what about (1,3) and (3,1)
#help-36
sleek quartz
the 3-sided spinner will never give you 3
so (3,1) will never happen
which means order does matter actually mb
gusty slate
oh so basically when the two things have the same outcomes
like rolling two dice
then you can have the flipped case
because the dice all have 1,2,3,4,5,6 on its sides
but if it were two dice but with different numbers on the sides it could not be flipped?
sleek quartz
for {1, 2, 3, 4, 5} and {2, 3, 6, 7} you can have both (2, 3) and (3, 2)
you can always have the flipped case as long as they have multiple numbers in common
sleek quartz
but to not confuse (or double count) (1, 7) and (7, 1) you should always keep order
👍
gusty slate
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tranquil pine
What's the notation for a and b both divisible by c? (c|a) ∧ (c|b) I used c | (a, b) but some people use (a, b) for GCD notation and read it as the GCD divided by c.
final saddleBOT
What's the notation for a and b both divisible by c? `(c|a) ∧ (c|b)` I used `c |...
pliant shore
What's the notation for a and b both divisible by c? `(c|a) ∧ (c|b)` I used `c |...
c | (a, b) is ambiguous notation
So you should just say (c|a) ∧ (c|b) or something
tranquil pine
what is the other something?
pliant shore
what is the other something?
Like a = mc and b = nc for $m, n \in \mathbb N$
soft zealotBOT
south
pliant shore
Oh right that's also possible
pliant shore
?
Yeah you're looking for $a \equiv b \pmod c$
Wait no
That's they leave the same remainder
So mod wouldn't be applicable here
empty junco
i think $a \equiv 0 \pmod c$
soft zealotBOT
rayn422
empty junco
same for b
pliant shore
Okay yeah that's how
tranquil pine
yeah but the point is I want a short notation premise for c divides both a and b
pliant shore
yeah but the point is I want a short notation premise for c divides both a and ...
It's useless to do that if other people don't know what you mean by your notation
Or worse, think you mean one thing when you mean something else
Just use words
c divides both a and b
There you go
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pliant shore
Nwnw
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royal schooner
what does the dx at the end of an integral represent
final saddleBOT
what does the dx at the end of an integral represent
@royal schooner Has your question been resolved?
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@royal schooner Has your question been resolved?
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@royal schooner Has your question been resolved?
lunar skiff
what does the dx at the end of an integral represent
short answer: derivatives and integrals are opposite to each other, thats why with derivatives u divide by dx, and with integrals u multiply by it
lunar skiff
short answer: derivatives and integrals are opposite to each other, thats why wi...
although not everybody would agree with that
royal schooner
hmm is there a much deeper answer behind that which isnt normally covered in calc 1
lunar skiff
it has to do with how u calculate the integral or the area under a curve by dividing the area into smalle rectangles (formally they would call it partitions)
lunar skiff
https://en.wikipedia.org/wiki/File%3ARiemann_Integration_and_Darboux_Upper_Sums....
so, u would divide the area into smalle rectangles and each rectangle has a width of ∆x and a hight of f(x)
now in order to have a more accurate results u need to increase the number of rectangles u have
lunar skiff
https://en.wikipedia.org/wiki/File%3ARiemann_Integration_and_Darboux_Upper_Sums....
as u can see in the gif, the more rectangles the more it will cover of the area
so basically u need an infinite number of rectangles, and when ever u want to use infinity u will use limits
(sorry for quality)
lunar skiff
@royal schooner , i think u have fallen from me 😂
royal schooner
sorry was afk
im wondering how is it acceptable to say when du/dx = 2x, write du = 2xdx and substitute that in the integral when neither dx or du are numerical quantities
lunar skiff
im wondering how is it acceptable to say when du/dx = 2x, write du = 2xdx and su...
well, the fact that it looks like fractions and u can use fractions operations on, doesn't mean that its fractions
lunar skiff
im wondering how is it acceptable to say when du/dx = 2x, write du = 2xdx and su...
it behaves like fractions cuz there is other theorys that u are actually using without knowing
lunar skiff
on khan academy they did just that
its fine for first derivative
royal schooner
it behaves like fractions cuz there is other theorys that u are actually using w...
this is exactly what i thought! if i were to learn this myself, what topics would i google
tho i will just drop it if its way beyond me
lunar skiff
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silent stirrup
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hybrid heath
,rccw
silent stirrup
So when they ask for FEH. I dont really know how to find that cause its such a wierd angle
soft zealotBOT
hybrid heath
It would be 360 - angle FDH
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stuck harness
Based on the trend the correct answer should be C right?
final saddleBOT
Based on the trend the correct answer should be C right?
stuck harness
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plain stone
I want to ask about 3
final saddleBOT
I want to ask about 3
plain stone
im sort of confused because it sort of seems obvious that this holds
im not sure how to reason about it
sorry that was a bad question
desert mantle
put into words why its obvious
plain stone
i really cant
it seems just very obvious
the right hand side is likely going to always be larger
unless y = x
in which both sides are equal
desert mantle
but why is it going to be bigger
plain stone
because you might be adding 2 diameters that exceed the diameter of x, z?
desert mantle
(d stands for distance)
plain stone
yeah sorry that makes a lot of sense, i mized them up
for instance the distance from x, y could be greater than the distance of x, z
desert mantle
d(x,z) stands for the shortest length of a walk from x to z
how can the right side represent a walk from x to z
plain stone
it looks like transitivity
technically it is only equivalent if
y = x
because then youre just doing 0 + the distance from x to z
desert mantle
well the graph could also look like
x --- y --- z
fiery pasture
If the graph is finite and connected (you can get to each vertex from any other) then either y lies on a path between x and z (in which case the LHS and RHS are equal) or it does not in which case the d(x,z) is closer than the walk involving vertex y.
mellow axle
technically it is only equivalent if
(there are more situations than just that when they're equivalent)
plain stone
If the graph is finite and connected (you can get to each vertex from any other)...
it does not in which case the d(x,z) is closer than the walk involving vertex y. -> this is where the greater comes in
fiery pasture
ehh i guess i need something like y lies on the shortest path between x and z or something.
could you just do the triangle by exhaustion? Case I: x-y-z, d(x,z)= 2 = d(x,y)+d(y,z) = 2. Case II: x-z-y, d(x,z)=1 < d(x,y)+d(z,y) = 2. insert vertices in appropriate places.
(oops I'm new, I see how channels work now. my bad)
desert mantle
d(x,z) is the length of the shortest walk from x to z
d(x,y)+d(y,z) is the length of the shortest walk from x to z which includes y.
clearly the first has to be <= the second, because every walk from x to z which includes y is at the very least a walk from x to z so it also counts for the first one
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@plain stone Has your question been resolved?
plain stone
d(x,z) is the length of the shortest walk from x to z
d(x,y)+d(y,z) is the lengt...
thank you
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flat fiber
im not sure what this is
final saddleBOT
im not sure what this is
fast heart
Well integration is just finding the area under the graph
If your curve is the rate of flow, and your integrating with respect to time
honest valve
Well integration is just finding the area under the graph
I don't think that's the best intuition for this question
fast heart
Well I would think about it as effectively multiplying flow of cars by time
If your graph was a straight line that would be how you find the area, integrating the curve is performing the same task
flat fiber
That makes me think that the area under curve for flow for certain amount of time would be the total number of cars then?
fast heart
So if rate of flow is cars per time, and you're multiplying by time, what is it showing?
fast heart
That makes me think that the area under curve for flow for certain amount of tim...
Exactly
flat fiber
got it ty
flat fiber
im confused whether its from 2 to 5 or just 3 hours
flat fiber
so that implies its 2am and 5am
fast heart
Yup
flat fiber
ty
honest valve
Can I suggest a different way to think about the problem?
flat fiber
sure
honest valve
When you see the flow "rate", you know it's how many cars per time (i.e number of cars/time) the integral essentially allows you to multiply the flow rate at any time by dt, so you'd end up getting #of cars/time X time
Getting just #of cars
Area doesn't always have to come into the picture
The limits tell you which times to look between
flat fiber
why is it time x time?
honest valve
Sorry (#of cars/time)xtime
honest valve
I just want to help you understand that the integral is essentially and infinite sum of very small things, it tends to be a lot better when people understand in this way, rather than "it's the area"
crisp epoch
can someone explain how to solve this
final saddleBOT
can someone explain how to solve this
crisp epoch
i tried finding the equation throguh P, the focus, and Q
and then setting it equal to y^2 = 4px
but then i got a really weird quadratic
final saddleBOT
@crisp epoch Has your question been resolved?
bitter monolith
My first solution in this channel, sorry if the form doesnt meet requirement
Basically, u apply the trick on ax^2+bx+c==0 with solutions x0+x1==-b/a && x0×x1==c/a
(the structure of the solution is messy but ideal for learner not just simply copying)
U need to understand and comprehend
bitter monolith
how did u get x1x0 = p^2?
Basically, u apply the trick on ax^2+bx+c==0 with solutions x0+x1==-b/a && x0×x1==c/a
crisp epoch
vieta's?
bitter monolith
This should be basic
crisp epoch
how do you even think of using that here 💀
bitter monolith
Math is either straightforward or magic
crisp epoch
where did the p^2 come from
i know you used vieta's but wouldn't it be equal to k^2p^2
honest valve
Wow that's really nice!
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verbal gale
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verbal gale
i feel like this is false but im confusing myself
formal trail
what is confusing you?
verbal gale
so the homogeneous solution set of the span thing is a subset of the solution ax=b right?
formal trail
the corresponding homogenous system is Ax = 0
so it's asking whether the set of vectors that satisfy Ax = 0 will also satisfy Ax = b
note that every solution to Ax = b *can *be constructed as the sum of one particular solution to Ax = b, with any solution to Ax = 0
verbal gale
hmm okay while the homegeneous set is not the only solution set of Ax=b, but Ax=b can be constructed with the sum of Ax=b and Ax=0?
I think i got it thank U
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distant prawn
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distant prawn
is this correct?
soft zealotBOT
static plinth
you can combine the sine together and multiply the constants together to make it more simplified like wolfram did here
reef canyon
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celest crane
reef canyon
I'm aware but is my derivative wrong?
celest crane
Yes.
reef canyon
why?
celest crane
You found the derivative of dz/dy, but not dy/dv.
And the derivative of ln(x^2 + 3xy)^-4 would become ^-5, not ^-3.
celest crane
Aside from the exponent, that part is correct.
reef canyon
I tried plugging (1,0) into it and I get a math error
celest crane
Yes, but you still need to multiply it by dy/dv.
reef canyon
I get that but computing the values of dz/dy gives a math error
celest crane
What answer are you giving?
reef canyon
celest crane
Do you have any idea why it would be giving you a math error?
reef canyon
no
celest crane
What does the ln(1) equal?
celest crane
Not part a?
reef canyon
unrelated
celest crane
No, #3 part a) part b)
Well, the first thing I would say is that z(1,0) DNE so it is not continuous at (u,v)=(0,0).
final saddleBOT
@reef canyon Has your question been resolved?
mellow owl
number 8, this has to do with exponential functions
final saddleBOT
number 8, this has to do with exponential functions
sleek quartz
have you tried setting up equations
sleek quartz
feel free to show what you tried
mellow owl
feel free to show what you tried
this is for a), the 18000 is supposed to be t/18000 as an exponent
sleek quartz
ok so for a) $\frac{18000}{5700} \approx 3.15$ half life periods have passed right
soft zealotBOT
b
mellow owl
yes
sleek quartz
and each time a half life period passes there's 0.5 times the last period's carbon
sleek quartz
that's it yeah
mellow owl
but i just don’t know how to get to it.
sleek quartz
well after 5700 years there's $y =.5^\frac{5700}{5700} = .5$ right
soft zealotBOT
b
sleek quartz
or after $2 \cdot 5700$ years there's $.5^2 = .25$ right
soft zealotBOT
b
mellow owl
yes
sleek quartz
so after 18000 years there's $.5^\frac{18000}{5700} = 0.113$ left
soft zealotBOT
b
mellow owl
times tht by 100 then boom my answer
sleek quartz
👍
sleek quartz
yeah you said something like $5700 \cdot .5^\frac{1}{18000}$
mellow owl
oh yeah
soft zealotBOT
b
mellow owl
what was it supposed to be? y= 0.5 ^18000/5700?
sleek quartz
yeah, if we take things one step at a time 18000/5700 = 3.15, so 3.15 periods have passed
$(.5)^{3.15} = 0.113$
soft zealotBOT
b
mellow owl
ohh okay
got it
so for b) it says how many years after an organism dies will the amount of C-14 that remains be 20% of its original amount
sleek quartz
try explaining to me why this gives you the correct answer
$(0.5)^\frac{t}{5700} = 0.2$
soft zealotBOT
b
sleek quartz
or $(0.5)^T= 0.2$ tbh
soft zealotBOT
b
mellow owl
**b**
how much life it remains 5700 years after its death,
if that makes sense
0.5 represents half a life, t represents the time period and then 5700 is like the start
and 0.2 is like 20% of it
sleek quartz
i would say that
- 0.5 is the half life
- t is the amount of years
- t/5700 is the amount of 5700 year periods
- 0.2 is the amount we want to have left
so in $0.5^\frac{t}{5700} = 0.2$, we want to know how many 5700 year periods get us to 0.2 left
soft zealotBOT
b
mellow owl
ohh okay
sleek quartz
at that point you just solve for t
sacred lodge
22
tough tiger
What is definition 2 ?
sacred lodge
And this is the definition
royal gust
They really want you to find the area of a √ function with pure Reimann sums, huh?
Well, you'll want to find the riemann sums expression for this function, first
mellow axle
They really want you to find the area of a √ function with pure Reimann sums, hu...
reread it lol, it says to just find the expression, not evaluate the sum
royal gust
Oh thank god
My brain was ready for a tough limit problem I guess
Anyway, I personally like to use a few rectangles to get a sense of the pattern
Because this is between 4 and 7, I suggest trying 3 rectangles, they should fit nicely
mellow axle
i think they want you to use the general expression
nvm i think you just meant use 3 as an example
sacred lodge
So what should I do
Do x1=4, x2=5, x3=6 left end points
Because I was thinking of changing the f(x) into sum of the big S
Or is that the wrong way
mellow axle
yeah your answer at the end will look like $\sum_{i=0}^\infty something$
soft zealotBOT
chebyshev's infinite pee norm
mellow axle
but you can go ahead and try a small example with 3 rectangles to start if you want
sacred lodge
For dx I got 3/n through (7-4)/n
mellow axle
Yep
sacred lodge
Right or left endpoints don’t matter right??
So it would be something like E f(x)3/n = lim n- infinity E (x^2 + sqrt(1+2x) 3/n
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@sacred lodge Has your question been resolved?
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@sacred lodge Has your question been resolved?
fluid oriole
I was on vacation and have to study off the notes we did in class
I understand a) but I do not understand b)
why did my teacher put the (1) ? how did he get there? did he just input an equation on his own so the answer would equal 4?
final saddleBOT
I was on vacation and have to study off the notes we did in class
I understand ...
blissful meadow
You want some x such that f(x) = -2.
Since we have a formula for f(x), that amounts to finding some x such that -3x + 7 = -2
Hmm
Nvm just realized it's from graph hhaha
fluid oriole
no worries lol
my friend just sent me a video i think i got it
its late asl i didnt think anyone would be on
blissful meadow
In short, draw a horizontal line at 4 on the y axis and see where it meets the curve.
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sterile trout
heres the question, i got the correct answer but my solution seems to be different than the one they've given in my book and google. so i want to know whether the solution ive used is logical
heres my solution
12y = x^3
y = x^3/12
y' = (x/2)^2
y' = 25 (at x = 10)
i.e, dy/dx = 25, meaning y cord changes 25 times faster than x cord
final saddleBOT
heres the question, i got the correct answer but my solution seems to be differe...
sterile trout
does this solution make sense logically?
faint crow
I would take the partial
sterile trout
well, tbh my solution is similar, but i dont get why they need to differentiate w.r.t to time
faint crow
You don’t need to
sterile trout
yeah
i have a doubt
how did they differentiate x^3 as 3x^2, this isnt possible right since we are differentiating w.r.t to time
faint crow
If you are using t you have to paramaterize it otherwise d/dt = 0
sterile trout
(t --> time)
sterile trout
If you are using t you have to paramaterize it otherwise d/dt = 0
wdym
sterile trout
hm yeah
sterile trout
so you cant differentiate a variable when its w.r.t to another variable?
sterile trout
Oh they did a implicit chain rule
blud what is this ☠️
faint crow
so you cant differentiate a variable when its w.r.t to another variable?
You can, it will just be 0
faint crow
blud what is this ☠️
They assumed but did not write that it was parameterized and applied chain rule assuming there was a function x(t)
Which is just wrong
sterile trout
They assumed but did not write that it was parameterized and applied chain rule ...
how do they do that
"applied chain rule assuming there was a function x(t)"
wdym
im a beginner sorryy
faint crow
Do you know chain rule
sterile trout
i do
faint crow
Eh I’m just trying to justify some weird thing they did
Bottom line is it’s incorrect
sterile trout
hm okay
faint crow
And the notation is wrong
sterile trout
um i have another doubt
how did they differentiate πr² when its w.r.t to dt!!!!
is this also rltd to implicit chain rule or whatever
indigo owl
chico
faint crow
how did they differentiate πr² when its w.r.t to dt!!!!
By assuming there is some undeclared parameterization r(t) and performing chain rule for an implicit differentiation on it
sterile trout
can you give like an example
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@sterile trout Has your question been resolved?
static pier
Seems like a Chinese remainder theorem problem
I’d start with looking at that
Honestly, it’s fairly straight forward, but takes a sec to figure out. A YouTube video will help significantly more than whatever I can type in here.
Alternatively ig u could do it by definition, but that’s not as nice. Will be easier to understand tho
Okay, so can you rearrange your question into equations? It’ll make it easier
You mean x=8n + 2 = 27k + 3?
Okay, so then just solve the equation 8n+2 = 27k + 3
Y?
Because they may not be the same number, remember that we are working mod 27 and mod8. So we really just want some multiple of 27 and some multiple of 8 the number of times you add it doesn’t matter
And in fact they are not the same number. If they were you’d get some weird decimal places but in fact u need integers
Yeah, x.
But x relies on both n and k when consider it this way
Right cos x=2mod8=3mod27
Which is also solveable, but I find it easier to put it in the other form first
Not according to what u said the question was
Either use Chinese remainder thrm or use trial and error or make an algorithm to solve it or use a graphing software to find where they intercept. Probably some other ways to that I can’t think of
They don’t have to be prime, just co prime
Here’s a really good resource that explains it better
Got to the bottom ish of page 2
Ye
Ahh, that’s beyond my pay grade. I think the solution is not solveable by crt in some cases
It will work in other cases
I’m sure there’s some way to tell
But I have no idea what it is
But yeah if crt can’t do it then it is not solveable at all tho
sturdy cypress
we can solve it, if there is a solution
static pier
Thanks, that’s a bit more concise
sturdy cypress
i don't know if it counts as doing crt, because it's not technically a method you do
sturdy cypress
But yeah if crt can’t do it then it is not solveable at all tho
this is false
static pier
Really? Shit, my bad.
sturdy cypress
well it depends on what you mean by "doing crt" i guess
i'm being vague because i don't know much about it
static pier
Well, yeah. I mean u should be able to re organise your system of equations to make something solveable by crt if it’s possible no?
sturdy cypress
coprime guarantees a solution
sometimes solution exists even if not coprime
that's all i know
29 56 83
are 27k+2
83 works
final saddleBOT
@tranquil pine Has your question been resolved?
sturdy cypress
i used nothing
that's the method, i listed 27k+2 and i noticed that 83 works
that's the smallest
because 80 is one of multiples of 8
it means 83 is 3 mod 8
so it takes a long time in general, you need to find remainder for every number
but in this case, it was very easy, because it happened to be the third number i checked, and not like fifteenth
no
the opposite
83 is already 2 mod 27
and i checked mod 8
you should choose the direction so the numbers grow faster
27 is larger than 8
so you add 27 to 2
instead of adding 8 to 3
ripe ingot
hi could somebody possibly help me understand this solution? this is the question
final saddleBOT
hi could somebody possibly help me understand this solution? this is the questio...
ripe ingot
i understand how to do this, it’s getting the equations for R and C and subtracting them to get to P
but i’m confused on how the given answer does it
this is how they solve it; it all makes sense to me except for the part where they multiply 2q by q+4/q+4
i’d really appreciate it if someone could explain to me why you can do that to get to the answer
tranquil pine
this is how they solve it; it all makes sense to me except for the part where th...
its how u add fractions
like
,, \4ab + \4cd
soft zealotBOT
tranquil pine
assume b and d are not the same
then, to add those fractions, you need to have a common denominator
easy way to do this is multiply both sides by the other denominator to make a common denominator. As in: [
\4ab +\4cd = \0r{\4dd\2}\4ab + \4cd\0r{\2\4bb}
]
soft zealotBOT
tranquil pine
here this process can be simplified as 2q has a denominator of 1, so you just need to multiply by q+4/q+4 for 2q
hope this is understandable
ripe ingot
ah ok i see i didn’t know you could do that! thank you :)
tranquil pine
ah ok i see i didn’t know you could do that! thank you :)
So to complete the process fully, it would be: [
\4ab +\4cd = \0r{\4dd\2}\4ab + \4cd\0r{\2\4bb} = \4{ad}{bd} + \4{cb}{bd} = \4{ad + cb}{bd}
]
Going back to your example: [
2q - \4{q^2 + 8q +4}{q+4} = \4{2q}1 - \4{q^2 + 8q +4}{q+4} = \0r{\4{q+4}{q+4}\2}\4{2q}1 - \4{q^2+8x+4}{q+4}\0r{\2\411}
]
soft zealotBOT
tranquil pine
you can try finishing the algebra by yourself from here
ripe ingot
yeah i see that makes sense
just didn’t know that was a rule hahah
thank you so much!
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gleaming bluff
how do you graph a tangent? I watched this guys video where he tries to graph f(x) = -2tan(x+π/2) i get it how he's doing but there are 2 things i dont get.
- he didnt include the two and adjusted the graph to it
- he found x-scale as π/2 but someone told me x-scale was found π/4 but it was a cosine/sine function so is it different for tan and cot?
Here's the video:
final saddleBOT
how do you graph a tangent? I watched this guys video where he tries to graph f(...
gleaming bluff
whos saying pi/4
someone on this channel told me that
final tangle
who
gleaming bluff
idk it was an arabish name
i asked this earlier
she found period π
no uhm,okay not period
i mean x scale
period/4
she said
the below line is the solution as we found
what is an x-scale btw i still dunno what does it represent she just added π/4 over these so is it the gao between those 2 dots
final tangle
the scale you choose can help it easier to show locations of key parts of the graph
gleaming bluff
oh ok
final tangle
they've chosen to use pi/4 for each gap since peaks/troughs occur every odd multiple of pi/4
and the function is 0 at even multiples of pi/4
tan doesn't have stuff like peaks/troughs so not much to gain to use a smaller scale, though you can if you want
final tangle
use whatever scale you want to show the info you want
gleaming bluff
so how do i do this -2 tan x + π/2 thing
final tangle
start with tan(x)
then do the horizontal shift
vertical stretch
then relfection
final tangle
,tex .transformation rules
soft zealotBOT
ℝαμΩℕωⅤ
gleaming bluff
oh but uhm isnt this a vertical translation here?
final tangle
no
gleaming bluff
huh
but
like
ive done it this way since the beginning of the chapter for sin and cos
final tangle
show those specific examples
gleaming bluff
okay wait
for example the picture i just sent u
has π/2
oh okay
okay
i meant horizotnal as well i was just confused
english is my second language thats why
sorry abt that
okay so what do we do with -2?
as i saw on the book
final tangle
vertical stretch
then relfection
gleaming bluff
when we add like a tanx and a > 0 then tanx startes to look more like a
line
yk
is this correct
and same for 2 but with -tanx
gleaming bluff
final tangle
ok. sure, on the same scale it'll start to resemble a straight line
final saddleBOT
@gleaming bluff Has your question been resolved?
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versed crater
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low yew
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low yew
severe hawk
BAD and DAM being congruent means that AD is the bisector of BAM, which means that BD = DM.
M being the midpoint of BC means that BM = MC = 2BD.
I would recommend giving names BD = x, and AD = y, and every other side needed as a function of either (x is probably better). With the relations of these you can compute the trig functions of the angles, and thus get the angles.
severe hawk
"easier" is not universal. Can you not use any of the things i've mentioned for some reason?
as in, have you not studied any of the mentioned things? or are you asked to do it with specific procedures?
low yew
I understand what you have mentioned
Its just that it gets more complicTwd than it had to be
I will try thanks
severe hawk
honestly? very likely. I have not solved the exercise, and i just told you the first way i could think of, which is the one i'd choose unless there's additional restrictions
For example, if you dont have a calculator avaliable, and the ratios dont correspond to usual known angles, you'd be forced to use other methods
low yew
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low yew
@severe hawk i found the easier way, interested ?
final saddleBOT
<@189916659417481216> i found the easier way, interested ?
severe hawk
sure
low yew
AM-bisector.
AD/DM=AC/MC
But DM=BD and MC=2BD so we get that AC=2AD
ADC is a 90° triangle
So the angle C is X (Angle A is 2x)
X=30
30,60,90
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severe hawk
that doesnt sound correct
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violet frost
how do you find the axis of symmetry for graphing quadratic equations
dapper torrent
You can use the vertex to find the axis of symmetry.
violet frost
how do i do that. like what equation do i use
dapper torrent
do you know the formula for coordinates of a vertex?
dapper torrent
violet frost
this helps thank you
dapper torrent
no problem
violet frost
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keen topaz
hi i need to do a standard deviation question but forgot exactly how the formula i used goes but ik it was similar to this to an extent. does anyone know if its right??
final saddleBOT
hi i need to do a standard deviation question but forgot exactly how the formula...
@keen topaz Has your question been resolved?
mint glen
hi i need to do a standard deviation question but forgot exactly how the formula...
almost
this is the variance
standard deviation will be the sqrt of this
keen topaz
ok thank you!
trim juniper
I have a question on how to prove a limit
I can do this if epsilon is provided but all the examples I've seen of doing it without epsilon being provided stump me
This is as far as I've gotten but I am not sure if it's correct at all
final saddleBOT
I have a question on how to prove a limit
I can do this if epsilon is provided b...
mellow axle
does that say $\lim_{x\rightarrow 3} 3x-7 = 2$
soft zealotBOT
chebyshev's infinite pee norm
trim juniper
Yes
mellow axle
so in the scratch work you get your delta
you can start from the end of that scratch work and undo all those steps to get the right direction
so start with |x-3| < epsilon / 3
then undo each manipulation
trim juniper
Wouldn't you end up with the original eq again?
mellow axle
yes which is what you're trying to prove
let $\epsilon > 0$ and choose $\delta = \epsilon/3$ then suppose that $|x-3|< \delta$ we get <undo the manipulations here>
soft zealotBOT
chebyshev's infinite pee norm
mellow axle
and you'll end up with $|f(x) - 2| = |3x - 7 - 2| < 3\delta = \epsilon$
soft zealotBOT
chebyshev's infinite pee norm
final saddleBOT
@trim juniper Has your question been resolved?
trim juniper
let $\epsilon > 0$ and choose $\delta = \epsilon/3$ then suppose that $|x-3|< \d...
| 3x - 7 - 2| < epsilon
oh wait
So we are describing epsilon in terms of delta?
mellow axle
wdym
trim juniper
if we choose delta as epsilon/3, then we define | 3x - 7 - 2| < 3delta
how did the = epsilon come?
mellow axle
no
we are not defining |3x-7-x| < 3delta
look at the definition of the limit
for all epsilon
there exists a delta
if |x-x0| < delta
then |f(x) - L| < epsilon
so starting the proof
let epsilon > 0
choose delta = epsilon / 3
and suppose |x-3| < delta
then multiply both sides by 3
and split the 9 into 7 and 2
what do you end up with
trim juniper
Epsilon?
mellow axle
choose delta = epsilon / 3
start from here
yes
so you've shown that if you start with |x-3| < delta
you end up with |3x - 7 - 2| < epsilon
trim juniper
| 3x - 7 - 2| > epsilon
3| x - 3| > epsilon
|x - 3| > epsilon/3
so if we take delta as epsilon/3
|x - 3| > delta
multiply both sides by 3
3|x-3| > 3delta
|3x-9| > 3delta
|3x - 7 - 2| > 3delta
trim juniper
Missed that
mellow axle
$\delta = \frac{\epsilon}{3}$
soft zealotBOT
chebyshev's infinite pee norm
mellow axle
what do you get when you multiply both sides by 3
trim juniper
|3x - 7 - 2| > 3(epsilon/3)
|3x - 7 - 2| > epsilon
Wait is it
|3x - 7 - 2| > 3delta = epsilon
or
|3x - 7 - 2| > 3delta = epsilon
like do you mean 3delta = epsilon or the entire inequality = epsilon
trim juniper
I am used to using > more than < so I make that mistake a lot
trim juniper
3dela = epsilon
That was the source of my problem!
My book wrote it the same way and I got confused how the entire ineq = epsilon
Now I understand it!
Tysm!
mellow axle
👍
trim juniper
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unreal flume
Isn't there a u sub in here
unreal flume
Ahm I don't think I did but sorry if I did
digital osprey
!original
final saddleBOT
Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
random rampart
^
sonic crystal
Ahm I don't think I did but sorry if I did
unreal flume
Can someone take one outta there I finished it
random rampart
The point of calculus is to try and see
If it works
U build experience
Try u sub n see what happens
digital osprey
U= sinx
smoky cedar
+C
digital osprey
[ \int (1 + \sin^2 x) \cos x , dx = \sin x + \frac{\sin^3 x}{3} + C ]
You should get this
soft zealotBOT
Gaza
unreal flume
I found it, solved the question on the website I took the question from and it said sinx+sin3x/3+C
Knew something was off
Anyways thanks
digital osprey
Lemme see what you did
digital osprey
lol do you know why it’s + C?
unreal flume
Ah fuck I always forget it
digital osprey
Yeah. So + C because it’s indefinite integral. When you have a definite integral C cancel out so no + C. Have a good time 🥰🥰
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pulsar wadi
Out of curiosity of k=1 then it would NOT be continous right
final saddleBOT
Out of curiosity of k=1 then it would NOT be continous right
pulsar wadi
because ln(1) = 0
digital osprey
Find the radius
pulsar wadi
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brazen tusk
I cant find the answer to this question, I've tried increasing 4 by the amount that 5 was increased by (7.5) but thats none of the answers. Another thing I tried is to divide 5 by 12.5 which gave me 0.4, I've tried adding and multiplying 4 by 0.4, which would give 4.4 and 2.0 respectively, which is none of the answers.
final saddleBOT
I cant find the answer to this question, I've tried increasing 4 by the amount t...
brazen tusk
so its fine if the width is larger than the length in this case?
I was thinking its a distraction to make the question more confusing, since its been added to alot of other questions before
but im not sure
yeah I know, so what did I do wrong?
proportions are two equal ratios
so am I meant to find the % increase from 4 to 5, then find what number is that % of 12.5?
alr let me try it
no sadly
is this part of percentage increase by any chance? this test is part of a chapter and I havent completely finished the chapter because of time constraints
percentage increase is a upcoming lesson in that chapter so
brazen tusk
so am I meant to find the % increase from 4 to 5, then find what number is that ...
already tried this by dividing 5 by 4 which gave 1.25, tried increasing 4 by 1.25% and it gave 5 which isnt any of the answers
let me try
I got 10, thats one of the answers!
I did 12.5x4 which is 50.0, then 50.0/5
which is 10
4/5 = x/12.5
alr tysm
yeah that could've worked too, but I prefer the way I tried
its something I learnt, just forgot to apply
alr ty gonna close it now]
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orchid hill
This feels wrong. How can two vectors be the basis for $\mathbb{C}^3$ surely you would need at least 3??
final saddleBOT
This feels wrong. How can two vectors be the basis for $\mathbb{C}^3$ surely you...
soft zealotBOT
Br;an
blissful meadow
I don't think it's claiming so. Isn't it just saying that U and V have a direct sum that is inside W, and not that W is said direct sum?
orchid hill
as in its a direct sum to some subspace of C^3
oh i guess mabey
im still curious would my assumption / intution be correct in that it is impossible to have basis for F^3 (where F is either C or R) with two vectors
blissful meadow
I think so yeah. That's why they don't go any further. The only requirement for the sum is that the intersection contains only the zero vector, and having both be l.i. means so.
orchid hill
huh fair
thank you so much man
sometimes i get tunnel visioned on these questions
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flat fiber
what do I do because theres x in the top and bottom?
final saddleBOT
what do I do because theres x in the top and bottom?
soft zealotBOT
Dyssrupt
digital osprey
!status
final saddleBOT
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
Idk what to do so I haven’t got anything rn
Oh wait there’s the equation
Ok ty
final saddleBOT
@flat fiber Has your question been resolved?
dense garnet
I have tried to google my way to finding an answer to this, but I haven't found any clear and concise proof that proves this claim. \
\textbf{Claim} Let $F\subset\mathbb R^n$ be a closed set and $x\in\mathbb R^n$. Then $d(x,F)=\inf{\lVert x-y\rVert :y\in F}$ is continuous. \
First of all, how does one prove this (I take anything, a sketch, a link...)? Also, is this true even without the assumption that $F$ is closed?
final saddleBOT
I have tried to google my way to finding an answer to this, but I haven't found ...
soft zealotBOT
Philip
dense garnet
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terse terrace
my friend got this question wrong and we're having trouble understanding what his teacher meant by "assumes equality"
final saddleBOT
my friend got this question wrong and we're having trouble understanding what hi...
static plinth
we're essentially trying to prove that the left is equal to the right hand side
that means we cant change anything on the right hand side
but we can change stuff on the left
cancelling out that 2 is assuming that the left is equal to the right
but it might not be
terse terrace
would it still not work if he wrote 2secx = secx + secx and replaced it with that
if i had to prove secx + a = secx + b given a = b then would it work to subtract both sides by secx and restate that a = b
lethal bronze
do you mean $\sec(x+a) = \sec(x+b)$ or $\sec(x) + a = \sec(x) + b$
soft zealotBOT
blahaquil
tough tiger
if i had to prove secx + a = secx + b given a = b then would it work to subtract...
It would be the wrong way to do it
Because here again you are assuming that sec x + a = sec x + b since you are starting with it to prove than a = b
To show that this is madness, here is an example :
I could start with
1=2
then I multiply by 0 on both side (wich I can do) and I would have
0=0 wich is true.
In your reasoning that would mean that 1=2
final saddleBOT
@terse terrace Has your question been resolved?
terse terrace
yeah i see it doesnt work if they both get multiplied by 0
terse terrace
if i had to prove secx + a = secx + b given a = b then would it work to subtract...
but there they're being added by sec x which is a constant when x is fixed
which is one to one i think
not multiplying by 0 though
tough tiger
not multiplying by 0 though
That just proves you that beginning with what you are supposed to prove is mathematically wrong
Let’s say we do your way
How are you going to end up the proof ?
When will it be good and you will be able to say « okay so the left part of the equation does in fact equal the right part »
@terse terrace
terse terrace
i think its hard for me to understand why its not an "if and only if" thing
since a function like f(x) = x+c is injective
like why
"sec(x) + a = sec(x) + b iff (sec(x) + a) - sec(x) = (sec(x) + b) - sec(x)"
wouldnt be true
tough tiger
That’s true
This an iff
But in general we don’t do that because as showed my example, if you use a non invertible fonction (multiplying by 0) it would be messed up
But your professor is a bit weird to tell you that you can do the first to line
But not the third
Because you did nothing wrong
The only wrong thing is using an equality a = b and doing stuff with it when you don’t know yet that this equation is true
terse terrace
ok i see so if its not clarified thats its a non invertible function then its not really proven
i think i understand now thank you
tough tiger
Yes but here you didn’t do that
Si it’s good
But don’t do that again in math
If you want to prove a = b. You can do
a = …. and play with a (factorise, dévelop..) and try to find b
Or you can do a-b= …. and try to find 0
terse terrace
oh so writing sec(x) + sin(x)/cos(x) + cos(x)/(1+sinx) - 2sec(x) would work?
terse terrace
okay thanks
terse terrace
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torn ether
what did i do wrong for b) ?
final saddleBOT
what did i do wrong for b) ?
torn ether
what approach should i take
final tangle
you seemed to have multiplied the (2x-3) by (2x-1) instead of just (x-1) in the bottom right
torn ether
oh
thats right
ill try again
ok the approach is wrong i think
i just end up with 0 = 0
what should i do
honest valve
Can I see your new working? I don't see why you'd get 0=0
honest valve
I see
What you want to actually do is set the first line equal to zero rather than the quadratic
torn ether
what do you mean
honest valve
And then following that you can say the quadratic is equal to zero
torn ether
like rather than setting the equations equal to eachother to just set one of them = 0?
honest valve
Yeah
torn ether
hm
let me try
but its non calculator question
itd be kind of hard to do it
i think
honest valve
You can solve quadratics without a calculator
honest valve
It's a skill that takes practice to get faster at
honest valve
sin2θ=-1/2 has solutions
torn ether
2θ=-pi/6, or 11pi/6
how did you get theses
honest valve
θ=11π/12
torn ether
im confused
honest valve
how did you get theses
The first one you should know, the second one is by adding 2π
As solutions for sin, cos or tan repeat every 2π
For tan it's every π
torn ether
As solutions for sin, cos or tan repeat every 2π
yes ik
but
isnt pi/3 is sqrt(3)/2 no?
honest valve
π/6 is 1/2
honest valve
Why π/12?
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tranquil pine
final saddleBOT
tranquil pine
what I'm confused about is how the hell I'd draw this
wanton pine
its not necessary to draw it
agile coral
tranquil pine
ah cosine law isn't working
I'm trying to do
$a^2=(b)^2+(c)^2-2bc \cdot \cos (A)$
soft zealotBOT
Remlis
tranquil pine
$a^2=(2)^2+(1)^2-2(2)(1)\cdot\cos (30^{\circ})$
soft zealotBOT
Remlis
tranquil pine
which gives be
me
a being 1.24
ok I see what I did wrong
it was the angle
but why can't I do 30 degrees
why is it 150
unique estuary
how and why does cos(-x/2) = cos(x/2)
final saddleBOT
how and why does cos(-x/2) = cos(x/2)
wanton pine
well u know what cos x looks like right?
unique estuary
yeah
wanton pine
well cos x is an even function
which means there is symmetry about the origin
that means the negative side of cos x looks like the positive side of cos x
therefore cos(-x) = cos(x)
unique estuary
therefore cos(-x) = cos(x)
ok so this is pi/2 and -pi/2 on the graph and beacuse it's symmetric their the same right
wanton pine
yep
unique estuary
so if in any equation it's cos(-x/2) it's best to simplify to cos(x/2)?
unique estuary
ok
unique estuary
but it can help
so if this is the textbook answer and instead of cos(x/2) i had cos(-x/2) it would sitll be right
wanton pine
it would be correct, but some people might specifically want cos(x/2) in order for u to get a mark on a test
so just be warey
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craggy bane
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craggy bane
Where did i went wrong ?
cerulean radish
Why do you say its wrong?
craggy bane
Why do you say its wrong?
well the ans given is positive
at the denominator
(2 sqrt 2 + 1 )/7
cerulean radish
are u sure about that
your working looks right
should be -7 in the denominator
,w (1+2\sqrt{2})(1-2\sqrt{2})
soft zealotBOT
craggy bane
aight
cerulean radish
ys
final tangle
you didn't square properly
cerulean radish
you have to square the whole right hand side
not individually
its a + b
so it will be (a+b)^2
white tiger
fell for the oldest trick in the book again
cerulean radish
fell for the oldest trick in the book again
aha
final tangle
no
cerulean radish
you don't need to take it to the right hand side
*left hand side
but you could do it that way
See what you come up with if u wanna do it that way
I suggest just keeping a and b to the right hand side and just doing (a+b)^2
craggy bane
craggy bane
I suggest just keeping a and b to the right hand side and just doing (a+b)^2
Get it
craggy bane
Well this part idk should I take down the 18 ?
final tangle
why did you set up your two equations like that
what are p and q supposed to represent?
craggy bane
why did you set up your two equations like that
what are p and q supposed to rep...
so just directly a and b ?
final tangle
square both directly then equation the rational and irrational parts
craggy bane
square both directly then equation the rational and irrational parts
make x = a sqrt 3 + b sqrt 6 ? same as just now
final tangle
no need to introduce x
craggy bane
no need to introduce x
if i square the big thing is be x^2 = 33 - 18 sqrt 2 tho