ok so they have the same steps as we do
#help-17
zinc quail
they also get -1 and -2 as roots
which our A,B,C values get too
still don't see the mistake :D
the quadratic is definitely the same
zinc quail
oh then maybe
do you perhaps use a different quadratic formula?
because the standard one uses b²-4ac
but for instance the German one uses (p/2)²-q
@past prairie yeah so just to be clear:
American discriminant = 36 is correct
past prairie
oh okok
zinc quail
and the calculations from before are correct too, they yield:
the right intersections
so everything was fine :D
past prairie
what did you use x)
zinc quail
just geogebra 3d to quickly do it
past prairie
ohh
past prairie
pretty nice
zinc quail
so I guess error was you have a different discriminant formula?
maybe some French version of it
if it's the same formula, then
is wrong
past prairie
so I guess error was you have a different discriminant formula?
and now for which value the line goes in the sphere ?
like i only need to know if an object is aiming toward that sphere
zinc quail
and now for which value the line goes in the sphere ?
well if discriminant >= 0
which it is
if Discriminant >= 0, then the object aims towards the sphere
past prairie
uh even when the object isn't aiming towards it saying it does : /
lmao
i forgot to square the radius x))
Uh it still strange
vocal sleetBOT
@past prairie Has your question been resolved?
vale frigate
what's the purpose of dF = dm * a
vocal sleetBOT
what's the purpose of dF = dm * a
solemn marten
what's the purpose of dF = dm * a
probably because small changes in force are not necessarily caused by changes in acceleration
for example if you apply a force dF on a block of mass m, there will be a small change in acceleration da
the mass will stay constant in cases like these
but for example if you had to measure the change in the force of a rocket
as its fuel is exhausted it continuously loses mass
so there is also a small change in force dF acting upon a change of mass dm whie it continues to accelerate at constant acceleraiton a
sullen shoal
probably because small changes in force are not necessarily caused by changes in...
it shld be dF = d(mv)
then it will become dF = dm*v + m*dv```@vale frigate Has your question been resolved?
solemn marten
```wait how can u say dF = dm*a
it shld be dF = d(mv)
then it will become dF = d...
mv is just impulse
m(v - u)/t
is the force
solemn marten
but here we have dm, ie mass is changing too
hm true both values are changing
sullen shoal
yes so u cant say dF/dt is only m(v-u)/t
solemn marten
yes so u cant say dF/dt is only m(v-u)/t
true but i didn't exactly say that
or at least that's not what i meant to say
i guess this is just a relation between changing mass as well as changing velocity
dF = dm * dv/dt
dF * dt = dm * dv
sullen shoal
nop
solemn marten
which i've never seen before honestly
sullen shoal
u+v rule dude
sullen shoal
u'v + v'u
solemn marten
u+v rule dude
you can't use that here though
sullen shoal
u can
solemn marten
we're not differentiating mv though
we're taking a small value dm and multiplying it with dv/dt
solemn marten
i suppose so
sullen shoal
but where did the actual guy go
vast shale
vocal sleetBOT
vast shale
These math jokes are annoying me so much
Can someone explain
OOOOOOOOOOOOOOOOOOOO
Dude
solemn marten
if x has a value it's no longer a variable
vast shale
if x has a value it's no longer a variable
This
solemn marten
you can't treat x as a variable and differentiate it using the power rule anymore
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scarlet bough
how would i do this question? like how do i get the values for y_0, y_1 y_1 etc?
winter thistle
how would i do this question? like how do i get the values for y_0, y_1 y_1 etc?
you will first have to figure out the x_0, x_1, ..., x_6
and then plug them into 1+cos(x) for y_0,...
scarlet bough
you will first have to figure out the x_0, x_1, ..., x_6
how would i do this bc i have no clue
winter thistle
how would i do this bc i have no clue
ok they want you to have 6 strips of equal width
6 strips mean 7 points
and the distance between each point has to be the same, and the first point is 0 and last point is 2pi because that's the bounds of your integral
so you can figure out the distance/width between every point?
scarlet bough
so you can figure out the distance/width between every point?
2pi?
winter thistle
that's the total distance from first point to last point
scarlet bough
2pi/6
winter thistle
yup
now x_1 = x_0 + 2pi/6 = 0 + 2pi/6
x_2 = x_0 + 2 * 2pi/6
or equivalently, x_2 = x_1 + 2pi/6
etc.
so can you work out all x_0, x_1, ..., x_5, x_6?
scarlet bough
so can you work out all x_0, x_1, ..., x_5, x_6?
x_1= pi/3 x_2=2pi/3 x_3=pi x_4=4pi/3 x_5=5pi/3 x_6=2pi
winter thistle
x_1= pi/3 x_2=2pi/3 x_3=pi x_4=4pi/3 x_5=5pi/3 x_6=2pi
nice
now y_0 to y_6 are the corresponding values when you plug x_0 to x_6 into the function given
winter thistle
btw would i have to do it manually every time?
do what exactly?
scarlet bough
do what exactly?
like to get x_n
winter thistle
like to get x_n
yes, if the question doesn't give them to you
scarlet bough
now y_0 to y_6 are the corresponding values when you plug x_0 to x_6 into the fu...
so i got y_1=3/2 y_2=1/2 y_3=0 y_4=1/2 y_5=3/2 y_6=2
scarlet bough
yes, if the question doesn't give them to you
i subbed into the trapizium formula and got 25.5 which isnt 2pi
winter thistle
i subbed into the trapizium formula and got 25.5 which isnt 2pi
how did you get 25.5?
scarlet bough
how did you get 25.5?
6/2((3/2 + 2) + 2(0.5 + 0 + 0.5 + 3/2))
winter thistle
why 6/2 at the beginning?
scarlet bough
why 6/2 at the beginning?
half the height using trapizium rule
winter thistle
6/2((3/2 + 2) + 2(0.5 + 0 + 0.5 + 3/2))
also did you forget to consider y_0?
winter thistle
half the height using trapizium rule
h is width not height, but anyway h is 2pi/6 here?
so half of that is pi/6
scarlet bough
also did you forget to consider y_0?
isnt y_0 my y_1?
winter thistle
no...
x_0 is 0
y_0 is 1+cos(x_0)
which is 1+cos(0) = 1
y_1 = 1 + cos(x_1)
where x_1 = pi/3
winter thistle
so is it (2pi-0)/6 * ((1 + 2) + 2(3/2 + 0.5 + 0 + 0.5 + 3/2))
1/2 * (2pi-0)/6 at the front
it's 1/2 * h where h is 2pi/6 here
scarlet bough
it's 1/2 * h where h is 2pi/6 here
oh ye, that still gives 11pi/6
scarlet bough
should be 7pi/6
(2pi-0)/6 * ((1 + 2) + 2(3/2 + 0.5 + 0 + 0.5 + 3/2)) is giving me 11pi/6
winter thistle
i know why
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worthy karma
Are the options incorrect
vocal sleetBOT
Are the options incorrect
worthy karma
Or do I not understand the question well
solemn marten
Are the options incorrect
i got dr/dt = 3/8
worthy karma
Howd you get that
solemn marten
dA/dt = 3pi
d (pi r^2)/dt = 3pi
2 pi r * dr/dt = 3pi
dr/dt = 3/2r
then they tell us area is 16pi
which means
16pi = pir^2
16 = r^2
r = +/- 4
in this case r = +4
so then
dr/dt = 3/2r
r = 4
dr/dt = 3/2(4) = 3/8
solemn marten
np but my answer isn't one of the options either lmao
solemn marten
no problem
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fast chasm
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wild saffron
If I have the function f(x) = x^2-6x+3, is there an easy way for me to get the range of it without using a calculator?
vocal sleetBOT
If I have the function f(x) = x^2-6x+3, is there an easy way for me to get the r...
peak matrix
If I have the function f(x) = x^2-6x+3, is there an easy way for me to get the r...
probably finding the vertex
if you find y-coordinate of the vertex Y, then depending on coefficient of x^2, the range is either [y, infinity) or (-infinity, y]
wild saffron
if you find y-coordinate of the vertex Y, then depending on coefficient of x^2, ...
ok thank you
.close
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foggy wave
I need to find vector MB but I can't seem to get the right answer 😭😭😭😭
foggy wave
It's a trapezoid and M cuts AC in half
Didn't say anything about BM but I'm assuming that its also cut in half
I thought it's supposed to be 0.5(b-2a) but that doesn't seem to be the case
vocal sleetBOT
@foggy wave Has your question been resolved?
stone gazelle
Find MA
foggy wave
why
stone gazelle
MA + AB = MB
foggy wave
AHH thank you I got it correct this time
Another question
I did it like this
But it's wrong
idk what's not right
I got 822 but it's supposed to be 522
stone gazelle
What is the question asking?
foggy wave
I need the scalar product
I covered the words cuz they're not in English anyway but all the needed info is there
stone gazelle
The calculations are correct, seems like you may have just put them together wrong
[ 0.5 \boldsymbol{a}^2 + \boldsymbol{a}\boldsymbol{b} - 3\boldsymbol{a}\boldsymbol{b} - 6\boldsymbol{b}^2 ]
[ = 512 + (-80) - (-240) - (150) = 522 ]
twin meteorBOT
shsgd
stone gazelle
@foggy wave
foggy wave
But
I got the last one -150
Wait
Bruh
I got it
I did the minus twice
I should've taken 6 instead of -6
stone gazelle
yeah]
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coarse wren
Can someone help me do this please? I’m just struggling with this one
chrome raptor
So what trigonometric functions do you know?
coarse wren
sin tan cos etc
coarse wren
So what trigonometric functions do you know?
how can i do this?
chrome raptor
So sin is just cos translated so what of sin and tan could be?
Look at that graph, there are some points of R which do not belong to the domain
coarse wren
Look at that graph, there are some points of R which do not belong to the domain
what is R?
chrome raptor
The real numbers
coarse wren
Look at that graph, there are some points of R which do not belong to the domain
i just stumbled across this online and for some reason i just want to find out the solution
im grade 10 so i dont really know a lot of this
coarse wren
do you know the parent functino
no, what's the parent function?
thats the entire problem
plucky trout
no, what's the parent function?
without transformation, sec(x)
or the other one
csc(x)
coarse wren
um idk
coarse wren
secant / cosecant
is it okay if you tell me how to do it?
chrome raptor
Ok what is the tangent of π/2?
chrome raptor
Is tan decreasing from 0 to π/4 and increasing from π/4 to π/2?
What is the value of tan(π/4)?
That's some helpful question
coarse wren
Is tan decreasing from 0 to π/4 and increasing from π/4 to π/2?
yea
chrome raptor
What is the value of the function in the graph for x=π/4?
coarse wren
What is the value of the function in the graph for x=π/4?
im not sure
maybe -inf or inf?
chrome raptor
You can't study trigonometric functions if you haven't mastered understanding a graph yet
chrome raptor
When x=π/4 what is the value of the function?
Look at the horizontal line at π/4 then through which point the graph passes
coarse wren
Look at the horizontal line at π/4 then through which point the graph passes
there is no pi/4 on the graph though
chrome raptor
π/4 is half π/2
coarse wren
-0.5
chrome raptor
That's right
So it'd be -1 if the function were tan but it is -0.5, so how can you change tan so as it is -1 instead of -0.5 at π/4?
coarse wren
So it'd be -1 if the function were tan but it is -0.5, so how can you change tan...
im not sure
oh wait
i figured it out
quartz beacon
The graph of the function ƒ : R → R is the line through the points (-2, 0) and (0, 1). Furthermore, the function g is given with g : R → R : x → g(x) = (ƒ(x))^3. Determine (g(8))^(-1).
vocal sleetBOT
The graph of the function ƒ : R → R is the line through the points (-2, 0) and (...
twin meteorBOT
quartz beacon
so id have 1/125 but answer says it should be 2
vocal sleetBOT
@quartz beacon Has your question been resolved?
solemn marten
why are you differentiating?
it's given that f(x) is a linear function
we can find the slope, which is 1/2
oh well oyou found f(x) anyway
solemn marten
did you do
g(x) = (x/2 + 1)^3 and then g(8) = (8/2+1)^3?
that's not what the question is asking
the question is asking g^-1(8)
not g(8)
quartz beacon
why are you differentiating?
that was a diff exercise
solemn marten
that was a diff exercise
my bad
quartz beacon
no worries
quartz beacon
the question is asking g^-1(8)
yeah i know but it should be 1/125 then
for what i did
solemn marten
yeah i know but it should be 1/125 then
no no
g^-1(x) != 1/g(x)
g^-1(x) denotes the inverse function
quartz beacon
okay how do i do that
solemn marten
well a property of inverse functions is
g(g^-1(x)) = g^-1(g(x)) = x
here we have g(x) = (f(x))^3
so we apply g^-1(x) on both sides
g^-1(g(x)) = g^-1(f(x)^3)
but like i said before, g^-1(g(x)) = x
so x = g^-1(f(x)^3)
you got it from here?
you should study inverse functions, it'll be helpful
solemn marten
damn
quartz beacon
but how do i know if theyre asking the inverse or 1/..
because doesnt that -1 mean the same
solemn marten
if they're asking 1/g(x) they'll probably put brackets around g(x) with the -1 exponent
like (g(x))^-1
if they're asking for the inverse function, they'll keep it right in front of g
like g^-1(x)
solemn marten
yeah basically
quartz beacon
.close
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peak raft
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peak raft
hi i think i understand what to do here but i'm struggling with the algebra
if we're finding the limit from first principles doesn't that mean that there is an h in the bottom? if so, why is there no h in the equation they showed in part a?
i understand that the numerator is just the difference of two squares and they didn't ask for the denominator but still
tidal dock
this is wrong, the x should go to a
tidal dock
if we're finding the limit from first principles doesn't that mean that there is...
not necessarily
there are two definitions
one where h (distance between the two x coordinates) goes to zero
and the other where the whole x coordinate approaches the other, i.e. x->a
they really are saying the same thing, in fact if you plug in a=x+h you'll get a more familiar form
tidal dock
it shouldn't
peak raft
sorry typo
tidal dock
why do you think the x should go to a tho
so that x goes closer and closer to a, and the distance between them also gets smaller
tidal dock
it's okay
peak raft
let me have a second to think about it myself and i'll ask again if i need help
but for now i close the channel
vocal sleetBOT
@peak raft Has your question been resolved?
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@peak raft Has your question been resolved?
oak magnet
what's f(a)
they probably want you to replace a with 3 therefore f(a) is f(3)
slim crest
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slim crest
ok so here in the 2nd condition I do not understand why it has to be less OR EQUAL
like why not only less than ..
if it is open interval on the top
loud walrus
!original
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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.
slim crest
determine all values of the parameter m, m belongs to R, for which the set of solutions of inequalities: .... is in the range of ...
the blanks are shown on my ss
ok so here in the 2nd condition I do not understand why it has to be less OR EQUAL
like why not only less than ..
if it is open interval on the top
<@&286206848099549185>
vocal sleetBOT
@slim crest Has your question been resolved?
strong ridge
What’s going on
slim crest
slim crest
determine all values of the parameter m, m belongs to R, for which the set of so...
.
translation
slim crest
seperate
strong ridge
Probably because it’s a different equation
So it’s setting up different initial conditions
To solve
slim crest
but it is the zero of a function
slim crest
but the condition is that the zero of the function must be lower OR EQUAL to 1
slim crest
The or equal to just makes you need to check the endpoints
but why do I need it
why not only less
than 1
strong ridge
It’s more practice so that you can work with the equal to sign
slim crest
;/
strong ridge
It’s for school right?
slim crest
yes but I doubt it is for more practise
I have been having the equations for 2 years now or so
- the conditions aren't given in the exercise you need to devise them on your own, but while checking I found out I have a mistake and it should be "also equal to 1", so that's why
strong ridge
+ the conditions aren't given in the exercise you need to devise them on your ow...
It’s not gonna be equal to one because interval is open
Might be practice so that you learn to work with endpoints?
slim crest
in answers it is m belongs to <-2;+infinity)
strong ridge
I wouldn’t worry about it tho
slim crest
every answer to this type of exercise is like this
strong ridge
in answers it is m belongs to <-2;+infinity)
If that is solution then it would be like [-2,1)
strong ridge
in answers it is m belongs to <-2;+infinity)
Is that interval notation?
slim crest
uh ye
slim crest
but I was talking about m solely
strong ridge
Ohhhhhh
slim crest
because one set must contain in another, and for the (-infinity; 1) the m must be <-2;+infinity)
slim crest
but I still can't figure out why it is closed interval ;c
slim crest
I thought you meant m is within both intervals
uhhh noo
slim crest
What do you want to tell me with this
strong ridge
The open interval bracket probably doesn’t matter
slim crest
How
it really does)::
because in some answers it is open and in some it is closed
so there must be a difference
it derives from this
but still dunno why it is also equal
vocal sleetBOT
@slim crest Has your question been resolved?
vocal sleetBOT
@slim crest Has your question been resolved?
vocal sleetBOT
@slim crest Has your question been resolved?
weary crow
How is this incorrect?
vocal sleetBOT
How is this incorrect?
heavy yoke
you set z = s and w = t but that isn't reflected in your solution
vocal sleetBOT
@weary crow Has your question been resolved?
weary crow
x3 is not equal to 0 always
I thought you set z and w as parameters as the bottom two rows are 0?
jagged cargo
Given two positive real number $x, y$ such that [
\log_2\frac{x^2+2x+2}{y^2-y+1} + 2x^2-y^2+4x +y+4=0
]
Calculate $6y-x$ as $-3x^2+y^2+2x-y+1$ reaches its maximum
vocal sleetBOT
Given two positive real number $x, y$ such that \[
\log_2\frac{x^2+2x+2}{y^2-y+1...
twin meteorBOT
jagged cargo
any idea? lagrange'ing it is pretty ugly
perhaps theres a cleaner way to approach this
brisk moss
vocal sleetBOT
@jagged cargo Has your question been resolved?
jagged cargo
hmmmge
maybe it could be written as a function?
f(a) = ...
ok big improvement
arranging the whole thing yields me this
both side is under the function $\log_2x+x$
twin meteorBOT
jagged cargo
it can be easily shown that the function is monotonically increasing
therefore
$2x^2+4x+4=y^2-y+1$
twin meteorBOT
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shell pagoda
Can someboddy help me?
vocal sleetBOT
Can someboddy help me?
vocal sleetBOT
@shell pagoda Has your question been resolved?
vocal sleetBOT
@shell pagoda Has your question been resolved?
vocal sleetBOT
@shell pagoda Has your question been resolved?
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@shell pagoda Has your question been resolved?
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@shell pagoda Has your question been resolved?
shell pagoda
I dont know what that means lol
quasi cosmos
hum
keep using letters, don't use numbers before getting an expression that you are ok with
i don't understand your question tho
fading willow
I need help doing that, been trying for an hour
he's just saying starting with inside out
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shell pagoda
yea bruh I got it sorted, you all could of told me how to carry out the multiplication 😭
.close
livid helm
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livid helm
how is part b done?
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night folio
I am so lost on this application problem my prof gave us for pre-calc. Trig has kicked my ass in general, but this problem is so far removed from anything we have done and I can't figure out how to even begin. I have read and reread through the text book. I have reviewed slides from class. I have searched YouTube and google far and wide. I have consulted with ChatGPT. I have asked my instructor for more help. I am honestly running out of options which brings here. I have pasted the actual problem for full context, but I am not necessarily just looking for an answer. I am not trying to cheat, I just really want to understand how to approach the problem.
I understand the sum and difference formula, but I don't understand how it is supposed to be applied in this scenario. I am also just very confused by the phrasing and framing of the question in general and have no idea how to connect it to what we learned in class.
I am really just at my wits end with this problem. I have spent 6+ hours on it and haven't made any progress and my prof isn't very helpful. He doesn't seem to understand the concept of a student simply not understanding something which is terribly frustrating.
vocal sleetBOT
I am so lost on this application problem my prof gave us for pre-calc. Trig has ...
mellow oyster
the whole concept here is about how pure sine and cosine waves representing sounds of a particular frequency and intensity can superimpose to make more complicated sound waves
so first to find the second tones waveform
you know the waveform will look like y = a*sin(b(t-c)) as per the formula provided
they also give you the amplitude of the second waveform, so there’s a
night folio
So from my understanding I wrote it like this, but I think I may be off
tone 1: -38cos(880πt)
tone 2: 38√3sin(880πt)
mellow oyster
b is related to the period, period is 2π/b. Observe that your second waveform has the same period as the first waveform
oh great
yeah that’s exactly what i was leading you towards
night folio
I just dont know how I am supposed to apply that to a sum and difference formula...
The example my prof gave looks like this, but this looks way different from that problem
mellow oyster
yeah so that’s what you’re doing here, maybe using the sine difference formula
so sin(a - b) = sin(a)cos(b) - cos(a)sin(b)
adding your two waveforms together, you get 38√3 sin(880πt) - 38cos(880πt)
now factor out a 38
and you get 38(√3 sin(880πt) - cos(880πt))
now look at the stuff inside the parentheses
notice that if you look at the sine difference formula, sin(a) matches with sin(880πt) and cos(a) matches with cos(880πt)
that means that you can let a = 880πt
now you need to find b
if you again compare the formulas, you see that cos(b) matches with √3 and sin(b) matches with 1
hmm that’s weird, there’s no b satisfying that
ohh
yeah okay you have 38(√3 sin(880πt) - cos(880πt)) = 76(√3/2 sin(880πt) - 1/2cos(880πt))
now with the same reasoning, let a = 880πt, then from the formula, cos(b) = √3/2 and sin(b) = -1/2
from the unit circle, this gives b = -π/6
night folio
Hmm yeah I wasn't sure how to handle apply that formula in this situation. I though we normally use that like to find 2 smaller simplier angles on a unit circle that add up to the bigger one. But I don't see that being the case here.
mellow oyster
which means the expression inside the parentheses is equivalent to sin(a-b) = sin(880πt + π/6)
and your overall waveform is 76sin(880πt + π/6)
did what i did make sense?
then part 3 is just rewriting this to fit their general form y = asin(b(t-c)), and part 4 is interpreting a and c
night folio
Hmm. Ok i need a minute to think through it
For context here is what I had done originally
sorry, got pulled away. Thank you so much for the help. I am just looking through it trying to understand it
how did you get 76(√3/2 sin(880πt) - 1/2cos(880πt))?
@mellow oyster How did you turn 38(√3 sin(880πt) - cos(880πt)) into 76(√3/2 sin(880πt) - 1/2cos(880πt))?
I mean it makes sense that if you multiple the outside by 2 you can divide the inside by 2 I guess, but what is the logic ?
mellow oyster
my logic was that i expected the √3 and -1 to be outputs of the cosine function, but that doesn’t really make much sense because of the range of cosine
like i guess i just recognized that if i divided by 2 i’d get recognizable cosine outputs
specifically i knew that i’d actually be able to find b unlike before i divided by 2
night folio
Ok yeah I guess that makes sense. When I tried to do it without that it did make it not make sense it would have basically cancelled itself out
Still very confused though lol
So with sin(a-b) = sinacosa-cosbsinb
I end up with 76(√3/2 sin(880πt) * 1/2cos(880πt)) - (1/2cos(880πt) * √3/2 sin(880πt))?
vocal sleetBOT
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desert glen
Why is the second derivative not postitive here
vocal sleetBOT
Why is the second derivative not postitive here
weary inlet
Why is the second derivative not postitive here
its concave down
desert glen
i just drew the purple tangent line, and that looks pretty positive to me
vast shale
i just drew the purple tangent line, and that looks pretty positive to me
That's tangent line
desert glen
its concave down
oh alr
weary inlet
concave down means second derivative is less than 0
vast shale
Does the slope of slope of Tangent line decrease
desert glen
That's tangent line
and isnt the slope of a tangent line the slope of function at that specific time which is basically the derivative at tha time?
desert glen
Does the slope of slope of Tangent line decrease
it doesnt seem like it
desert glen
concave down means second derivative is less than 0
yes
but how would i figure out whether the second derivative is increasing/decreasing graphically tho
weary inlet
from min to max of function its decreasing
from max to min of function its increasing
since max is concave down so its negative
and min is concave up so its positive
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violet compass
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strange crater
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serene terrace
we know that all three terms on the left hand side must add up to be zero
if you observe, you will reallize thats only possible when x is 1,y is 3 and z is 4
hence x + y + z should be 8
vocal sleetBOT
hence x + y + z should be 8
As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.
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spring sigil
Not a homework problem but I'm trying to find the Laplace transform of sin(t) via the definition and got
L(sin(t)) = Im[ (s+i)/(s^2 +1) ] = Im[ s/(s^2 +1) +i 1/(s^2 +1)] but what I'm not sure about is I saw on Wikipedia s is actually a complex number so I'm not sure what to do.
vocal sleetBOT
Not a homework problem but I'm trying to find the Laplace transform of sin(t) vi...
@spring sigil Has your question been resolved?
spring sigil
<@&286206848099549185>
vocal sleetBOT
@spring sigil Has your question been resolved?
vocal sleetBOT
@spring sigil Has your question been resolved?
warm path
Ok. Im(z) means imaginary part of the complex number z, I hope you get that
and the imaginary part of that is basically what after $i$ (i.e., coefficient of $i$)
twin meteorBOT
warm path
so you got it already.
If you don't understand it. Suppose I have a complex number $z = a+bi$, then $\text{Im}(z) = b$.
twin meteorBOT
vocal sleetBOT
@spring sigil Has your question been resolved?
spring sigil
If you don't understand it. Suppose I have a complex number $z = a+bi$, then $\t...
I do understand that but isn't s a complex number or am I mistaken something
spring sigil
not really.
Why does it say this then
warm path
it is often treated as real number when we evaluate the integral. But if we want to do it in a more rigorous way,
$\mathcal{L}{\sin(t)} = \mathcal{L}\left{\frac{e^{it} - e^{-it}}{2i}\right} = \frac{1}{2i}\left(\frac{2i}{s^2+1}\right)$
twin meteorBOT
warm path
Here, you still can use definition to evaluate the laplace transform of exponential function
In that case, $s$ is treated as complex.
twin meteorBOT
warm path
In your case, $s$ is treated as real in order to use the notion of imaginary part for the part that involved with $\sin(t)$ only.
twin meteorBOT
warm path
but yes, $s$ is complex. it is just that we treated $s$ as real for the computation (when it is equivalent)
twin meteorBOT
vocal sleetBOT
@spring sigil Has your question been resolved?
spring sigil
but yes, $s$ is complex. it is just that we treated $s$ as real for the computat...
I guess I'll have to take your word for it, if you take the real part of it you get s/s^2 +1 which is L(cost) which makes sense
warm path
like, in stats, that is treated as real (not all the time, but most of the time).
albeit, we use two-sided laplace transform in stat more often
in control, it is more useful to treat that as complex
spring sigil
like, in stats, that is treated as real (not all the time, but most of the time)...
Is the reason it can be treated as a real number because it works nicely or it's like a standard convention
warm path
in stats, that is called moment generating function
and most of the random variables are real
so... expectation of the exponential function of the random variable should be real
spring sigil
.close
vocal sleetBOT
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amber knoll
finding simplest form of the quotient, dont know where to go from here
vocal sleetBOT
amber knoll
.reopen
vocal sleetBOT
✅
amber knoll
still lost
lone tartan
you can convert 80 root 4 to (80)^1/4
because any root is essentially just the number in the root raised to the reciprocal of the root
for example the square root of 3 is just 3 raised to the 1/2 power
amber knoll
so would 2 remain in the 4th root?
loud walrus
Yes
loud walrus
Subtract -2 in both sides yes
amber knoll
adn then square root both sides, correct?
loud walrus
No need to do that
I want you to do in a different way
First, tell me the equation after subtraction
amber knoll
square root of x-6 = 5
amber knoll
yes
loud walrus
To know if your solution is correct use it in your original equation
amber knoll
so what do i do if it is not a perfect square
loud walrus
Give me an example
robust python
Do you agree that 5 and sqrt25 is the same?
they only works cause 5^2=25
at which point
just square both sides?
amber knoll
loud walrus
they only works cause 5^2=25
at which point
just square both sides?
I am giving alternative ways, this is to improve vision
amber knoll
becasue you end up with square root of 2x+8 = 2
loud walrus
Ok so now
robust python
I am giving alternative ways, this is to improve vision
its effectively the same exact method
loud walrus
In that example we are gonna try the same
Add 6 to both sides
And tell me what you got
loud walrus
its effectively the same exact method
What is your point
robust python
"What number has a square root of 5?" is the same as "What is the inverse square root of 5?" which is p much the same as "What is the square of 5?"
which ends up doing the same exact thing as squaring both sides
youre overcomplicating this
loud walrus
But that is not true
robust python
But that is not true
what isnt
amber knoll
And tell me what you got
square root 2x+8 = 2
loud walrus
This is for slim channel
robust python
Only for porsitive numbers
okay, so add 'only for positive numbers' and move on with it?
amber knoll
i put it in
robust python
You forgot the root
its right there
robust python
square root 2x+8 = 2
,,\sqrt{2x+8}=2
twin meteorBOT
ᚸᚸᚸ᛫ᛈᚢᛋᛠᛞᛖᛋᛏᚱᚩᛁᚱ᛫ᚸᚸᚸ
robust python
that, right?
loud walrus
Oh u wrote
I didnt read sorry
Ok so first solve it yourself and tell if you have any problems
amber knoll
x=-2
loud walrus
How did you do?
loud walrus
Perfect
Now asking your question
We can do the same like we did before here
Sqrt(2x+8)=2
Sqrt(2x+8)=sqrt(4)
Sqrt(2x+8)=sqrt(2(-2)+8)
Can you see that x is -2 here too?
loud walrus
Nice, just one thing you have ti have in mind before using your method of quarr
Is, be careful when there is a negative number
Because if u square it, it will become positive
And u will lose that information forever
loud walrus
that makes perfect sense but why do that with no perfect squares
Just for fun
amber knoll
okay, so just like a short cut
loud walrus
You can solve things usually in more than one way, its good that u explore all the ways
So you have always tools in case y forget one of the others
amber knoll
cool, thanks for the trick
now
when writing a an equation in logramithic form how do you put a fraction in as the base
so
for example
,,124^4/3=625
not that
124^4/3=625
just that
becasue that 4/3 is the base but can i wtite it as a fraction
amber knoll
yes
amber knoll
125^(4/3)=625
loud walrus
Oh 125 yes
amber knoll
typo sorry
loud walrus
Ok so what is the question
amber knoll
write that equation in log form
loud walrus
Well, not sure if this is what u mean but you could take log in base 5 to both sides of the equation
Log_5(125^(4/3))=log_5(625)
amber knoll
quite simply just write the equation as a log
\so
2^5 = 32
is log base 2 32=5
right?
amber knoll
ohhh
yeah
solving using natural log what would be the first step?
missed in my notes
??
please help
wide summit
hmmm
amber knoll
4x+8ln8=15?
signal saddle
How to get help
amber knoll
4x + 8 ln 8 = ln 15
wide summit
4x + 8 ln 8 = ln 15
sumtin like that
wide summit
wait give me a sec
wide summit
so im solving for x
ln = log base e
so 8e would be ln 8e or log e 8e
since (4x + 8) ln 8e = ln 15 , you divide the equation with ln 8e
4x + 8 = 0.87939...
amber knoll
arent e and ln the same
wide summit
arent e and ln the same
no
amber knoll
whats the difference?
wide summit
so am i not solving for x
you are solving for x
e has a value
which is 2.7182... , you can find it on your calculator
amber knoll
so i have 4x+8 = .8794
wide summit
so i have 4x+8 = .8794
yea
wide summit
the answer is -1.78 ...
wide summit
subtract 8 and divide by 4
correct !
wide summit
np
amber knoll
so when solving this is it the same concept?
twin meteorBOT
slim shadey
amber knoll
10
amber knoll
how did you get that?
wide summit
how did you get that?
when there is log , 3 is a power
log 10 ^3
im not good at explaining sorry
wide summit
after that you can cancel both logs
amber knoll
and be left with a normal equation?
wide summit
leaving you with (4x + 10) = 10^3
wide summit
and be left with a normal equation?
yea
amber knoll
so then x= 247.5?
wide summit
so then x= 247.5?
mhm
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fresh tide
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fresh tide
how did they get x+1 in here?
and where did the 5x+-3 even come from
this is a integration by partial fractions problem by the way
inner osprey
how did they get x+1 in here?
looks like they performed polynomial division
fresh tide
oh u mean long division?
inner osprey
yeah
fresh tide
Im terrible at that
inner osprey
that's normally the first or second step in integrating a rational function with higher (or just as high) degree numerator than denominator
fresh tide
ah I see
so if the numerator is higher( or just as high) than denominator do we just assume its long division
@inner osprey have I done this right?
strong grove
no?
fresh tide
yea I thought so
strong grove
i dont really know how id explain , but let me know if anything seems difficult
ill try to explain
next step
and then we convert it into fraction and get
$$(x+1) + \frac{5x-3}{x^2 -x-6}$$
twin meteorBOT
JustToPro
strong grove
Q + R/D
where Q is the quotient , R is reminder and D is divisor
strong grove
hope u understand
fresh tide
Yea it made a lot of sense
I also watched a quick example
it helped too
but thanks
.close
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whole gate
HI
sterile radish
Are you getting the other terms rather than x^2
whole gate
is it a right thought trying to make it match the blue equation?
desert hornet
should be O(x^3)
vocal sleetBOT
@whole gate Has your question been resolved?
vocal sleetBOT
@whole gate Has your question been resolved?
kind light
???
sharp lynx
Cheese is just a loaf of milk
hallow plover
Cheese is just a loaf of milk
Beer is liquid bread
kind light
a table is just a big chair thats not meant for sitting
versed bane
if you let a shopkeeper get eaten by a mantrap, the mantrap will usually let the shopkeeper go, but if it dies with the shopkeeper inside, the shopkeeper no longer escapes
dark kiln
a ligma is just non lethal aneurysm
hallow plover
a ligma is just non lethal aneurysm
Who's Steve Jobs?
kind light
any toilet has a possibility of a gen alpha kid coming out of it and singing skibidi toilet
vast shale
Who's Steve Jobs?
Apal see ee oo
dark kiln
Who's Steve Jobs?
as an AI I'm not aware of who's Steve Jobs because people are complicated
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lethal heart
I have to study the convergence of this sequence of functions on the interval $I=(0,1)$ $f_{k}(x)=\frac{k}{(1+kx)^2}$ I understand that punctual convergence exists, it is actually the sequence of functions tends to $f(x)=0$ .But now it says that it does not converge uniformly because the function is balanced.I don't know what this last statement means
vocal sleetBOT
I have to study the convergence of this sequence of functions on the interval $I...
twin meteorBOT
sofficino
fiery scroll
Açı ortay meaning Angle
lethal heart
?
lethal heart
this is my help for now , I don't know what you are writing regarding my question
lethal heart
Why are you sending me these things
If you need to ask for help, you should write to a help room that is not already occupied
versed bane
not sure what they mean by balanced. do you just want a proof that f_n doesn’t converge uniformly?
like you’re right that it converges pointwise to 0
lethal heart
On the PDF it says that it does not converge uniformly because they are equilimited
What does that mean?
A family of functions F on an open set A ⊂ C is said to be compact equilimited (or even locally equilimited) if for every compact K ∈ A there exists a constant M such that for every f ∈ F and for for every x ∈ K we have |f(x)| ≤ M.
In theory it's this thing but I don't know what it means
versed bane
i have no clue lol can you send the pdf, it would be way easier to do this with context
lethal heart
That would be my exercise above
It is written that fk is not equibounded in (0,1) therefore it does not converge uniformly
versed bane
yeah that is standard terminology
so what we require for uniform convergence is that the supremum of the function go to 0 (actually the supremum of the difference of the function and the thing they converge pontwise to, but we’re dealing with the 0 function so we’re good)
twin meteorBOT
sofficino
lethal heart
Because it is always 0 when all x varies
versed bane
Because it is always 0 when all x varies
what is always 0
versed bane
yeah the pointwise limit. so now we can try to evaluate this expression more and see that we can’t
twin meteorBOT
sofficino
lethal heart
Ok
versed bane
and then we can see that $$\sup_{x\in I} f_k(x) \geq k$$
twin meteorBOT
smay
versed bane
so the supremums can’t converge to 0
lethal heart
But how did you find the sup
lethal heart
But to find the sup in general, do we fix an x and make k tend to infinity?
No
Wait
versed bane
well, it depends on what we are considering. we are allowed to fix k and let x vary, or we are allowed to fix x and let k vary, or whatever else we want
lethal heart
The limit of k to infinity applies to the largest difference
lethal heart
well, it depends on what we are considering. we are allowed to fix k and let x v...
Its indifferent?
versed bane
Its indifferent?
usually it’ll be clear from context. you have to specify what set you’re taking the supremum of (remember that the supremum is about subsets of R)
so what i really was saying was $$\sup {f_k(x) \mid x\in I} \geq k$$
twin meteorBOT
smay
versed bane
but some people will shorten/change the notation to what i said earlier
versed bane
so what i really was saying was $$\sup \{f_k(x) \mid x\in I\} \geq k$$
we get one statement like this for each k
but we wanted, when considering the uniform convergence of the $f_i$, to hope that the sequence $$\lim_{k\to\infty}\sup{f_k(x)\mid x\in I}$$ converged to $0$. but the statement earlier means that that can’t happen
twin meteorBOT
smay
lethal heart
Practically the sup tends to infinity
versed bane
yep
lethal heart
Because if I fixed k=9000 and x=1/k, squaring x would give k=9000 which does not tend to 0
versed bane
Because if I fixed k=9000 and x=1/k, squaring x would give k=9000 which does not...
it would give f_k(1/9000) = 9000, that was the point, but yeah since we got something big there we suspect the convergence to not be uniform
lethal heart
So this conclusion necessarily meant that x was equal to something that depended on k
Because otherwise if I thought of x as a constant I found that the upper tended to 0
versed bane
yeah we had to consider stuff that depended on k, and yes, the second thing is because f_k actually does converge pointwise to 0
lethal heart
So what does the concept of equilimitability refer to?
versed bane
same as uniform boundedness pretty much(PROBABLY). if i have a family of functions F (could be a sequence), then it’s equilimited if the supremums are all bounded by the SAME constant M
in particular, if a sequence of functions converges uniformly, it has to be equilimited
lethal heart
And what would be the fixed constant here?
versed bane
And what would be the fixed constant here?
we couldn’t find one because the sequence doesn’t converge uniformly
unless it literally said “the sequence IS equilimited”
lethal heart
same as uniform boundedness pretty much(PROBABLY). if i have a family of functio...
Here i mean
versed bane
which would be weird
versed bane
Here i mean
oh well we can do an example
and we can see what M would be for that particular example
lethal heart
Ok
versed bane
so the family of functions f_k(x) = 1/k, constant functions
lethal heart
Ok
versed bane
that is a sequence of functions that is uniformly bounded/equilimited, since the supremums are all bounded by 1
lethal heart
It converges both punctually and uniformly
versed bane
so M would be 1 in this case
https://en.m.wikipedia.org/wiki/Uniform_boundedness here are a few examples as well
i get the impression that equilimited is the same thing as uniform boundedness