can i just do this for like all of them
#help-33
runic temple
those three equations should be all you need
runic temple
like for that for example
do you need help with this one?
humble river
So
E can be
oh i did that one
bu ti just used the 2 formulas seprately
not solve by substition for them
So E can be 4.87x10^5
humble river
o
runic temple
So E can be 4.87x10^5
that energy seems really high
humble river
o
i converted it to J
Kj to J
I thought E in these formulas are usually representing the E of 1 single photon
J is TOtal
humble river
kj/ml photon
oH
Mol
we want to convert ml to photon atom then?
mol*********
then get a new value
runic temple
yes, convert mol of photons to numbers of photons
humble river
Ok perfect
runic temple
that way you can get the amount of energy per photon
humble river
sry i went afk
so the 4.87x10^5
get divided by avegado
and we got
big number
like 8.09x10^-19 if ihad to round it
runic temple
,calc 4.87/6.022
elfin berryBOT
Result:
0.80870142809698
i believe it
humble river
now
i have 8.08x10^-19 = those 2 big htimes c number divided by our uknown
and what i did is
8.08 crossed with the unknown
so we divide
1.986478E with 8.08x10^-7
= 2.456x10^-7
meters
then convert it
so 246nm
marsh citrusBOT
@humble river Has your question been resolved?
dusty shoal
I don't understand why A is true
marsh citrusBOT
I don't understand why A is true
dusty shoal
because
this has a trivial solution
but this has no solution
nvm im bad at english
.close
marsh citrusBOT
cold pecan
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Available help channel!
humble river
do i assume that its mj/mol?
marsh citrusBOT
do i assume that its mj/mol?
fervent rampart
no, it's just mJ
humble river
oh ok
fervent rampart
you can calculate the energy per photon separately
humble river
so
E=hc/wavelength
wait no
i already have all thos
lol
c=wavelength times v
get v
put it in e=hv
and thats the answer
fervent rampart
E=hc/wavelength
you can also use this form directly
fervent rampart
no
humble river
o
fervent rampart
they give you the total energy of every photon, E = hc/λ gives the energy per photon
marsh citrusBOT
@humble river Has your question been resolved?
marsh citrusBOT
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marsh hazel
finding dy/dx via implicit differentiation, did I do this right?
marsh citrusBOT
finding `dy/dx` via implicit differentiation, did I do this right?
wary kite
$\frac{dy}{dx} = -\frac{2xy + y^2 - 2}{x^2 + 2xy}$
elfin berryBOT
knief
wary kite
consider $F(x,y) = x^2y + xy^2 - 2x$ then $\frac{\partial F}{\partial x} + \frac{\partial F}{\partial y} \frac{dy}{dx} = 0$
elfin berryBOT
knief
jagged relic
If you can choose any value for n and p, 4p-n could also be any value
Do you understand why?
Does there exist an m greater than any value?
Really? What would it be?
That's not the same order
That would be "for all x there exists y such that y > x"
Yes
Could also be "exists m, exists p"
The point is that if you already set the values for m and n, then you can choose a value for p
tawdry isle
confused on this..
marsh citrusBOT
confused on this..
ember prism
So you can look at each point individually
Look at A for example, it's (0,1)
And it changes to (1,0)
wary kite
Look at A for example, it's (0,1)
you mean B
tawdry isle
So you can look at each point individually
oo okok
ember prism
But yea, if you look at it, u can see that it's rotating 90 degrees, as it's switching to a different axis
And it's going from up to right, so 90 degrees clockwise
If u can't see it, then I think u can use the formula
(x,y) after a 90 degrees clockwise rotation becomes (y,-x)
Which is true for this point and for all 3 points but u don't rlly have to verify them
tawdry isle
i think i get it
tawdry isle
im going over one of my quizzes that i didnt pass so i might need help actually
i think this might be right but im not rlly sure
ember prism
So I think this is a silly mistake lol, u might have confused A and A prime
wary kite
yup
tawdry isle
oh LMFAO
ember prism
Ye, u got the right idea tho
tawdry isle
im kind of 50/50 on this one aswell but i think its correct ?
ember prism
That's right
tawdry isle
what about this?? the questions r like weirdly similar so
tawdry isle
this one is kind of hard cause theres not alot to go off of but i think its just a me problem
tawdry isle
some of the questions r kinda easier then others tbh? like i rlly struggle on things like rotations cause i struggle with math in general
tawdry isle
confused on this..
for this i think its 270 around the origin but i could be wrong
ember prism
some of the questions r kinda easier then others tbh? like i rlly struggle on th...
Everyone does at some point
ember prism
for this i think its 270 around the origin but i could be wrong
That's right
Just specify counter clockwise
ember prism
some of the questions r kinda easier then others tbh? like i rlly struggle on th...
Have you learned the formulas? For rotations?
tawdry isle
Have you learned the formulas? For rotations?
somewhat but i still get confused about it sometimes
ember prism
If it helps, use ez points to verify if you got the formula right
Like take (0,1)
And try the formula for say 90 counter clockwise
tawdry isle
ooo okay
tawdry isle
that might help aswell
ember prism
Ye, and if that takes too much time, just try it best to memorize them
tawdry isle
im not rlly sure about this either since theres no like lettering to determine which is the original shape
tawdry isle
yeah thats what i assume most of the time
ember prism
Yea, but u have wrong answers ticked
tawdry isle
oh LMFAO
ember prism
Ye, it should just be reflection in the y axis I believe
Since translating to the left gets you farther away from the orgin
And reflection axis gets u to the 4th quadrant
ember prism
Since translating to the left gets you farther away from the orgin
Nvm
That's also r
I assumed the lighter one is original lol
tawdry isle
LOL
i think this might be correct aswell
weridly confusing though since im split between a and b for this one
ember prism
i think this might be correct aswell
U might want to try again
tawdry isle
i think it might be c actually ??
ember prism
U got it
dapper dagger
tawdry isle
shapes in general confuse me alot
ember prism
Lol no worries, just match each shape to the original one, and see if it looks like they're split in the middle
tawdry isle
this one aswell .. like i get confused abt line symmetry cause its overwhelming to me
ember prism
So it's not D
Bcs try adding 2 lines, that goes through the middle
From the 2 vertices
That's 2 lines of symmetry
tawdry isle
mhm
ember prism
Try again, see if you can find the answer
tawdry isle
im assuming it could be c but im mixed on a aswell
ember prism
im not rlly sure on this one either
U almost got it
It is a reflection
But u could also go 10 units down
tawdry isle
i didnt rlly think of that LMFAO
im prtty sure this one is correct aswell but i cant rlly be sure yk
ember prism
Ye lol, happens so many times to me as well, u have to look at all the options
ember prism
im prtty sure this one is correct aswell but i cant rlly be sure yk
U can be sure
Bcs that absolutely right
tawdry isle
i think this might be right but im not rlly sure
im kind of confused on this one aswell
im pretty sure its just me mixing it up tho
ember prism
Ye, if you look at the darker triangle
And go 3 units down from there
I get to the lighter triangle
ember prism
Np
tawdry isle
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marsh citrusBOT
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brazen hull
I need help!
The problem says I have to find the values of the 6 trig. functions with the given information.
I have NO CLUE where to begin in solving this, so I apologize in advance.
marsh citrusBOT
I need help!
The problem says I have to find the values of the 6 trig. functions...
sick walrus
It looks like you know the relationship between cot and tan
All of the functions are related similarly
Cotangent is cos/sin
brazen hull
:o
Okay I think something just clicked as I read that.
Thank you so much!
Pretty sure I can figure it out from here, I just have to look into the relationships of the functions.
sick walrus
Great
brazen hull
.close
wispy ledge
how do i put -3-3i in exponential form?
marsh citrusBOT
how do i put -3-3i in exponential form?
desert dirge
z=-3-3i
z=|z|e^{i(arg(z))}
desert dirge
lengths cant be negative
r is essentially the distance of the number from 0 in the complex plane
wispy ledge
r is essentially the distance of the number from 0 in the complex plane
also is r called the modulus?
desert dirge
it is yeah
hollow spear
marsh citrusBOT
marsh citrusBOT
@hollow spear Has your question been resolved?
hollow spear
<@&286206848099549185>
marsh citrusBOT
@hollow spear Has your question been resolved?
hollow spear
<@&286206848099549185>
tender wasp
ok
tender wasp
Right
marsh citrusBOT
@hollow spear Has your question been resolved?
hollow spear
<@&286206848099549185>
river estuary
Do you know how to solve the integral you wrote down? So far everything looks good.
hollow spear
Do you know how to solve the integral you wrote down? So far everything looks go...
nope, thats where im kinda stuck
I thought of maybe doing by parts but that doesnt seem to help
river estuary
it does, but it's annoying. you need to do it twice and then you'll get back to the original integral
$$I = \int \sin(3x)e^{-5x} dx = -\frac13\int e^{-5x} d\cos(3x) = -\frac13\cos(3x)e^{-5x} + \frac13\int \cos(3x)d e^{-5x}$$
elfin berryBOT
ENStucky
hollow spear
when I find the derivative of sin(3t) I get 3cos(3t) and then the second time I get -9sin(3t)
river estuary
right
$$=\frac13\cos(3x)e^{-5x} - \frac59\int e^{-5x} d\sin(3x) = \frac13\cos(3x)e^{-5x} - \frac59 \sin(3x)e^{-5x} -\frac{25}{9} I$$
hollow spear
oh I guess u could just take that out of the integral?
elfin berryBOT
ENStucky
river estuary
so assuming I did my algebra right, notice that if we take the far-left and the far-right sides, we get $I= [stuff] - \frac{25}{9}I$, and so $\frac{34}{9}I = [stuff]$.
elfin berryBOT
ENStucky
river estuary
Which lets us solve for I, the integral we wanted.
river estuary
okay yeah you're on the right path but I think you're dropping constants a bunch, shouldn't that middle term be divided by 5? And also just leave -153 out of this for now; it got lost early on
hollow spear
which middle term do u mean?
river estuary
the one with cosine
oh yeah you accidentally multiplied by 5 in the right side of the table instead of dividing by 5 again
so the cosine term should be divided by 25
river estuary
it's still wrong because of the -153 issue
but after you fix that i'm not sure
working on it
hollow spear
wait why's the -153 wrong?
river estuary
the LHS* is -153( Su + 9/25 Su), not -153 Su + 9/25 Su
Yeah after that, I think our answers agree except for a sign but your sign work looks good so I trust you 🙂
oh yeah i see where i lost it
hollow spear
oh I see, thanks
one more thing, do you have sort of a general way of deciding which way to solve a second order non homogenous ode to get a particular solution?
like picking between the method of undeterminant coefficients and this method (I forgot the name)
river estuary
this is variation of parameters
and yes: use undetermined coefficients every time it is at all possible
sorry XP
"at all possible" means anything of the form [polynomial]*[exponential]*[sin/cos]
so like x^2e^(-5x)
or (x^4-2x)cos(6x)+ e^(3x)sin(x)
etc.
for the forcing term
common dick move is to do like DE = tan(x); that's a trig function but only works with variation of parameters
hollow spear
I see, so anytime there isnt anything in the form [polynomial][exponential][sin/cos] you're forced to use variation of paramters
otherwise always use undetermined coefficients
river estuary
right
yeah i'm trying to think of any overlap when VoP is easier but you're almost always going to get these integrals where you need IBP or some trig formula nonsense.
one advantage is that VoP always works, so you technically don't ever have to use undetermined coefficients. but on a timed exam that's not a good enough reason to reach for it imo
hollow spear
makes sense, thanks alot!
marsh citrusBOT
@hollow spear Has your question been resolved?
civic veldt
If I have this ineqquality, why can't I erase both x+3 by multiplying each term?
civic veldt
So that I get (x^2)-1 < 3
median kindle
One small question -3, -10, -1 arrange in order please
Perfect way to put
I am confused but this is correct way to get the inequalities
In simple why she didnt simplified eqn is not to miss any eqn during the simplification of the inequality
So she wont disturb the sign change or wont miss out term to get the excat range.
tawdry rampart
If I have this ineqquality, why can't I erase both x+3 by multiplying each term?
Cos raid then you will have to specify some things like x is not -3 and x is greater than -3
Do you understand why these conditions must be put?
However, my suggestion is that just take the RHS to LHS, then you would not have to worry about all of this
marsh citrusBOT
@civic veldt Has your question been resolved?
civic veldt
Do you understand why these conditions must be put?
Meh
cosmic raptor
marsh citrusBOT
elfin berryBOT
cosmic raptor
Is it like this?
marsh citrusBOT
@cosmic raptor Has your question been resolved?
marsh citrusBOT
@cosmic raptor Has your question been resolved?
marsh citrusBOT
@cosmic raptor Has your question been resolved?
marsh citrusBOT
@cosmic raptor Has your question been resolved?
eternal matrix
bit difficult to visualize it since you have to draw in 3d! but i guess its possible
errant condor
how do u do b
marsh citrusBOT
how do u do b
errant condor
why differentiate?
errant condor
also find the entire series from a first
yea i have
main idol
try it and see
main idol
yea but i need a reason for it 😭
work backwards from the expression
$\int (n+1)x^n = ?$
elfin berryBOT
riemann
errant condor
what?
errant condor
what do i do after i differentiate the power series?
errant condor
if two functions are equal, then so are their derivatives
they arent equal tho?
main idol
they arent equal tho?
wot
show your derivative of the power series
and show your power series
errant condor
main idol
...
those are two different functions, why are you setting them equal
or why do you think they're equal
errant condor
if two functions are equal, then so are their derivatives
this
errant condor
what does (1-x)^(-1) equal?
1 + 2x + 3x^2..
main idol
main idol
if two functions are equal, then so are their derivatives
now do this
errant condor
-2/(1-x)^2 = 1 + 2x + 3x^2
ohh right so the part on the left is the answer
on the ms it says its 1/(1-x)^2 tho
marsh citrusBOT
@errant condor Has your question been resolved?
main idol
on the ms it says its 1/(1-x)^2 tho
you differentiated (1-x)^(-1) wrong
,tex .diff rules
elfin berryBOT
riemann
main idol
use power rule and chain rule
marsh citrusBOT
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Available help channel!
wicked bramble
for this question
marsh citrusBOT
for this question
cunning fiber
Standard procedure is to let $z=a+bi$ for real numbers $a, b$ and then compare the real/imaginary parts
elfin berryBOT
Civil Service Pigeon
limber hearth
Adding the modulus of complex number, you can get it from all of this
marsh citrusBOT
@wicked bramble Has your question been resolved?
mystic gazelle
Not sure if I did number b right
marsh citrusBOT
Not sure if I did number b right
marsh citrusBOT
@mystic gazelle Has your question been resolved?
marsh citrusBOT
@mystic gazelle Has your question been resolved?
marsh citrusBOT
@mystic gazelle Has your question been resolved?
cosmic raptor
marsh citrusBOT
cosmic raptor
how come this function is not contiunous?
wary kite
consider $\lim_{x \to 0} f(x)$
elfin berryBOT
knief
marsh citrusBOT
@cosmic raptor Has your question been resolved?
fresh lynx
Have 2 other questions
1 to make sure I did it right and the other I need help on
fresh lynx
rancid geode
First one
I think good
Second one, remember that discontinuity can be removed by factoring and cancelling
@fresh lynx
fresh lynx
Ok I got it thanks
fresh lynx
Second one, remember that discontinuity can be removed by factoring and cancelli...
Wait does it have to be x-4 to cancel
Cause vertical is only the denominator
rancid geode
All of the choices except one of them will have no (x-4) in the denominator
When u finish cancelling
feral shale
can you kindly send a clear image?
fresh lynx
can you kindly send a clear image?
I think it should be B right?
feral shale
I think it should be B right?
will the equation be valued 0?
fresh lynx
will the equation be valued 0?
it only says theres avertical at x = 4
marsh citrusBOT
@fresh lynx Has your question been resolved?
fresh lynx
<@410867220718157834> the top and bottom completely cancel
so its a since theres a 4-x left?
rancid geode
Yes
If there’s a (4-x) or (x-4) left
In the denominator
There’s a vertical asymptote at x=4
cosmic raptor
marsh citrusBOT
cosmic raptor
why can't the min-max theorem apply on this one?
glass silo
What's the min-max theorem, in full?
glass silo
Is f continuous on the interval given to you?
true parcel
is it continuous on 3?
marsh citrusBOT
marsh citrusBOT
Available help channel!
quartz sun
this dilation is really confusing me i keep going off track but i can’t seem to find what im doing wrong
marsh citrusBOT
this dilation is really confusing me i keep going off track but i can’t seem to ...
quartz sun
i found the points of a b & c and counted from point A to the other points and divided it by 3 but it didn’t work out
marsh citrusBOT
@quartz sun Has your question been resolved?
white maple
i found the points of a b & c and counted from point A to the other points and d...
think of like shifting A to the origin
then just multiplying all points by 1/3
then shifting back
so yes, find all your points, if A were at the center
what did you get
quartz sun
think of like shifting A to the origin
i also cannot shift any of the points because they are printed onto the paper the most i can do is shift the label
white maple
i also cannot shift any of the points because they are printed onto the paper th...
ig i mean like imagine they are
white maple
everytime i retry i get something new
and not being mean at all but that's kinda your fault right :x
quartz sun
yes i am aware of that
white maple
i mean that bc it happens to me too
if you know what to do
you just have to sit down and do it slow, don't rush it
quartz sun
it’s like complex for me because when he the lesson was supposed to happen my teacher wasn’t here and he played some random rushed video
im going to change the label names so the a’s aren’t confusing
white maple
ye i don't mean like it's your fault you don't have the knowledge
but more just like
if you subtract numbers, there is a single answer
so if you do it a few times and get a few answers
it's just cause you're going too quick
white maple
im going to change the label names so the a’s aren’t confusing
and yeah that sounds good xd
quartz sun
mkayyy
so from what i know of what he briefly told me is i have to count from point A to point J etc
quartz sun
and from what i’ve counted i multiply by 1/3 or divide by 3
white maple
so from what i know of what he briefly told me is i have to count from point A t...
yes. or a "shortcut" is just subtract off the point A from every triangle point
white maple
and from what i’ve counted i multiply by 1/3 or divide by 3
yes
then finally shift back (add back what was subtracted off the first step)
quartz sun
down and left are subtract
up and right are add correct?
white maple
rightt
quartz sun
yes. or a "shortcut" is just subtract off the point A from every triangle point
how would i do this btw
white maple
A = (-4, -2)
quartz sun
yeah
white maple
how would i do this btw
so like take I = (-1, 7)
then I - A = (-1, 7) - (-4, -2)
= 3, 9
like you subtract x and y separately
quartz sun
the three is because of the subtracting the negative numbers right
white maple
rightt
quartz sun
same with 9
white maple
yeah perfect
white maple
so yeah, in short, add 4 to every x and add 2 to every y
(bc it's subtract -4 and subtract -2)
white maple
and you said subtract them too?
this is the first step, that was done for I
quartz sun
ohh okay
white maple
J = (-1, 7) becomes (3, 9) (sry i guess that's a J, thought it was I)
white maple
k becomes (6,3)
yes perfect
quartz sun
L (-3,3)
white maple
thats L i confused them
haha sry
quartz sun
irs fine i confuse myself too
quartz sun
it is late at night anyway
white maple
true true
quartz sun
so from the first step where do we go now
white maple
ok
so that was the "shift" - now it's as if we moved point A to the center
so with that done
now we divide verything by 3
all numbers, x and y
white maple
yep
quartz sun
J (1,3)
K (2,1)
L (-1,1)
white maple
looks goood
ok just one more step then
which is just the reverse of the first step
add back A = (-4, -2)
quartz sun
im guessing i add them now
white maple
yes perfect
quartz sun
okay
white maple
and again they're negative but yeah
white maple
yeahh sounds right to me
quartz sun
okay so
there might be a problem
it created a straight line instead of a triangle
of
one second
white maple
no i think it looks good
white maple
nice yeah perfect
quartz sun
it doesn’t matter if point A is outside of it right
white maple
as an interesting check, it's like, all corners moved toward A, in a straight line
yeah that's ok
quartz sun
okay because my teacher is kinda new and doesn’t really cover a lot of things correctly but he’s pretty lit
white maple
it actually would be bad if it switched between inside and outside here
quartz sun
how do if i may ask
so*
and this works for any dilation right ?
that involves a half or a third etc
white maple
yeah it does
right, or 2x or 3x
yeah
there also might be a quicker way to do it with drawing once you get good at it
but this way will always work with the math
white maple
so*
ah i guess for any dilation overall, it should be like zooming in or out with a camera. but it shouldn't change whether one point is in a triangle or outside
that's the idea
quartz sun
he kinda js confuses me a little yk
white maple
(it can change if some things are dilated and some others aren't - but if everything is dilated the same, then yeah)
i feel that
quartz sun
the ones that aren’t around the origin are what confuse me but i’ll def keep this in mind and if he marks it wrong it’s fine because i learned a new technique
white maple
yeah true. the whole shifting thing was to "pretend" A really was the origin
and nicee good mindset haha
quartz sun
yeah true. the whole shifting thing was to "pretend" A really was the origin
thank you he might wonder where i got this from because this is a method we haven’t used but as long as the work is done he should be grateful
im going to try the other 2 and if i get stuck i’ll probably just open another one
thank you so much for your help
white maple
yeah def 👍 and he might know the faster way you can do with just drawing/graphing, you could ask him about other ways if you want
sweet though, glad i could help
quartz sun
yeah def 👍 and he might know the faster way you can do with just drawing/graphi...
def will ask abt it sometime
.close
marsh citrusBOT
white maple
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Available help channel!
stable axle
can somebody help me out?
marsh citrusBOT
can somebody help me out?
stable axle
idk where to get started
nimble prairie
What does |x| describe?
nimble prairie
Like in general, what would you say the absolute value of x describes?
stable axle
i mean, idrk
all i know is that its a positive and negative version of a number
of a solution
a range
i think idk
i just started learning this stuff
nimble prairie
Have you gone over the intuitive notion of what it describes?
stable axle
no 😔
nimble prairie
I see, well would it make sense if I said that |x| describes the distance from x=0?
nimble prairie
So if we instead had |x + 8|, what would you say it described?
nimble prairie
Let me maybe change letter
So |b + 8|
And you know |x| described the distance from x = 0
What would x be here?
nimble prairie
Sorry that wasn’t clear of me, like comparing |b+8| and |x|, then I was trying to say that x = b+8
So |b+8| describes the distance from b+8 = 0
I.e |b+8| describes the distance from the point b=-8 on the number line
Makes sense?
stable axle
yes
nimble prairie
So if we know want to solve for b in the inequality |b+8|<5
Then you’re essentially asking
“For what b are we within a distance of 5 from the point -8?”
stable axle
what
okay i got told from the teacher that absolute value is basically how far a number is from zero
thats all i learned about them
and then we learned how to solve them
stable axle
So if we know want to solve for b in the inequality |b+8|<5
you would do |b| < (5 - 8)
not like that youd just subtract them
i mean yes like that but
you wouldnt write it like that
im just doing that to show you what i would do
idk
nimble prairie
Okay let’s maybe step back and view this more systematically
So say you were solving |x| < b, for b > 0
stable axle
im still here, just waiting
nimble prairie
Then if x >= 0, we have that 0<= x = |x| < b.
So 0<= x < b in that case.
Makes sense?
stable axle
0 <= x < b
b > x => 0
nimble prairie
I’m taking advantage of the definition of absolute value if you’ve gone over that
stable axle
b is more than x and x is more than or equal to 0
nimble prairie
0 <= x < b
b > x => 0
Yes
So this is only when x >= 0, but what if x < 0?
How can we rewrite |x| in that case?
nimble prairie
Let’s focus on rewriting |x| first
stable axle
if x < 0
nimble prairie
I’m trying to guide you towards the answer
nimble prairie
Right now we’re separately dealing with how to rewrite |x| since it’s not immediately obvious how to deal with it in our inequality.
So let me rephrase it like this instead:
if x is negative, what is |x|?
nimble prairie
What did |x| describe again?
stable axle
OH
nimble prairie
Always go back to the definition!
nimble prairie
Yes! So can that be negative?
nimble prairie
Indeed, so if x is negative, then |x| is positive
stable axle
and vice versa?
nimble prairie
If x is positive then |x| positive
More exactly |x| just makes our number positive
So |4| = 4
|-3| = 3
Right?
nimble prairie
So if x is negative then how can we rewrite |x| without the absolute value notation?
stable axle
x > 0
nimble prairie
its a whole number thats always positive?
Even a non negative real number, note that |0| = 0
stable axle
yes x is less than zero
nimble prairie
So how do we make it so it’s positive again?
nimble prairie
A hint: you can multiply by a specific number to make it positive again
nimble prairie
Oh that’s infact true, but there’s an easier one that will help us solve the inequality! Want me to spoil it?
nimble prairie
What happens if we just multiply by -1?
nimble prairie
Yes and it’s positive!
nimble prairie
So we can rewrite |x| as -x if x is negative!
nimble prairie
So to quickly recap before we solve |x| < b.
If x >= 0 then |x| < b was the same as 0<= x < b. Remember?
stable axle
why not just b > x >= 0
nimble prairie
Sure! It’s subjective
So let’s now consider the only other case, when x < 0
Then we know that |x| = -x
So 0 < -x = |x| < b
Would you agree?
stable axle
yes
nimble prairie
Nice so we have that |x| < b is the same as 0 < -x < b if x < 0
stable axle
-x > 0
nimble prairie
Also recall that we noticed that -x is positive so it must be true
nimble prairie
Nice so we have that |x| < b is the same as 0 < -x < b if x < 0
Given that 0 < -x < b, how would you “isolate” x?
stable axle
bro this is so confusing to me im sorry
ive never done this stuff at all before and its just really confusing
nimble prairie
I see, that’s fine don’t worry. So i wanted you too see that we can multiply by -1 again to get
0 > x > -b
So in summary taking all x’s into account
We have that 0<= x < b or -b < x < 0
That is |x| < b is the same as writing
-b < x < b
And this should make intuitive sense
nimble prairie
Since all the x’s in-between -b and b are within a distance of b
nimble prairie
Indeed
So with this knowledge, how would we solve the question you were asked, I.e
|b + 8| < 5?
stable axle
so
you need to do
b + 8 < 5
b + 8 > -5
and then
b + 8 < 5
b < 5 - 8
b < -3
and then
b + 8 > -5
b > -5 -8
b > -13
and then
-13 < b
-13 < b < -3
i think i did it
nimble prairie
stable axle
yippee
nimble prairie
Correct!
marsh citrusBOT
marsh citrusBOT
Available help channel!
vocal wyvern
I don't quite understand how should I tackle the 100 digit natural number that is divisible by its own digit
marsh citrusBOT
I don't quite understand how should I tackle the 100 digit natural number that i...
vocal wyvern
I'm very confused and can't do it entirely
unborn condor
when it says that the 100 digit natural number has nonzero digits, does it mean "at least one digit is not zero" or "no zeroes at all"? if it's the first case, it's extremely simple
vocal wyvern
when it says that the 100 digit natural number has nonzero digits, does it mean ...
No zeroes at all
<@&286206848099549185>
marsh citrusBOT
@vocal wyvern Has your question been resolved?
marsh citrusBOT
@vocal wyvern Has your question been resolved?
novel juniper
marsh citrusBOT
novel juniper
I was thinking of direct proof
I've gotten this far :
We want to prove that if $a,b \in \Z$, then $gcd(a,b) = gcd(a-b,b)$
let $gcd(a,b)=l; l \in \Z$.
so $a=k_1l; b=k_2l$, so, $a-b=l(k_1-k_2)$
so from this we can conclude that $l$ is a common factor of $a-b,b$
elfin berryBOT
Veni, vidi, perii is not f(wai)
dense crater
Hint:
Show (\gcd(a, b)) divides both (a - b) and (b).
Then show (\gcd(a - b, b)) divides both (a) and (b).
elfin berryBOT
Trenton
novel juniper
Hmm
Okay
Okay, so I've already shown that gcd(a,b) divides (a-b), (b)
We want to prove that if $a,b \in \Z$, then $gcd(a,b) = gcd(a-b,b)$
let $gcd(a,b)=l; l \in \Z$.
so $a=k_1l; b=k_2l$, so, $a-b=l(k_1-k_2)$
so from this we can conclude that $l$ is a common factor of $a-b,b$
Let $m$ be the gcd of (a-b,b). Then $(a-b)=q_1m; b=q_2m$. Adding them we get $a= (q_1+q_1)m$, which proves $m$ divides both of them
elfin berryBOT
Veni, vidi, perii is not f(wai)
quaint elm
marsh citrusBOT
marsh citrusBOT
Available help channel!
jaunty urchin
what do i do? i tried to get the derivative but was wrong
marsh citrusBOT
what do i do? i tried to get the derivative but was wrong
snow palm
what do i do? i tried to get the derivative but was wrong
use the product rule
jaunty urchin
i did 💀
snow palm
what did you get?
jaunty urchin
-6xsinx + 6cosx
snow palm
yeah
jaunty urchin
what
jaunty urchin
how do i get b?
snow palm
how do i get b?
put the value of x, y and m to get b
jaunty urchin
got it. thanks
snow palm
nice
jaunty urchin
.close
marsh citrusBOT
rich anchor
uh
marsh citrusBOT
Available help channel!
tropic bolt
marsh citrusBOT
tropic bolt
solving
limpid pond
multiply both sides by 4 it’ll make it easier to solve for z
tropic bolt
ok whats next
limpid pond
ok whats next
try isolating the z's in one side and the numbers in the other by either adding or subtracting
marsh citrusBOT
@tropic bolt Has your question been resolved?
tropic bolt
.close
half dagger
can i have some help with this?
marsh citrusBOT
can i have some help with this?
half dagger
I know how to find the gradient (i think) of this function, but im not quite sure what to do past that
wait, if its parellel to the z axis then that the change in x and y should be 0. so, would i just plug in x and y=0?
that feels wrong because my derivitive for z would still change. since its 268y+8z
which depends on y still
marsh citrusBOT
@half dagger Has your question been resolved?
half dagger
wait, it intersects the xy plane right? so that means if i were to do a dot product it would be 0?
half dagger
i think im going at this wrong
so i have the gradient is (2x+7, x+4y+268z, 268y+8z)
if i want it to be parallel to the z plane, then i need the change in x and y to be 0. So, the directional vector should be <0,0,1> yes? multiplying the gradient by this gives 268y+8z. Since this is supposed to be the change in z, and everything else is 0.
I'm just not sure where to go past this point
amber birch
so i have the gradient is (2x+7, x+4y+268z, 268y+8z)
not sure where you get that gradient from
the gradient is (2x + y, 4y + x + 4z, 8z + 4y)
now, that's the direction of steepest increase, so that is the normal vector to the surface
half dagger
not sure where you get that gradient from
oh wait i think i see what i did
thanks for pointing that out
amber birch
the normal vector of the tangent plane would be (0, 0, k) for some real number k
so that implies 2x + y = 0 and 4y + x + 4z = 0
half dagger
right
amber birch
yeah so the intersection of these two planes is a line actually
weird
anyway if we skip to part b
we have this equation for the tangent plane
half dagger
yeah so the intersection of these two planes is a line actually
it is?
amber birch
it is?
yeah, the intersection of two planes is a line in most cases
the exceptions are if the two planes don't intersect at all
or if they are the exact same plane
half dagger
hm
im not really sure how that would help me find the equation for the plane it intersects
amber birch
yeah I don't know either what it wants
half dagger
i mean. its just the xy plane. but i doubt that simply stating that would give a correct answer
oh wait
the equation of the plane it intersects should be z=0
amber birch
oh wait
yeah you just have to find constants (there should be two) say a, b
such that z = a and z = b are the tangent planes
great
ahhhh
yeah honestly a is trickier than b lol
half dagger
well now that i realized the plane's equation is just z=0. it makes more sense
amber birch
so that implies 2x + y = 0 and 4y + x + 4z = 0
so that implies (x, y, z) = (t, -2t, (7/4)t) if you work it out
,w t^2 + 2(-2t)^2 + 4((7/4)t)^2 + t(-2t) + 4(-2t)(7/4 * t) = 67
elfin berryBOT
half dagger
so that implies (x, y, z) = (t, -2t, (7/4)t) if you work it out
how is this what you got?
amber birch
nah I don't get that
amber birch
how is this what you got?
let x = t, then 2x + y = 0 implies y = -2t
then sub those x, y into 4y + x + 4z = 0
amber birch
then sub everything into the equation of the ellipsoid
amber birch
from this, remember that z = 7/4 * t, so those are the two z-values
so $z = \pm 2 \sqrt{67/21} \cdot \frac{7}{4}$
elfin berryBOT
higher's secret twin brother
amber birch
oh wait did I find the wrong thing
ah shit I didn't do the tangent plane is vertical sorry
the normal vector will be parallel to the x-axis
half dagger
Wait what?
elfin berryBOT
amber birch
,w (16z)^2 + 2(-2z)^2 + 4z^2 + (16z)(-2z) + 4(-2z)z = 67
elfin berryBOT
amber birch
yeah
half dagger
So that’s the plane?
amber birch
yep!
half dagger
Hm
I’ll have to go back in a bit and look through your work more
Since I didn’t see all of it
half dagger
I’ll try and work on b later too
amber birch
I’ll try and work on b later too
here's an example for question b with different numbers
half dagger
i appreciate it. If i have any questions, ill probably ask
amber birch
okok no worries!
half dagger
in case you cant help me later if i do have more questions, thank you for your help
amber birch
nwnw
marsh citrusBOT
@half dagger Has your question been resolved?
marsh citrusBOT
@half dagger Has your question been resolved?
marsh citrusBOT
@half dagger Has your question been resolved?
half dagger
x^2+2y^2+4(x^2 + 2y^2 + 4z^2 + xy + 4yz)^2+xy+4y(x^2 + 2y^2 + 4z^2 + xy + 4yz)=67.
which gives us the absolutely disgusting function of:
2x+8(x^2 + 2y^2 + 4z^2 + xy + 4yz)*(2x+y)+4y(2x+y)=0
half dagger
If I were to instead not plug in the equations, I would have 2x+8•f(x,y)•(df/dx)+y+4y•(df/dx)=0
this gives me 8•f(x,y)•(df/dx)+4y•(df/dx)=-2x-y
Then I get (df/dx)•(8•f(x,y)+4y)=-2x-y.
which gives (df/dx)=(-2x-y)/(8•f(x,y)+4y).
If I now plug in (-1,2). I get (2-2)/30.98386. Which gives me 0
if anyone could confirm that i did this correctly, it would be greatly appreciated
silk sable
If I were to instead not plug in the equations, I would have 2x+8•f(x,y)•(df/dx)...
I think you made this more complicated than it needs to be.
You're looking for this.
When you observe the ellipsoid from above and just look at the xy-plane, you will see this view.
For any point along the edge, the k component of the Normal vector will equal zero.
The plane that goes through all of these points will be your k component equal to zero.
half dagger
I think you made this more complicated than it needs to be.
this isnt how i shouldve solved part b?
silk sable
That's for solving 2a, not 2b.
silk sable
If I were to instead not plug in the equations, I would have 2x+8•f(x,y)•(df/dx)...
This looks correct.
half dagger
ok cool. i should be able to do fy without issue then
marsh citrusBOT
@half dagger Has your question been resolved?
buoyant jetty
Let $B = {(4, 1, 2), (0, 1, 0), (-3, -1, -1)}$ be a basis of $\mathbb{R}^3$.
Find all vectors of $\mathbb{R}^3$ such that their coordinates in the canonical basis are double their coordinates in the basis $B$.
marsh citrusBOT
Let $B = \{(4, 1, 2), (0, 1, 0), (-3, -1, -1)\}$ be a basis of $\mathbb{R}^3$.
...
elfin berryBOT
938c2cc0dcc05f2b68c4287040cfcf71
marsh citrusBOT
@buoyant jetty Has your question been resolved?
deep anvil
Let $B = \{(4, 1, 2), (0, 1, 0), (-3, -1, -1)\}$ be a basis of $\mathbb{R}^3$.
...
Let's try this in Español, perhaps you'll have an easier time understanding it.
Para comenzar, sabes cual es la base canonica?
deep anvil
Y dado un vector X, digamos X=(1,2,3), sabes escribir las coordenadas de X con respecto a esa base canonica?
Sabes que son las coordenadas de un vector con respecto a una base?
marsh citrusBOT
@buoyant jetty Has your question been resolved?
buoyant jetty
Y dado un vector X, digamos X=(1,2,3), sabes escribir las coordenadas de X con r...
(1,2,3)
buoyant jetty
Sabes que son las coordenadas de un vector con respecto a una base?
si hermano
deep anvil
Ok, entonces, podemos transformar este problema en un problema de resolver una ecuación vectorial.
deep anvil
Let $B = \{(4, 1, 2), (0, 1, 0), (-3, -1, -1)\}$ be a basis of $\mathbb{R}^3$.
...
Puedes reescribir el problema en un problema de ecuaciones vectoriales?
De un lado de la ecuacion vectorial vas a tener una combinacion lineal de vectores en la base B, del otro lado vas a tener una combinacion lineal de vectores en la base canonica.
Okay, antes de seguir, Que significa que las coordenadas en la base canonica sean el doble de las coordenadas en la base B?
Digamos que las coordenadas en la base B son (a,b,c)
buoyant jetty
(a,b,c)=2(4,1,2)+2(0,1,0+2(-3,-1,-1)
@deep anvil
,w (a,b,c)=2(4,1,2)+2(0,1,0)+2(-3,-1,-1)
elfin berryBOT
deep anvil
(a,b,c)=2(4,1,2)+2(0,1,0+2(-3,-1,-1)
Hay un error
deep anvil
Okay, antes de seguir, Que significa que las coordenadas en la base canonica sea...
Primero responde esta pregunta
Okay, dejame reformular la pregunta
Dada que las coordenadas de el vector X en la base B son (a,b,c), escribe las coordenadas de el vector X en la base canonica (dado que son el doble de las coordenadas en la base B)
deep anvil
no
deep anvil
Dada que las coordenadas de el vector X en la base B son (a,b,c), escribe las co...
Hay un malentendido, por eso estoy pidiendo que resuelvas esta pregunta mas sencilla antes de proceder.
buoyant jetty
(2a,2b,2c)
deep anvil
si
exacto
Entonces, dado que las coordenadas de el vector X con respecto a la base B son (a,b,c), escribe X como una combinacion lineal de la base B.
buoyant jetty
2a+2b+2c=0