i wouldnt say "so multiply both sides by lim h-> 0(h)" because thats the same as saying "multiply both sides by 0". But your claim and idea is correct. just a little nitpick
#help-23
restive niche
copper shadow
Ok so the overall logic is at follows. We have a f' which is piecewise. But regardless since f' exists for x>0 then f is continuous for x>0. Thus we need the f 's we obtain from f' both follow the same limits. And since continuity implies the limit at a point is the same as the value at point then we can find the values at 0. The only thing is that if you graph both functions it isn't continuous
copper shadow
i wouldnt say "so multiply both sides by lim h-> 0(h)" because thats the same as...
How should we phrase it then?
verbal cloud
Ok so the overall logic is at follows. We have a f' which is piecewise. But rega...
no, you need f' to exist at 0 for f to be continuous at 0
restive niche
How should we phrase it then?
f is differentiable at x so lim_{h -> 0}(f(x+h)-f(x) - hf'(x))/h= 0 so lim_{h -> 0}(f(x+h) - f(x) - hf'(x)) = 0 so lim_{h->0}f(x+h) = f(x) so f is continuous at x
copper shadow
no, you need f' to exist at 0 for f to be continuous at 0
Hmm but we never had it defined there
copper shadow
f is differentiable at x so lim_{h -> 0}(f(x+h)-f(x) - hf'(x))/h= 0 so lim\_{h ...
Where did your h in the denominator go in the second limit?
verbal cloud
I think you have the right idea but I will still clarify:
- you have two expressions for f: one when x <= 0, the other when x > 0
- you KNOW that f' is continuous on R (the expression is given), so it's defined at 0 (i.e. f diff at 0), thus f is continuous at 0
- knowing f is continuous at 0, the limits from both sides must match: since you have the expression of f on each side of 0, you get an equation that you can solve
restive niche
Where did your h in the denominator go in the second limit?
i did the same thing as you did. im just not saying "multiply by lim h-> 0 (h)" because i dont want to say "multiply by 0". you can write it out in terms of epsilons and deltas and the definition of limits if you want. anyways this isnt the main point of the problem
verbal cloud
I think you have the right idea but I will still clarify:
- you have two expres...
you don't really need f' continuous but it's the case here
as long as it's defined , f will be continuous
copper shadow
I think you have the right idea but I will still clarify:
- you have two expres...
I'm don't see where we have f' is continuous. Since it is piecewise
verbal cloud
it just so happens that both sides coincide at x=1 / u=0
but again, it's not important
copper shadow
i did the same thing as you did. im just not saying "multiply by lim h-> 0 (h)" ...
Tell me if I'm understanding. You split your limit into $\lim_{h\to0} \frac{f(x+h)-f(x)}{h} -\lim_{h\to0}\frac{hf'(x)}{h}=0$
flat frigateBOT
BigBen
copper shadow
it just so happens that both sides coincide at x=1 / u=0
Oh I see. So we have that it is continuous at 1 which is u=0. Which means it is continuous at 0
verbal cloud
it's ok to multiply both sides by 0 btw, but remember that you can't undo it (no reverse direction)
copper shadow
@verbal cloud isn't this an issue?
fathom jewel
That wasn't my concern but whether if the function was differentiable in the first place
verbal cloud
<@700680837603000350> isn't this an issue?
the piecewiseness for f is at u = 0 remember
copper shadow
Oh wait so the bounds should be x≤0 for the first and x>0 for the second
restive niche
That wasn't my concern but whether if the function was differentiable in the fir...
thats even more clear, since
f' is clearly continuous so obviously f will be differentiable once you integrate. (infact it will be continuously differentiable.)
safe radishBOT
@copper shadow Has your question been resolved?
fathom jewel
thats even more clear, since
f' is clearly continuous so obviously f will be dif...
.I wasn't sure because of the log(x) but yeah
errant parcel
Im looking for the euler angles that can be applied in XYZ order to rotate the orthonormal basis<1, 0, 0> <0, 1, 0> <0, 0, 1> (left) into <-0.577, 0.577, -0.577> <0.577, 0.577, 0.577> <0.577, 0, -0.577> (right)
safe radishBOT
Im looking for the euler angles that can be applied in XYZ order to rotate the o...
errant parcel
Also I guess in general how to see the euler angles that bridge 2 orthonormal basis
safe radishBOT
@errant parcel Has your question been resolved?
errant parcel
.close
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wild viper
Yes?
round notch
safe radishBOT
Yes?
Asking the actual question right away is more likely to get responses.
Asking "Can I ask...?" or "Does anyone know about...?" doesn't give people enough information to decide whether they can help, and answering can feel like a promise to help with the actual question, which they might find themselves unable to.
wild viper
.close
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pearl field
For example cos(pi-v)=-cos(v)
safe radishBOT
For example cos(pi-v)=-cos(v)
pearl field
Will be either in first or second quadrant
So it can be minus or plus cause in first sin and cos are +
safe radishBOT
@pearl field Has your question been resolved?
pearl field
Fucking useless
pearl field
Bitches i figured it myself
azure harbor
!done sighs
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azure harbor
Please dont clog up help channels with these kinds of messages.
left gyro
.close
distant mirage
for part D can someone explain why we set (m-2)=0 after we substitute into the diff eq and get 2x-mx-b=m
safe radishBOT
for part D can someone explain why we set (m-2)=0 after we substitute into the d...
tardy mango
for part D can someone explain why we set (m-2)=0 after we substitute into the d...
$(2-m)x-b=0x+m$, now compare coefficients.
flat frigateBOT
Civil Service Pigeon
tardy mango
(Note this implicitly uses the fact that {1,x} is a linearly independent set.)
distant mirage
so the right side is a constant which is why it is 0x+m?
distant mirage
i see and then u can match the terms kinda
so m=-b
wait but why does the coefficient for x on the right have to be 0
tardy mango
wait but why does the coefficient for x on the right have to be 0
as you said, m is a constant
If the coefficient was anything else, then it would vary with x
aka not be constant
safe radishBOT
tardy mango
ok thanks mate
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cosmic plover
for #14 i dont understand how the amswer is .34 isn't the problem saying the probability of the event given X is less than or equal to 1? which would mean its p({x=-2}U{x=-1}U{x=0}U{x=1})
safe radishBOT
for #14 i dont understand how the amswer is .34 isn't the problem saying the pro...
tawdry plover
for #14 i dont understand how the amswer is .34 isn't the problem saying the pro...
it's saying what the probability that the absolute value of X is less than or equal to 1
in which case, it would be $P(X = -1 \cup X = 0 \cup X = 1)$
flat frigateBOT
blanketism
cosmic plover
tawdry plover
<a:scrajj:1007709010033643540>
what's confusing you?
outer hull
It's Pr( |X| <= 1), not Pr( X <= 1)
cosmic plover
yea i mean i understand the absolute value but wouldnt the probably of that be AFTER i calculate
tawdry plover
yea i mean i understand the absolute value but wouldnt the probably of that be A...
no, its just asking when the absolute value of X is less than or equal to 1
tawdry plover
so take whatever values you have for X first
tawdry plover
cosmic plover
Thank you sorry my brain is fried
tawdry plover
<:ohbow:1275683618097856512> Thank you sorry my brain is fried
happens to the best of us, its okay 
as an aside, you should maybe take a break if you've been studying/working for a while
solar hazel
tawdry plover
(if you have the time, that is)
cosmic plover
ahh i just got started on this stuff i was working on my econ
tawdry plover
<a:pi_melopeace:1270126175305597009>
cosmic plover
solar hazel
cosmic plover
2 midterms on friday!
solar hazel
cosmic plover
gotta get it done
tawdry plover
good luck twin
safe radishBOT
tawdry plover
thank you! and ty for you help
anytime! 
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burnt notch
, prova
burnt notch
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gaunt seal
this is only a quewtsion the pair of linear equations that show up in the 8ottom half of the screenshot
- do matrix manipulation rules work for angles. the aswer column is in newtowns but the coefficients are in degrees. Should i change the cosines into num8ers 8efore i start matrix manipulation or can i leave them as degrees?
also just generally what would 8e the strategy to solve the unknown force magnitudes. the 8ook doesnt show any working out. Thank you!
safe radishBOT
this is only a quewtsion the pair of linear equations that show up in the 8ottom...
frosty crow
You seem to have stated all the correct ideas. The force on the wires is in the direction each wire pulls, which is the cosine of each coordinate (and also tbe other compementary / supplementary angle). You do have to compute the cosines into numbers first.
I think the setup is off the vertical, which might be important. So 90° - 70°= 20°, the weight is pulling down here. The partial derivative appears to be changing the angle numbers using "direction cosines"
Is your question how to solve a system like Ax = y?
safe radishBOT
@gaunt seal Has your question been resolved?
heady plume
one day
ill become smart enough to answer this question
i plan to study mechanics over the summer
frosty crow
Pick up hibbler's mechanics
or may8e hi88ler's mechanics, it seems.
safe radishBOT
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pearl lynx
i dont even know where to start im very lost
safe radishBOT
i dont even know where to start im very lost
pearl lynx
periodic functions btw
craggy tree
it's helpful to take points where the hour hand is at 9, 12, 3 and 6 since those are the easiest to calculate
craggy tree
for part a yeah
at 9 and 3, the height is 0
at 12, the height is +12 cm
and at 6, the height is -12 cm
so for part a you get points like (0h, 0cm) (3h, 12cm) (6h, 0cm) (9h, -12cm) and so on
craggy tree
yeah at t = 0 it would be 12cm
pearl lynx
for b are the points (0,12) (3,0) (6,-12)
heady plume
Pick up hibbler's mechanics
Yes i just need to finish calc 2
near sky
what have you tried so far
near sky
what do i do to find this
try drawing a diagram
vague phoenix
Lowest point is?
vague phoenix
right
pearl lynx
now wat?
pearl lynx
f(x)=sine(x)?
flat frigateBOT
Minλ
vague phoenix
Now let's start with amplitude
warm warren
general sine formula was what was meant
vague phoenix
If diameter is 18, then what is the radius?
pearl lynx
9
vague phoenix
And this is ? of the function?
pearl lynx
amplitude
vague phoenix
Now, let's do vertical shift
You knew max and min height
The vertical shift is the mean of their sum
pearl lynx
12
pearl lynx
im really lost here
vague phoenix
Recall sine formula as shown, which starts at half-point between highest and lowest point, now the player starts at very bottom part, you need to shift the function
what can you do here to make the min point starts at y=0?
pearl lynx
not really
near sky
When is the peak of sinx
pearl lynx
90
near sky
Right, when is sinx =0
pearl lynx
0
near sky
So what is the difference you need shift
pearl lynx
+90?
near sky
No, you need to shift the lowest point
near sky
Recall sine formula as shown, which starts at half-point between highest and low...
If you recall this, the peak is at the same distance from sinx =0 as the lowest point, sin90 = 1, sin(-90)=-1
For your function to start at lowest point you need to shift the lowest point towards y=0, so you shift to the right
90 is correct not the sign
pearl lynx
so -90
near sky
Yes
pearl lynx
f(x)=9sin(X-90)+12
vague phoenix
Dude didn't think and already typed the answer
pearl lynx
i did
near sky
f(x)=9cos(x+180)+12
That is correct, but you can simply more by using cos( 180 +x) =-cosx
pearl lynx
i forgot i could do that
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wind laurel
Hello! I get so confused when trying to solve these types of questions (a). Solving the actual ode is quite easy but whenever i get asked to show that the situation described is modelled by that ode i have no clue where to start and how to answer (┬┬﹏┬┬)
safe radishBOT
Hello! I get so confused when trying to solve these types of questions (a). Solv...
wind laurel
Transcribed:
Two different colours of paint are being mixed together in a container.
The paint is stirred continuously so that each colour is instantly dispersed evenly throughout the container.
Initially the container holds a mixture of 10 litres of red paint and 20 litres of blue paint.
The colour of the paint mixture is now altered by
* adding red paint to the container at a rate of 2 litres per second
* adding blue paint to the container at a rate of 1 litre per second
* pumping fully mixed paint from the container at a rate of 3 litres per second.
Let r be the amount of red paint in the container at time t seconds after the colour of the paint mixture starts to be altered.
a) Show that the amount of red paint in the container can be modelled by the differential equation $$\frac{dr}{dt} = 2 - \frac{r}{\alpha}$$
where $\alpha$ is a positive constant to be determined.
round egret
think of it as the change in red paint = (Rate of red paint entering) - (Rate of red paint leaving)
so $\frac{dr}{dt} = \text{Rate In} - \text{Rate Out}$ basically
flat frigateBOT
MxRgD
wind laurel
think of it as the change in red paint = (Rate of red paint entering) - (Rate of...
I did, and for the rate of red paint entering its simple, but i dont understand what the rate of red paint leaving is
would it just be r/V? but sometimes its expected to have an equation in t in the denominator, and sometimes it isnt
round egret
I did, and for the rate of red paint entering its simple, but i dont understand ...
In the concentration, there are r litres of red paint in a total of 30 litres right? So the redness of the mixture is r/30
and you are pumping it out 3 litres of the mixture every second
wind laurel
where do you get 30 from? Is that the intially stated volume so 10 + 20?
round egret
and you are pumping it out 3 litres of the mixture every second
basically how much red is in that 3L is being pumped out?
wind laurel
sorry im back
wind laurel
basically how much red is in that 3L is being pumped out?
ohh
so rate out is like "how much red is being pumped out?" which is the rate of volume being pumped out multiplied by the concentration of red in that volume?
round egret
so rate out is like "how much red is being pumped out?" which is the rate of vol...
yep
round egret
I can read it fine
wind laurel
because here i have a function of t in the denominator
wind laurel
1000 - 20t + 25t?
wind laurel
ohh
round egret
and you know there are x grams of pollutant in V litres of water
so what do you think is the concentration then?
at that specific time t
(it's basically the other question but you include time t as well)
round egret
yep
wind laurel
so a general equation for the rate out would be:
(rate volume out)*concentration, or (rate volume out)* x/V(t)
round egret
so a general equation for the rate out would be:
(rate volume out)\*concentratio...
yeah sort of like that
I prefer breaking it down
safe radishBOT
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Available help channel!
fickle mantle
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gentle edge
what ya need help wit
gentle edge
3rd one looks correct to you?
fickle mantle
My teacher told me the tete sign starts at the 360 line and when u move anti clockwise it’s positive and clownish is negative
fickle mantle
3rd one looks correct to you?
Mb
winter whale
What is the g11441?
fickle mantle
The what
gentle edge
My teacher told me the tete sign starts at the 360 line and when u move anti clo...
✅
you know how 90 degree change the quadrent?
fickle mantle
Yes
gentle edge
how you check the sign of sin(180-theta)
golden nebula
What is the g11441?
Its probably a reference to a video
fickle mantle
how you check the sign of sin(180-theta)
Won’t it be sin
gentle edge
Won’t it be sin
no no like will it be +ve or -ve
fickle mantle
What’s ve
gentle edge
positive
fickle mantle
Oh
gentle edge
and negitive
fickle mantle
It’ll be positive
fickle mantle
It moved baward
Erm
So you go to the 180 line
And move backwards into the 2nd wharf
Wharf
Quard
fickle mantle
Yes
gentle edge
ok then thats the basics
gentle edge
rest all on this seem correct
fickle mantle
I struggling s the ones that say, theta - degree
fickle mantle
rest all on this seem correct
Oh Ty
gentle edge
I struggling s the ones that say, theta - degree
first check the borderline with the degree
and go clockwise or anti accrd to theta
for example 270 degree - theta
fickle mantle
Ohh
gentle edge
which quardent would it be
gentle edge
No no I mean when it’s rather theta - 270
do this one first
gentle edge
I don’t understand what your saying
find
sin(270 degree + theta)
fickle mantle
-cos the the
gentle edge
correct
fickle mantle
On the 270 line we move forward into the fourth quard
fickle mantle
Yes
gentle edge
so now
what -ve degree turns do
is instead of going anticlockwise
you go clockwise
fickle mantle
On the line?
gentle edge
just how we were going 270 degree
fickle mantle
Oh
gentle edge
wait i got paint
gentle edge
yep
my drawning sucks
:/
now for -270 degree
we do this
we take 270 degree angle
but from clockwise direction
fickle mantle
I don’t understand this one
fickle mantle
my drawning sucks
It’s ok
fickle mantle
Ooo
fickle mantle
So +ve is normal one which moves anti clockwise
gentle edge
thats what making a -ve angle means
gentle edge
So +ve is normal one which moves anti clockwise
yeah
fickle mantle
And the opposite is -ve
gentle edge
yep
fickle mantle
Can u help me label them when it’s clockwise
Idk how to label the degrees for the lines
gentle edge
give an example
fickle mantle
Will the line labeling be the same for -ve
Brb I’m going for a book to write this down
gentle edge
Will the line labeling be the same for -ve
270=-90
180=-180
90=-270
0=0
for -ve angles
fickle mantle
Back
fickle mantle
<@1354133211499860008> dont forget to type ".close" when you're done
Yes I, writing stuff down but Ty
I’ll close it soon
gentle edge
kk nw
fickle mantle
Still in use sorry
gentle edge
Sorry
it would be -theta in 4th quad
-theta - 90 in 3rd
and so on
but ig you can also write theta - 90
both mean the same thing
fickle mantle
.close
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fringe dock
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fringe dock
wait
okay, i dont really know how to do this question
all i have is a graph i drew
but its useless
suppose a Bell Shaped Normal Distribution curve
oof idk
plucky elk
do you know how to write the z scores in terms of the weights of the pigs
fringe dock
do you know how to write the z scores in terms of the weights of the pigs
the question doesnt expect me to use that so no
i think thats something i can use on my calculator to work out
plucky elk
Do you know empirical rule
fringe dock
Do you know empirical rule
no, all ive been taught to be able to answer this question is the basics like subsituting standard deviation and mean onto the calculator to get the % below the graph
plucky elk
i think thats something i can use on my calculator to work out
No one else has your calculator so don't know what buttons you have access to
fringe dock
No one else has your calculator so don't know what buttons you have access to
i was able to use ai to help me break down the working out on the solution because it didnt make any sense in the worked solution
so its something i have to memorise
plucky elk
the question doesnt expect me to use that so no
what
fringe dock
<:monkey:651816857153699840> what
plucky elk
no, all ive been taught to be able to answer this question is the basics like su...
And what
fringe dock
<:monkey:651816857153699840> what
because i havent learnt that
plucky elk
This is using the empirical rule
plucky elk
i was able to use ai to help me break down the working out on the solution becau...
So is this
fringe dock
So is this
oh okay
just one thing
do u know a graph online
that shows this data
i dont know how to draw it as i havent learnt it
prisma wren
You mean a visual? This can work, just be careful about the shaded vs nonshaded area https://www.desmos.com/calculator/zxtwpjsjen \ https://www.desmos.com/calculator/np9cg4ftbn
If you set Z = -1 (meaning one standard deviation below the mean), it will shade the region below that value. 16% will compose the shaded area, 84 the non-shaded area
safe radishBOT
@fringe dock Has your question been resolved?
stiff nest
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upbeat swan
stiff nest
doesnt help much
upbeat swan
Wdym
It's the only thing you need
Just write f (x) and g(x) and see what was added and what was subtracted
stiff nest
mapping
f(x) = 2(x + 1)^2 + 7
onto
g(x) = 2(x - 2)^2 + 4x - 3
i have another 4x term in g(x)
upbeat swan
Dude just because it asked you to write f(x) in another form doesnt mean you cant use the original form for part c
Use the original f(x)
upbeat swan
Well plug x-2 into f(x) and see if you get g(x)
safe radishBOT
upbeat swan
okay its a translation by (2, -4) i see
What does this mean
(2, -4) is a point in the coordinate system
stiff nest
What does this mean
shifted 2 units right in the x axis and 4 units down in the y axis
stiff nest
(2, -4) is a point in the coordinate system
i cant write vectors on a keyboard
upbeat swan
Its wrong
stiff nest
its write i checked on the mark scheme
upbeat swan
Yeah its correct sorry
near sky
dam a blud spotted
upbeat swan
It's a fake blud 🥀
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jolly yacht
hi im working on some langrangian stuff and i have this pulley problem but i feel like it doesnt make sense to have y_4''(t) = g because that would mean its freefalling (T_4 = 0) and i cant see how that would be physically possible, can someone check my work to make sure i did it correctly?
safe radishBOT
hi im working on some langrangian stuff and i have this pulley problem but i fee...
jolly yacht
<@&286206848099549185>
jolly yacht
discord.gg/physics
oh i didnt know this existed ty
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jolly yacht
.close
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celest glen
me
celest glen
need help with this
burnt notch
Which of them?
celest glen
all of them starting from 1.
burnt notch
!status
safe radishBOT
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
opaque fern
need help with this
Is this an exam by any chance?
celest glen
something like that
my classmate took a picture while i was absent
and forwarded it
ok i know the first one its pretty easy
wait no
wait yes
ok onto the next one
quiet plume
something like that
Is it an exam or is it not an exam?
celest glen
yes
quiet plume
It doesn't sound like you're meant to have this available to you then, let alone ask for help on it.
fathom jewel
try to guesd an integer root
if ur teacher is sane you will find one and easily apply long div
celest glen
15x^2+34x+15=0
and then quadratic formula x2 and x3 since we already have x1=1
well x, x2 and x3
fathom jewel
[ \prpl \polylongdiv{15x^3+19x^2-19x-15}{x-1}]
ionic blaze
and then quadratic formula x2 and x3 since we already have x1=1
u can make it easier just so it's less technical
notice that 34=25+9=5^2+3^2, so there's 3x * 3 and 5x * 5 and you can get (5x + what is multiplied to the 3x)(3x +smth else multiplied to the 5x)
ionic blaze
takes like 3 minutes to solve
ur choice if u wanna do sqrt(34^2-4(15)(15))
ionic blaze
\[ \prpl \polylongdiv{15x^3+19x^2-19x-15}{x-1}\]
they divided it right
fathom jewel
thx for confirming
celest glen
ok now this
ionic blaze
for 2., since y is just a lone variable involved here, x+y=5, then subtract x from both sides to get y=5-x, then substitute y=(5-x) into x^2-8x-y+15=0 then solve
note that since the first one is quadratic and it's an intersection between a line and a quadratic equation there are at most 2 possible points, but there is still a chance of 0 or no points at all
celest glen
so x^2 -8x - (5-x) +15
ionic blaze
so x^2 -8x - (5-x) +15
yes
celest glen
so after finding x1 and x2 i just need to type in x1 + y1 = 5 right
and x2 + y2 = 5
so y1 =5- x1
and the same for y2
ionic blaze
and the same for y2
yep
ionic blaze
it is
ionic blaze
for 3 I'm pretty sure there's some sort of division involved
let the top equation be equal to eq 1 and the bottom equation be eq 2
notice that in eq 1, it's pretty hard to factor, matter of fact, it's not really factorable
so we have to substitute values of y, you can get these from the bottom equation
to factor, you may opt to let y equal to an arbitrary number, for example, if you let y=1, eq 2 turns into 0=15x^2-13x+2=(5x-1)(3x-2) now you just place y back in, so you can notice that it's equal to (5x-y)(3x-2y) the above, now since that is equal to 0, then 5x-y=0 OR 3x-2y=0
so that gives you y=5x, and 2y=3x => y=(3/2) x
now you take those and substitute them into eq 1
kind of like earlier let y=(3/2) x or 5x
celest glen
wait
ionic blaze
feel free to ask questions I'm not really the best explainer
celest glen
eq2 is factorable no?
ionic blaze
eq2 is factorable no?
eq2 is factorable, I was talking about eq1 not being factorable
celest glen
15x^2 - 15xy +2xy +2y^2=0
ionic blaze
15x^2 - 15xy +2xy +2y^2=0
try referring to the way to factor I mentioned in the 2nd paragraph
celest glen
and then t=x/y
ionic blaze
need help with this
for no 4. it also doesn't look factorable, but I suppose you can multiply eq 1 by 2, and then subtract eq 2 from eq 1 to get 13=x^2+y^2 and you can also get -2x^2+2xy=4 from just subtraction of the normal eq1-eq2,
so y^2=13-x^2 and 2xy=4+2x^2, now substitute these into either eq 1 or eq 2 to get values for x and y since you have some 2xy and y^2 to get x values, do remember to see if these are possible values
ionic blaze
need help with this
for no. 5
square both sides to get x^2-3x+2>(x-2)^2=x^2-4x+4
subtracting x^2-4x+4 to both sides
ionic blaze
what to do from this
t=x/y
x/y=2/3. x/y=1/5
ionic blaze
multiply y to both sides and substitute x as the value of whatever you get for
celest glen
yeah wait
ionic blaze
2/3*y=x is a line
celest glen
3x=2y
ionic blaze
t=x/y
x/y=2/3. x/y=1/5
for this one you can get 5x=y, either substitute y for 5x or x for y/5
celest glen
y=5x
ionic blaze
t2 is better
but t1 also has a solution
celest glen
do i need to do both
celest glen
ah
i will try and solve it now
i applied y=3x/2 to the first equation and got that x=sqr(36/31)
ionic blaze
i applied y=3x/2 to the first equation and got that x=sqr(36/31)
isn't it sqrt(36/37)
ionic blaze
yeah but its still a messy number
it is unfactorable anyways
now let y=3/2 x
also note that I think negative solutions would also work
they should
so x=-6/sqrt37 also works
celest glen
y=9/sqr(37)
ionic blaze
ya
ionic blaze
y=9/sqr(37)
you can double check the values if you're suspicious
ionic blaze
so x=-6/sqrt37 also works
since it involves x^2
ionic blaze
x2=3/sqrt(39)
yep
now note that those can also be negative since eq 1 would only yield positive if x and y are both negative
celest glen
so +-
celest glen
i can also multiply eq1 by 5 and eq2 by 9 to get 0
ionic blaze
don't listen to this,
celest glen
hahah
ionic blaze
ok
celest glen
whos dumb enough to send a mrbeast scam thingy in a math server
ionic blaze
i can also multiply eq1 by 5 and eq2 by 9 to get 0
yeah that also works but that makes it gigantic, and it would yield you a line(y=mx)
celest glen
i got 30x^2 -26xy +4y^2 = 0
which looks very factorable
and then i divide by y^2
and 30t^2 -26 t+4 =0
t=x/y
celest glen
did i do something wrong
celest glen
its like the 3. task
ionic blaze
yep
ionic blaze
eh i wont even finish it
you probably should
celest glen
why
ionic blaze
to be sure
ionic blaze
for no. 5
square both sides to get x^2-3x+2>(x-2)^2=x^2-4x+4
subtracting x^2-4x+...
.
and note that for x<2, it is negative so strictly less than the square root of any positive number
and that x^2-3x+2 is positive for x<1
celest glen
for no. 5
square both sides to get x^2-3x+2>(x-2)^2=x^2-4x+4
subtracting x^2-4x+...
i get this
celest glen
and note that for x<2, it is negative so strictly less than the square root of a...
why tho
ionic blaze
why tho
x-2 for x<2 is negative
for x<1, the sqrt(x^2-3x+2) is positive
positive > negative
celest glen
ah makes sense
actually
eq >0-2
since x<1 x can be 0
and also -1>-2 so it isnt strictly positivew
wait
what am i talking
ionic blaze
and also -1>-2 so it isnt strictly positivew
?
celest glen
nevermind
ionic blaze
square roots give you positive by convention
celest glen
x-2 for x<2 is negative
for x<1, the sqrt(x^2-3x+2) is positive
positive > negat...
may you reexplain this
ionic blaze
may you reexplain this
x-2 for any x value less than 2 is negative
any value of sqrt(x^2-3x+2) which is real, is positive
we know positive>negative
celest glen
any value of sqrt(x^2-3x+2) which is real, is positive
why is it positive
ionic blaze
so what values of x makes x-2 negative, but sqrt(x^2-3x+2) real
ionic blaze
why is it positive
usually, by convention
ionic blaze
usually, by convention
in this case, there is the involvement of a radical from the get-go so you can't really disregard that it would only yield positive solutions(since if it were negative, it wouldn't make sense)
celest glen
ah
what do i do form this
from
(x-2)(x-1) > (x-2)² /:(x-2)
x-1 >x-2
right
yes
i get it now
any number less than 2 would be wrong
so its above 2
yes it makes sense
what now
safe radishBOT
@celest glen Has your question been resolved?
celest glen
<@&286206848099549185>
potent crag
can u clarify what u wanna ask
celest glen
i know
potent crag
yas so your doubt is?
potent crag
yupppp
celest glen
oh my God
celest glen
if x=0 then its 2>4 which is false
potent crag
when x<2
the root is defined (x<=1)
so -infity to 1
celest glen
minus infity to 1
.
celest glen
.
potent crag
the final soln is (-infity , 1] U (2, infinity)
potent crag
yes....
potent crag
how did (x-2)^2 come
celest glen
refer to the . message
potent crag
THE FIRST STEP IS WRONG
because you squared without checking the sign of both sides
u are missing x<=1
celest glen
yes you are right
celest glen
just a sec
what do you mean by you have to account for both signs
sign of both sides
potent crag
u must consider whether x-2 is negative and non negative
celest glen
can you explain it from the start
celest glen
how did you get that
potent crag
if x<2
then
non negative > negative
celest glen
why 0
potent crag
(x-1)(x-2)
potent crag
hm
celest glen
oof
why did the signs get flipped at one point
<>
but square root cant be negative
wait
youre right
why must the signs be flipped
tardy mango
celest glen
because
the signs may flip
so its 'or'
and we got that in both cases it cant be 2
hence x=/=2
and the other condition is
so its 1 and less
and above 2 to infinity
i get it now
spasibo bratan
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Available help channel!
shut willow
yo is this correct?
safe radishBOT
yo is this correct?
last heath
looks good to me
tardy mango
,w x^3+y^3=6xy
shut willow
hmm
mixed up some working there
ok yeah I mixed up the x terms
$$\begin{align}
\frac{d}{dx} (x^{3}+ f(x)^{3}) &= \frac{d}{dx} 6x f(x) \
3x^{2}+ 3f(x)^{2}f'(x) &= 6f(x) + f'(x)6x \
\text{i.e. } x^{2}+ f(x)^{2}f'(x) &= 2f(x) + 2f'(x)x \
\text{i.e. } 2f'(x)x - f(x)^{2}f'(x) &=x^{2}- 2f(x) \
f'(x)[2x - f(x)^{2}] &= x^{2}- 2f(x) \
\therefore f'(x) = \frac{x^{2}- 2f(x)}{2x - f(x)^{2}}
\end{align}$$
flat frigateBOT
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calm cloak
mixed up while taking out common terms here
shut willow
alr ty
i have one more problem im stuck on
Find $f'(x)$:
$$\begin{align}
f(x) = x^{x^{x^{x^{\dots}}}}
\end{align}$$
$$\begin{align}
\text{i.e. } f(x) = x^{f(x)} \
\text{i.e. } \ln (f(x)) = f(x) \ln x \
\text{i.e. } \frac{d}{dx} \ln f(x) = \frac{d}{dx} f(x) \ln x \
\text{i.e. } \frac{1}{f(x)} f'(x) = f'(x)\ln x + f(x) \frac{1}{x} \
\text{i.e. } \frac{f'(x)}{f(x)} - f'(x) \ln x = f(x) \frac{1}{x} \
\text{i.e. } f'(x)\left[ \frac{1}{f(x)} - \ln x \right] = \frac{f(x)}{x}
\end{align}$$
flat frigateBOT
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shut willow
I dunno what to do past this point, do I just divide the other side by that factor of f'(x)?
calm cloak
I dunno what to do past this point, do I just divide the other side by that fact...
yes I guess
shut willow
Find $f'(x)$:
$$\begin{align}
f(x) = x^{x^{x^{x^{\dots}}}}
\end{align}$$
Response
$$\begin{align}
\text{i.e. } f(x) &= x^{f(x)} \
\text{i.e. } \ln (f(x)) &= f(x) \ln x \
\text{i.e. } \frac{d}{dx} \ln f(x) &= \frac{d}{dx} f(x) \ln x \
\text{i.e. } \frac{1}{f(x)} f'(x) &= f'(x)\ln x + f(x) \frac{1}{x} \
\text{i.e. } \frac{f'(x)}{f(x)} - f'(x) \ln x &= f(x) \frac{1}{x} \
\text{i.e. } f'(x)\left[ \frac{1}{f(x)} - \ln x \right] &= \frac{f(x)}{x} \
\text{i.e. } f'(x)\left[ \frac{1 - f(x)\ln x}{f(x)} \right] &= \frac{f(x)}{x} \
\therefore f'(x) &= \frac{f(x)}{x} \cdot \left[ \frac{f(x)}{1- f(x)\ln x} \right] \
&\boxed{= \frac{f(x)^{2}}{x - xf(x)\ln x}}
\end{align}$$
flat frigateBOT
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shut willow
.close
safe radishBOT
safe radishBOT
Available help channel!
stiff vine
how u do this
safe radishBOT
how u do this
golden nebula
If a vector is subtracted from a vector and leaves magnitude 0, try imagining how it looks on a graph
I don't have a notebook, but try observing these images (for a)
stiff vine
i just did algebra for a to turn u-v=0 to u=v
golden nebula
K. So are you stuck on b?
stiff vine
K. So are you stuck on b?
i mean so i can just remove the magnitude signs?
stiff vine
i just did algebra for a to turn u-v=0 to u=v
So how come this is right
stiff vine
Are you stuck on b tho?
Yeah
golden nebula
So 2 vectors can have the same magnitude
But their direction is different
So v is not necessarily an inverse of u
Or even equal to u
golden nebula
In here, the 2 vectors have same magnitude, but v is not equal to u or the inverse of u
safe radishBOT
@stiff vine Has your question been resolved?
solar rapids
Hello, I have had some difficulty solving this question:
safe radishBOT
Hello, I have had some difficulty solving this question:
solar rapids
I understand that I must first compute T(b), that is, applying T to each of the basis vectors. After that, do I solve for T(b) as a linear combination of the basis vectors in B?
delicate shore
yep
solar rapids
So then the solution would be, by column, [2 2 5], [3 -1 1], [4 -1 2]?
tiny veldt
its an inverse model
tiny veldt
So then the solution would be, by column, [2 2 5], [3 -1 1], [4 -1 2]?
yea thats P
get inverse of that
get the product of a and that column and multiply with the inverse
solar rapids
Sorry... what do you mean by multiply with the inverse?
delicate shore
there is a shortcut to finding the matrix representation wrt to another basis
but I don't think they mean for you to use that here
solar rapids
Okay. So the matrix I wrote out (by column) above is still correct?
(Additionally, I am a little curious about this shortcut?)
honest perch
So then the solution would be, by column, [2 2 5], [3 -1 1], [4 -1 2]?
no
not unless u just chose the bases elements as columns
delicate shore
(Additionally, I am a little curious about this shortcut?)
,, M_{mathcal B} = P^{-1} M P quad text{where} quad P = (mathbf b_1\ cdots\ mathbf b_n).
flat frigateBOT
delicate shore
but like to understand this formula it would help to do the process by hand at least once
which is what they're making you do here
solar rapids
but like to understand this formula it would help to do the process by hand at l...
Right, so conceptually, this involves first computing T(b), which sends each of the basis vectors to their transformed vector under T, and then converting T(b) with respect to B?
delicate shore
yes
you need to compute Tb1, Tb2, Tb3 by matrix multiplcation
this will give you Tb1, Tb2, Tb3 in terms of the standard basis, then you need to find their new coordinates in terms of the new basis
solar rapids
this will give you Tb1, Tb2, Tb3 in terms of the standard basis, then you need t...
So I should obtain, by column, [-8 -64 53], [-6 -47 39], [-8 -64 53]?
solar rapids
A matrix calculator online.
delicate shore
I guess you punched in some numbers wrong
but also the numbers are so small here I recommend you just compute them by hand
solar rapids
but also the numbers are so small here I recommend you just compute them by hand
I intend to do so once I can confirm that I'm getting the correct answer. I'll try again online
solar rapids
I guess you punched in some numbers wrong
I recomputed: [2 5 -5], [-6 -47 39], [-8 -64 53]?
delicate shore
err not quite I don't think
solar rapids
What should I be getting?
I must be doing something incorrectly, or utilizing a calculator incorrectly
delicate shore
(-1, 4, 3) for Tb1
solar rapids
Yes, and (3 -4 1) for T(b2), while T(b3) is (4 -5 2).
Then I proceeded to solve c1b1 + c2b2 + c3b3 = T(bi), for i in {1, 2, 3}.
delicate shore
Yes, and (3 -4 1) for T(b2), while T(b3) is (4 -5 2).
yes these look correct now
delicate shore
Then I proceeded to solve c1b1 + c2b2 + c3b3 = T(bi), for i in {1, 2, 3}.
correct
solar rapids
Yet when I solved in the calculator online, I obtained the columns I provided above?
delicate shore
are those like your final answers
solar rapids
Yes
solar rapids
are those like your final answers
Are they incorrect?
delicate shore
,w [[2,3,4],[2,-1,-1],[5,1,2]]^-1 * [[-2, -6, 3], [3, 9, -4], [-2, -4, 3]] * [[2,3,4],[2,-1,-1],[5,1,2]]
flat frigateBOT
flat frigateBOT
delicate shore
Then I proceeded to solve c1b1 + c2b2 + c3b3 = T(bi), for i in {1, 2, 3}.
I solved for c1, c2, and c3 for Tb1 here
you can see they are 4, 25, and -21 resp.
this corresponds to column 1 of the final matrix
you can similarly find the c1 c2 c3s for Tb2 and Tb3 and you should get the numbers in that matrix
solar rapids
Ah, I believe I'm getting it now.
Thank you so much!
Do I keep this channel open if I have other questions (not immediately, but in a bit of time)?
delicate shore
err I guess you can open another help channel when the time comes
solar rapids
err I guess you can open another help channel when the time comes
Sounds good. Thanks for all your help!
delicate shore
solar rapids
.close
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Available help channel!
spiral saddle
safe radishBOT
flat frigateBOT
Renato
molten acorn
!status
safe radishBOT
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
@spiral saddle Has your question been resolved?
spiral saddle
it is reflexive
because any f in X is a function so under same input the function gives same output
it is NOT symmetric
counterexample
g(x) = {x if x != 2 or 3 if x = 2}
f(x) = x
we get that f(1) = g(1) and f(2) <= g(2) but not that g(2) <= f(2)
it is transitive, take an arbitrary f,g,h in X , then
f R g <=> f(1) = g(1) and f(2) <= g(2)
g R h <=> g(1) = h(1) and g(2) <= h(2)
we get that f(1) = g(1) and g(1) = h(1)
since we know the equality relation is an equivalence relation, we know that equality is transitive
we get that f(1) = g(1) = h(1) and so f(1) = h(1)
we also get that f(2) <= g(2) and g(2) <= h(2)
similarly, since we know that the <= inequality relation is transitive
we get that f(2) <= g(2) <= h(2) and so f(2) <= h(2)
so f(1) = h(1) and f(2) <= h(2) <=> f R h .
molten acorn
counterexample
g(x) = {x if x != 2 or 3 if x = 2}
f(x) = x
we get that f(1) = ...
i think you meant f(1) = g(1)
but everything else seems alright
spiral saddle
yes, thankiu, let me edit
molten acorn
how about antisymmetric?
spiral saddle
the def says that if x R y , y R x => x = y
loosely speaking, yes, this is indeed the case
no
wait no
because a function can be f(1) = g(1) aswell as f(2) = g(2) but the other 8 elements in his domain are unrestricted to where they map
so it is not necessarily true that f = g
molten acorn
because a function can be f(1) = g(1) aswell as f(2) = g(2) but the other 8 ele...
yep
f(1) = g(1) *
spiral saddle
yes, let me edit
molten acorn
**Renato**
can you do part b?
spiral saddle
so this shit is not antisymmetric, all there is left for a) is giving a counterexample to prove that R is not antisymmetric
let me brush up on the definition of injectivity
f(a) = f(b) => a = b
the idea would be to count first how many injective functions on X exist?
molten acorn
f R g means that
f(1) = g(1) = 1
and
f(2) ≤ g(2) = 2
which means f(2) = 1 or 2
but because f is injective
f(2) must equal 2
the rest is just simple combinatorics
molten acorn
f(2) ≤ g(2)
g(2) = 2
spiral saddle
f R g means that
f(1) = g(1) = 1
and
f(2) ≤ g(2) = 2
which means f(2) = 1 or 2
so we need to count the number of functions in X that
f(1) = g(1) and f(2) = g(2)
molten acorn
so we need to count the number of functions in X that
f(1) = g(1) and f(2) = g(...
yeah
specifically f(1) = 1 and f(2) = 2
spiral saddle
but f needs to be injective
molten acorn
but f needs to be injective
yes
spiral saddle
but that doesnt matter, because we used that already
molten acorn
but that doesnt matter, because we used that already
it still matters
spiral saddle
damn
molten acorn
we cant let f(3) = f(4), for example
spiral saddle
ok so f(n) < f(n+1)
molten acorn
ok so f(n) < f(n+1)
not necessarily 😭
spiral saddle
this is hard
molten acorn
well, we alr know what f(1) and f(2) are
just need to find f(3) till f(10)
f(3) is in {3, 4,...,100}
molten acorn
f(3) is in {3, 4,...,100}
(because it cant be 1 nor 2 due to injectivity)
then, f(4) is in {3, 4, ..., 100}\f(3)
f(4) cant be f(3) because injectivity
molten acorn
f(3) is in {3, 4,...,100}
so there are 98 possibilities for the value of f(3)
97 possibilities for f(4)
96 for f(5)
and so on
spiral saddle
98 x 97 x 96 x 95 x 94 x 93 x 92 x 91
f3 f4 f5 f6 f7 f8 f9 10
so 98! / 90!
molten acorn
yep
spiral saddle
idk maam
dont get it at all
I do understand that f3 has 98 possibilities but what about the condition on g(n)?
oh, I see, fRg only restricts inputs of 1 and 2
I am deadass stupid
I do get it now
now that I see it with this perspective it wasnt that hard, I really appreciate the help
like R only imposes a restriction on the first 2 inputs
and the domain needs to be mapped to something for the rest of the 10 - 2 = 8 inputs
and since we need injectivity we cant repeat same output twice because if we do f(3) = 4 and f(4) = 4 for example then 3 != 4
it certainly looked more complicated at first sight
,w 98! / 90!
flat frigateBOT
spiral saddle
though doing this type of combinatorics certainly feels like dark magic sometimes, ngl
.solved
safe radishBOT
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Available help channel!
high perch
help i don't have a graphing calculator
safe radishBOT
help i don't have a graphing calculator
high perch
what i can i do
if possible can anybody with a graphing calculator give me the answer
azure harbor
if possible can anybody with a graphing calculator give me the answer
sure use ,w <expression>
,w integrate x
ionic blaze
help i don't have a graphing calculator
manually integrate just like our forefathers before us 💪💪💪(kidding)
u can prolly go on wolfram and input it there in case ur confused about how to do it here
high perch
ok yeah i think i can probably do it manually
ionic blaze
,w integrate x
this is also the same thing
ionic blaze
ok yeah i think i can probably do it manually
NO THAT WAS A JOKE DON'T DO IT
high perch
why
i don't see why it wouldn't work if i just start doing engineering math
oh wait
now i see
ionic blaze
i don't see why it wouldn't work if i just start doing engineering math
I'm ngl, you could manually integrate but it'd just be equivalent to hell so just use wolfram
ionic blaze
sure use `,w <expression>`
what this guy said
high perch
.close
safe radishBOT
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Available help channel!
lavish magnet
https://www.youtube.com/watch?v=xrK4LCL8oZY
In this video at time stamp 12:22 he defined acceleration as change in speed over time, and then rewrote the equation and said it was a more accurate version of newtons second law, could anyone here take a look at that and explain to me what he means? I thought acceleration was mathematically defined as a=(v-u)/t where v and u are final and inital velocity since acceleration is a vector quantity.
safe radishBOT
https://www.youtube.com/watch?v=xrK4LCL8oZY
In this video at **time stamp 12:22*...
covert rain
a = dv/dt
its the definition
and this formula describes how the speed changes as time passes
lavish magnet
its the definition
what does d in the formula mean, or do you mean delta the little triangle?
lavish magnet
and this formula describes how the speed changes as time passes
But doesn't the acceleration formula talk about velocity not speed since acceleration is a vector?
covert rain
i use translator
rare olive
But doesn't the acceleration formula talk about velocity not speed since acceler...
a=(v-u)/t is correct but thats about values, not vectors
tiny veldt
But doesn't the acceleration formula talk about velocity not speed since acceler...
only velocity bro
lavish magnet
a=(v-u)/t is correct but thats about values, not vectors
whats the difference
tiny veldt
if u derivate speed u get 0
rare olive
v-u is the change in speed, which are only numbers
dv means change in velocity but velocity is speed+direction
You can think of speed as the magnitude of velocity
lavish magnet
only velocity bro
but isn't acceleration also a vector?
rare olive
yes it is
lavish magnet
dv means change in velocity but velocity is speed+direction
ohhh that makes a lot more sense
tiny veldt
but isn't acceleration also a vector?
yes
tiny veldt
ohhh that makes a lot more sense
i meant speed is not a vector .
tiny veldt
i have one more question
sure
lavish magnet
https://www.youtube.com/watch?v=xrK4LCL8oZY
In this video at **time stamp 12:22*...
in the video he rewrites newtons second law like this what does that mean and what is he trying to show?
azure harbor
<@&268886789983436800>
tiny veldt
in the video he rewrites newtons second law like this what does that mean and wh...
delta mv is is momentum
2nd law
lavish magnet
<@&268886789983436800>
lavish magnet
delta mv is is momentum
so newtons second law can be rewritten to show change in momentum also?
<@&268886789983436800>
tiny veldt
I thought it was p=mv or thats how my book shows it
F=ma just substitute a=dv/dt
thats it
lavish magnet
okay that answers all my questions, thank you for your help.
tiny veldt
np 👍
safe radishBOT