but idk what that means 😭
#help-26
hollow void
potent silo
In Part A, you can say:
When c = 10, we have one real root and no stationary points
When c = 0, one real root and one stationary point
When c = 1/2, one real root and two stationary points
so on and so on
potent silo
but idk what that means 😭
It means that you have to prove your observations mathematically, with the help of calculus
hollow void
mhm mhm
hollow void
It means that you have to prove your observations mathematically, with the help ...
would it mean like derivative calculations?
potent silo
For example, imagine you have the question:
Let f(x) = x^3 - 3cx + 2, where c is constant strictly greater than 0. Show that f has one stationary point and one root.
Answering that question would solve the first part
potent silo
Yes
hollow void
if c > 0, then it has to be atleast 3x^2 - 3 and onwards
wait what about decimals?
should i care about that
wait nvm let me restart
f'(x) = 3x^2 - 3c
let f'(x) = 0
0 = 3x^2 - 3c
0 = x^2 - c
c = x^2
x = +/- sqrt c
thats one root
and no stationary point
true hamlet
What is the definition of a stationary point?
potent silo
^
potent silo
Yes, and how does it translate here?
hollow void
to my working out? um
cant i substitute the real root into the originial f(x) and solve?
hollow void
yes
potent silo
Ok so fixing c > 0
f'(x) = 3x^2 - 3x = 3(x^2 - c)
And it does have two solutions, as you said, x = sqrt(c) or x = -sqrt(c)
hollow void
f'(x) = 3x^2 - 3c
let f'(x) = 0
0 = 3x^2 - 3c
0 = x^2 - c
c = x^2
x = +/- sqrt ...
did i already like mention it in my working out?
potent silo
This means f has two stationary points in this case, so in other words your answer is part A is wrong 😬
hollow void
what 😭
hollow void
f'(x) = 3x^2 - 3c
let f'(x) = 0
0 = 3x^2 - 3c
0 = x^2 - c
c = x^2
x = +/- sqrt ...
so this is wrong?
potent silo
No, this part is right, but the part A is wrong
hollow void
oh thiss
potent silo
When c = 1, for example, you have f(x) = x^3 - 3x + 2, and the graph looks like this
hollow void
When c = 1, for example, you have f(x) = x^3 - 3x + 2, and the graph looks like ...
it has two real roots and two stationary points right
potent silo
You have two stationary points here
Yes
But why is c = 1 included in two cases then? I didn't see carefully the 4th line here:
Even when 0 < c < 1, it falls into two categories
It should not happen 🙂
hollow void
wait so i cant have double ups?
potent silo
Well, in this case, no, because there are contradictions
hollow void
can i change it to like 0 < c < 1.01
potent silo
When c = 1/2, for example
First line tells there are no stationary points
while third line tells there are two stationary points
potent silo
can i change it to like 0 < c < 1.01
No, you have to distinguish all the possible outcomes such that there aren't any confusion like the one I just described
potent silo
I would suggest to first focus on the stationary points
And see for which values of c you have 0, 1 or 2 solutions to the equation f'(x) = 0
When c > 0, you have two solutions, for example
hollow void
wait i wrote it wrong
when c is negative, there's only one real root and no stationary points
hollow void
wait im so stupid 😭
sorry
my fault
i was trying to do something like approaching +/- infinity but i didnt know how to do it
potent silo
It's ok, don't worry, typos happen
hollow void
potent silo
Since f'(x) = 3(x^2 - c)
This proves that f has no stationary point if c < 0
one stationary point if c = 0
two stationary points if c > 0
hollow void
yeah
potent silo
And now, you can solve the question of the number of real roots for each cases
potent silo
You're welcome!
indigo comet
Given a rectangle ABCD where |AB| : |BC| = 2 : 1.
On its sides AB, BC, CD, DA are given the points K, L, M, N such that KLMN is a rectangle,
in which |KL| : |LM| = 3 : 1. Calculate the ratio of the contents of the rectangles ABCD and KLMN.
I thought I could calculate the white part, but I don't know if that's the right way and I don't know how I should do it
topaz sinewBOT
Given a rectangle ABCD where |AB| : |BC| = 2 : 1.
On its sides AB, BC, CD, DA ar...
indigo comet
I tried using Pythagorean theorem but it got me nowhere
topaz sinewBOT
@indigo comet Has your question been resolved?
steady escarp
What does it mean by the contents of the rectangles?
indigo comet
What does it mean by the contents of the rectangles?
area I put in in the translator my bad
indigo comet
and CML is equal to AKN
Yea I thought of that but didnt know what to do with that info
sterile wave
So DM/NA = MN/NK = DN/AK = 3
steady escarp
Well I think you can make use of similar triangles then
U can take BC = x and ML = y or something to help you if necessary
They should cancel out in the end
sterile wave
The idea is to calculate the ratio of DA/NK, and you can get the result because ABCD = 2DA^2 and KLMN = 3NK^2
I can give you the process directly but I think it is better to go through step by step
indigo comet
Oh I see
sterile wave
and we also need Pythagorean theorem here
indigo comet
and we also need Pythagorean theorem here
Yeah I thought that
I will try to do it I know the general direction now
Thank you
sterile wave
Yeah I thought that
cool, then you can try it by yourself at first, I can provide more details if you still need help.
topaz sinewBOT
cool, then you can try it by yourself at first, I can provide more details if yo...
@sterile wave Has your question been resolved?
sterile wave
.close
neon iron
A typical ring has charge $\dd Q$, inner radius r, and outer radius $r + \dd r$. Its area is approximately equal to its width $\dd r$ times its circumference $2\pi r,$ or $\dd A = 2\pi r \dd r$
topaz sinewBOT
A typical ring has charge $\dd Q$, inner radius r, and outer radius $r + \dd r$....
thorny flameBOT
neon iron
so this is my sketch of the situation
i know its simple geometry
but how is width*circumference = area in here?
split igloo
dumb physics notation
split igloo
but how is width*circumference = area in here?
it's an approximation
there's a (dr)^2 term you ignore
topaz sinewBOT
@neon iron Has your question been resolved?
radiant torrent
for the first one, show that $\mathcal{P}(M \cap N) \subseteq \mathcal{P}(M) \cap \mathcal{P}(N)$ and $\mathcal{P}(M) \cap \mathcal{P}(N) \subseteq \mathcal{P}(M \cap N)$
thorny flameBOT
AlphaNull
odd pagoda
no
radiant torrent
you're forgetting emptyset
odd pagoda
missing the emptyset
dont write emptyset as {}
very bad for readability
yes
yes
yes
well for this specific example
not in general
well in general for set equality you show double inclusion
take an element from the left side and show that it is an element of the right side
then take an element from the right side and show it is an element of the left side
odd pagoda
for the first one, show that $\mathcal{P}(M \cap N) \subseteq \mathcal{P}(M) \ca...
in symbols, this
well depends on what you mean by knowing
what are the elements of P(M n N)
I mean speaking generally
using the definition of powerset
the set of all subsets but yes
so if X is an element of P(M n N), that means X is a subset of M n N
pastel oracle
you need to prove it for arbitrary sets M and N
btw I think the second equality is not true
yes
pastel oracle
only one containment is true, P(M) u P(N) is a subset of P(M u N)
odd pagoda
to prove that something is wrong you have to give a counterexample
pastel oracle
the containment in the other direction is not true, so they are not equal
that is true
odd pagoda
those things are not mutually exclusive
pastel oracle
do you know what set equality means, in terms of subsets/containments?
$A = B \iff (A \subseteq B) \land (B \subseteq A)$
thorny flameBOT
cwatson
pastel oracle
I wouldn't say "trivially", but yes it is false
if you are looking for a counterexample, you could choose what M and N are. But no, what you wrote is not true. with those examples, P(M) n P(N) would be a set with only the empty set, sure
for the intersection equality, which is true, you have to prove it for general/arbitrary M and N
you prove it by showing both $\mathcal{P}(M \cap N) \subseteq \mathcal{P}(M) \cap \mathcal{P}(N)$ \emph{and} $\mathcal{P}(M) \cap \mathcal{P}(N) \subseteq \mathcal{P}(M \cap N)$
thorny flameBOT
cwatson
pastel oracle
how would you prove any other set equality/containment?
do you know the definition of $A \subseteq B$?
thorny flameBOT
cwatson
pastel oracle
yes, so you choose an element in A, and show it must also be in B
...you are not doing it correctly. you don't choose what A "looks like"
in this case, A is $\mathcal{P}(M \cap N)$ and B is $\mathcal{P}(M) \cap \mathcal{P}(N)$. you need to use the definitions of those to prove it. that's all
thorny flameBOT
cwatson
pastel oracle
let $x \in \mathcal{P}(M \cap N)$ arbitrary. use the definition of the power set to show also $x \in \mathcal{P}(M) \cap \mathcal{P}(N)$
thorny flameBOT
cwatson
pastel oracle
what does it mean for some x to be in the power set of some set
then I suggest you consult your textbook and/or lecture notes to understand these definitions. otherwise you won't be able to prove statements about them
no
$X \in \mathcal{P}(A)$ is equivalent to $X \subseteq A$
thorny flameBOT
cwatson
pastel oracle
no... study the material carefully
while it is true that A is an element of P(A), that's not how I suggest thinking about it
yes
topaz sinewBOT
topaz sinewBOT
Available help channel!
green sierra
Hi
fast fulcrum
I need help calculating value of tan(-150).
fast fulcrum
the period of tan? what do you mean
i'm supposed to use the unit circle, but i mess up on the paper calculations
fast fulcrum
alright, how does that help me solve this? this is my second week so im still learning alot
fast fulcrum
oh okay so theyre supplementary i think its called
but it says the correct answer for the value of tan(-150) is sqrt3/3
i just dont know how to get there
neon iron
they are the same thing
sqrt3/3=1/sqrt3
they just multiplied both denominator and numerator by sqrt3
frosty axle
tan(x) = tan(180°+x)
fast fulcrum
is it the same thing for secent? to be clear i dont need to look at a unit circle to solve this:
frosty axle
sec is just 1/cos
fast fulcrum
how does that help me solve this problem if i dont have cos
i need to find cos first?
frosty axle
uh
fast fulcrum
!close
frosty axle
type close but with .
real sierra
why are we subtracting?
topaz sinewBOT
why are we subtracting?
real sierra
a(x-x0) etc..
odd pagoda
so that if we plug in (x0, y0, z0) all the terms are 0
real sierra
but i don’t get get how or why this is an eqn of a plane
like how can having a point and a normal vector create a plane
topaz sinewBOT
@real sierra Has your question been resolved?
candid kelp
I'm having an argument about a video game, and how it calculates damage reduction.
Everyone claims the formula is NOT diminishing returns - 100/(100+Rating)
topaz sinewBOT
I'm having an argument about a video game, and how it calculates damage reductio...
candid kelp
That formula plots to DR obviously
The stat in the video game is PRR
They claim - Every 100 prr is as valuable as the previous 100. While it gives a smaller number, it's effect is your base hp applied again. If you have 900 prr, you take 10% of normal damage - and it's 10 hits now.
If you had 1900 prr, you'd take 5% of normal damage and it's 20 hits. EVERY 100 prr is another hit. Every 100 prr is your base hp added again. It continues like that.
"Not only did that not clarify anything (it is smaller, but the same? Is that you Theodore Logan?) but like much of the DDO player base, you seem to not know what "diminishing returns" actually means. It is not a term that is used to describe something that has a linear return on investment, which PRR most certainly does. Funny enough...nobody says that about melee/ranged/spell power and they use the same exact mechanic."
"With 0 MRR/PRR/power you are at 1:1. Every point of MRR/PRR/power adds .01 to the ratio.
So at 0 PRR every 1 point of incoming damage does 1 hit point. Or a 1:1 ratio.
At 100 PRR every 2 points of incoming damage does 1 hit point, or a 1:2 ratio.
At 400 PRR every 5 points of incoming damage does 1 hit point, or a 1:5 ratio."
"You can literally have a first grader put them on a simple line graph and see how it linearly scales."
This is the thread FYI - https://www.reddit.com/r/ddo/comments/l8cpfi/prr_vs_con_set_bonus_augments/
topaz sinewBOT
@candid kelp Has your question been resolved?
candid kelp
@cold wagon ping
topaz sinewBOT
@candid kelp Has your question been resolved?
topaz sinewBOT
@candid kelp Has your question been resolved?
candid kelp
@cold wagon ping
candid kelp
.close
topaz sinewBOT
topaz sinewBOT
Available help channel!
neon iron
im trying to graph this rational function, just wanting to check if its right
neon iron
topaz sinewBOT
@neon iron Has your question been resolved?
regal kiln
An object is moving along the number line; its position at each time t≥0 is given by: s(t)=At3−Bt2+1, where A and B are real numbers. This object's velocity when t=1 is 10, and its acceleration at this time is 3. Find the position of this object at t=1.
topaz sinewBOT
An object is moving along the number line; its position at each time t≥0 is give...
regal kiln
Well turns out I need more help
pseudo sonnet
you can differentiate once to get the velocity function, then again to get the acceleration function
you can solve for the constants A and B given the information that v(1) = 10 and a(1) = 3
pseudo sonnet
Its basically a system of equations right
yeah it is
regal kiln
I did that but I was confused
pseudo sonnet
you have 3 unknowns, 3 eqs
pseudo sonnet
I did that but I was confused
show what you did
regal kiln
Well, when we differentiate we get:
v(t) = 3At^2 - 2Bt
a(t) = 6At - 2B
Or plugging in:
3A - 2B = 3
6A - 2B = 10
Right
pastel oracle
looks right
regal kiln
The simplest thing would be canceling the B's right
So if I just multiply by 1 for the first eq, and -1 for the second then we get
3A - 2B = 3
-6A + 2B = -10
pastel oracle
it's up to you
pseudo sonnet
-3A = -7
A = 7/3
wait how
pastel oracle
you didn't add them correctly
regal kiln
Oh
Im dumb
Oops
There we go
Then we plug in A into our eq right. The first is easier
So 3(7/3) -2B = 3
21/3 - 2B = 3
7 -2B = 3
-2B = -4
B = 2
Then plug into s(t)?
pastel oracle
do what you think is right
regal kiln
(7/3)(1)^3 - (2)(1)^2 + 1
7/3 - 2 + 1
4/3?
But the problem with that is
I believe all my answer choices were fractions over 6
🥲
pseudo sonnet
I believe all my answer choices were fractions over 6
how?
regal kiln
I
Am not sure
Maybe its none of the above
I think that was also an option
For reference, this was the problem. All the choices are over 6. And the answer isnt none of the above.
I also just tried to solve this other one and also got it wrong. Not sure what Im missing or doing wrong here
<@&286206848099549185>
scenic crow
Ok here I can help you
scenic crow
For this one
regal kiln
Well, when we differentiate we get:
v(t) = 3At^2 - 2Bt
a(t) = 6At - 2B
From here
scenic crow
you didn't add them correctly
This was the issue
regal kiln
Wha
scenic crow
You're adding (-6A+2B)+(3A-2B)=-10+3
regal kiln
Yes
scenic crow
No actually you did it right
regal kiln
🥲
scenic crow
A=7/3 is correct
regal kiln
So plug that into 3A - 2B = 3
scenic crow
So 3(7/3) -2B = 3
21/3 - 2B = 3
7 -2B = 3
-2B = -4
B = 2
This is as well right B=2
regal kiln
But
As you can see the answer choices are wrong
Or maybe the answer is wrong
And none of the above is definitely not right
scenic crow
The answer would be F) none of the above
regal kiln
It isnt
scenic crow
And none of the above is definitely not right
It can be
regal kiln
Bc unfortunately, I already tried it
scenic crow
Bc unfortunately, I already tried it
Ah OK I see
scenic crow
I can try plugging the question into this app I have its called sizzle ai
It answers questions in math, chemistry, physics, bio and economics
regal kiln
Interesting
scenic crow
I got this for A and B
@regal kiln
You can @ mentoin me when you see this to make sure I see your message
regal kiln
Im looking at it now
Im just confused
Why did you have to find 2B like that
I understand how you did this, but its just odd that the problem would require you to find 2B instead of just B alone
Basically, why cant you use the normal way of solving a system of equations instead of substituting alot
scenic crow
I just wanted to try a different way of finding B because apparently the idea of canceling wasn't working and so I substituted 2B because it was there in both equations
cyan plaza
Let's solve the given system of equations step-by-step:
(10 = 3A - 2B) (Equation 1)
(3 = 64 - 2B) (Equation 2)
(3 + 2B = 6A) (Equation 3)
(2B = 6A - 3) (Equation 4)
To solve the system, we'll start by isolating A in terms of B and substituting it into the other equations:
From Equation 3, we can rewrite it as:
(2B = 6A - 3)
Rearranging:
(6A = 2B + 3)
Dividing through by 6:
(A = \frac{2B + 3}{6}) (Equation 5)
Now we'll substitute Equation 5 into Equation 1:
(10 = 3\left(\frac{2B + 3}{6}\right) - 2B)
Simplifying this expression:
(10 = \frac{6B + 9}{6} - 2B)
Multiplying through by 6:
(60 = 6B + 9 - 12B)
Combining like terms:
(6B - 12B = 60 - 9)
(-6B = 51)
Dividing through by -6:
(B = -\frac{17}{2})
Now that we have the value of B, we'll substitute it into Equation 5 to find the value of A:
(A = \frac{2\left(-\frac{17}{2}\right) + 3}{6})
Simplifying this expression:
(A = \frac{-17 + 6}{6})
(A = \frac{-11}{6})
Therefore, the solution to the given system of equations is:
(A = -\frac{11}{6}) and (B = -\frac{17}{2}).
To check the solution, we can substitute these values back into the original equations and see if they hold true.
thorny flameBOT
Spooderman
cyan plaza
Is this correct
regal kiln
Unfortunately, I cant check the right answer
cyan plaza
This is only my opinion
regal kiln
I will determine what ends up right once I try doing a similar problem again
But I think Sarah did it correct with less work
Although I appreciate both of your assistance
scenic crow
No problem 🙂
fleet summit
Find an improper integral where the intergal with limits a to b f(x)dx have an infinite discontinuity at c where a<c<b
topaz sinewBOT
Find an improper integral where the intergal with limits a to b f(x)dx have an i...
topaz sinewBOT
@fleet summit Has your question been resolved?
sick basalt
given that L1//C1, the angles a and b are equals, right? but the calculations doesnt say that
topaz sinewBOT
given that L1//C1, the angles a and b are equals, right? but the calculations do...
restive inlet
well the issue here actually is that this setup isn't possible
graceful leaf
i keep getting mixed up plz
topaz sinewBOT
i keep getting mixed up plz
graceful leaf
-b/2a is for vertex or completion of the square
or (b/2a)^2
I happen to be doing both lessons and it keep mixin g me up
keen matrix
-b/2a is the vertex
completing the square is b/2
$x^2+bx+c=x^2+bx+c+(\tfrac{b}{2})^2-(\tfrac{b}{2})^2=(x-\tfrac{b}{2})^2+c-(\tfrac{b}{2})^2$
thorny flameBOT
FancyBredFries
keen matrix
THnxs very much
try complex the square with $x^2+\tfrac{b}{a}x+\tfrac{c}{a}=0$ and watch some magic happen! You may recognize the result :)
thorny flameBOT
FancyBredFries
topaz sinewBOT
@graceful leaf Has your question been resolved?
graceful leaf
Yes excellently thnx very much
topaz sinewBOT
topaz sinewBOT
Available help channel!
neon iron
bonjour j eparle français et j'ai besoin d'aide urgente
topaz sinewBOT
bonjour j eparle français et j'ai besoin d'aide urgente
neon iron
flat kindle
junior acorn
bonjour j eparle français et j'ai besoin d'aide urgente
De quoi avez-vous besoin ? Envoyez l’image
neon iron
De quoi avez-vous besoin ? Envoyez l’image
merci pour votre aide mon dm est en pdf vous preferez une capture d'écran?
j'ai besoin d'aide pour le 3 de l'exercice 1 et le 1 de l'exercice 2
junior acorn
Envoyer l’image s’il vous plaît
neon iron
junior acorn
Qu’avez-vous fait jusqu’à maintenant ?
neon iron
tout sauf le 3) de l'exercice 1 et le 1) de l'exercice 2
junior acorn
qu'avez-vous fait pour 3) de l'exercice 1
neon iron
qu'avez-vous fait pour 3) de l'exercice 1
en fait j'ai besoin d'aide uniquement pour le 1) de l'exercice 2 j'ai trouver les coordonnées de tout les points sauf B et C
junior acorn
en fait j'ai besoin d'aide uniquement pour le 1) de l'exercice 2 j'ai trouver le...
qu'avez-vous fait pour 1) de l'exercice 2
neon iron
j'ai placer tout les points dans un repère
j'ai trouve
D:0;0
A:0;1
F:1;0
K:0.5; 1
E:1;1
mais pas B ni C
@junior acorn
topaz sinewBOT
@neon iron Has your question been resolved?
topaz sinewBOT
@neon iron Has your question been resolved?
topaz sinewBOT
@neon iron Has your question been resolved?
neon iron
@junior acorn
topaz sinewBOT
@verbal bane Has your question been resolved?
zealous dock
did i do this right and if so should I simplify the answer more or should this be fine
topaz sinewBOT
did i do this right and if so should I simplify the answer more or should this b...
upper portal
topaz sinewBOT
@zealous dock Has your question been resolved?
topaz sinewBOT
@zealous dock Has your question been resolved?
topaz sinewBOT
@zealous dock Has your question been resolved?
zealous dock
hello
zealous dock
.close
visual jewel
Wondering if someone can help teach me a simple way of simplifying sqaure roots.
For example sqaure root 96 simplified is according to my booklet is 4 multiplied by the sqaure root of 12
topaz sinewBOT
Wondering if someone can help teach me a simple way of simplifying sqaure roots....
visual jewel
long stirrup
that's wrong
visual jewel
Hi, I'm.just trying fmto wrap my head around simplifying these non perfect square numbers
long stirrup
sqrt(6×16)
sqrt(6)sqrt(16)
4sqrt(6)
visual jewel
Sqrt?
long stirrup
square root
visual jewel
K
long stirrup
i don;t know any simple ways, just saying 4 root 12 is the wrong answer
visual jewel
Well this is what they give me as an.answer
Number 13 is the answer but trying to get there is something I do nit understand
If divided the answer and for to 12 but it doesn't seem right
Nvm sorry to bother you, im just wasting your time with this simple stuff
radiant tapir
nah that's why we're here as helpers
radiant tapir
you have the right idea here
clear lodge
sqrt(96) is not 4*sqrt(12), it's 4*sqrt(6)
radiant tapir
you do the prime factorization, and any even powered prime factors can be simplified
clear lodge
the solution given by the book is wrong there
radiant tapir
Mainly
clear lodge
you'd get it by doing the next step that you're doing here
radiant tapir
$\sqrt{96} = \sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3} = \sqrt{2^4 \cdot 2 \cdot 3} = 4 \sqrt{6}$
thorny flameBOT
MellowDramaLlama
clear lodge
yes, the answer key is wrong
visual jewel
I'm now confused on why the 2 to the power of four becomes a 4 on the left side if the radical
radiant tapir
So you can do the following
$\sqrt{96} = \sqrt{2^4 \cdot 6} = \sqrt{2^4} \cdot \sqrt{6}$ Then we know, by definition that $\\ \sqrt{a} = a^\frac{1}{2}$, so $\sqrt{2^4} = (2^4)^\frac{1}{2} = 2^{4 \cdot \frac{1}{2}} = 2^2 = 4$
thorny flameBOT
MellowDramaLlama
visual jewel
So an easier way to understand it for me might be that though 2^4 is 16 I can simplify it to ITS square root and that is still true
Soo
visual jewel
Yes?
Full disclosure I have a NVLD so I often don't see the same patterns others do so easily
Thank you though, this changes everything
placid cedar
.
neon iron
how do i know how to graph functions with two x’s
topaz sinewBOT
how do i know how to graph functions with two x’s
neon iron
like square root (x+4)(x-4)
noble laurel
how do i know how to graph functions with two x’s
!occupied
topaz sinewBOT
how do i know how to graph functions with two x’s
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).
topaz sinewBOT
.
What step are you on?
1. I don't know where to begin
2. I have begun but got stuck midway
3. I got an answer but I'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above
2
noble laurel
!show
topaz sinewBOT
Show your work, and if possible, explain where you are stuck.
neon iron
how do i graph
actually how to find transformations
one says right 4 one says left 4
do i do both?
pls help
topaz sinewBOT
@neon iron Has your question been resolved?
topaz sinewBOT
@neon iron Has your question been resolved?
hearty goblet
Hello I’m trying to prove the converse of 17. I know the answer is when s and t are coprimes and I know that implies st=lcm(s,t), but im not sure how I can go from that to a = b (mod st)
topaz sinewBOT
Hello I’m trying to prove the converse of 17. I know the answer is when s and t ...
hearty goblet
<@&286206848099549185>
topaz sinewBOT
@hearty goblet Has your question been resolved?
hearty goblet
.close
stray prairie
topaz sinewBOT
broken niche
The first one is saying find f(g(x)), so to find the rule, replace the x in f(x) with g(x)
stray prairie
The first one is saying find f(g(x)), so to find the rule, replace the x in f(x)...
yeah ik i just dont really understand how to get the domains
broken niche
Find the points that are undefined, and exclude them. For f(g(x)) if x=-1 then you would be dividing by zero
for g(f(x)) you get the denominator is x^2+x+1 so whenever that equals 0 needs to be excluded from domain
frosty axle
for more information: ||x^2 + x + 1 NOT EQUAL zero||
topaz sinewBOT
Available help channel!
hardy vector
.
hardy vector
nah i was here first
topaz sinewBOT
nah i was here first
hardy vector
theres other open channels dude
hardy vector
like 3
covert onyx
ll
hardy vector
if you have a vector (2,2), is this a point? or a line from (0,0) to (2,2)
covert onyx
if you have a vector (2,2), is this a point? or a line from (0,0) to (2,2)
a point where the 0,0 meets
till 2,2
hardy vector
alr
nimble copper
Heyo i need help with complex numbers
topaz sinewBOT
Heyo i need help with complex numbers
lethal copper
solve for side c. every time i use law of cosines i get a ridicuously large number (51.25 specifically). the actual answer is 7.16 but i am truly stumped on it
topaz sinewBOT
solve for side c. every time i use law of cosines i get a ridicuously large numb...
drifting swift
show work!
lethal copper
alright 1 sec
lethal copper
show work!
plugging into c^2 = a^2 + b^2 - 2ab cos C
3^2 + 6^2 - 2(3)(6) cos 100
45 - (36 cos 100)
45 - (-6.25...)
51.25
i have it in degree mode (between you and me i still have no idea when im supposed to be in which one) which gets me around 13-ish, which is better but not the answer lol
drifting swift
the law is c**^2** = a^2 + b^2 - 2bc cos(C).
51.25 is the value of c^2 and not of c itself.
topaz sinewBOT
drifting swift
also you're supposed to use degree or radian mode according to which unit your angles are in
and here, the angle is in degrees
topaz sinewBOT
Available help channel!
lucid junco
So I'm not sure how to integrate $\pi\int_{0}^{2}\left(\frac{1}{x^{2}-6x+9}\right)dx$. I thought of splitting fractions but that wouldnt work, any ideas?
topaz sinewBOT
So I'm not sure how to integrate $\pi\int_{0}^{2}\left(\frac{1}{x^{2}-6x+9}\righ...
thorny flameBOT
water beam
lucid junco
I originally had $\pi\int_{0}^{2}\left(\frac{1}{3-x}\right)^{2}dx$
thorny flameBOT
water beam
opal vault
the antiderivative of (x-a)^n is pretty easy to find
if it's too hard to see, do a variable change
lucid junco
What, like u-sub?
opal vault
u = 3-x
junior acorn
**water beam**
just do u = x-3
lucid junco
so then u^-2?
junior acorn
yeah
thorny flameBOT
water beam
lucid junco
$-\pi\int_{0}^{2}u^{-2}du$
thorny flameBOT
water beam
lucid junco
is this correct so far?
junior acorn
you forgot to switch the bounds of the integral to correspond to u instead of x
other than that, yes
lucid junco
you forgot to switch the bounds of the integral to correspond to u instead of x
could you elaborate
junior acorn
wait wot
u = 3 - x, so, if the integral goes from x = 0 to x = 2, then that corresponds to u = 3 to u = 1
lucid junco
o
junior acorn
which you would have to multiply by -1 to flip around to the correct order of 1 to 3 instead of 3 to 1
which is why i said u = x - 3, since that's easier to work with if you dont understand this switching of bounds
lucid junco
ohhhhhhh
lucid junco
no yeah i remember now
its this thing $\int_{g\left(a\right)}^{g\left(b\right)}f\left(u\right)du$ i havent done definite u-sub in a few days so forgot that part
thorny flameBOT
water beam
lucid junco
but now i remember
@junior acorn hey im having some calculation issues maybe its me plugging in things wrong but I'm getting -2pi/3 when i should be getting positive
i get pi/3 - pi
which is weird
lucid junco
**water beam**
@junior acorn its this step
i cant factor out the negative from the du
so the negative pi messed with my calculations
junior acorn
i cant factor out the negative from the du
yeah the negative goes into flipping the bounds, remember?
lucid junco
wai wut
junior acorn
you have -du
junior acorn
which you would have to multiply by -1 to flip around to the correct order of 1 ...
and i mentioned this
you have to multiply by -1 again
so
it goes away
lucid junco
which you would have to multiply by -1 to flip around to the correct order of 1 ...
why does it reverse
junior acorn
why does it reverse
cause if it doesnt then your integral goes from a higher bound to a lower bound
kinda makes no sense
basically how it works is
lucid junco
i dont get how like the -1 * (insert integral) changes the stuff
i think i have an idea maybe
junior acorn
Let $F(x) = \int f(x)$ $dx \Rightarrow \int_a^b f(x)$ $dx = F(b) - F(a)$
thorny flameBOT
nalin
junior acorn
so $\int_b^a f(x)$ $dx = F(a) - F(b) = - (F(b) - F(a))$
thorny flameBOT
nalin
lucid junco
$\int_{a}^{b}f\left(x\right)dx\ ↔\ -\int_{b}^{a}f\left(x\right)dx$
thorny flameBOT
water beam
junior acorn
yeah
lucid junco
ohh ok
junior acorn
that's what im conveying
lucid junco
i see
so multiply by -1
can i multiply the final answer by -1 and itll still work?
or do i have to do it from the start
drifting swift
$\int_{a}^{b}f\left(x\right)dx\ ↔\ -\int_{b}^{a}f\left(x\right)dx$
$\leftrightarrow$ should just be $=$ here.
thorny flameBOT
Ann
junior acorn
$\leftrightarrow$ should just be $=$ here.
yeah thanks for pointing that out
lucid junco
can i multiply the final answer by -1 and itll still work?
so can anyone answer this question
i guess it doesnt matte rthe order
junior acorn
can i multiply the final answer by -1 and itll still work?
yeah it will
the place you do it doesnt really matter in this case, although technically you should be doing it before you evaluate the integral but it's fine
thick oyster
we have
thick oyster
(x^3+4)^2 - (x^3+4) [second to last step] -> [last step]
why did they take it out of the denominator
sleek galleon
anyone can come to help-19?
neon iron
How to prove that a real symmetric matrix with real and distinct eigenvalues has linearly independent eigenvectors?
topaz sinewBOT
How to prove that a real symmetric matrix with real and distinct eigenvalues has...
neon iron
More than the proof method
Or idea
I want an intuition as to why this should be true
worthy storm
it holds without the real and symmetric assumptions, btw
distinct eigenvalues suffices
For intuition, you might consider an example where two eigenvectors are linearly dependent. Then one is just a scaled version of the other, so the matrix has to do essentially the same thing to both vectors
In particular, it can't apply different scale factors to them
topaz sinewBOT
@neon iron Has your question been resolved?
quiet niche
is it possible to find a closed form of the nth order taylor expansion of polynomials of any degree around a non-zero point? i realised you can express nth derivates of monomials as n!/(n-k)! * a*x^(n-k) (which when inserted in the taylors expression yields a binomial coefficient) but i'm not sure if this can come into use with this aim
topaz sinewBOT
is it possible to find a closed form of the nth order taylor expansion of polyno...
cursive thorn
The taylor expansion of a polynomial just gives the original polynomial
topaz sinewBOT
@quiet niche Has your question been resolved?
topaz sinewBOT
@quiet niche Has your question been resolved?
quiet niche
j
quiet niche
<@&286206848099549185>
topaz sinewBOT
@quiet niche Has your question been resolved?
knotty geyser
How do i invert this equation?
topaz sinewBOT
How do i invert this equation?
neon venture
hmm wdym
knotty geyser
wait a second let me find the page of the exercise
How do i turn it into the answer below? like instead of f(x) to f^-1(x)?
neon venture
hmm solve for x f(x) = y🤔
$$f(x) = \frac{3}{2x²+1} = y$$
$$\frac{2x²+1}{3} = \frac{1}{y}$
thorny flameBOT
Herels
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
neon venture
$y(2x²+1) = 3$
thorny flameBOT
Herels
neon venture
2x²+1 = 3/y
2x² = 3/y -1
x² = 3/(2y) -1/2
knotty geyser
oh and the y becomes the x got it
neon venture
yes
knotty geyser
yeah this concept was a bit shady for me. Thanks for making me understand it!
neon venture
np
knotty geyser
.close
wraith thicket
hi
sweet shard
what do i do when bases are different
,tex .log rules
thorny flameBOT
riemann
sweet shard
Base change formula
foggy bronze
Base change property
acoustic tangle
Where's x = 5 coming from?
wraith thicket
thats what i did
well i got +- 5
but x can't be negative in this case right
<@&286206848099549185>
stable rivet
thats what i did
log(25)/log(x) isnt equal to log(25/x)
wraith thicket
isnt it log25 - log x
stable rivet
yeah
stable rivet
but you never had log(25/x)
you started with log(25)/log(x)
if you're trying to solve for x just bring it to the other side
so you have
log(25)=log(x)^2
wraith thicket
ohhh okay
wraith thicket
ahh
so you can just square root logs?
say i had log 16 = 8
could i just say log 4 = sqrt (8)
stable rivet
no
stable rivet
so you can just square root logs?
you are applying the square after you take the log
so its valid to square root
stable rivet
you get rid of the log after you square root
wraith thicket
so you can square root logs?
stable rivet
so we have log(25)=log(x)^2 right
we first take the square root of both sides so we can get log(x) by itself
wraith thicket
wait
wraith thicket
i thought we had 2 = log5(x)^2
stable rivet
sqrt(log(25)) isnt equal to log(sqrt(25))
wraith thicket
yes
stable rivet
yes
wraith thicket
but why cant you sqrt after
stable rivet
well cuz
if i have 2 = log(x)^2
then i raise to the 5th
so 25 = 5^(log(x)^2)
it doesnt really help to solve for x
wraith thicket
But
stable rivet
i just made it more complicated
stable rivet
sorry if i was being unclear with it before
but we have log(x) then square
not square then log
stable rivet
np
wraith thicket
.close
rare wadi
proof that for every positive number not equal to 0 n:
|x| < a et |y| < n implies |(x + y)/2| + |(x - y)/2| < n
rare wadi
can someone please help
the method i used is the triangulare thing
i inverted the thing so it becomes
.close
crude socket
hi plizz help
topaz sinewBOT
hi plizz help
crude socket
sweet shard
.close
raw solstice
A car travels 1/3rd of it's total path at the speed of 15km/h, another 1/3rd at speed of 20km/h and another at 50km/h. Find it's middle speed across the entire path.
topaz sinewBOT
A car travels 1/3rd of it's total path at the speed of 15km/h, another 1/3rd at ...
raw solstice
I am not given the length of the path or the time of traveling
cinder rock
can you try to find an average
raw solstice
the professor specifically said that's not the correct answer
topaz sinewBOT
@raw solstice Has your question been resolved?
thorny trellis
Did he say what “middle speed” is
raw solstice
Did he say what “middle speed” is
this
im not a native english speaker so I went with a literal translation
thorny trellis
That computes the change in distance over change in time, aka velocity
If youre looking for the average velocity, it will be the total distance traveled divided by total the amount of time
idle hazel
it seems that i'm a little confused on the terminology
topaz sinewBOT
it seems that i'm a little confused on the terminology
topaz sinewBOT
@idle hazel Has your question been resolved?
molten crystal
topaz sinewBOT
molten crystal
isnt this supposed to be like this?
sweet vector
yes
weak bluff
can anyone help me im stuck on how to start this
topaz sinewBOT
can anyone help me im stuck on how to start this
neon iron
can someone help me with this?
noble laurel
can someone help me with this?
!occupied
topaz sinewBOT
can someone help me with this?
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).
noble laurel
can anyone help me im stuck on how to start this
!show
topaz sinewBOT
can anyone help me im stuck on how to start this
Show your work, and if possible, explain where you are stuck.
neon iron
!occupied
topaz sinewBOT
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).
noble laurel
!occupied
Don't spam this persons channel. Open a new one for your question, instead of sending that alert again back to yourself.
weak bluff
!show
that's where im stuck on i don't know how to begin
but it looks like a rotation 90 degrees i think
topaz sinewBOT
@weak bluff Has your question been resolved?
topaz sinewBOT
@weak bluff Has your question been resolved?
topaz sinewBOT
topaz sinewBOT
Available help channel!
west wing
Genuinely unsure where to start on this.
topaz sinewBOT
Genuinely unsure where to start on this.
west wing
Actually wait, I do have some idea in regards to it, but I assume it involved integrating some probability function from a to b?
Wait, I have a strong idea actually regarding to a) do I just integrate 1/(b-a) from a to b?
But what about b?
Actually nvm regards to my thoughts on a) I believe that's not the case
Okay, I thought about it for a bit, and I have a strong idea regarding b)
OH
I think I get how...
I just need someone to confirm
That sampling from a uniform distribution ends up with the data having a normal distrubiton
<@&286206848099549185>
Essentially I'm still somewhat stuck on it, but I have some idea in regards to solving it.
topaz sinewBOT
@west wing Has your question been resolved?
west wing
Well, I can solve the problem, but I need to know some information regarding distributions and whatnot first:
First, what would P(X,D) be for a normal distribution of X, and a uniform distribution of D?
west wing
<@&286206848099549185> I assume I can ping again if I haven't received help in another 15 minutes(Been 40 minutes)
topaz sinewBOT
@west wing Has your question been resolved?
prime forge
topaz sinewBOT
prime forge
can someone please tell me how I could've seen this?
bounding the number of products in k!
I found this on mathstackexchange
but in my example above, why do they take -1 in the exponent?
what's the point
sweet shard
but in my example above, why do they take -1 in the exponent?
They explain it in the next sentence
Say k=11. How many of these are at least 11/2? 1,2,3,4,5,6,7,8,9,10,11
prime forge
Say k=11. How many of these are at least 11/2? 1,2,3,4,5,6,7,8,9,10,11
5.5-1 = 4.5?
What does it mean 4.5 factors lol
non integer
!
prime forge
5 factors
sweet shard
Yea so at least 5 of them are greater than or equal to 5
sweet shard
Yea and 6 > 11/2-1
prime forge
or did you just like
sweet shard
k is arbitrary
sweet shard
You can prove with induction if you want
prime forge
and go like "yep, probs will work with larger cases"
"can go by induction if we want"
sweet shard
I have no idea what you're asking
prime forge
the claim that there are at least (k/2)-1 factors of (k/2) in k!
for any positive integer k
^^ is that a common fact?
i.e., have you seen it a lot before
sweet shard
No idea
prime forge
is this your first time seeing a claim like this?
sweet shard
No
prime forge
ah ok
my question was basically
how are you so convinced that it's true for all k
so quickly
prime forge
Pretty silly mistake for a textbook if it's wrong
wait so, why do they do -1 in the exponent?
isn't this a "nicer" lower bound?
sweet shard
Who cares?
Follow whatever proof makes sense to you
As long as you can make sense of either one
prime forge
Who cares?
because I think there might be gaps in my knowledge otherwise
because I don't think I would've been able to invent one of these in an exam
unless I've seen and understoof something similar beforehand
sweet shard
Well pick and follow one proof rather than look for other ones
prime forge
Well pick and follow one proof rather than look for other ones
would you give that as general advice
I often get stuck at lecture's proofs, which often use slightly differnet methods to a textbook
which make much more sense to me
i.e. Abstract Algebra
topaz sinewBOT
@prime forge Has your question been resolved?
neon iron
topaz sinewBOT
neon iron
How do I find the domain of this?
I'm learning towards it being [-3,0)u(0,inf) because
9-3x=0
-3x = -9
x = 3
since its a fraction and cant be 0 and since its in a sqaure root it cannot be a negative
so [-3,0)u(0,inf)
but when I put it in mathway, it says its (-inf, 3) and i dont get why
keen matrix
the range of a function is the domain of it's inverse :)
neon iron
well first what would make this function undefined? <:thonk:406575732563902485>
since it has a square root
it cannot be a negative number
keen matrix
well not quite, right idea, wrong execution
the value of the sqrt cannot be negative
keen matrix
if we have $\sqrt{a}$ then $a>0$
thorny flameBOT
FancyBredFries
keen matrix
oh wait, you solved your inequality in correctly
neon iron
?
keen matrix
$9-3x>0$
thorny flameBOT
FancyBredFries
topaz sinewBOT
topaz sinewBOT
Available help channel!
topaz sinewBOT
<:helper_heart:735615641184239666>
topaz sinewBOT
keen matrix
it did it again 
icy sky
_>
topaz sinewBOT
Available help channel!
west temple
Could someone explain step 2 and 3
topaz sinewBOT
Could someone explain step 2 and 3
@west temple Has your question been resolved?
Available help channel!
rose horizon
topaz sinewBOT
neon iron
how to get radical 3?
$$\cot(x) = \frac{\cos(x)}{\sin(x)}$$
thorny flameBOT
<:F_button:1095679234497843251>
