How do I got about doing that again? Sorry it’s been a while since I’ve done math and English isn’t my first language
#help-13
crimson sedge
rugged palm
x² + 6x + [something] can be written as (x + [something else])²
crimson sedge
Yeah that’s what I’m trying to do. Though I don’t know what’s the proper way of coming up with the remaining term
rugged palm
Do that intuitively
rugged palm
Otherwise you can find it with $\frac{[ \text{coef. of x}] ^2}{4 \cdot [[ \text{coef. of x²}]}$
crimson sedge
No problem I appreciate it
rugged palm
Otherwise you can find it with $\frac{[\text{coef. of x}]^2}{4 \cdot [\text{coef. of x²}]}$
wraith daggerBOT
Nonna
rugged palm
But it's not worth it most of the times
rugged palm
**Nonna**
By the way, this gives you that term you are missing, remember that when you get that in the form of (x ± __ )² you have to take the root of that number
rugged palm
How did you get that?
rugged palm
the coef. of x is 6
crimson sedge
Assuming we take that as a reference
rugged palm
the coef of x² is 1
crimson sedge
Oh I see
rugged palm
$\frac{6^2}{4\cdot 1}$
crimson sedge
So it should be six
wraith daggerBOT
Nonna
crimson sedge
I see thank you very much the formula is going to be very useful for me
Yeah I can work it out from here I was struggling with that bit
rugged palm
I see thank you very much the formula is going to be very useful for me
It was for me too, it's great! But most of the time it's easier and faster with the intuitive method
(x + a)² = x² + 2ax + a²
so to find a² you have to think about the coef. of 2ax and divide it by two, and just then to square it
crimson sedge
This is what I got in the end
I don’t know about the radius is being the square root of six
rugged palm
,rotate
wraith daggerBOT
silent bobcat
@rugged palm You got the help? Else I can help the helper 
rugged palm
I think I'm ok, but if you want to help feel free to ^^
silent bobcat
Hell yeeeee
rugged palm
You didn't complete the square
silent bobcat
Forgot about the sign but here is what I have in the end
Let's not forget that a radius is a distance, so it has to be positive. Does a negative length make sense geometrically? 
rugged palm
from what we did before we found the complete square:
x² + 6x + 9
let's get it into (x + 3)²
crimson sedge
Oh I’m stupid
silent bobcat
Sorry I’ve not done math in a while
We're here for the help no problem take your time!
rugged palm
from what we did before we found the complete square:
x² + 6x + 9
let's get it i...
But we just did something very bad: we summed something that is not 0, so our equation is not true anymore
silent bobcat
Yeah it is your a² in this (x+a)²
crimson sedge
Radius should be 3 in the end
rugged palm
To make it easy:
n = n
But if we add something to a side only
n + k ≠ n k is not 0
So you have two choices:
Add the number to both sides:
n + k = n + k
Subtract it immediately:
n + k - k = n
crimson sedge
rugged palm
The completed square of x² + 6x is x² + 6x + 9
silent bobcat
Yeah if you want to make something appear additively, just add it and subtract it. If it's multiplicatively though, multiply and divide by the number you want to see appearing. That generally is very helpful!
rugged palm
(You forgot the + between 9 and y)
If you add something to an equation, remember to subtract it too, to make the equation true again
crimson sedge
hot pivot
It' not a question, Im just making a program in python where I need to know the time it takes if I have the (x,y) position pf start and goal position, as well as the angular and linear velocity needed to get to end goal
silent bobcat
And don't hesitate to take time to make things step by step, in order not to feel that moment where you would feel lost because of all of the mathematical information
rugged palm
In the second step:
(x²+ 6x + 9) - 9+ (y² - 2y + 1) - 1 = -6
crimson sedge
Oh I see
rugged palm
It' not a question, Im just making a program in python where I need to know the ...
This channel is occupied, if you need help with an old question, ask it again in another empty channel
crimson sedge
I believe the values I got from your formula were positive. So I guess I’ll have to flip the signs of whatever I get with the formula now?
silent bobcat
It' not a question, Im just making a program in python where I need to know the ...
You can use one of the help channels above, which are available (written in the name of the category)
rugged palm
What do you mean?
crimson sedge
hot pivot
You can use one of the help channels above, which are available (written in the ...
Yeah I just did thanks!
rugged palm
Remember that with that formula, the result is always positive because it's the result of a square root
silent bobcat
It's at these moments that I really want to be able to help in vc 
rugged palm
To check whether in (x ± _)² it's + or -, look at the coef. of x, if it's positive keep the +, otherwise switch it to -
crimson sedge
I see. So why do we change their signs when we put the 9 and 1 onto the rest of the equation?
rugged palm
To make it easy:
n = n
But if we add something to a side only
n + k ≠ n ...
To keep the equation true
rugged palm
,rotate
wraith daggerBOT
crimson sedge
Yes!
rugged palm
We can't, in an equation, just add something to both sides
so we have to subtract it immediately
crimson sedge
Oh, I forgot about that step
crimson sedge
I see. I think it’s best that I watch some lessons and do it all over again
I really appreciate your help throughout all this
I’m sure it was somewhat frustrating so I really appreciate it
.close
cedar kilnBOT
rugged palm
I’m sure it was somewhat frustrating so I really appreciate it
Not really, ahah, it's just a weird topic and in chat it's super hard to explain
I recommend this video
silent bobcat
Just try to get to know a bit better how to use the relation (x+a)²=x²+2ax+a²
crimson sedge
Thank you, I'll check it out
crimson sedge
Just try to get to know a bit better how to use the relation (x+a)²=x²+2ax+a²
Yeah, it's important to understand the basis of it regarding that
silent bobcat
And not only the way you might automatically think
crimson sedge
It's just been so long and I've forgotten a lot
cedar kilnBOT
Available help channel!
silent bobcat
From right to left too, in the factorisation way
cedar kilnBOT
From right to left too, in the factorisation way
crimson sedge
Of course, thank you guys so much! I'll be sure to take everything into consideration and try my best to get it right
silent bobcat
I'll close guys no worriiiies
crimson sedge
I was doing these right but the second step on this problem is what dragged me down and confused me
silent bobcat
The video will help you for sure
rugged palm
If you need more help feel free to ask
crimson sedge
Alright, I'm going to go eat. Have a good one guys!
silent bobcat
.close
cedar kilnBOT
silent bobcat
```.close```
Please be patient 
obsidian coral
I mean, I was just correcting you
cedar kilnBOT
Available help channel!
crimson sedge
I am not looking for an answer I'm looking for the working of this through d/dx
compact junco
The derivative?
crimson sedge
Yeah
compact junco
what
crimson sedge
Oh lmao i just googled and Quotient rule is d/dx
just the y ain't written
maths is soothing when you understand it
thank you @compact junco
how does one close this?
.close
cedar kilnBOT
crimson sedge
this is my work so far and not sure if it’s correct
fy(y) > 0, for that does that mean i just find derivative of
-3y^2 +2y +1 and set it to 0 to find roots?
elfin hemlock
sure
crimson sedge
so i got fy(y) > 0. when y is 0 < y < 1/3
does that sound right or am i terribly wrong
gleaming cloud
fy(y) > 0, for that does that mean i just find derivative of
-3y^2 +2y +1 and s...
Why would you find its derivative
crimson sedge
dont know what else to show fy(y) > 0
elfin hemlock
he needs where f_Y(y)>0, which isnt whole interval [0,1]
gleaming cloud
dont know what else to show fy(y) > 0
You just need to know when that polynomial is > 0 in the interval
It's just a parabola so should be easy to sketch
Or find its roots
gleaming cloud
I don't see exactly how the derivative helps with finding roots
crimson sedge
is it [0,1) ? im kinda getting confused with the derivative thing now
elfin hemlock
it is
crimson sedge
how would you be able to figure it out with the derivative just wondering
elfin hemlock
test 0 see its >0
its increasing from 0 to 1/3
and decreasing after
so check 1
crimson sedge
isnt it increasing from 0 to 1/3
elfin hemlock
well same
gleaming cloud
In a sense, the derivative loses information about the original function, upto a vertical shift. So the derivative alone can't tell you much about the roots
ivory moss
cedar kilnBOT
ivory moss
having a time with this one
as usual, I'm sure it's something simple I'm just not seeing
I know the ultimate answer is (2 * 4^n)/n+1 but I'm not sure how I get there from what I have
I thought I had it for a second separating out 2^1^2 * 2^n^2
oh wait I think I might see it now
ok nope, still confused
elfin hemlock
Expand (2^(n+1))^2
ivory moss
sorry to be dense, but can you elaborate on that.. I think I'm making like a basic algebra mistake or something when I try to expand
elfin hemlock
Expand it, what do you get?
elfin hemlock
Huh
ivory moss
I know I'm missing something super basic
elfin hemlock
What is (n+1)^2
ivory moss
as in the foil result? n^2 + 2n +1
elfin hemlock
So use that to expand
ivory moss
ahhhhh
so 2^n^2 cancels with the denominator, 2^2n = 2^2 * 2^n = 4^n, and of course 2^1 = 2
giving me my denominator of 2*4^n
I get it now
I knew it was some basic algebra stuff I just wasn't seeing
thanks!
.close
crimson sedge
cedar kilnBOT
crimson sedge
I'm trying to come up with the h,k values for the circle
And what I got didn't seem to align with what the graph says
I got (1,-1) and it shows it should be (-1, 1)
I don't know where in my process i did something wrong
I would appreciate it if anyone could help me out 🙂
Ping me once you're here, thank you
cedar kilnBOT
@crimson sedge Has your question been resolved?
obsidian coral
Equation of a circle is $(x - h)^2 +(y - k)^2 =r^2$
wraith daggerBOT
dldh06
crimson sedge
Oh I see, let me check
so in the fourth step
I guess that's not really the end of the formula
Because h and k need to be negative
So I would just have to write the formula one more time but with the signs switched?
obsidian coral
It means the signs of the center are the opposite based on the equation
So the center is (-1, 1)
crimson sedge
Yeah, but why didn't I get that?
obsidian coral
Your equation is correct, it's just that you need to take in account that for the equation of the circle, you need to take the opposite signs
crimson sedge
Alright I see it
So basically I have to run over it again like this?
Because the step prior does look like the formula I needed
But I guess that wasn’t the final step
obsidian coral
No, your solution for the equation of the circle is correct
crimson sedge
Or I don’t know when I’m supposed to change the signs
obsidian coral
As I mentioned, to get the center of the circle, you need to take in account of the opposite signs because of the equation of the circle
crimson sedge
Alright
Also the radius confuses me
From my calculations it shows it should be square root of negative one
And I don’t know what that even is lol
obsidian coral
That's wrong
crimson sedge
Though in the graph it appears to have a radius of 2
I see
Where did I go wrong?
crimson sedge
I've seen the video, I'll be sure to watch it again
But I'm still confused
I moved the +1 from the left side into the right so it'd be a -1
And ad the end that was the radius
obsidian coral
And this might help too
https://youtube.com/watch?v=cR7N2gIAPOo
The thing you keep forgetting is when you complete the square, you're adding a value on the left side, whatever you do on the left you need to do on the right side as well, to balance the sides
So when you completed the square for x^2 + 2x, you added 1 to make x^2 + 2x + 1, when you added one on the left side, you have to add 1 on the right side as well
crimson sedge
OH, you're absolutely right
That's what's been dragging me down
I'll give it another try
obsidian coral
Just remember that whatever you do on one side, you need to do to the other, to balance the sides
They're equations so the left equals the right side, so if you're adding 1 on the left side and do nothing on the right, the sides are unequal now
crimson sedge
Of course, I'll be sure to remember that
So in the end
the radius is
Square root of one
obsidian coral
Yes
crimson sedge
That's the bit that confuses me now, the graph makes it seem like its 2 instead
What even is square root of one? +-1?
So would I have two radiuses? of +1 and -1?
Though again the graph shows it being 2 perhaps?
obsidian coral
Radius is from the center to the edge of the circle
crimson sedge
Oh I am stupid
It is one
the graph was split in 0,5 increments
So if the radius is the radical of a number what should I do?
Always assume it's positive which was this case?
obsidian coral
What exactly do you mean?
crimson sedge
So the radious that i got was square root of 1
obsidian coral
Yes
crimson sedge
Isn't that supposed to mean both positive one and negative 1?
crimson sedge
Oh I see
Alright that clears things up
Thank you very much
I was struggling with this one all day!
Have a good day
.close
cedar kilnBOT
dire geode
Show what you've tried
What part? Do you know how to multiply (x-y)^4?
What do you get
lament leaf
cedar kilnBOT
cedar kilnBOT
Available help channel!
native cosmos
cedar kilnBOT
native cosmos
can someone explain this
buoyant latch
shouldn't it be arcsin the right side
lone garden
yes it should be arcsin of the whole thing
buoyant latch
buoyant latch
sin(a+b) = sin(a)sin(b) + cos(a)cos(b)
this looks sus
fiery whale
💀
lone garden
back to the gulag with me
buoyant latch
it's ok it happens
cedar kilnBOT
@native cosmos Has your question been resolved?
native cosmos
https://www.math-only-math.com/arcsin-x-plus-arcsin-y.html
ty
lilac birch
small question : if f(x) is a polynomial , The factor is x - 1 and x + 5 , then is (x - 1)(x + 5) a factor of f(x)?
cedar kilnBOT
small question : if f(x) is a polynomial , The factor is x - 1 and x + 5 , then ...
@lilac birch Has your question been resolved?
cedar kilnBOT
cedar kilnBOT
Available help channel!
crimson sedge
where does the formula for the determinant of a 3x3 matrix come from?
dire geode
crimson sedge
yea i know what a determinant is
but like
why is it positive then negative then positive
for a 3x3 matrix
dire geode
It's like that for all n by n and that's just the definition
dire geode
https://en.m.wikipedia.org/wiki/Determinant#:~:text=In%20mathematics%2C%20the%20...
Read the geometric meaning here if you want more intuition
cedar kilnBOT
vapid widget
Break it down
Take each percentage separately
So take 30%, 2% and 0.5% and then just add your answers
copper peak
ignore them
tell me what percentage means
what does 50% mean?
<@&268886789983436800> ^
wicked helm
yow, you know fractions?
rustic coyote
banned
wicked helm
because a percent is "per hundred"
copper peak
@crimson sedge are u here?
cedar kilnBOT
@crimson sedge Has your question been resolved?
cedar kilnBOT
@crimson sedge Has your question been resolved?
sharp surge
cedar kilnBOT
sharp surge
The given here is the two slope-intercept forms, it asks for value of (m), rearranged; (y=1-x/m) what do I do now? Where do I even allocate the acquired slope?
Also the first equation is perpendicular to the second hence the slope of the second
sharp surge
Yes
sick orbit
and u know the gradient is -1/3
sharp surge
Yes
sick orbit
so the coeffecient before the x
sharp surge
Is 1?
sick orbit
yes
cedar kilnBOT
@sharp surge Has your question been resolved?
crimson sedge
Does this look correct to anyone, I was a bit stuck on the 2nd line. Originally I extended the line but then I realized it wants x>-2 which technically means anything to the right of it, so I erased the line extending out
crimson sedge
just a little odd looking to me
cedar kilnBOT
@crimson sedge Has your question been resolved?
crimson sedge
hmm
long arrow
isn’t that the vertical line?
no, it represents x = 4
long arrow
yes, because y (output) should be equal -1 for every x > 3
crimson sedge
ohhh
long arrow
so f(4) = -1, f(5) = -1, f(10000) = -1 etc.
crimson sedge
ohh okay! so that needs to be horizontal, but the rest are good as they are
it’ll intercept a few points i think but
long arrow
rest looks good
cedar kilnBOT
cedar kilnBOT
Available help channel!
wintry tulip
what does all that symbol mean
cedar kilnBOT
what does all that symbol mean
unborn valve
what does all that symbol mean
So y= f(x)
and y = x(1 - x)
The domain of y is [0, 1]
latent blaze
what does all that symbol mean
this ones < >?
wintry tulip
The domain of y is [0, 1]
what is "the domain"
wintry tulip
this ones < >?
yes
unborn valve
what is "the domain"
The domain is what the “x” portion of the graph have to be
The x portion belongs a certain interval
In this case being [0, 1]
wintry tulip
what does ""x" portion of the graph" mean?
unborn valve
Now extend that idea from a 1d perspective into a 2d perspective
latent blaze
Its called greater than and less than
unborn valve
And the limitations of the y axis (vertical) is called a Range
latent blaze
9 is greater than 8 so 9 > 8
unborn valve
😂
wintry tulip
I am not understanding any of this
latent blaze
And the limitations of the y axis (vertical) is called a Range
we are not talking about the same thing
latent blaze
I am not understanding any of this
Is your native language portuguese?
unborn valve
I’m just giving basic info
wintry tulip
Is your native language portuguese?
spanish
wintry tulip
ye
latent blaze
So 9 is bigger than 8 right?
wintry tulip
ye, 9 > 8
latent blaze
Exaclly
wintry tulip
but I dont understand what does this mean in math, I use it for coding but
latent blaze
But for coding is the same thinhg
wintry tulip
, 0 < x < 1
is it just saying that 0 is less than x and x is less than 1?
I dont see a conditin there
latent blaze
X has to be bigger than 0 and less than 1
latent blaze
is it just saying that 0 is less than x and x is less than 1?
Not about 0 anymore
Its about whats in the midle
wintry tulip
why is 0 in the middle
latent blaze
why is 0 in the middle
No
x its in the middle
And we do not know what is x
But it has to be bigger than 0 and less that 1
wintry tulip
ok
latent blaze
U got it?
wintry tulip
kinda
latent blaze
I can send u a tutorial if you would like
wintry tulip
yes please
latent blaze
but it was just weird to see there
yeah it drived me crazy for a long time
spare meadow
Is this the correct way to take the antiderivative of an expression with a negative exponent?
dire geode
Yup
bold vine
@spare meadow ^^
spare meadow
ok, thank you
bold vine
ok, thank you
Close if you’re done
spare meadow
.close
storm junco
hey how would i go about solving this
cedar kilnBOT
hey how would i go about solving this
storm junco
i tried plugging it into photomath and it just showed a massive fraction and i feel like thats a bad way to solve it
livid hound
if you have a calc, just use that
storm junco
naw im not allowed
well when im going to be tested im not allowed
this is a khan academy sat prep thing
gaunt hamlet
The sat has a calculator section
storm junco
yeah ik but what if its on the non calc
livid hound
then multiply it out by hand
gaunt hamlet
I highly doubt you'll get something like this on the non calc section
storm junco
which... is the answer if i add 22 to it
storm junco
I highly doubt you'll get something like this on the non calc section
how much like multiplication and stuff is on the non calc section
division ect
cosmic steppe
hey how would i go about solving this
Are you required to simplify
If not, you can leave it like that
gaunt hamlet
Last time I took the sat, the non calc section wasn't tedious. This problem isn't hard and can be done without a calculator, but it's tedious
livid hound
do you know what it means to square soemthing...?
gaunt hamlet
The sat is testing your knowledge of concepts and algebra, not arithmetic. Not really, anyway
livid hound
are you implying you don't know how do evalute
$2 \times 3^2$
wraith daggerBOT
ℝamonov
cosmic steppe
How exponent :(
storm junco
Are you required to simplify
also yes
storm junco
are you implying you don't know how do evalute
$2 \times 3^2$
ig i dont but idk why this is the case
i feel like this would work out if there was -22
because 39.22*39.22 is massive, .74.74 is 54
cosmic steppe
hey how would i go about solving this
$$0.74^2 = 0.74\cdot 0.74$$
Which is the same as:
$$74^2\cdot 0.0001$$
wraith daggerBOT
Umbraleviathan
storm junco
sorry 39*53
cosmic steppe
Which is how I would do it because I hate decimals with a burning passion
storm junco
decimals are fine
its just nothing adds up
nomatter how i multiply this i dont get 51
cosmic steppe
What is (0.74)^2
storm junco
54.70
cosmic steppe
No
cosmic steppe
**Umbraleviathan**
Read this
storm junco
huh
cosmic steppe
It's just decimal multiplication
$(0.74)^2$ is not 54.70, I don't even know how you got 54.70
wraith daggerBOT
Umbraleviathan
storm junco
.74*.74
cosmic steppe
Yeah
cosmic steppe
**Umbraleviathan**
Which is what I have up here
livid hound
is less than 1
livid hound
why 2?
storm junco
or do you not do that with them both being decimals
gaunt hamlet
Whyre you moving the decimal over
gaunt hamlet
For example?
livid hound
why 2
storm junco
because 20*.7
cosmic steppe
thats = .5476 move the decimal digit over 2
You move the decimal to the left, not right
storm junco
u move dec over to
livid hound
wdym by because
20*.7
livid hound
wheres 20*.7 coming from
storm junco
becase its dec multiplied by dec
storm junco
wheres 20*.7 coming from
my brain with an example wdym
gaunt hamlet
No, you just calculate 20 * 0.7. You only move the decimal if you're changing that to 20 * 7
storm junco
ye then u move the decimal after u do that
cosmic steppe
@storm junco you're right about moving the decimal placement but it's the wrong direction
You move it to the left, not to the right
gaunt hamlet
<@618992877846134804> you're right about moving the decimal placement but it's t...
0.74² = 0.5476, so they shouldn't move the decimal at all
cosmic steppe
.5476
livid hound
here, if you're considering
74 * 74, you'd shift the decimal place 4 places to the left ( 2 from each)
storm junco
and then multiply that by 53
cosmic steppe
0.74² = 0.5476, so they shouldn't move the decimal at all
No they meant like from@the original (0**.**74)
You move that first factor's decimal place over
Not the answer's decimal placement
As a visual for where the decimal should be for the answer
You don't move the answer's decimal
That's what I meant
storm junco
so you get 289910
from .5470*53
and the decimal would go after 28
so 50.89910
then round and u get 51
mistyped
anyways
how many digits would u move decimals over and why
1*2 u move it 0
2*.2 u move it?
1 time?
compact junco
yes
cosmic steppe
from .5470*53
Buddy how did you get a number that's larger than 53
storm junco
ok so just however many non 0s are to the right of the decimal
cosmic steppe
From multiplying 53 by a number that is less than 1
cosmic steppe
53(0.74)^2 ≠ 289910
storm junco
28.9910
storm junco
ye
compact junco
then ur done
storm junco
50 round up
cosmic steppe
It rounds to 51
livid hound
,calc 53*0.74^2
wraith daggerBOT
Result:
29.0228
your multiplication is a bit off
storm junco
but for regards to decimal digits
cosmic steppe
Eh slight error
compact junco
u should probably review multiplication
cosmic steppe
^
storm junco
just if it is to the right of a digit once
compact junco
its not good to go into sat without knowing how to multiply
livid hound
consider how many places you originally shifted and then shift it back
compact junco
this question is just multiplication and addition, so i dont see where its confusing
livid hound
that's part of the issue
storm junco
but yeah so just however many dec is to the right i shift it to the left that many
livid hound
yeh, that's what i mean
storm junco
eh its what i was taught and it works if i do it right
yeah but thats all my questions
ty guys that clears it up
my main issue was the dec movin
livid hound
if you have 0.74 * 0.74
from 0.74 → 74 (you're shifting it to the right twice)
for the second 0.74 → 74 (you're also shifting it to the right twice)
and after doing 74*74, you shift it to the left four times
crimson sedge
cedar kilnBOT
gaunt hamlet
!15m
cedar kilnBOT
Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
crimson sedge
help please
cosmic steppe
Well you understand that warning right?
compact junco
ok first look at the right side
can you write taking the fourth root as raising it to an exponent?
cosmic steppe
And the fact that 100 is in the denominator
cedar kilnBOT
@crimson sedge Has your question been resolved?
compact junco
bruh
arctic bluff
cedar kilnBOT
arctic bluff
how do u differentiate this?
hazy minnow
cedar kilnBOT
@arctic bluff Has your question been resolved?
main shadow
cedar kilnBOT
crimson sedge
hi i need some help with question b
cedar kilnBOT
hi i need some help with question b
crimson sedge
is this the right PDF? and if so, how do i justify it?
cedar kilnBOT
@crimson sedge Has your question been resolved?
crimson sedge
<@&286206848099549185>
cedar kilnBOT
@crimson sedge Has your question been resolved?
cedar kilnBOT
crimson sedge
.reopen
cedar kilnBOT
✅
cedar kilnBOT
@crimson sedge Has your question been resolved?
crimson sedge
.close
delicate ember
cedar kilnBOT
delicate ember
is there a way to say $a^2tan^{-1}(a) = b^2tan^{-1}(b) \implies a=b$
wraith daggerBOT
ホタル
jovial smelt
Do you know of any criteria for real functions to be injective?
maybe even visually
delicate ember
horizontal line test
jovial smelt
which is?
delicate ember
if a horizontal line intersects the graph only once its injective
delicate ember
if f(a) = f(b) when a != b
jovial smelt
lets consider only continuous functions
Imagine my function has a peak at some x0
lets say its increasing before x0 then decreases after that
will my function be injective?
delicate ember
nope
ah so if a function is only strictly increasing/decreasing on R then its injective?
jovial smelt
if its strictly increasing or decreasing, then you can be sure that it is injective, yeah
but it isn't necessary
in some extreme cases
if the function's derivative changes sign, then you can be sure it isn't injective
but consider x^3
delicate ember
I see
jovial smelt
the derivative of x^3 at 0 is 0
delicate ember
yup
jovial smelt
it increases less and less, then kind of stagnates at 0, then starts increasing again
delicate ember
mhm
jovial smelt
you want to consider the derivative
delicate ember
d/dx(x^3) is always >= 0
elfin hemlock
Is 0 injective?
delicate ember
Is 0 injective?
huh?
elfin hemlock
Derivative of 0 is 0 and we have 0>=0
jovial smelt
so as long as the derivative never changes sign (we can allow a 0 derivative), then the function is injective
jovial smelt
yeah, we allow a 0 derivative if it is only at one point
if there exists a neighborhood where the derivative is 0, then clearly your function is constant on that neighborhood
delicate ember
anyway d/dx(x^2tan^-1x) is 2xtan^-1x + x^2/(1+x^2)
x^2/(1+x^2) is always positive
2xtan^-1x is also always positive right?
positive or zero*
jovial smelt
well tan^-1 is an odd function so tan^-1(-x) = -tan^-1(x)
delicate ember
yup
jovial smelt
since x is also odd, then the product of the two is even, yes
delicate ember
so it is increasing, but not strictly
delicate ember
mhm
jovial smelt
and as ScapeProf pointed out, if you have a 0 derivative on an interval (that is not just a single point like [a]), then your function won't be injective
but, if you can show that the derivative of the function at your stationary point is a strict minimum (or maximum), then any point close to the stationary point will necessarily have a larger derivative, so your function would be injective
jovial smelt
that would work, but its not necessary
you just want to show that x = 0 is the only point where f'(x) = 0
delicate ember
ah that is easy
jovial smelt
since your derivative is positive, that would mean 0 is a strict minimum
delicate ember
2xtan^-1(x) = -x^2/(1+x^2)
from this we can directly imply that x=0 right?
since both the quantities are non-negative
jovial smelt
I would probably look at both terms individually first
x^2/(1+x^2) = 0 implies x^2 = 0 implies x = 0
similary reasoning for the second term
and yes, since they are non-negative, if there sum is 0 then both of them have to be 0
jovial smelt
Do you understand why, if the derivative at x0 is a strict minimum and x0 is the only point where f'(x0) = 0 then the function is injective?
delicate ember
yes
jovial smelt
great
delicate ember
how about surjectivity
jovial smelt
again, your functions is continuous so it makes life a lot easier
yeah, an application of IVT would work well here
delicate ember
but how do i do it
jovial smelt
well you want to say that for any real number a, you can find x0 and x1 such that f(x0) is less than a and f(x1) is greater than a to apply IVT, no?
What does IVT tell you?
delicate ember
f is continuous on R, so for any y in R (codomain) between f(-inf)=-inf and f(+inf)=+-inf there exists an x in R (domain) such that f(x) = y, thus f is surjective
delicate ember
ic
jovial smelt
it depends how rigorous you're meant to be, but yep that's the idea
delicate ember
$\lim_{x \to \infty} x^2tan^{-1}(x) = \infty \times \pi/2$
wraith daggerBOT
ホタル
jovial smelt
so there you go, surjective and injective
steep meadow
I'm a bit stuck on a question so help would be appreciated. Suppose I have an Linear Programming Problem. I can obviously draw the feasible area if the problem only uses x_1 and x_2. Now the question is what exactly is the dualcone of every corner of the feasible area? I'm supposed to draw it but I don't even know it
cedar kilnBOT
I'm a bit stuck on a question so help would be appreciated. Suppose I have an Li...
@steep meadow Has your question been resolved?
tulip river
From a point P outside a circle of center O and radius 1 cm, draw the tangents to this circle that meet it at points A and B. The area of the polygon PAOB is √ 3 cm2 if the distance of P from O is
tulip river
from my understanding, the the geometry figure in question is a triangle.
If that was the case, we would have sqrt3 = b*h/2
where b is equal to 2 and h is unknown
but h, to have sqrt3 as the result, has to be sqrt3 itself
but i dont find this solution from the possible results... i believe there must be something wrong with my reasoning
cedar kilnBOT
@tulip river Has your question been resolved?
tulip river
<@&286206848099549185>
tulip river
Oh, i considered all three points A O B parallel
calm agate
If you do that then the lines that pass through A and B are going to be parallel
calm agate
If you do that then the lines that pass through A and B are going to be parallel
P is the intersection of those 2 lines right?
tulip river
I see thanks, but i have to say that even with this new information im having trouble understand what to do
I know that both OA and OB are 1cm
if i knew the angle of AOB i could find AB but i dont
Wait, its 90
calm agate
Not necessarily
calm agate
if i knew the angle of AOB i could find AB but i dont
That's what I was thinking too
So, what is the formula to find the area of a shape like PAOB?
calm agate
Something similar to that, but the formula is the same
calm agate
Are you sure there are no other angles or sides given?
tulip river
From a point P outside a circle of center O and radius 1 cm, draw the tangents t...
Thats a direct translation of the problem
i didnt change anything
I believe AB has to be sqrt2
calm agate
If AOB is 90°, but how do you know that?
tulip river
well that's the only possibility left
tulip river
Are you sure? we have two sides equal to each other
it can either be 45 45 90
60 60 60
or something strange like 30 75 75
calm agate
or something strange like 30 75 75
There are infinitely many combinations like this one
tulip river
I see
Well, for whats it worth i tried to find the value of PO by doing 1/2 times sqrt2 times x = sqrt3
and found a non possible solution
calm agate
Hey I solved it, if you want I can try to explain it, it's pretty long
Maybe there's a shorter way, idk
@tulip river
tulip river
Hey I solved it, if you want I can try to explain it, it's pretty long
Well i found a solution online, but it wasnt that long
basically its as if there are 2 equal right triangles
we know that the surface of the right triangle is 1/2 times OA times AP
so the surface of the whole thing is two times that
calm agate
Oh yeah that's actually really easy that way
calm agate
I calculated the length of a few segments in terms of AB using Pythagoras theorem and then solved the equation. If you want I can send you my solution, but I need to rewrite it properly before
tulip river
Sure, it's always interesting to see other points of view
tulip river
Oh i see what you did, you pretty much did the same thing
calm agate
Yeah
tulip river
Honestly, i didn't even know that since A and B are both tangent they would make an angle of 90
So i would have failed this exercise 100%
calm agate
Yeah, tangent lines to a circle make a 90° angle with the radius
tranquil bay
cedar kilnBOT
tranquil bay
I tried to multiply the parantesis with conjugate fraction and ended up with some crazy numbers and i still couldnt evaluate the fraction because i got infinity-infinity/2
The answer is supposed to be -2/3
We dont use that at highschool level
long arrow
I'll try multiplying by conjugate as you did, wait
whole surge
Sorry it's wrong
cedar kilnBOT
@tranquil bay Has your question been resolved?
cedar kilnBOT
tranquil bay
.reopen
cedar kilnBOT
✅
cedar kilnBOT
@tranquil bay Has your question been resolved?
cedar kilnBOT
@tranquil bay Has your question been resolved?
reef steeple
i did it like this
take n common out and write it as a 0/0 form and then L hopital (or however it’s written)
reef steeple
maybe that works too but no i did a bit differently
$n\left(\sqrt[3]{1+\frac{1}{n}+\frac{1}{n^2}+\frac{2}{n^3}}-\sqrt{1+\frac{2}{n}+\frac{5}{n^2}}\right)$
wraith daggerBOT
Alpha Kappa
reef steeple
send that n to denominator then its 0/0
tranquil bay
how do I send that to denominator?
reef steeple
ah like a = 1/(1/a)
tranquil bay
Oh
cedar kilnBOT
@tranquil bay Has your question been resolved?
cosmic seal
cedar kilnBOT
wraith daggerBOT
muted bear
this requires knowledge of trig identites, how good are you with those?
unborn valve
Which one are you solving for the top or bottom
cosmic seal
this requires knowledge of trig identites, how good are you with those?
Yup@pretty decent
Do you reckon this is correct?
muted bear
looks good i think
wraith daggerBOT
cosmic seal
,rotate
wraith daggerBOT
cosmic seal
,rotate
wraith daggerBOT
cosmic seal
How do you do thiss😭😭
muted bear
you would use the sine double angle formula for 4theta
unborn valve
how do i make sin4 theta into sin2 theta
Sin4theta turns into 2sin2theta cos2theta
Omg it turns into tacos
😂
cosmic seal
Omg it turns into tacos
😭😭
cedar kilnBOT
@cosmic seal Has your question been resolved?
cosmic seal
Yes
hardy torrent
16 divided by 120 = 0.13
For example if quotient of some division is 0.102 them how does 0 come here but 0 doesn't come when I do 16 divided by 120
cedar kilnBOT
@hardy torrent Has your question been resolved?
hardy torrent
<@&286206848099549185> help me
cedar kilnBOT
@hardy torrent Has your question been resolved?
sage forge
16 divided by 120 = 0.13
For example if quotient of some division is 0.102 t...
Not sure if you know it. But 16/120 = 0.13333333.... with Infinitly many 3s
sage forge
<@&286206848099549185> help me
Can you formulate your question clearer?
hardy torrent
Can you formulate your question clearer?
I'm saying that how can 0 appear after decimal point in quotient
If the value is less than divisor you write zero in quotient there that's what i did there to get same answer as calculator
But
When 160-120 = 40
Now adding 0 to 40 it's 400
But we also need to write 0 in quotient right ?
So answe should be 0.10... something why is it 0.13 on calculator
@sage forge
sage forge
We're not just adding 0
hardy torrent
How would zero exist after decimal point then
sage forge
Those are 2 different questions. Both of them can be answered
hardy torrent
Answer plz
sage forge
Can we talk about it in a voice channel?
sage forge
Don't ping
hardy torrent
I'm in mathemathic voice channel
I could not speak tho cuz it's night
??
U can explain here
sage forge
Streaming my pc isn't allowed in those channels.
hardy torrent
U can just type
sage forge
Ok. Anyways. You seem to not really understand how this written division works
sage forge
And you're applying them correctly
So we start with the first digit: 16 < 120. So it is 0
hardy torrent
Got it
sage forge
Then we go to the second digit. 16 is basically 160 / 10.
160 is 120 + 40. So it's a 1 and 40 remaining
hardy torrent
1?
sage forge
The second digit 0.1
sage forge
That's basically what you're doing above
hardy torrent
What I wrote is specilation of answer from calculator
Thats why I think they did this divison
It's not my answr
Answer
sage forge
So each subtraction step is basically a digit of the number
hardy torrent
Yes
sage forge
160 / 120 = 1 R 40. So the second digit is 1
You add a 0 because you're going to the next digit after the comma, which is 10 times "less" than the previous one.
160/120 = 120/120 + 40 / 120
160/120 = 120/120 + 40 / 120 = 1 + 40/120 = 1 + 400/ 120 x0.1 = 1 + (3+40/120) x 0.1 = 1.3 + 40/120 x 0.1
sage forge
160/120 = 120/120 + 40 / 120 = 1 + 40/120 = 1 + 400/ 120 x0.1 = 1 + (3+40/120) x...
You may wonder what am I doing here. But this is exaktly what you're doing in written division without actually thinking about it
Do you understand why you are multiplying by 10 now before switching to the next digit?
hardy torrent
Do you understand why you are multiplying by 10 now before switching to the next...
Isn't that how it works
Thats how we got zero in first place
sage forge
Isn't that how it works
Yes. We're multiplying the remainder by 10 after going to the next digit because the following digit is 10 times less "worth" than the previous one. The justification for that is above. But that does not mean that there needs to be a 0 after the comma
hardy torrent
Dividing 2 by 5
We first write 0 as quotient zero because 2 is less than 5 and 2-0=0
Now we add decima point l and write 0 after 2 in remainder so it becomes greater than 5
20 divided by 5 is 4
So our answer is 0.4
sage forge
Yeah
hardy torrent
So so u are saying u should not multiply by 10 because it's less than zero ?
sage forge
I'll give you an example for an 0 after the comma. Let's say we devide by 17 and we have remainder 1 from the previous digit
hardy torrent
Less than one *
sage forge
You basically did the multiplying by 10 by writing a 0 after the 2 to get 20
hardy torrent
You basically did the multiplying by 10 by writing a 0 after the 2 to get 20
Isn't that correct ?
sage forge
I'll give you an example for an 0 after the comma. Let's say we devide by 17 and we have remainder 1 from the previous digit
1 remainder. When we're going to the next smaller digit we have to multiply by 10 or add a 0 after the 1. So we get 10 /17. But 10 is still smaller than 17
You can follow that?
hardy torrent
Yes
sage forge
So 10 is smaller than 17, so we have a 0 as the digit. Going to the next digit we get 100, which is now bigger than 17. So in this case we get a 4
hardy torrent
Can u explain how I'm supposed to solve this one from the beginning ?
sage forge
Can you accept my call?
hardy torrent
.close
humble plank
Hello!
I have 2 points and they have an angle. I would like to know at which coordinates they intersect. I tried to do some research but i can't seem to find anything! This for a game on roblox but it will be 2 dimensional so third axis will not be a problem. Ask me for me details if needed! All the angles will be in increments of 45 if that helps
cedar kilnBOT
Hello!
I have 2 points and they have an angle. I would like to know at which coo...
@humble plank Has your question been resolved?
cedar kilnBOT
cedar kilnBOT
Available help channel!
junior merlin
factor 9/4y^4 - 3/4y^2(y^2-y) please