feral light
pls
#help-17
white acorn
Find the highest value in terms of n for y(1+x) for 0<y<1 and x+2y=n
white acorn
(gonna be gone for a while but ping me if u know how, I'm actually confused)
There's probably a really easy method I can't see
My visualization is not working today
vocal sleetBOT
@white acorn Has your question been resolved?
blazing depot
i do not understand this
vocal sleetBOT
i do not understand this
viral copper
Differential
viral copper
yeah but that's d(tan2x)/dx
mild flower
this is doing the chain rule in an... interesting way
viral copper
$$\dv{\tan 2x}{x} = 2\sec^2 2x$$
$$\dd{(\tan 2x)} = 2\sec^2 2x \dd{x}$$
twin meteorBOT
jan Nejon
viral copper
You can "multiply"*** the dx on both sides
The only reason I feel comfortable saying that is because Hayley is right here
viral copper
yeah basically that
viral copper
wdym
mild flower
d(thing) only makes sense with respect to some other thing, often that's dx
$\dv{y}{x}$ is the same as saying $\dv{x} y$
twin meteorBOT
jan Hayley
viral copper
isnt it dy/dx
Yeah but y isn't defined here
You can let y = tan2x and find dy
You end up with the same thing
blazing depot
thanks alot
i am taking AP calculus this year
self studying
gonna have tough time
mild flower
this is a weird way to write it i'll say that
you might see like $(\tan 2x)' = \sec^22x\cdot(2x)' = 2\sec^22x$ as you said earlier
twin meteorBOT
jan Hayley
blazing depot
👍
mild flower
dx just marks the variable we're differentiating with respect to
vast shale
Hello, id need help with this to visualize and construct this
"Construct a section of the cube along the plane given by the points. Describe the construction of the cut."
vocal sleetBOT
Hello, id need help with this to visualize and construct this
"Construct a secti...
vocal sleetBOT
@vast shale Has your question been resolved?
hasty lark
Draw a plane that intersects all three points
Or, on every face that shares two points, connect those points with a line
Then connect the lines with a plane
hasty lark
Well, it's a rough drawing but the idea is there
I think describing the cut is more important than a hand drawn thing
vast shale
uh ill be honest im lost
i have no idea about geometry
i drew this
as all three points are on the same side
could i just simply connect them
or no
?
@hasty lark
sorry if the pings bother lmk
but like i cant extend it to make it 4 points
cuz they would connect
vocal sleetBOT
@vast shale Has your question been resolved?
vast shale
no
vocal sleetBOT
@vast shale Has your question been resolved?
vocal sleetBOT
@vast shale Has your question been resolved?
vocal sleetBOT
vocal sleetBOT
Available help channel!
vagrant moat
$x$ and $y$ are two real numbers such that:
$1 \leq x^2 + y^2 - xy \leq 2$
Show that: $\frac{2}{9} \leq x^4 + y^4 \leq 8$
vocal sleetBOT
$x$ and $y$ are two real numbers such that:
$1 \leq x^2 + y^2 - xy \leq 2$
Show ...
twin meteorBOT
gkeocog
vocal sleetBOT
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@vagrant moat Has your question been resolved?
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Available help channel!
small pumice
Can anyone explain to me why the answer is -2r and I got -32r
vocal sleetBOT
Can anyone explain to me why the answer is -2r and I got -32r
small pumice
?
iron flame
that picture is realllyy blurry
i cant discern what those exponents are but yeah it should -2
just looking at the constants, its (8)*(1/2)^2
small pumice
could you explain this in a way I could understand, please?
outer warren
a clearer image is needed for a proper explanation
outer warren
yes, can you show your work leading to -32r
outer warren
why are you multiplying
r^7 with 2r^4
small pumice
after I move the 2r-4 to the numerator
outer warren
$\frac{1}{2r^{-4}} \redneq 2r^4$
small pumice
I need to multiply them
twin meteorBOT
ℝam()n()v
outer warren
$\frac{1}{2r^{-4}} = \frac 12\cdot \underbrace{\frac{1}{r^{-4}}}_{r^4} = \frac{r^4}{2}$
twin meteorBOT
ℝam()n()v
small pumice
I don’t understand
outer warren
which part
small pumice
Don’t we need to move the 2r^-4 from the denominator I to the numerator
small pumice
Don’t we need the exponent to be positive?
outer warren
the power of -4 only applies to the r
and from negative exponent laws, that 1/r^{-4} = r^4
the 2 in the denominator is not affected
there is no negative power being applied to the 2 and remains there
as demonstrated above
1/(2r^(-4)) is the product of 1/2 and 1/r^(-4)
small pumice
I don't get this
outer warren
$\frac{1}{ab} = \frac1a \times \frac1b$
twin meteorBOT
ℝam()n()v
twin meteorBOT
ℝam()n()v
outer warren
do you have any issues with the two above images
small pumice
um
small pumice
could you try explaining how to solve the problem step by step, so it will be easier for my to understand the mistake I made?
outer warren
i'm litereally pin pointing the exact part you're making the mistake
your mistake was turning $\frac{\text{stuff}}{2r^{-4}}$ into $\text{stuff} \times 2r^{4}$
twin meteorBOT
ℝam()n()v
outer warren
the exact contents of the numerator isn't really important here, so I'm focussing on just
$$\frac{1}{2r^{-4}}$$
twin meteorBOT
ℝam()n()v
small pumice
**ℝam()n()v**
How come, don’t we need to turn the negative into positive?
outer warren
and as I've asked earlier, do you have any issue with
$$\frac{1}{ab} = \frac1a \times \frac1b$$
twin meteorBOT
ℝam()n()v
small pumice
**ℝam()n()v**
I understand this
outer warren
ok applying that to your question
$$\frac{1}{2r^{-4}} = \frac12 \times \red{\frac{1}{r^{-4}}}$$
twin meteorBOT
ℝam()n()v
outer warren
do you have any issue with this
small pumice
No
outer warren
the negative exponent law allows you to express $\frac{1}{r^{-4}}$ as $r^4$, \
that has no effect on the $\frac12$
twin meteorBOT
ℝam()n()v
vocal sleetBOT
@small pumice Has your question been resolved?
frozen carbon
vocal sleetBOT
frozen carbon
i didn't get it right could someone explain the correct steps
vocal sleetBOT
@frozen carbon Has your question been resolved?
frozen carbon
<@&286206848099549185>
shy willow
not sure what your working is- i cant rlly understand
what did you do for the first transformation?
reflection on x?
frozen carbon
ya
frozen carbon
yes but what did you do to f(x)
wait wdym
in the transformation
i substituted f(x) into the transformation
shy willow
i substituted f(x) into the transformation
ok assume y = fx)
i transform f(x) via a reflection along the x axis. what is the new function?
frozen carbon
ok assume y = fx)
i transform f(x) via a reflection along the x axis. what is th...
-f(x)
shy willow
correct
frozen carbon
what abt horizontal stretch via factor 2 added to that
-f(1/2x)
frozen carbon
-f(1/2x-3)
shy willow
-f(1/2x-3)
close. the shift is applied to x, so it would be -f(1/2(x-3))
frozen carbon
oh yes i was going to fix it to that
shy willow
oh yes i was going to fix it to that
what abt a shift up by 1?
frozen carbon
-f(1/2(x-3)+1
shy willow
-f(1/2(x-3)+1
what is this?
frozen carbon
oh wait so the transformation isn't square rooted?
shy willow
oh wait so the transformation isn't square rooted?
wdym?
the sqrt() is just an operation applied to x
just like + 5, or any other arithmetic
frozen carbon
oh i see
shy willow
so what is your answer?
frozen carbon
what is this?
i'm trying to figure it out but i'm still kind of confused
y = -x+1
y = -1/2(x+1)
y = -1/2x+1/2
y = -1/2(x-3)+1/2 - 5
that's my progress so far
shy willow
y = -x+1
y = -1/2(x+1)
y = -1/2x+1/2
y = -1/2(x-3)+1/2 - 5
that's my progress ...
1/2 only applies to x, not (x+1)
also dont forget the square root.
also wheres the -5 in the other lines?
shy willow
y = -x+1
y = -1/2(x+1)
y = -1/2x+1/2
y = -1/2(x-3)+1/2 - 5
that's my progress ...
also note the negative applies to the whole function, so to the sqrt() and the -5
for a function fx), x is the input and this is the ouput:
when a transformation is applied inside the f(), jts applied to x
if applied outside, its applied to the whole function
frozen carbon
okay i'll try to do it again
shy willow
okay i'll try to do it again
-f(1/2(x-3)+1
this answer you gave me is correct- you can apply this directly
whats inside the () used to be x but now is 1/2(x-3)
so you can substitute that directly
as for what was applied to f(), you made it negative (multiply by -1) and added 1, which can then be done to the whole equation
frozen carbon
ok wait so
frozen carbon
as for what was applied to f(), you made it negative (multiply by -1) and added ...
is this starting point good
or do i add a - 5 at the end
oh wait hold on
shy willow
is this starting point good
yes but the -5 has to be there and also multiplied as part of f(x)
shy willow
is this starting point good
yes good starting point. dont forget the -5
remember it changed when f(x) -> -f(x)
shy willow
sorry for taking a while
dont be dw
shy willow
is this right?
the negative doesnt go into the square root. treat the square root like a bracket when multiplying by numbers (like -1 in this case)
so it would be 2x + 1 in the sqrt in the first step
in the second step you forgot the 1 all together
you seem to have a good grasp on the way transformations work, but work on your algebraic manipulation and application of the transformations to functions
minor mistakes tho
frozen carbon
ohh okay let me fix it then
frozen carbon
you seem to have a good grasp on the way transformations work, but work on your ...
is this better
i just noticed that i forgot the +1
i also have one more math question if you don't mind helping me with that
shy willow
is this better
correct!
frozen carbon
thank you! it's just question b and c
the letter i tho i dont need help w the description
shy willow
thank you! it's just question b and c
they want you to rewrite f(x) as a single square root
so take the constant (1/3) into the square root
do you remember how to do that?
frozen carbon
is it like the steps we just did?
shy willow
is it like the steps we just did?
not really. they want it to be equivelant to f(x) so theres no transformations
its just using properties of roots to rewrite somethng like Asqrt(b) as sqrt(c)
frozen carbon
do you remember how to do that?
im not sure how to get it into the square root actually
shy willow
im not sure how to get it into the square root actually
numbers can only be combined into single square roots if they are both square roots
like thjs
see now theyre both in the square root
but you have 1/3 which isnt a square root
so you need to re write it as one
frozen carbon
could you like give me an example similar to the equation so i can see how it would work
shy willow
could you like give me an example similar to the equation so i can see how it wo...
square root operations square roots numbers, so when removing the square root ( or taking it out of the sqrt), the number needs to be square rooted. the opposite for when putting it back in. this is thr reverse of square rooting, so you need to square it
generally:
shy willow
could you like give me an example similar to the equation so i can see how it wo...
does this make sense?
shy willow
sorry for the late reply i had to do smth
nw
frozen carbon
generally:
in the square root it wild be smth like 1/3^2(4x) then if it's like this?
shy willow
so like sqrt 4/9x?
yup!
shy willow
and for b isn't the equation already in that form?
no because they want only x in the square root
so you need to seperate sqrt(x)
frozen carbon
so is it like 1/3(4)^2 sqrt x?
ouu i dont think i did. it right
oh wait i got. it now
1/3 times sqrt 4 times sqrt x
and that equals 2/3 sqrt x
okay i got. it now tysm for the help!
.close
naive ore
vocal sleetBOT
twin meteorBOT
naive ore
Hi can someone just help me on this because I have no idea how or where to start
waxen hawk
for sure
naive ore
tysm
next bronze
mint remnant
Let me try
I’m not a helper lol but I can do some calculus
How would we figure out A?
The blades are 65m long so that’s involved somewhere
naive ore
i have no idea i used the formula
but i don’t think i’m right
cus we want dP/dt? right
so we would need dA/dt
i’m going to cry
mint remnant
I doubt we need dA/dt
naive ore
why?
naive ore
does the A not change with time?
mint remnant
Why would it silly
The size of the wind turbine does not change
Sorry
A equals 0.01285
Let me try with 6am
Okay it’s confirmed, A is 0.012755
It varies since the power produced is not 100% efficiency
But the size remains relative the same in all times
We can substitute it into the power equation
mint remnant
Hold on A might not be needed
naive ore
what does that do
mint remnant
We might be able to substitute that into the power equation
Make it power as a function of time
Power=… with t instead of v involved
naive ore
oh
mint remnant
A equals around 0.0012755
naive ore
how did u get that
mint remnant
You can check that by substituting the values from the table into the power equation
Small p=1.225
Power=0.5
v=4
At 6am ^^
naive ore
wouldn’t you have to derive the power equation first though?
mint remnant
dP/dv?
naive ore
yes bc ur looking for the rate of change in power
and since Area isn’t constant?? wouldn’t you have to do dA/dt?
mint remnant
dA/dt means rate of change of area with respect to time
You would need to get a function of A in terms of time for that
Which I doubt is possible. But, I’m certain you can assume the area of the wind turbine does not change over time
naive ore
No 😭 it’s just an assignment
mint remnant
Ooof
Okay so what I did is, integrate the 0.0015
You get v=0.0015t+c
Now we will make it so that ‘t’ is the time since 6am in hours
we can substitute v=4 and t=0
We get that c=4
Now we have v=0.0015t+4
Substituting this into the power equation, we get:
Power=0.5x1.225x0.012755x(0.0015t+4)^3
We can differentiate it so that it becomes dP/dt
Then we substitute t=0 into dP/dt
Which turns out to be 0.0005625 megawatts per hour
Not sure if that’s what they’re asking for but here it is
naive ore
wait i think the area is just A = pi * r^2
mint remnant
Yes but
No info on that
Just gives us the length of the blades :p
Wait no
0.0005625 megawatts per second
You can convert that into hours
2.025 megawatts per hour
That makes more sense
naive ore
ohh
mint remnant
Again, I made a lot of assumptions but that uses all the information given
The 0.0015, the table and the equation
vocal sleetBOT
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vocal sleetBOT
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charred socket
is this correct?
vocal sleetBOT
is this correct?
strong grove
yeah so far correct
charred socket
is this right and how do I work out c?
vocal sleetBOT
is this right and how do I work out c?
@charred socket Has your question been resolved?
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buoyant ravine
This is false right
vocal sleetBOT
buoyant ravine
.close
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long flame
i couldn't understand how to solve this problem, any advice
vocal sleetBOT
i couldn't understand how to solve this problem, any advice
long flame
i tried plotting different coordinates and couldn't see any obvious relationship, i tried to relate it to pythagoras but the equation i got was completely wrong
above is textbook answer, below is my attempt lol
vocal sleetBOT
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@long flame Has your question been resolved?
long flame
<@&286206848099549185>
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@long flame Has your question been resolved?
crisp saddle
Try drawing the scenario. Fix points A and B. Since angle B (meaning B is the ``corner") is fixed, the point C's that satisfy the scenario should form a line. Find the slope of that line using the slope of line connecting A and B. Hope the relationship will emerge there.
Note that for perpendicular lines x and y, $m_x = -\frac{1}{m_y}$ where $m_x$ and $m_y$ are slopes of x and y, respectively.
twin meteorBOT
poypoyan
vocal sleetBOT
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buoyant ravine
Is it correct to say: Char(R) = Char(Q(R))
vocal sleetBOT
Is it correct to say: Char(R) = Char(Q(R))
@buoyant ravine Has your question been resolved?
lapis bough
vocal sleetBOT
regal slate
,rccw
lapis bough
twin meteorBOT
hybrid flicker
factor the denominator
lapis bough
Cosx-sinx?
hybrid flicker
i'm not asking you to revert back to cos2x
hybrid flicker
cos^2 x - sin^2 x
without using formulas related to cos or sin, factor this
hybrid flicker
Not at all
lapis bough
No it’s not
lapis bough
Of cos2x?
hybrid flicker
you have (a+b)^2 = ...
you should have been taught three identities like this
find the one that corresponds here
lapis bough
(A-b)^2 = a^2-2ab+b^2
(A+B)^2 = a^2 +2ab+b^2
A^2-B^2 = (a+b)(a-b)
So is it (sinx+cosx)(sinx-cosx)?
vocal sleetBOT
@lapis bough Has your question been resolved?
vast shale
yooo
lapis bough
Hello
vast shale
what do you need help with?
vast shale
oh, trignometry
lapis bough
Yh exactly
lapis bough
Take your time
vast shale
thanks
swift dock
vocal sleetBOT
@lapis bough Has your question been resolved?
hybrid flicker
So is it (sinx+cosx)(sinx-cosx)?
this is it yes
now bring the left hand side to a single fraction
then, notice the 2sinxcosx on the numerator (looks a lot like 2ab)
hybrid flicker
(A-b)^2 = a^2-2ab+b^2
this should, again, remind you of one of the identities you wrote here
vocal sleetBOT
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Available help channel!
sinful magnet
I need to complete this using completion of squares
vocal sleetBOT
I need to complete this using completion of squares
outer warren
start by factoring out 9 from
9x^2 - 30x
or the whole expression
sinful magnet
start by factoring out 9 from
9x^2 - 30x
or the whole expression
do i apply the same method
or is it gonna be tough
sinful magnet
im supposed to get this as an answer
sinful magnet
same method is applicable
where did i go wrong
outer warren
you didn't distribute the 9 to the 18/81
sinful magnet
so i do a double parentheses
sinful magnet
and how does it simplify to this
outer warren
distribute the 9 and first simplify a few fractions
sinful magnet
do i expand
outer warren
distribute the 9 only
sinful magnet
i did
outer warren
what do you have now?
sinful magnet
it gives 9(...) ^2+2
outer warren
whats inside those ()
sinful magnet
x-15/9
outer warren
simplify the fraction
sinful magnet
outer warren
note that 9 = 3^2
which can then be multiplied into the (x - 5/3)^2
applying the exponent law
p^n * q^n = (pq)^n
outer warren
which denominator
vocal sleetBOT
@sinful magnet Has your question been resolved?
outer warren
did you apply the property i mentioned?
sinful magnet
yes
outer warren
show your work for that
sinful magnet
outer warren
no
sinful magnet
ahh
outer warren
$p^2q^2 \redneq pq^2$
twin meteorBOT
ℝam()n()v
outer warren
yes
full vessel
vocal sleetBOT
full vessel
help 😭
tidal umbra
do you know anything about rationalising the denominatior.
full vessel
THANKS
vocal sleetBOT
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full vessel
.close
kind light
how do you even start? idk what do do eith the area of the triangle BDF
vocal sleetBOT
how do you even start? idk what do do eith the area of the triangle BDF
@kind light Has your question been resolved?
kind light
<@&286206848099549185>
vocal sleetBOT
@kind light Has your question been resolved?
vocal sleetBOT
@kind light Has your question been resolved?
worn forum
vocal sleetBOT
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'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above
1
thin vale
What have you tried?
worn forum
i got part a
but idk how to do part b
without manually adding the 20 numbers together
vapid trout
I want to find the price after adding 42.12% to 3.16 = A
vocal sleetBOT
I want to find the price after adding 42.12% to 3.16 = A
thin vale
42.12% of what?
vapid trout
3.16
thin vale
please post a picture of your original question
vapid trout
It does not have an original picture, it is for own purposes
I am bit confused of it
thin vale
what is the purpose?
vapid trout
Businesses
thin vale
Okay
so
3.16 is the price
you want to know what the price is after you add 42.12% of 3.16 to 3.16
is that correct?
thin vale
??
vapid trout
Why
formal yew
oh
ye
17.12% + 25%= 42.12%
idk why u included that but sure
yk. if i were i would just use a calc.
vapid trout
Both 5.46 and 3.81 are based on 3.16
formal yew
vapid trout
I will enter 5.46 to the system, and deduct 25%. But it result 4.10 not 3.81
formal yew
bro what are u yapping about
formal yew
I will enter 5.46 to the system, and deduct 25%. But it result 4.10 not 3.81
the result is 4.095
idk how u got 3.81
vapid trout
I mean I wanted it to be 3.81
formal yew
so u just wanted it to be
vapid trout
How can I calculate based on 5.46
vapid trout
yes
vapid trout
How do you calculate?
formal yew
1.65 is what percent of 5.46
1.65 divided by 5.46
thats equal to 0.30219780219
0.30219780219 times 100% = 30.219780219%
vapid trout
Is there any other way without using 1.65
vapid trout
In excel, I cant get 1.65
formal yew
i guess u could do
3.81 is what percent of 5.46
and then do 100% - 3.81 is what percent of 5.46
3.81 is 69.7802198% percent of 5.46
so 100%- 69.7802198% = 30.2197802%
vocal sleetBOT
Available help channel!
stiff ocean
how do i convert this to polar form?
vocal sleetBOT
how do i convert this to polar form?
stiff ocean
i’m evaluating z^5 btw
flat whale
,tex .demoivre
twin meteorBOT
riemann
flat whale
i’m evaluating z^5 btw
Use ^
regal bane
stiff ocean
how could i do this?
vocal sleetBOT
how could i do this?
paper depot
dead ringer for de moivre's lol
flat whale
how could i do this?
Didn't I already give you that formula
flat whale
how could i do this?
stiff ocean
Didn't I already give you that formula
oh i use that again
ah damn ok got it
thanks
.close
visual oracle
f is a map from X to Y with $U, V\subseteq X$. Check if $f(U\cup V)=f(U)\cup f(V)$
vocal sleetBOT
f is a map from X to Y with $U, V\subseteq X$. Check if $f(U\cup V)=f(U)\cup f(V...
twin meteorBOT
ed
vocal sleetBOT
@visual oracle Has your question been resolved?
visual oracle
I know you have to show that the first is a subset of the other and vice versa but I kinda want to see a written proof so I can know what is right
winter hawk
try it yourself first. that will draw out errors and misconceptions
visual oracle
try it yourself first. that will draw out errors and misconceptions
i don't know where to start is the problem
i start with x is an element of f(UuV)
and then idk what to do
it seems obvious to me why its true but idk how to mathematically word it 💀
wintry quiver
💀
what just happened
winter hawk
@wintry quiver channel occupied
wintry quiver
<@1104660405088948364> channel occupied
oh ok which channel can I ask questions im new here
visual oracle
#❓how-to-get-help @wintry quiver
winter hawk
visual oracle
anyways did you see my previous messages
i already tried
literally wrote one line
x is an element of f(UuV)
or maybe the question is just like is this statement true because its so obviously true?
idek
winter hawk
lets use y instead of x
visual oracle
true
winter hawk
let y be in f(U cup V)
visual oracle
yes
winter hawk
by definition what does that mean?
visual oracle
f(x), x \in set
winter hawk
state that more carefully
winter hawk
ik what you mean to say but say it carefully
visual oracle
like y belongs in the set {f(x), where x \in set}?
winter hawk
there exists x in set st (such that) y=f(x)
winter hawk
now go back
visual oracle
U and V are nonempty
winter hawk
what does y in f(U cup V) mean?
visual oracle
so that means there exists x in U st y=f(x)
winter hawk
no
visual oracle
oh wait
wait what
wdym
go back?
ok i'll go back
y in f(UcupV) means that there exists x in UcupV st y=f(x)
winter hawk
keep in mind we eventually wanna say y is in f(U) cup f(V)
visual oracle
that means x is in U or x is in V
visual oracle
cases?
winter hawk
ok sure
winter hawk
pick a case
visual oracle
say x is in U
winter hawk
what can we say in this case?
winter hawk
what justifies that?
visual oracle
uh
oh wait
ok so what do we have
we have there exists x in UcupV st y=f(x)
we assume x is in U
wait that doesn't justify that
bruh
wait what why am i so confused
winter hawk
you sure?
visual oracle
this is hurting my brain
winter hawk
how is f(U) defined?
visual oracle
wait no thats why we took cases
nvm it does
its just for this case it's true
f(U) is the set {f(x), x in U} right
visual oracle
bruh
im so dumb
ok true
and then similar for the other one
hence its true
bruh
well actually we have to do the other way
winter hawk
hold up we need y to be in the union
visual oracle
huh
winter hawk
i left off at y is in f(U)
winter hawk
remember we need to say y is in f(U) cup f(V)
winter hawk
you need to finish here in each case
winter hawk
in each case we need to say y is in f(U) cup f(V)
visual oracle
yeah ok
yes?
thats what you meant
wait yeah you're right that makes sense
you're actually a good teacher
this is good
ok
winter hawk
from y being in f(U), how can we say y is in f(U) cup f(V)?
visual oracle
truth table
that was proved earlier i think
A implies A or B
i think
wait
im having as troke
i need to think
winter hawk
X is a subset of X cup Y
visual oracle
yeah thats right isn't it
winter hawk
which on the truth level is the same as what you said
visual oracle
yes it is
i thought i said something wrong
nvm
the statements are just like x is in X and x is in Y
and then its true
right
yes
yes
ok
winter hawk
the thing is that fact (and basically all others) can and should be proven without truth tables
visual oracle
oh
shit
wait but like isn't it obvious 💀 i hate this rn
like if something is in X of course its in X or some other thing
winter hawk
tables are kinda fine for the first few weeks but eventually they just get in the way of how logic naturally flows
winter hawk
eg i define "P or Q" as true precisely when at least one of the statements is true
winter hawk
"P implies Q" is true precisely given: whenever P is true, Q is also true
to prove P implies Q, temporarily assume P then show Q must also be true
winter hawk
no
visual oracle
yeah i think theres a fundamental misunderstanding with me
winter hawk
well if you mean P being false implies the implication then sure
visual oracle
yeah thats what i meant
visual oracle
but its also true when P is false
winter hawk
yes but those arent the only cases where implies holds
visual oracle
yeah
winter hawk
when i define things i state precisely when they hold
so heres the theorem P implies P or Q
proof: assume P is true. then no matter whether Q is true or false, P or Q holds
visual oracle
yeah
winter hawk
no tables here
winter hawk
i dont even case about the case P is false
im following the structure of proving implications
to prove P implies Q, temporarily assume P then show Q must also be true
visual oracle
yes but the structure of proving implications works like that
because if P is false anyways
the logical value of Q doesn't matter (in P implies Q)
right?
the last sentence
thats why we prove A implies B by assuming A is true because everything else is true anyways
then you have to show B is true if A is true thats just it
winter hawk
if youre concerned about truth tables, then the result holds anyway if P is false
visual oracle
yes thats my point
winter hawk
the only case we care about establishing is the one where P is true
winter hawk
thats what justifies the more natural approach to proving implications
winter hawk
only care about the case where the "if" is true
visual oracle
yeah
winter hawk
and hopefully you can start shifting away from tables very soon
now on the set level, this is just "X is a subset of X cup Y"
we can just apply that here
we had y is in f(U), and f(U) is a subset of f(U) cup f(V), so y is in f(U) cup f(V)
winter hawk
NOW the first case is done
winter hawk
is the other case clear to you?
visual oracle
what yeah
its the exact same thing
just with V
if x is in V
then y is in f(V) because y=f(x)
f(V) is a subset of f(U)cupf(V)
hence y is in f(U)cupf(V)
winter hawk
yeah
visual oracle
yeah
ok now lemme try other one
Does this work
Didn’t do the other case but yeah
winter hawk
,rccw
twin meteorBOT
winter hawk
some bits are weird
twin meteorBOT
e
winter hawk
im gonna go through the whole thing anyway so you see what i find weird
visual oracle
thats good
visual oracle
im 17 i don't have anyone to teach me
what that means y is in f(U) or y is in f(V)
i have to write that.
winter hawk
thats an "or", how do you handle it?
winter hawk
pick a case
visual oracle
thats what i did right
y it in f(U)
y is in*
i have to write all that ig
hmm
yes
winter hawk
yeah what you should do is collect what we said and format that into a single paragraph. then compare it with the first attempt
ok now we're in the case y is in f(U). what does it mean?
visual oracle
there exists an x in U st y=f(x)
visual oracle
yeah so im saying if there exists an x in U
there exists an x in U cup V because U is a subset of U cup V
winter hawk
ok proof language
once we say there exists x st blah
from then on we can just talk about x without "there exists"
both you and the reader know x is a specific object with some property
for example you can look back to how i talked about x when we were proving the other subset relation
visual oracle
oh right
x_0 and x_1 is stupid basically
all i have to do is just say x is in Ucup V now
yes?
winter hawk
ok and why is that true?
visual oracle
because U is a subset of UcupV
wait what i think im gonig off of intuition too much
winter hawk
yes but ALONG with x being in U
winter hawk
i think at the stage youre at itd help to put justification for each line
so this line goes "since x is in U and U is a subset of U cup V, x is in U cup V"
winter hawk
now what do you say next?
visual oracle
oh
winter hawk
dont forget to bring in the justification in the same line
visual oracle
i just say y is in f(UcupV)
winter hawk
justify that
visual oracle
because x exists where y=f(x) with x in U cup V
winter hawk
nope no "exists"
visual oracle
oh because i already said that earlier
because y=f(x) with x in U cup V
thats it
winter hawk
yep so the line is "since y=f(x) and x is in U cup V, y is in f(U cup V)"
winter hawk
thats this case done
winter hawk
im very glad this helped, and ik it can feel pretty nitpicky in some bits but its kinda expected in intro logic/proofs
visual oracle
its good
ifl i have the ideas i just can't express it
this helped
communication is important
so yeah
winter hawk
its definitely the grammar of an entire other language youre learning that can be hard
not just the vocab
visual oracle
wise words
winter hawk
anyway once you finish the other case you should transcribe what we said. dont forget to follow the format ive suggested (dont repeat "exists", justify each new statement in the same line, clearly label cases and indicate when you start/finish one, etc)
vocal sleetBOT
@visual oracle Has your question been resolved?
merry prawn
hello!
I've asked this yesterday but i still cannot solve it
vocal sleetBOT
hello!
I've asked this yesterday but i still cannot solve it
merry prawn
in the video they show that you can recognize it as a taylor series and go from there.
But what if you don't recognize it?
Ann told me this:
so i was trying the second method
and i got this far:
2/n*(4/n*Sum(i^2) + Sum(-1))```i found out about the sum of first n squares which i was not aware of. and i get this for the Sum(i^2)
(n^2-n)(2n-1)/6
ad for Sum(-1) it should be
-(n-1)
but now i have this big expression and i'm not sure what this represents: i think I can take the limit for n>infinity
vocal sleetBOT
@merry prawn Has your question been resolved?
cyan lagoon
like multiplying 2 equations together to solve the system basically
maybe i learned this a long time ago but i forgot lmao
river minnow
Why should it be not allowed?
vocal sleetBOT
vocal sleetBOT
Available help channel!
vast shale
is it possible to integrate cos^2(x) by parts
vocal sleetBOT
is it possible to integrate cos^2(x) by parts
hard atlas
well try it
hard atlas
yes. so you can't
vast shale
but you can with sin^2(x)?
i swear i could remember seeing it somewhere
so basically i need to memorize trig identities to do this integral
hard atlas
yeah
vocal sleetBOT
vocal sleetBOT
Available help channel!
sweet crown
vocal sleetBOT