\end{pmatrix}$.
woven radishBOT
OurBelovedBungo
wanton aspen
ah ok ,i understand that now
wanton aspen
**OurBelovedBungo**
ah ok i understand this too
wicked turtle
So you're OK with the answers from your original screenshot now?
wanton aspen
with ab, yea. I'll work out ba just to verify
wicked turtle
Cool
wanton aspen
oops, sorry for not using TeX
wicked turtle
No worries, it's clear enough either way
wanton aspen
ok, thanks for the help. I think i got confused with the cycle notation
wicked turtle
No problem, it is confusing at first
wanton aspen
so is the general idea to write out that mapping , then construct the notation from there?
wicked turtle
If by "that mapping" you mean work out where each element goes, yes, that's the idea, and you build the result based on that
As far as I know there's no secret trick to do it faster, except in certain special cases.
lone wind
How come sqrt(3) ^ 3 = sqrt(3) * 3
devout snowBOT
How come `sqrt(3) ^ 3` = `sqrt(3) * 3`
lone wind
isn't there some property I can do with 3 ^ (0.5 * 3)?
deft hemlock
How come `sqrt(3) ^ 3` = `sqrt(3) * 3`
$\sqrt{3}^3 = \sqrt{3}^2 \cdot \sqrt{3}$
woven radishBOT
Yottachad
graceful cosmos
Because you have 3 different √3s, and you can combine two of them to make a 3
deft hemlock
isn't there some property I can do with `3 ^ (0.5 * 3)`?
yes exactly you can split the power to 3 = 2 + 1
lone wind
I can't tell if you're being sarcastic
deft hemlock
im not
deft hemlock
**Yottachad**
what i described is just this
deft hemlock
yes
lone wind
3 ^ (0.5 * 3) = 3 ^ (0.5 * (2 + 1)) = 3 ^ ~~(0.5 * 2)~~ + 3 ^ (0.5 ~~* 1~~) = 3 + sqrt(3)
that's
pretty wack
deft hemlock
not + its multiplicatoin
devout snowBOT
@lone wind Has your question been resolved?
restive river
why is the solution for (x+1) (x+2) ≥ 0
devout snowBOT
why is the solution for (x+1) (x+2) ≥ 0
odd kestrel
and not x ≥ -2
What is the value of that quadratic polynomial function at x = -3/2?
restive river
you solve (x+1) (x+2)
and put the last term on the other side of the equality
I think that was the question
odd kestrel
I am asking another question, to help you understand
restive river
ah sorry
exotic stump
you can't just divide off the terms, you don't know the sign of x
odd kestrel
Let's return to this question: What is the value of that quadratic polynomial function at x = -3/2?
exotic stump
and put the last term on the other side of the equality
ya did here
restive river
.5 squared?
restive river
negative point five squared
odd kestrel
Hence not all real numbers not less than -2 satisfies the given inequality
odd kestrel
and not x ≥ -2
Does that answer your question?
restive river
nah i don't get it
restive river
is there any rule that prevents me from doing logical algebra to get the right answer
exotic stump
yes
odd kestrel
I gave an example of a real number, not less than -2, which does not satisfy (x+1) (x+2) ≥ 0
exotic stump
you don't know the sign of x
restive river
okay
exotic stump
if you divide by (x+2), it could be positive or negative, depending on x
restive river
yeah i understand that
restive river
but how do you get the correct answer
without drawing a graph
or guessing values
by pure algebra I mean
odd kestrel
You do "logical algebra"
A product of two real numbers is nonnegative if both are nonnegative, or both are nonpositive
exotic stump
essentially do the algebra as if there were an equal sign there, then create a number line
restive river
wait hold on
y=xsquared + 3x ≥ -2
exotic stump
why is the solution for (x+1) (x+2) ≥ 0
go back to this equation
odd kestrel
The solution (x <= -2) or (x >= -1)
is really the simplified form of (x <= -2 and x <= -1) or (x >= -2 and x >= -1)
restive river
how are they the same?
exotic stump
overlap
restive river
x^2 + 3x ≥ -2
exotic stump
x must be less than or equal to -2, but x must be less than or equal to -1
restive river
from solving it considering both cases, you have conflicting answers
okay I get that now but lets say i did not draw a number line or a graph
how would I solve this x^2 + 3x ≥ -2
with the rules of algebra
exotic stump
the number line uses algebra
restive river
x^2 + 3x ≥ -2
x^2 + 3x + 2 ≥ 0
(x+1) (x+2) ≥ 0
x≥-1
x≥-2
what is the problem in here^^
exotic stump
x must be greater than or equal to -2 but it also must be greater than or equal to -1
so you need to consider the option that allows all x to satisfy the equation
restive river
yeah
so that means i just randomly change the inequalities to make my answer more sensible?
restive river
only by looking at a graph i can make a logical choice
however whenever i put this in symbolab it gives me this
exotic stump
x must be greater than or equal to -2 but it also must be greater than or equal ...
consider this sentence
restive river
however whenever i put this in symbolab it gives me this
so it does mean that its possible to solve it using just algebra
exotic stump
x must be => -2 but also => -1
restive river
consider this sentence
i completely understand that
restive river
so pick the logical option
and if were to choose an option it would break the rules of solving inequalities
exotic stump
symbolab is a calculator, it has access to literally any type of problem solving it wants
you're not breaking any rules by choosing the logical option
restive river
you're not breaking any rules by choosing the logical option
nah you're flipping the symbols around
exotic stump
?
restive river
you're turning ≥ to ≤
exotic stump
x^2 + 3x ≥ -2
x^2 + 3x + 2 ≥ 0
(x+1) (x+2) ≥ 0
x≥-1
x≥-2
if you want to talk about breaking rules, this is breaking rules
zero product property doesn't apply to inequalities
restive river
apparently it does tho
exotic stump
it doesn't
again, symbolab is a calculator
it's not applying the zero product property
restive river
what is it doing then?
exotic stump
it can plug in points, it can use an imaginary number line, it can use the graph
restive river
surely after a couple of millennia of advancements in maths, they must have come up with a rule or something to solve it
like you have one for the quadratic equation
exotic stump
considering cases 
restive river
you don't have to draw rectangles every time to solve one
any way thanks for the help @exotic stump
.close
devout snowBOT
devout snowBOT
Available help channel!
finite cosmos
How do i calculate the derivative on question 7:a?
devout snowBOT
How do i calculate the derivative on question 7:a?
copper mango
power rule
finite cosmos
Not sure if i calculated it correctly, is this correct?
<@&286206848099549185>
may someone please confirm if my calculations are correct
.close
true kettle
Quick physics question for anyone that can help
devout snowBOT
Quick physics question for anyone that can help
true kettle
So I’m helping a friend with simple harmonic motion
They have a box on a spring
The teacher wrote that at equilibrium the spring potential energy (Us) was a value other than 0
They also wrote that at equilibrium, x=0.294
Is this possible???
I swear that equilibrium had to be x=0 and that Us is 0
dim spindle
F_Gravitation = F_Spring
<=> m * g = k * s
<=> s = m * g / k = 3kg * 9.81m/s^2 / 100N/m = 0.294m
@true kettle
true kettle
F_Gravitation = F_Spring
<=> m * g = k * s
<=> s = m * g / k = 3kg * 9.81m/s^2 /...
Where does the Fg=Fs come from again?
dim spindle
uhh,
for Equilibrium to exist, the sum of all forces has to be 0
a.k.a F_Grav - F_Spring = 0;
F_Spring must be negative because its pointing in the different direction
and then just solving
dim spindle
Isn’t the equilibrium position when Fs=0
no. Equilibrium is reached when the sum of all forces = 0
the sum of all forces on the body we are observing
that is
balmy jasper
devout snowBOT
balmy jasper
I multi by the highest power in the denominator right ?
supple knot
try it and show your work
leaden aurora
Carry out the long division
devout snowBOT
@balmy jasper Has your question been resolved?
devout snowBOT
devout snowBOT
Available help channel!
pearl token
devout snowBOT
exotic stump
• After 15 minutes, feel free to ping @Helpers.
• Show your work, and if possible, explain where you are stuck.
exotic stump
• Show your work, and if possible, explain where you are stuck.
pearl token
> • Show your work, and if possible, explain where you are stuck.
these question
exotic stump
is this a quiz/test?
pearl token
is this a quiz/test?
homework
vagrant mural
you can send pics
devout snowBOT
@pearl token Has your question been resolved?
proven schooner
I am working with transformations of variables
devout snowBOT
I am working with transformations of variables
proven schooner
Give me a moment
That is the equation I am working with, I am only concerned with the spatial coordinates.
These transformations are used
To get the arguments of this exponential function in this integral.
If it helps, I am working with a three dimensional classical system, and this is the spatial part of the partition function in statistical mechanics
I guess I didnt exaclty ask the question. I dont know how this transformation works
I have worked through it backwards to see if I could get anywhere but to no avail
<@&286206848099549185>
devout snowBOT
@proven schooner Has your question been resolved?
proven schooner
<@&286206848099549185>
devout snowBOT
@proven schooner Has your question been resolved?
storm crater
To get the arguments of this exponential function in this integral.
Which integral did we start with before the transformation?
proven schooner
the argument of the exponential would be -B(three last terms of H) where the integral is a triple integral over x_1, x_2, and (x_1-x_2)
should I write it out and send a picture?
storm crater
should I write it out and send a picture?
yes frankly
I am very rusty with physics.
storm crater
huh
The integration does not make sense though.
What do you mean by d(x_1 - x_2)?
proven schooner
I dont know. That is my problem. The guide that I am looking at used that transformation to get rid of that coupled term. That is what I assume it would be prior to the transformation.
I guess not
Maybe it is only a double integral before that point
storm crater
Likely
proven schooner
It is just a lot worse and doenst collapse into a gaussian integral
storm crater
Otherwise as a mathematician I totally clueless.
I know physicists sometimes write funny things but no idea what this would mean.
proven schooner
As my professors always say. Mathematicians hate Physicists versions of math.
We just do wonky shit
storm crater
yep
proven schooner
Just focusing on the argument. Is there anyway to get from the last 3 terms, to the argument of the exponential with the transformation from an algebra standpoint?
Ignore the integral
storm crater
Have you computed it?
proven schooner
I have tried foiling it out, backward and forward, tried wolfram alpha, mathmatica
nothing
My guide just throws that transformation at me and says that it is a simple transformation
I dont feel comfortable moving forward without understanding it
storm crater
,wolf simplify -k((a + b)/2)² - (k / 4 + K/2)(a - b)²
woven radishBOT
proven schooner
Thank you for your help.
storm crater
as in positions
so dx_1 dx_2
which also makes sense as a mathematical expression lol
no dx_1 - x_2
proven schooner
Yeah that was my mistake
storm crater
Then transofrmation formula
to see how the integral element changes we need the determinant.
proven schooner
Oh god not linear algrebra
storm crater
Do you know the transformation formula?
storm crater
Oh no
I meant "transformation formula"
as the general theorem called "transformation formula"
how to apply multidimensional variable transforms in integrals.
proven schooner
Guess not, I am not that far in my second major
storm crater
Let me see if Wikipedia is comprehensible.
I think this might be tough to read.
Let me summerize it.
If you have a differentiable function f with n dimensional input and output
then you can compute the determinant of the matrix of all derivatives
in all directions
e.g. if f is 2 dimensional
$$s = \det \begin{pmatrix} \frac{\partial f_1}{\partial x_1}&\frac{\partial f_1}{\partial x_2}\\frac{\partial f_2}{\partial x_1}&\frac{\partial f_2}{\partial x_2} \end{pmatrix}$$
woven radishBOT
M8732
proven schooner
Damn, what class does this appear in?
storm crater
Mathematically: $$\int_A g(x) s d(x_1, \ldots, x_n) = \int_{f^{-1}(A)} g(f(x)) d(x_1, \ldots, x_n)$$
woven radishBOT
M8732
proven schooner
Linear Algebra 3 or numerical analysis?
storm crater
I am from Europe
I had no class with either name.
I think physicists learnt it first semester in a course called "calculation methods" (very quick)
and mathematicians in analysis 3 aka measure theory
proven schooner
Here I took mathematical physics, which was like 2 years of math in 1 semester. It was rough
storm crater
Calculus 3 maybe in the US?