#help-39
midnight haven
pearl pondBOT
midnight haven
The negation of this statement would be
There is atleast one $\epsilon >0$,for which there is a positive number $\delta$ such that $|x-a|< \delta$ implies $|f(x)-f(a)| \geq \epsilon$
jolly parrotBOT
midnight haven
if this statement was true,then the original statement would be false
and thus we can say its the negation of the originals tatement
but then again
arent we supposed to be using demorgans law ?
p would be there is a positive number $\delta$ such that $|x-a|< \delta$
jolly parrotBOT
midnight haven
and q would be $|f(x)-f(a)| < \epsilon$
jolly parrotBOT
pearl pondBOT
pearl pondBOT
Available help channel!
silver moat
Can someone help me understand why the limits of the integration switched places but the minus before the integration is still there? Is this a mistake ?
unkempt yacht
an error perhaps?
silver moat
Its done for simplification purposes
But wouldnt thwt change
The sign
In the integral ?
pearl pondBOT
@silver moat Has your question been resolved?
silver moat
No
tough bobcat
Hi can someone help me with the last part please?
pearl pondBOT
Hi can someone help me with the last part please?
tough bobcat
If you could send over the working for that lsat part that'd be great
I've got the entire q till now only that part I'm stuck on
cobalt hinge
Hi can someone help me with the last part please?
have you, by chance, heard of the properties of the roots of unity?
anyways
i think i have an idea, let me see if it works
yeah, no, this polygon is a really odd shape
i suppose you could use right triangles to solve for the area, but it doesn't look nice at all
tough bobcat
have you, by chance, heard of the properties of the roots of unity?
i know what roots of unity is but not sure abt properties
cobalt hinge
i know what roots of unity is but not sure abt properties
wait, how did you solve part b)?
tough bobcat
wait, how did you solve part b)?
i used shoe lace method
cobalt hinge
okay
why don't you try using shoelace method for a general polygon?
$z^k=2^k\qty(\cos(\frac{\pi k}{n})+i\sin(\frac{\pi k}{n}))$
jolly parrotBOT
;(
tough bobcat
why don't you try using shoelace method for a general polygon?
hmm wdym
tough bobcat
i meant polygon, sorry
could you send working lol
i'm confused 😭
cobalt hinge
could you send working lol
i'm confused 😭
i sketched the graph out by hand, but are graphing utilites allowed on this test?
i double checked it using desmos
tough bobcat
i sketched the graph out by hand, but are graphing utilites allowed on this test...
I think it's a non calc paper unfortunately (I'm not sure ) but usually complex number qs are
tough bobcat
i double checked it using desmos
but it's alr ig we can bend the rules in this case because I'm j trying to understand lol
cobalt hinge
but it's alr ig we can bend the rules in this case because I'm j trying to under...
ok
tough bobcat
https://en.wikipedia.org/wiki/Shoelace_formula#Shoelace_formula
hmm
pearl pondBOT
@tough bobcat Has your question been resolved?
pearl pondBOT
pearl pondBOT
Available help channel!
left dock
is 103993/33102 tge most accurate FRACTIONAL form of pi?
pearl pondBOT
is 103993/33102 tge most accurate FRACTIONAL form of pi?
dusty jungle
no
pi being irrational, theres always a fraction more accurate for any fraction you choose
left dock
pi being irrational, theres always a fraction more accurate for any fraction you...
do u know the most accurate one yet?
dusty jungle
there is no "most accurate" one
there can never be a rational number that is most accurate to pi
cus if there was
there would be one more accurate than it
and thats a condradiction
left dock
there is no "most accurate" one
i mean, the one that's been discovered the latest
but i get wgat youre talking about!
dusty jungle
i mean, the one that's been discovered the latest
trillions of digits of pi have been calculated
thats probably not the most accurate one discovered
left dock
im guessing this one is the latest discovered fraction yet
dusty jungle
trillions of digits of pi have been calculated
.
dusty jungle
glad to help
left dock
.close
pearl pondBOT
pearl pondBOT
Available help channel!
fluid path
The cross-section of a channel is a rectangle with a semicircle attached at the bottom.
Choose the dimensions of this rectangle such that, for a given perimeter U of the channel's cross-section, its area is maximized.
fluid path
What is the formula for U?
Half circle at the bottom and a rectangle that is open at the top
pearl pondBOT
@fluid path Has your question been resolved?
pearl pondBOT
Available help channel!
bright tartan
Hello! I have just understanding problem. The screenshots are in german but I will try to translate the most important things and where i am stuck. I was doing this exercise and i solved it correctly but I dont understand where the numbers to t1 and t2 came from. Can i just take random number and put is as t1 and t2 or is there something im missing? I dont understand why its t1=1 and t2=0 and also later t1=-3 and t2=5
The exercise: Determine the line 𝐺 that passes through the two given points 𝑝 and 𝑞. (screenshot)
My problem: After the first representation of G i dont understand the further representations (how do i get values for t1 and t2 or can i just think of any number in this example?)
I also included screenshot of chatgpt translation of the german screenshot
bright tartan
t1=-3 and t2= 5 seem very specific for me thats why im not sure if its only example/i can choose the numbers myself
rough forge
Hello! I have just understanding problem. The screenshots are in german but I wi...
Deutsch auch gut
Tatsächlich kann eine Geradengleichung unterschiedlich aufgestellt werden
bright tartan
very nice from you but i would like to understand this in english (Ich studiere auf Deutsch, aber meine Rechtschreibung ist katastrophal.) deswegen schreibe ich auf english lol
aber du kannst gerne auf deutsch es mir eklären :) danke
rough forge
Du kannst ja immer ein Vielfaches des Richtungsvektors nehmen
Und der Ortsvektor kann ja p oder q sein
Also was ich meine ist ob du von q startest und zum Punkt p läufst oder umgekehrt
Es läuft aufs Gleiche hinaus
bright tartan
aaaah ok
Und was ist mit t1 und t2 später? Kann ich einfach beliebige Zahlen nehmen?
rough forge
Achso ja
Ich musste erstmal verstehen was da gemacht wurde
Ja die Sportsfreunde
die haben auf den Ortsvektor ein Vielfaches des Richtungsvektor drauf addiert
Damit die quasie zeigen, dass die Gerade G, wie erwähnt, anders aufgestellt werden kann
bright tartan
ah oki ich verstehe jetzt
rough forge
Das kannst du mit einem beliebigen t machen ja
bright tartan
dankeschön :)
rough forge
thank you too
bright tartan
.close
pearl pondBOT
pearl pondBOT
Available help channel!
visual canyon
for the prompt - Explain how you know when you can factor to solve quadratics as opposed to using the square root method.
Did I explain thoroughly and clearly?-
You know when you can factor to solve quadratics vs using the square root method by looking at the structure of the equation.
you should factor when the quadratic equation can be rewritten as the product of two binomials. This means you can easily break it down into two simpler expressions that multiply together.
use the square root method when you can rearrange the equation to isolate the squared term (x^2) on one side, or if the equation is a trinomial.
pearl pondBOT
for the prompt - Explain how you know when you can factor to solve quadratics as...
@visual canyon Has your question been resolved?
edgy wren
for the prompt - Explain how you know when you can factor to solve quadratics as...
" or if the equation is a trinomial." isn't that the case for completing the square I don't see how the sqrt method would work on trinomials
(unless you are referring completing the square as the square root method)
visual canyon
" or if the equation is a trinomial." isn't that the case for completing the squ...
yes, I am sorry for the confusion
to my understanding they are the same thing??
edgy wren
yes, I am sorry for the confusion
technically yea for the ax^2+bx+c=0 in case of b=0 then
it can be solved using the square root method
and for the b not equal to 0 you complete the square
and just so I can be on the same boat
you are trying to compare factoring vs completing the square ?
visual canyon
yeah essentially
edgy wren
<@1132371071975309322> Has your question been resolved?
btw click the X reaction here so the room doesn't automatically close
visual canyon
i've interpreted the sort of vague (imo) prompt "Explain how you know when you can factor to solve quadratics as opposed to using the square root method." to be asking when it is appropriate to pick factoring vs when you should solve by completing the square/ square root method
edgy wren
i've interpreted the sort of vague (imo) prompt "Explain how you know when you c...
the problem with the method I'm trying to state is the fact that I always generalize for all cases including cases for non-real solutions so I'm not sure how you could improve your statement srry
visual canyon
the problem with the method I'm trying to state is the fact that I always genera...
thats fine, does it look good enough?
like-- passable
edgy wren
thats fine, does it look good enough?
I could see that it's good but you could add something like "completing the square is the method used for trinomials where it can be reshaped by adding or subtracting to perfect squares"
pearl pondBOT
pearl pondBOT
Available help channel!
orchid token
can someone explain how come this is just 10! / 5!
pearl pondBOT
can someone explain how come this is just 10! / 5!
orchid token
I designated 5 spaces
and said the first can be any number from 0-9 so 10 possibilities
and then second has to be 9 third 8 and so on
so i js got 10x9x8x7x6
which is right too
but how is that formula being come up with
brave sluice
10! / 5! ← i just think of this as 10x9x8x7x6
cause the 5! cancels out the rest
orchid token
yeah
but i mean i woudln't be able to do that for larger problems so how can i do it the way they had?
unkempt yacht
why wouldn't you?
cobalt hinge
can someone explain how come this is just 10! / 5!
let’s say you have n-numbers to choose from and k numbers you need to choose.
unkempt yacht
that's literally what the factorial notation is for
brave sluice
for example 100x99x98x...x32
it can be written as 100! / 31!
orchid token
let’s say you have n-numbers to choose from and k numbers you need to choose.
n choose k
bruh
is it combinatorics
cobalt hinge
is it combinatorics
yeah, no shit, it says choose
orchid token
this is supposed to be FCP and permutations homework 💀💀💀
bro im cooked
thanks anyway
.close
pearl pondBOT
cobalt hinge
this is supposed to be FCP and permutations homework 💀💀💀
permutations...is combinatorics?
orchid token
permutations...is combinatorics?
no i meant nCr
they called it something else
i forget
cobalt hinge
no i meant nCr
you don’t use nCr here.
orchid token
no?
cobalt hinge
you use nPr.
jolly parrotBOT
;(
cobalt hinge
yw
pearl pondBOT
Available help channel!
tranquil badger
tranquil badger
bitter lodge
!status
pearl pondBOT
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
4?
bitter lodge
,w xx + 2(b/2)x + (b/2)^2 + c = (x + b/2)^2 + c
jolly parrotBOT
tranquil badger
https://cdn.discordapp.com/attachments/905286165316403271/1330345829243944960/20...
- The substitution, why specefically 2(6/2)? And does the squaring work because of the devisor of two or?
cobalt hinge
1. The substitution, why specefically 2(6/2)? And does the squaring work because...
isn’t that a b/2?
cobalt hinge
regardless, yes, they wanted to preserve the original form of the quadratic and use completing the square for it.
cobalt hinge
i thonk soemone explain it before, but recall that $x^2+2ax+a^2=(x+a)^2$
jolly parrotBOT
;(
bitter lodge
$$x^2 + bx$$
$$x^2 + 2\qty(\frac b2)x + \qty(\frac b2)^2$$
$$\qty(x^2 + \frac b2x) + \qty(\frac b2x + \qty(\frac b2)^2)$$
$$x\qty(x + \frac b2) + \frac b2 \qty(x + \frac b2)$$
$$\qty(x + \frac b2)\qty(x + \frac b2)$$
$$\qty(x + \frac b2)^2$$
tranquil badger
10^2 = 100 20^2 = 400
and because the factors of 100 add up to 10 which is half of what I want this proves the rule
cobalt hinge
10^2 = 100 20^2 = 400
yeah, if you mean what i think.
tranquil badger
why beehive
jolly parrotBOT
King Leo
tranquil badger
cobalt hinge
why beehive
lmfao
cobalt hinge
**King Leo**
yeah, this is a pretty valid method
tranquil badger
https://cdn.discordapp.com/attachments/905286165316403271/1330345829243944960/20...
why do they have 2 behind it?
bitter lodge
**King Leo**
Note: the first and second expressions arent always equal to each other, which is why you must add (b/2)^2 to the other side of the equation too
cobalt hinge
why do they have 2 behind it?
again, preserving the original quadratic form; $2\cdot\frac{b}{2}=b$
jolly parrotBOT
;(
tranquil badger
again, preserving the original quadratic form; $2\cdot\frac{b}{2}=b$
?????? what is the original form
oh to undo the division?
so just to keep b as b?
🅱️
also if im slower than usual of which i am typically alr slow its bc im sick (tbh this is 100x faster cuz i know 90% of whats going on this time)
so uhh... yea
tardy reef
There is an idea in mathematics called the teakettle principle. It goes as follows:
A mathematician asked a physicist how, given an empty teakettle and an unlit gas stove, water could be boiled. The physicist conjectured that water could be boiled by filling the kettle, lighting the stove, and placing the kettle on the stove. The mathematician agreed.
The mathematician then asked how water could be boiled given a filled kettle and a lit stove. The physicist proposed that the kettle should be placed on the stove. The mathematician said, "Wrong! It would be simpler to empty the kettle and turn off the stove. Then, we would have a problem that we already have solved!"
Thats how this problem works too.
You reduce the quadratic equation to one of the known forms, aka the factorization of a^2 - b^2
And we know a^2 - b^2 = (a+b)(a-b)
so thats why we wanna reduce the efforts and just convert the problem into a simpler one
cobalt hinge
oh to undo the division?
👍
unborn escarp
pearl pondBOT
unborn escarp
How do I turn this into an exponential function
cloud plume
y+4 = 2log_4 (1/8x+2)
log_4 (1/8x+2) = (y+4)/2
now consider (1/8x+2) as a one whole term, (y+4)/2 as another term
unborn escarp
How do they apply
pearl pondBOT
pearl pondBOT
Available help channel!
keen lichen
exponent rules, simplifying this equation i ned help
pearl pondBOT
exponent rules, simplifying this equation i ned help
pearl pondBOT
Available help channel!
surreal dove
pearl pondBOT
surreal dove
what is the answer
clever zenith
where did you get stuck?
pearl pondBOT
@surreal dove Has your question been resolved?
cobalt hinge
what is the answer
!noans
pearl pondBOT
what is the answer
The purpose of this server is to help you learn, not to hand out answers. Do not ask someone to give you the answer directly.
cobalt hinge
do you know what a right and left-hand limit are?
clever zenith
the guy is not here
cobalt hinge
?
clever zenith
he asked it 30 min ago and never responded
cobalt hinge
he asked it 30 min ago and never responded
maybe he's sleeping, maybe he got in a car accident; we don't know why he's offline, so i'll just leave a question for him to respond to
clever zenith
why is car accident the first thing that comes to mind?
pearl pondBOT
surreal dove
do you know what a right and left-hand limit are?
no
pearl pondBOT
Available help channel!
cobalt hinge
so, basically, a right hand limit is represented by $\lim_{x\to a^+}f(x)$; it represents the limit as the graph is approaching $a$ from the right
jolly parrotBOT
;(
cobalt hinge
a left-hand limit is similar; $\lim_{x\to a^-} f(x)$, so the limit as the graph is approaching $a$ from the left
jolly parrotBOT
;(
open rivet
sometimes called from above and below aswell
hollow swan
open rivet
!occupied
pearl pondBOT
Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).
@cobalt hinge Has your question been resolved?
tidal stone
Yo guys
tidal stone
i ned hepl
open rivet
!da2a
pearl pondBOT
No need to ask “Can I ask…?” or “Does anyone know about…?”—it’s faster for everyone if you just ask your question! See https://dontasktoask.com/
tidal stone
SD = squart root of VAR
VAR = take mean
minus all the data given by the mean,
times the data by themselves,
find mean again and boom
Is this correct?
my friend juts summarized this for me
and i have no idea if hes right
SD = standard deviation
VAR = variance
jolly parrotBOT
mmmm7
quick star
take each data and subtract it from the mean and square it -> this is your work for each data point
and keep a counter called sum or whatever
and then add your work for each data point
to get a new sum
finally divide by the number of data points
that's for the population variance
for the sample variance, you have a correction
pearl pondBOT
@tidal stone Has your question been resolved?
wet sky
pearl pondBOT
wet sky
Can someone explain why it's important to reverse the order of factors in this derivation and in general? What is the difference in equation 1.3.14 between <V|i><i| and <i|<V|i>?
pearl pondBOT
@wet sky Has your question been resolved?
wet sky
<@&286206848099549185>
plush bramble
Can someone explain why it's important to reverse the order of factors in this d...
bras and kets relate to each other. do you know what the difference is
should be explained earlier in whatever text this is
wet sky
bras and kets relate to each other. do you know what the difference is
bras are conjugate transpose of a ket
It's Shankar quantum mechanics
plush bramble
for matrices sure
adjoint is more general
take the conjugate transpose of 1.3.13 to get 1.3.14
which equation are you talking about here
wet sky
1.3.13 to 1.3.14
plush bramble
take the conjugate transpose of 1.3.13 to get 1.3.14
did you do this
wet sky
did you do this
Isn't it just |i> <V|i>? Since <V|i> is a dot product and produces a scalar the order of a scalar and a ket shouldn't matter?
plush bramble
bras are conjugate transpose of a ket
.
wet sky
It should matter? Thats what I am confused by.
jolly parrotBOT
riemann
wet sky
Adjoint of $| i \rangle = \rangle i|$
jolly parrotBOT
doomfletcherpoe
wet sky
Woops wrong angle bracket
plush bramble
take the conjugate transpose of 1.3.13 to get 1.3.14
yea try this again
wet sky
yea try this again
I'm missing something obvious here because I don't get it.
plush bramble
do a concrete example with just vectors in C^n
.
$|i \rangle = (1, i, 1+i)$ what's the adjoint?
jolly parrotBOT
riemann
plush bramble
left side i is a vector. right side i is imaginary unit
wet sky
$|i \rangle = (1, i, 1+i)$ what's the adjoint?
A column vector (1, -i, 1-i)
jolly parrotBOT
riemann
plush bramble
Isn't it just |i> <V|i>? Since <V|i> is a dot product and produces a scalar the ...
like you're asking here
wet sky
No it equals <i|. Oh I see I should of switched the ket to a bra.
plush bramble
then use commutativity of scalar and vectors to get 1.3.14
wet sky
So <i|<V|i> = <V|i><i| right?
wet sky
then use commutativity of scalar and vectors to get 1.3.14
Here the order doesn't matter right?
plush bramble
that's what commutativity means yes
pearl pondBOT
pearl pondBOT
Available help channel!
supple roost
How can I select 4 letters from AAAA LL H B D
pearl pondBOT
How can I select 4 letters from AAAA LL H B D
supple roost
What cases should I make?
plush bramble
is that the original question
plush bramble
do you know stars and bars
supple roost
A little bit
hang on let me revise it
Hmm fine. So how can I use it here?
4 A
2 L
H
B
D
So
A L H B D
4|0|0|0|0
3|0|0|0|1
3|0|0|1|0
3|0|1|0|0
3|1|0|0|0
2|1|1|0|0
2|1|0|1|0
2|1|0|0|1
2|0|0|1|1
2|0|1|0|1
2|0|1|0|1
1111
I guess i missed some 2 cases
Let me use permutations
Hang on
The number LL will not affect?@rie.mann
@plush bramble
supple roost
No but if you guide me i will read and learn@clever zenith
clever zenith
ok,
the idea of a generating function is that if you have a series $a_n$ you can make a generating function of that series like this "$f(x)=a_0+a_1 x+a_2 x^2+a_3 x^3+...+a_n x^n$".
jolly parrotBOT
Horsi135
clever zenith
wait
if the series is finite then what you do is add infinite 0 after the last $a_n$
jolly parrotBOT
Horsi135
clever zenith
one important thing is that the generating function is not actually a function it's more of a string (like computer scince string).
supple roost
@plush bramble are they trolling?
clever zenith
about this functions like multiplication and addition
clever zenith
the idea is not to give an answer but to explain how to get to the solution
midnight haven
oh, yeah.
dusty jungle
horsi can you keep explaining please
clever zenith
sure
I will now write generating functions like this $\sum_{n=0}^{\infty} a_n x^n$
jolly parrotBOT
Horsi135
clever zenith
$f(x)=\sum_{n=0}^{\infty} a_n x^n$ and $g(x)=\sum_{n=0}^{\infty} b_n x^n$ the addition $f(x)+g(x)=\sum_{n=0}^{\infty} (a_n+b_n) x^n$
jolly parrotBOT
Horsi135
midnight haven
A L H B D
4|0|0|0|0
3|0|0|0|1
3|0|0|1|0
3|0|1|0|0
3|1|0|0|0
2|1|1|0|0
2|1|0|1|0
2|1|0|0|1
2|0|0|1|1
2|0|1|0|1
2|0|1|0|1
1111
I think it is wrong.
dusty jungle
**Horsi135**
wait whats bn
clever zenith
the multiplication $f(x) * g(x)=\sum_{n=0}^{\infty} [\sum_{i+j=n} (a_i*b_j) ] x^n$
clever zenith
wait whats bn
another series
clever zenith
Shouldnt it be ai + bj
yes
jolly parrotBOT
Horsi135
clever zenith
but the way we can solve this problem
midnight haven
A L H B D
4|0|0|0|0
3|0|0|0|1
3|0|0|1|0
3|0|1|0|0
3|1|0|0|0
2|0|0|1|1
2|0|1|0|1
2|1|0|0|1
2|0|1|1|0
2|1|0|1|0
2|1|1|0|0
2|2|0|0|0
1|2|0|0|1
1|2|0|1|0
1|2|1|0|0
1|1|0|1|1
1|1|1|0|1
1|1|1|1|0
0|2|0|1|1
0|2|1|0|1
0|2|1|1|0
0|1|1|1|1
And you have to consider the order here.
@supple roost
clever zenith
is by creating this functions and multiplying $(1+x+x^2+x^3+x^4)(1+x+x^2)(1+x)(1+x)(1+x)$ each function represent one of the letters
jolly parrotBOT
Horsi135
clever zenith
and the power is how many times we chose the letter
clever zenith
for example the letter 'A' we can take 0 times 1 times all the way to 4 times
this is why its from x^0 to x^4
clever zenith
if we will carry the multiplication we will get $(1+x+x^2+x^3+x^4)(1+x+x^2)(x^3+3x^2+3x+1)$
jolly parrotBOT
Horsi135
clever zenith
we will take for a second $(1+x+x^2)(x^3+3x^2+3x+1)$
jolly parrotBOT
Horsi135
dusty jungle
x^9 + 5 x^8 + 12 x^7 + 19 x^6 + 23 x^5 + 23 x^4 + 19 x^3 + 12 x^2 + 5 x + 1
is the expanded form
clever zenith
of all?
clever zenith
now if you will look each one of the powers tells us how many letters we used and the coefficients tells us how many ways we got to that amount
for example we can get 9 letters only by taking all of them so only in 1 way
dusty jungle
yah
clever zenith
this is why the coefficient of $x^9$ is 1
jolly parrotBOT
Horsi135
midnight haven
x^9 + 5 x^8 + 12 x^7 + 19 x^6 + 23 x^5 + 23 x^4 + 19 x^3 + 12 x^2 + 5 x + 1
what is x?
dusty jungle
x does not matter here
its the generating function that does
its kind of how like the binomial theorem lets us find terms of the pascal's triangle
midnight haven
how can i solve the problem if x does not matter?
dusty jungle
cus we dont need to find x
clever zenith
this is not an equation
dusty jungle
we need to see the coefficients
dusty jungle
this is not an equation
continue explaining
clever zenith
we can see that the coefficient of x^4 is 23 so that tells us the the amount of ways we can choose 4 letters from the letters we were given is 23
dusty jungle
so 23 is the answer?
midnight haven
I think the answer is 285.
pearl pondBOT
@supple roost Has your question been resolved?
midnight haven
P(4,4)/P(4,4)+P(4,4)/P(4,3)*4+P(4,4)/P(2,2)*12+P(4,4)/P(2,2)/P(2,2) + P(4,4)*4 = 285
@dusty jungleWhat do you think my answer?
@clever zenith
clever zenith
is this with or without order? and how?
midnight haven
with order.
pearl pondBOT
pearl pondBOT
Available help channel!
wind lagoon
Let $f:U\to\mathbb R^d$ be differentiable on the open subset $U\subset\mathbb R^d$. We then have $$\lim_{h\to0}\frac{|f(h+a)-f(a)-Df(a)(h)|}{|h|}=0,\quad \forall a\in U.$$ Here $Df(a)$ is the derivative at $a$, i.e. the linear transformation. Let now $K\subset U$ be compact. Under what conditions on $f$ does the fraction above converge uniformly to $0$ on $K$?
jolly parrotBOT
psie
wind lagoon
I'm partly reading this answer (and also my own textbook, where this claim is made) and it clearly holds for d=1 (with K=[a,b]) and with the help of the mean value theorem and the assumption f being continuously differentiable. For d>1, I'm not sure anymore which mean value theorem/inequality applies and how to prove the above claim, if we assume f is continuously differentiable.
pearl pondBOT
@wind lagoon Has your question been resolved?
pearl pondBOT
@wind lagoon Has your question been resolved?
pearl pondBOT
pearl pondBOT
Available help channel!
hollow cliff
yo
jovial zenith
hollow cliff
4y³ - 9y??
U switch the 9 and 12 around?
Alr
This right?
@jovial zenith
Alr I checked the answer it's right
But how do I do b 💀
I figured it out
How do I do c tho?
Seriously
<@&286206848099549185>
midnight minnow
whatcha got so far?
midnight minnow
Sounds good 
midnight minnow
good grouping for sure, factor what you can from those and see
hollow cliff
Factor what?
midnight minnow
Factor what?
why did you decide to put 2n^4+n^3 and 16n+8 in brackets?
midnight minnow
ok, i thought they were good groupings because 16 is twice 8 and 2 is twice 1, so we can hope that those two bracketed sections work nicely with each other
hollow cliff
True
midnight minnow
try factoring out some stuff based on that grouping
midnight minnow
like you can factor out n^3 from (2n^4+n^3)
midnight minnow
midnight minnow
don't forget the poor 2n factor 
hollow cliff
Oh
midnight minnow
but other than that, the n^3-8 can be factored a bit more, then you're done
midnight minnow
8 is also a cube 🧊
hollow cliff
How do I factor that?
How?
What cube?
Difference of cubes
Idk the formula ngl
midnight minnow
n^3-8 is a difference of cubes, you probably have a formula for it :)
hollow cliff
I forgot
midnight minnow
$a^3-b^3=(a-b)(a^2+ab+b^2)$
jolly parrotBOT
maskyboi
midnight minnow
b here is 2
hollow cliff
Oh cuz 2³ is 8
Okok
I'll solve this then I'll show u my answer
Give me a min
Right?
@midnight minnow
Alr I checked the answer it's right
I gtg now
Thanks for ur help
.close
pearl pondBOT
pearl pondBOT
Available help channel!
hoary blade
is the derivatve of my first pic this? if so, which x values will make the derivative equal to 0? is it 0, 1, and 3/7?
versed mica
,w derivative x^3(x-1)^4
jolly parrotBOT
versed mica
<@834774466290253834>
yes
pearl pondBOT
@hoary blade Has your question been resolved?
hoary blade
wait so then @versed mica does that mean that 0 is an x value where the tangent line has a slope of 0?
versed mica
wait so then <@834774466290253834> does that mean that 0 is an x value where the...
what
0 is a number
zeros of the first derivative also called critical points are values of x that make the first derivative zero
and they’re important for determining whether or not a function has a max/min at that particular point
versed mica
yes since x = 0 makes f’ = 0
hoary blade
ok ok so then the answer key is wrong i think?:
cause a point should also include 0 i think
and for this last question, what is the derivative?
is it (x +8 / 2 root (4-x))
@versed mica
versed mica
cause a point should also include 0 i think
wdym
the thing is with x = 0 here it doesn’t really do anything since the factor is x^2
it doesn’t change sign
versed mica
since f’ does not change sign at x = 0 f will not have a max or min at x = 0
and it’s essentially an insignificant critical point
hoary blade
for this question we just have to see if it changes direction at any of these critcial points. how can u just tell that f' does not change sign at x=0 though? @versed mica
versed mica
because x^2 is always positive
for x < 0 it’s still positive
likewise for x > 0 it’s positive
so it doesn’t change sign at x = 0
hoary blade
but thats just one part of the derivative
how can u just look at that one term and tell it dont change sign @versed mica
hoary blade
yes
versed mica
can i see your sign chart
hoary blade
i see that it continues to be positive
i am just wondering how u can tell just by looking at the term
ik u just ssaid but i dont really get i t
cause like even though x squared is always positive, there is still other sign changes when looking at different points other than x = 0 @versed mica
versed mica
i am just wondering how u can tell just by looking at the term
because it’s x^2
even power
always positive
hoary blade
but then why is there parts of the function that are negative
versed mica
if it has an even power then it doesn’t cross the x axis
versed mica
but then why is there parts of the function that are negative
because the function isn’t x^2?
x^2 is simply a factor the derivative
hoary blade
but like when u look at x = 0 dont u have sub in 0 for all x's? So like if that is positive how do u know if the other parts will be positive or negative without looking at them @versed mica
versed mica
yes you do but the thing is since the other factors don’t change sign at zero it will change nothing
they will still have the same sign until their "critical point"
so since the other factors have the same sign until then and x^2 doesn’t change sign either, it’s all the same in that interval
so you can essentially neglect it
hoary blade
and for this last question, what is the derivative?
.
versed mica
,w differentiate xsqrt(4-x)
jolly parrotBOT
versed mica
product rule
hoary blade
o wait i think ik where i went wrong
versed mica
$x \cdot \frac{-1}{2\sqrt{4-x}} + \sqrt{4-x}$
hoary blade
lemme see if i see my error
jolly parrotBOT
knief
versed mica
then common denominator
hoary blade
can u also get rid of fractions?
versed mica
$\frac{-x + 8-2x}{2\sqrt{4-x}} = \frac{8-3x}{2\sqrt{4-x}}$
jolly parrotBOT
knief
hoary blade
wait so a restrtion for this is it has to be equal or less than 4 right?
so my critical point intervals shoudl go to posityive infinity? @versed mica
versed mica
wait so a restrtion for this is it has to be equal or less than 4 right?
what?
you mean for the domain
yes x < 4
versed mica
like restriction for x values
yes domain restriction
versed mica
well not here no since it’s in the denominator
because it would make the denominator zero
for sqrt(x) the function yes
hoary blade
ohh ok
wait but les say this was in numerator then u could? so one interval would be (infinity, 4] ?
@versed mica
like if it was just that in numerator
versed mica
all good don’t need to ping every 5 seconds though
hoary blade
wait so it would be 4 with square brackets?
versed mica
if your function was sqrt(4-x) yes
but since it’s in the denominator we have the added condition that it can’t be zero
since you can’t divide by zero
hoary blade
but like how come in like all the questions i have done on this topic, i have never used square brackets for the intervals?
versed mica
you mean for increasing/decreasing?
hoary blade
like all these
hoary blade
you mean for increasing/decreasing?
yes
versed mica
because f’ = 0 is technically neither increasing nor decreasing
what i just said above was related to the domain
which was what i thought you were asking me
hoary blade
so it includes 4 in this case because its not 0 slope
and it coudl still be increasing/decreaasing?
hoary blade
no like the hypothetical thing i said where the function was sqrt 4-x
versed mica
the function or the derivative?
versed mica
if f’(x) = sqrt(4-x) then f’ > 0 for all x for which the function/derivative is defined
since sqrt(x) is defined to be the nonnegative number whose square is the nonnegative number x
versed mica
well
f’(x) = sqrt(4-x)
sqrt(anything) is always nonnegative
and we must have anything >= 0
hoary blade
ok wait but cant u get a negative after sqrting
hoary blade
but like -2 squared and and 2 squared both are 4
versed mica
im aware
hoary blade
that is smth i get confused about becuase sqrting i thought can sometomes have positive
versed mica
but the sqrt itself is defined to give the nonnegative number
but of course the equation x^2 = 4 has two solutions
x = 2 and x = -2
hoary blade
so when do ik when to account for the negative answer
like ik for Pythagorean theorem u take negative nunmber too
based on the quadrant
(from my pre calc class)
@versed mica
o sorry
versed mica
so when do ik when to account for the negative answer
when you’re solving an equation with something like x^2 = a
that has two solutions
but the sqrt itself is defined to be nonnegative
jolly parrotBOT
hoary blade
yes yes
versed mica
notice that if it returned both positive and negative values it wouldn’t be a function
hoary blade
yes
versed mica
,w plot x^2
jolly parrotBOT
versed mica
here it’s still a function
and we are only solving for values of x whose square is a particular number
x^2 = y
that gives two values
a positive and a negative root
but the sqrt on its own is defined to only give the positive root
we just defined it to be that way
hoary blade
wait so if it was x squared then the x can be negative or positive is what ur saying
?
hoary blade
so its like two different directions
ok
but then in the pyhagorean theorem, why do we look at - and +
versed mica
whoever came up with and agreed on these rules decided that we would only take the positive root to avoid problems
hoary blade
we are square rooting a number
versed mica
but then in the pyhagorean theorem, why do we look at - and +
well in that context everything is positive
since we’re looking at side lengths
hoary blade
no it can also be negative
like if we look at a triangle in quadrant 3
horizontal line of triangle is negative
and also vertical line
just not hypotenuse
hoary blade
oh
versed mica
it’s entirely consistent with the usual xy plane
x is negative to the left of the y axis and positive to the right of it
and y is negative below the x axis and positive above it
what’s the confusion there?
hoary blade
ok ya
versed mica
the unit circle doesn’t trace out a function
versed mica
it’s really the combination of sqrt(1-x^2) and -sqrt(1-x^2)
hoary blade
it’s really the combination of sqrt(1-x^2) and -sqrt(1-x^2)
? wdym
versed mica
well x^2 + y^2 = 1 is the unit circle
but a circle isn’t a function
by vertical line test if you want to hear it that way
but we usually write functions as y = some function of x
so if you solved for y
hoary blade
so like functions only have the positive value?
versed mica
you’d have y^2 = 1 - x^2
but here as i said before
we have something squared equals something
so we take both the positive and negative roots
,w plot sqrt(1-x^2)
jolly parrotBOT
versed mica
,w plot -sqrt(1-x^2)
jolly parrotBOT
versed mica
looks terrible but you get the point
desmos is better
this makes it look like a parabola
hoary blade
like positive and negative roots of the y?
versed mica
what
hoary blade
wait roots are like value of the variable that make the equation true right?
hoary blade
you’d have y^2 = 1 - x^2
so like in this, would the roots be the x values, can u technically say the roots are y values? or no
versed mica
wait roots are like value of the variable that make the equation true right?
no i meant sqrt
root <—> square root
that’s what i meant here
sorry for confusion
hoary blade
so we take both the positive and negative roots
are u talking about the x or y values
versed mica
$y^2 = 1-x^2$
jolly parrotBOT
knief
versed mica
$y = \pm \sqrt{1-x^2}$
jolly parrotBOT
knief
hoary blade
and if it is, depending on the way its asked, then we look for either both or positive value
hoary blade
when you’re solving for something via getting rid of a square
would the last iimage u posted be an example of that
versed mica
but getting rid of square root is the opposite?
i’m not sure what you’re asking
versed mica
yes
assuming there’s no negative coefficient in front of course
like -sqrt(x) is always negative/ nonpositive
hoary blade
and both of these cases only go for if it is a function?
cause we have examples like pythargreean theorem
versed mica
i mean we don’t have to consider it in that context
hoary blade
excuse my spelling
versed mica
like if someone walked up to you on the street and said “hey man i’ll give you a million dollars if you can tell me what the square root of 4 is”
if you said -2 you’d lose out on a million bucks
hoary blade
wait so what is the context you are talking about =
hoary blade
when eliminating square root it is positive right?
versed mica
not sure what you mean
hoary blade
cause we have examples like pythargreean theorem
no but i was saying these rules dont always apply when not talking about functions like pythagrean theorem
hoary blade
not sure what you mean
like y = sqrt x
versed mica
well for the pythagorean theorem it’s because what does a negative side length mean?
hoary blade
ok ya true
versed mica
like y = sqrt x
yes that’s a function and to qualify as a function it must be true that for all values of x in its domain there is a unique value of y that it gets mapped to
we don’t want one input returning two or more different outputs
vertical line test
draw a line through any given x value
it should only cross once
(any given x value in its domain of course)
hoary blade
so y = sqrt x, x value u get is only positive
and x^2 = y, the x values positive and negative that u look at
versed mica
well the x value must be >= 0 but yes it only returns nonnegative y values (>= 0)
versed mica
and x^2 = y, the x values positive and negative that u look at
yes
hoary blade
well the x value must be >= 0 but yes it only returns nonnegative y values (>= 0...
ya otherwise no solution
right
versed mica
well yea because we are only considering real values of y
sqrt(negative number) makes no sense for real numbers
hoary blade
square root of number that is less than 0 is just no solution isnt it? i mean thas what my teachers said
mayeb not in higher level math tho
versed mica
yes at this level
later on you might see i = sqrt(-1)
it’s how we defined sqrt for negative inputs
versed mica
we just called it something because it was useful and it shows up a lot
hoary blade
it is very much appreciated
versed mica
well thanks for all the help
you’re welcome
hoary blade
very helpful u were
versed mica
i try
pearl pondBOT
pearl pondBOT
Available help channel!
rapid wagon
Is my desmos broken? My teacher just said it's not 0
pearl pondBOT
Is my desmos broken? My teacher just said it's not 0
rapid wagon
He had to define it like this:
$\begin{cases} e^{\frac{-1}{x^2}} & x \neq 0 \ 0 & x = 0 \end{cases}$
jolly parrotBOT
prograce
rapid wagon
Is my desmos broken? My teacher just said it's not 0
e^-1/x^2 is never 0 so how?
sharp vigil
desmos sometimes treats things as defined even when they aren't, strictly speaking
internally it is probably evaluating it as something like "e^(-infinity) = 0" which is true in the limit but doesn't make sense as an arithmetic statement
rose robin
in desmos 1/0 might be infinity
and 1/(-0) is -infinity
based on how it works in floating point math
it then shows non numbers as undefined
1/(1/0) also shows 0
pearl pondBOT
pearl pondBOT
Available help channel!
lime river
Could someone explain how do we get this condition
pearl pondBOT
Could someone explain how do we get this condition
@lime river Has your question been resolved?
rough forge
Could someone explain how do we get this condition
You basically want to minimize the distance between $f$ and its asymptote as $x \to \infty$ so that we say the limit or desired distance at infinity should be 0, so [ \lim_{x \to \infty} f(x) - (mx+c) = 0 ]
which can be transformed as [ \lim_{x \to \infty} f(x) - mx = c. ]
jolly parrotBOT
anti-algebraist 𝔸dωn𝓲²s
lime river
You basically want to minimize the distance between $f$ and its asymptote as $x ...
Ahhh I get it now
Thanks alot ❤️
.close
ashen lance
Is this weight average?
pearl pondBOT
Is this weight average?
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.
hot canyon
"Do your best to provide a translation"
hot canyon
Is this weight average?
From what I can make sense off, given are four groups A, B, C, D of 15, 20, 10, 18 students and their respective average heights
ashen lance
Yes.
hot canyon
Yes.
that implies, the sum of heights of all students in group A + that of group B + ... is 15*1.62 + 20*1.48 + ...
(why? hint: average = sum of data / frequency)
now, to get the total average use formula again: total avg = [total sum of all heights]/[total frequency]
ashen lance
So 15 * 1,62 and do that each one and then divide by the total amount of students?
.closw
.close
pearl pondBOT
pearl pondBOT
Available help channel!
prime bane
pearl pondBOT
prime bane
Why am i getting the wrong answer
Isnt speed in 1’st part + 9 = speed in 2’nd part
Btw im stuck in C
hallow remnant
no
speed in 1st part=speed in 2nd part +9
the first part of the journey is the one with the bigger speed
prime bane
Huh
They said soeed for the first part is 9 faster than second part
Which means 1’st + 9 = 2’nd
hallow remnant
Which means 1’st + 9 = 2’nd
no?
hallow remnant
no it means 1st=9+2nd
prime bane
Lol let me try
hallow remnant
because the speed in 1st is 9 km/h faster
hallow remnant
let me see
pearl pondBOT
pearl pondBOT
Available help channel!
tepid geyser
Forgot who to do this, I need to find the angle measurement of LPR and also pls explain step by step🙏
light helm
stary by looking up parallel line theorems
unborn shuttle
Forgot who to do this, I need to find the angle measurement of LPR and also pls ...
Use the Intern altern angles
PLN angle = MPL angle
pearl pondBOT
@tepid geyser Has your question been resolved?
pearl pondBOT
pearl pondBOT
Available help channel!
prime bane
Why am i getting +35000 instead of -35000 i checked everything
prime bane
<@&286206848099549185> sorry for the early ping but i need help quick
iron stream
<@&286206848099549185> sorry for the early ping but i need help quick
Pump b takes 21 mins longer
iron stream
Pump b takes 21 mins longer
So you would probably add 21 to the other side of the eqn
iron stream
<@&286206848099549185> sorry for the early ping but i need help quick
Also you must be aware that 3500/(x-10) > (3500/x) when x>0
trim seal
what did i do wrong?
pearl pondBOT
what did i do wrong?
trim seal
cobalt hinge
what did i do wrong?
why did you suddenly take the i^2 thing
eternal tulip
you added 9/100 when you should've subtracted
cobalt hinge
you can just do $\qty(x-\frac{3}{10})=\sqrt{\frac{-29}{100}}$ without those shenanigans
jolly parrotBOT
;(
cobalt hinge
so yeah, you also went wrong there
trim seal
aright thanks
pearl pondBOT
@trim seal Has your question been resolved?
cunning compass
WTH
cunning compass
How did my teacher get 43 and 44
Ur only suppose to find arc po then add all the arcs apart from mp no?
Make it make sense pla
Pls
gaunt surge
For 7 she wants the measure of the angle not the arc
cobalt hinge
How did my teacher get 43 and 44
can u get a better picture of the diagrams pls!
cunning compass
Sure
cobalt hinge
for 9
gaunt surge
Same with 9 I’m pretty sure
cunning compass
cunning compass
For 7 she wants the measure of the angle not the arc
Ohhh
Omg
I DIDNT SEE GHAT
😭😭😭
Thanks chat
cobalt hinge
yeah cause theres no way an angle is existing like that in a triangle
sanity check!
cunning compass
.close
gaunt surge
Same with 9
pearl pondBOT
pearl pondBOT
Available help channel!
rugged jewel
this is a rookie question and i think the answer should be -3xyz but the question paper says 0
unborn abyss
just let x=1 y=2 z=-3 and see where you end up
rugged jewel
i mean is -3xyz the correct answer
sly plover
i mean is -3xyz the correct answer
yeah
inland ivy
i mean is -3xyz the correct answer
it is