#help-26
topaz sinewBOT
cinder oxide
it looks like I picked the wrong u twice
how do you know to only pick u = sin(x) and not u = sin(x)^2?
my answer should be sin^3(x)/3 + C
not cos^4(x)/4 + C
cinder oxide
why won't it work with sin^2(x) tho?
golden mesa
because you can't get a good substitution for cosx then
cinder oxide
I managed to make the integrand = sin^3(x)
cinder oxide
oh i was wondering about that
empty sail
And how did this work?
cinder oxide
this is illegal?
golden mesa
(this is why i never understand people who teach mixing variables)
sturdy oracle
The derivative of sin^2(x) isn't even tan(x$
empty sail
maybe i am mistaking that for cos^2(x)
No because the derivative for cos^2(x) isn't tan(x) either
cinder oxide
or integral.. one of those..
something squared equals tan
trig is the worst for me
i gotta review them
empty sail
I think you mean the derivative of tan(x) being sec^2(x)
cinder oxide
alright so how do you know just by looking at this, what u should be?
cinder oxide
I think you mean the derivative of tan(x) being sec^2(x)
that could be it, so the integral of sec^2(x) = tan(x) + C
but I'm not even doing integral so I messed that up badly
alright, so I will take sin(x) = u
but why sin(x)?
golden mesa
$\frac d{dx}sinx=cosx$ which works nicely
cinder oxide
I tried with cos(x) and it didn't work
thorny flameBOT
chlamydia
empty sail
alright, so I will take sin(x) = u
Because if you did u = cos(x) then the derivative is du = -sin(x) dx then when you sub that in, you get $\int u * -sin(x) * dx$ because you only replaced one of those sines, I believe
cinder oxide
wouldn;t $\frac d{dx}[sin(x)]^2=2cos(x) * sin(x)$ also eliminate a term tho? (cos(x))
thorny flameBOT
avidrunner
cinder oxide
oh
I gotta try that out
before I state that
alright so let's say you choose the wrong u sub, happens to us all from time to time..
at what point do you realize this?
sometimes it takes you 2 tries, 3 tries, 4 tries to get it right?
at this point you will realize the u sub is wong?
or you gotta plug it in first to see what cancels out
empty sail
When you sub it back in and it makes your life harder than it needs to be
cinder oxide
OK
and if u sub just doesn't work, no matter what you make u equal to
integration by parts instead?
topaz sinewBOT
@cinder oxide Has your question been resolved?
rough barn
topaz sinewBOT
rough barn
hows it one
topaz sinewBOT
@rough barn Has your question been resolved?
topaz sinewBOT
@rough barn Has your question been resolved?
topaz sinewBOT
@rough barn Has your question been resolved?
gray ridge
did you try dividing x on both numerator snd denominator?
topaz sinewBOT
topaz sinewBOT
Available help channel!
pale kestrel
topaz sinewBOT
pale kestrel
!status
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
1
drifting swift
$2 \sin(\alpha) = \sin(120\dg - \alpha)$
thorny flameBOT
Ann
drifting swift
this?
pale kestrel
yes
drifting swift
do you know angle sum/difference identities? i'd try to start with those.
pale kestrel
I need to find a general solution
pale kestrel
i don't know how to implement them here
drifting swift
ok can you list them
pale kestrel
drifting swift
ok
which of these can be applied in your equation?
just tell me which one, don't apply it just yet
pale kestrel
2\4?
drifting swift
what does the backslash mean
drifting swift
2 sin(α) = sin(120° - α)
look at the right side of this
sin(difference)
that's identity #2 in your list
apply it
topaz sinewBOT
topaz sinewBOT
@cerulean sun Has your question been resolved?
sly arch
Is everything else above correct?
topaz sinewBOT
topaz sinewBOT
Available help channel!
nocturne remnant
topaz sinewBOT
drifting swift
!status
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
I need help solving it
drifting swift
i mean duh that's why you're here
but you still gotta tell us how far you got
if you don't know where to begin that's called status 1
nocturne remnant
Yes
drifting swift
ok
so then let's say i gave you the following equation:
x/9 = 30
could you solve it
drifting swift
right
so for your equation $\frac{x}{7} = \frac{\sqrt{2}}{2}$ why not do the same
thorny flameBOT
Ann
nocturne remnant
So do I multiply by 7 to get x by itself
drifting swift
what's stopping you from doing that?
nocturne remnant
Is that what I do?
nocturne remnant
I just wanted to confirm
winged moat
Can't forget you are still dividing by 2
nocturne remnant
7 radical 2 divided by 2
winged moat
Yes
winged moat
No problem
nocturne remnant
.close
meager grotto
\textbf{Question:}
Simplify
[ \frac{2k^3 + 15k^2 + 37k + 24}{24} ]
into
[ \frac{k+1}{24} \cdot \left(2k^2 + 13k + 24\right) ]
thorny flameBOT
aight
meager grotto
yeah so basically idk how to do this
i searched online and i saw someone solving it using the synthetic division method
is there any other way to do it?
dense rain
If you already know that k+1 is a root of the polynomial, you can multiply it out with an arbitrary quadratic polynomial ak^2 + bk + c and solve for a, b, c
meager grotto
If you already know that k+1 is a root of the polynomial, you can multiply it ou...
lemme try doing that
meager grotto
yeah
meager grotto
If you already know that k+1 is a root of the polynomial, you can multiply it ou...
\documentclass{article}
\usepackage{amsmath}
\begin{document}
\textbf{Solution:}
Given the expression:
[ \frac{2k^3 + 15k^2 + 37k + 24}{24} ]
We aim to simplify it to the form:
[ \frac{k + 1}{24} \cdot (2k^2 + 13k + 24) ]
\textbf{Step 1:}
We begin by factoring out the common factor of ( \frac{1}{8} ) from the numerator:
[ = \frac{1}{8} \cdot \frac{2k^3 + 15k^2 + 37k + 24}{3} ]
\textbf{Step 2:}
Next, we factor the cubic polynomial (2k^3 + 15k^2 + 37k + 24). Since (k = -1) is a root (as (k + 1 = 0) implies (k = -1)), we perform polynomial division:
[
\begin{array}{c|ccc}
& 2k^2 - k + 24 \
\hline
k + 1 & 2k^3 + 15k^2 + 37k + 24 \
& 2k^3 + 2k^2 \
& 13k^2 + 37k \
& 13k^2 + 13k \
& 24k + 24 \
& 24k + 24 \
\hline
& 0
\end{array}
]
This yields (2k^3 + 15k^2 + 37k + 24 = (k + 1)(2k^2 - k + 24)).
\textbf{Step 3:}
Substituting this back into the original expression, we have:
[ = \frac{1}{8} \cdot \frac{(k + 1)(2k^2 - k + 24)}{3} ]
\textbf{Step 4:}
Finally, after canceling common factors, we arrive at:
[ = \frac{(k + 1)(2k^2 - k + 24)}{24} ]
This matches the desired form:
[ \frac{k + 1}{24} \cdot (2k^2 + 13k + 24) ]
Therefore, the original expression simplifies to the desired form.
\end{document}
thorny flameBOT
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meager grotto
yeah i need to relearn polynomial division to do this
dense rain
You know that a must be 2 and c must be 24 by inspection, so all you really need is the b, which you get from looking at bk + 24k = 37k so b = 13.
meager grotto
You know that a must be 2 and c must be 24 by inspection, so all you really need...
\documentclass{article}
\usepackage{amsmath}
\begin{document}
\textbf{Given Expression:}
[ \frac{2k^3 + 15k^2 + 37k + 24}{24} ]
\textbf{Desired Simplified Form:}
[ \frac{k+1}{24} \cdot (2k^2 + 13k + 24) ]
\textbf{Solution:}
-
\textbf{Factor Out the Common Factor}:
[ \frac{2(k^3 + 7.5k^2 + 18.5k + 12)}{24} ] -
\textbf{Simplify the Fraction}:
[ \frac{k^3 + 7.5k^2 + 18.5k + 12}{12} ] -
\textbf{Divide Each Term by 12}:
[ \frac{k^3}{12} + \frac{5}{8}k^2 + \frac{37}{24}k + 1 ]
Now, since (k+1) is a root, we can make the following observations:
- The coefficient of (k^2) term is 2, which implies (a = 2).
- The constant term is 24, which implies (c = 24).
- To find (b), we compare coefficients:
[bk + 24k = 37k]
Simplifying, we get:
[(b - 13)k = 0]
For this equation to hold true for all (k), it must be the case that (b - 13 = 0), which means (b = 13).
Hence, the simplified form is:
[ \frac{k+1}{24} \cdot (2k^2 + 13k + 24) = \frac{k^3}{12} + \frac{5}{8}k^2 + \frac{37}{24}k + 1 ]
\end{document}
thorny flameBOT
aight
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meager grotto
**aight**
Compile Error! Click the <:errors:504266556054962196> reaction for mor...
this should be it right?
i think learning polynomial division real quick would be simpler
dense rain
Well, at a glance, not really. It kind of looks like a half edited mess where you are taking information from different sources without understanding what is going on
meager grotto
😐
meager grotto
Well, at a glance, not really. It kind of looks like a half edited mess where yo...
\documentclass{article}
\usepackage{amsmath}
\begin{document}
\textbf{Given Expression:}
[ \frac{2k^3 + 15k^2 + 37k + 24}{24} ]
\textbf{Desired Simplified Form:}
[ \frac{k+1}{24} \cdot (2k^2 + 13k + 24) ]
\textbf{Solution:}
-
\textbf{Factor Out the Common Factor}:
[ \frac{2(k^3 + 7.5k^2 + 18.5k + 12)}{24} ] -
\textbf{Simplify the Fraction}:
[ \frac{k^3 + 7.5k^2 + 18.5k + 12}{12} ] -
\textbf{Divide Each Term by 12}:
[ \frac{k^3}{12} + \frac{5}{8}k^2 + \frac{37}{24}k + 1 ]
Now, given that (k+1) is a root, we can deduce the following:
- (a = 2) because of the (2k^2) term.
- (c = 24) because of the (+24) term.
To find (b), we compare coefficients:
[bk + 24k = 37k]
Simplifying:
[(b - 13)k = 0]
For this to be true for all (k), we must have (b - 13 = 0), which implies (b = 13).
Hence, the simplified form is:
[ \frac{k+1}{24} \cdot (2k^2 + 13k + 24) = \frac{k^3}{12} + \frac{5}{8}k^2 + \frac{37}{24}k + 1 ]
\end{document}
thorny flameBOT
aight
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lucid moat
topaz sinewBOT
lucid moat
currently working on this proof
here is what I have so far
I feel like this proves that gcd(a, n) = gcd(b, n) but not <=
i may be wrong that it even proves = though
tbh im quite lost here
cursive thorn
Well youre right that it implies gcd(a, n) = gcd(b, n) since a == b mod n implies b == a mod n
lucid moat
Well youre right that it implies gcd(a, n) = gcd(b, n) since a == b mod n implie...
how would I go about showing its <= though?
cursive thorn
Well, a = b directly implies a <= b
If 4 = 4, then 4 <= 4
Your proof can be simplified by using gcd rules btw
odd pagoda
your proof so far only shows d | gcd(b,n). which means d<= gcd(b,n)
cursive thorn
Have you ever seen this rule: gcd(x + y, y) = gcd(x, y)?
lucid moat
Have you ever seen this rule: gcd(x + y, y) = gcd(x, y)?
I havent
lucid moat
your proof so far only shows d | gcd(b,n). which means d<= gcd(b,n)
well if d divides both doesnt that mean that both show d<=gcd ?
odd pagoda
that's what I said. d<=gcd(b,n)
which is what you wanted to show
so you are done
and then by symmetry, like jelle said, you get the other direction and the gcd's are the same
topaz sinewBOT
topaz sinewBOT
Available help channel!
broken condor
topaz sinewBOT
uncut gale
have you worked out the area of the trapezoid and/or the triangle
what have you done so far
broken condor
what have you done so far
i dont even know what to do
uncut gale
wait do you understand what the question is asking you first
uncut gale
$A = \frac{1}{2} \cdot (a + b) \cdot h$
thorny flameBOT
chef wang gang
uncut gale
this is the formula for the area of a trapezoid
oh yikes ur too young to be here then
<@&268886789983436800>
uncut gale
bro why would you joke about being underage
broken condor
so you would make it easier so that even a 9 year old would understand
urban grove
how old are you
broken condor
19
urban grove
aight
uncut gale
so you would make it easier so that even a 9 year old would understand
you couldve just asked me
broken condor
alright
uncut gale
to try and explain it easier
broken condor
well know you know
uncut gale
well what are you confused about
with the formula
applying it?
or is there something else
broken condor
im confused on how im supposed to calculate it when the top and bottom have different measurements ( because its a trapezium, im only used to rectangular measuring)
uncut gale
unique rose
Hey hey, so I'm doing what I assume is functions, but I don't know how to start with this as this seems to be completely foreign to me
topaz sinewBOT
Hey hey, so I'm doing what I assume is functions, but I don't know how to start ...
@unique rose Has your question been resolved?
topaz sinewBOT
@unique rose Has your question been resolved?
oak lodge
Hey hey, so I'm doing what I assume is functions, but I don't know how to start ...
ahh so this is basically the kinematic equations
unique rose
ahh so this is basically the kinematic equations
I think so yeah, i think my teacher combo'd this with the kinematics class
oak lodge
yes mate
vcostheta is the horizontal component of velocity
and v sintheta is the vertical component
unique rose
Yeah, but I'm just really stuck on what to do with these
oak lodge
well are u aware of the three kinematic eqns
v = u + at
s = ut + (1/2)at^2
v^2 = u^2 + 2as
unique rose
Yep yep
oak lodge
s = ut + (1/2)at^2
can u find the similartity between this eqn and the second eqn
unique rose
Ahh ok the u is v sintheta
unique rose
Yeh
oak lodge
but there is a negative sign as g acts downwards
so dont u think the time of flight would be how much ever time the particle takes to come back to the reference level ( ground )
unique rose
Hmm i don't have the formulas hand so I can't say (currently at the gym)
I do get what you mean so I'll try to work it out later, thanks!!
oak lodge
😁 👍
topaz sinewBOT
Available help channel!
amber shard
topaz sinewBOT
topaz sinewBOT
@amber shard Has your question been resolved?
undone zinc
you only did half the work actually
undone zinc
you showed if X is true, Y must be true. You also need to show if Y is true, X is true.
amber shard
OH
so like
i proved v(a^ negationb) = y
but now i need
v(negation (a impliesb)) = x
?
undone zinc
that works 👍
undone zinc
sorry I gotta go now, hope somebody else help you
topaz sinewBOT
@amber shard Has your question been resolved?
amber shard
<@&286206848099549185>
topaz sinewBOT
@amber shard Has your question been resolved?
random ridge
suppose you wish to retire at age 60 with $70,000 in savings. determine your monthly payment into an IRA if the APR is 8.5% compounded monthly and you begin making payments at 30 years old. round your answer to the nearest cent if necessary
topaz sinewBOT
suppose you wish to retire at age 60 with $70,000 in savings. determine your mon...
elfin sparrow
can you show what you did?
neon iron
random ridge
https://tenor.com/view/side-eye-dog-suspicious-look-suspicious-doubt-dog-doubt-g...
scared
random ridge
can you show what you did?
uhhh yas
topaz sinewBOT
@random ridge Has your question been resolved?
wise mesa
The x intercept of a line is -12 and it’s slope is 2.5, what Is the y-intercept of this line??
topaz sinewBOT
The x intercept of a line is -12 and it’s slope is 2.5, what Is the y-intercept ...
wise mesa
The x intercept of a line is -12 and it’s slope is 2.5, what Is the y-intercept ...
Grade 10th level math btw
half edge
What have u tried
wise mesa
I tried placing in in the y=ax + b equation but got lost
half edge
Ok let’s go with that
If x-intercept is -12
What’s the actual point where the line crosses the x-axis
Gimme the coordinate pair
wise mesa
They just gave us this 😭
half edge
No I want u to figure it out and tell me
wise mesa
What’s the point associated with that
-12..?
thorny flameBOT
half edge
-12..?
Gimme an ordered pair, a coordinate pair whatever
For example (6,5)
(7,4)
What’s the point associated with x intercept of -12
No
How’d u get that
Also, do u know what an x intercept is
wise mesa
Where the line crosses on the x axis
Im sorry I think my brain is cooked after 5 hours of math
I feel so stupid
half edge
Nah u good
half edge
Where the line crosses on the x axis
That’s right
And so in a coordinate pair (x,y)
The point of the x intercept will be (x,0) because y=0 on the x-axis
So what’s the point of the x intercept in our situation
@wise mesa
wise mesa
So what’s the point of the x intercept in our situation
To find the y intercept
half edge
The point of the x intercept will be (x,0) because y=0 on the x-axis
As I did here
Any point on the x axis will be in the format
(x,0)
And it’s given where our line intersects the x axis
topaz sinewBOT
@wise mesa Has your question been resolved?
carmine whale
topaz sinewBOT
carmine whale
how would this look like?
the vectors are <-3,2,3> & <-7,-4,2>
do i plot the vectors and then draw a line from the origin to it?
topaz sinewBOT
@carmine whale Has your question been resolved?
topaz sinewBOT
@carmine whale Has your question been resolved?
carmine whale
odd jacinth
the last point of the parallelogram is B+u+v
let's call it D
you place it and then draw BADC
u = (-3, 2, 3) and v = (-7, -4, 2)
so D coordinates are (-5, -1, 7)
carmine whale
the D coordinate is for what?
carmine whale
how did u get those coordinates?
odd jacinth
the last point of the parallelogram is B+u+v
^
carmine whale
for the image i used the coordinates i have are GDEF
odd jacinth
you parallelogram doesn't have a single vertex at the origin since none of the vertices have coordinates (0, 0, 0)
you drawing doesn't have any meaning relative to what's asked
start by placing the points A, B and C
then I told you how to find the last vertex
and gave its coordinates
odd jacinth
yes
odd jacinth
vector BA + vector BC = (-10, -2, 5)
odd jacinth
hmm it's weird to call BD as BA+BC
like, when you put the arrows above
and it's a vectorial equality
that's true, bc parellelogram
but if you write like lengths
it's not true
carmine whale
how would i label the vector BA and vector BC?
carmine whale
i need to label vector
vector BA + vector BC on the graph thats why
and also vector BA - vector BC
carmine whale
<@&286206848099549185>
chilly flume
whats hard about it?
yeah you can label 2 vectors
You can call it BAD or DAB or ADB or ABD
no wait, BAD BDA DAB DBA ADB ABD
carmine whale
i just wanted to know if the orange line that i made for the graph is considered vector BA + vector BC
chilly flume
it looks BD
I might be silly and ignorant il look and see
ah I see now
its like literally BA and BC, I think
but you only have 1 vector
which is either not labled right or BD
B connected to U and V yes
why would you do that?
carmine whale
wut do u mean?
chilly flume
wait howd you get D?
carmine whale
i have to label vector BA + vector BC on the parallelogram
i got D from B + vector BA + vector BC
chilly flume
I see
chilly flume
it looks like you drew the paralelagram
chilly flume
A vector is a mathematical quantity that has both magnitude and direction.
well thats a weird question at first glance
carmine whale
its connected with the first image i posted
chilly flume
id love to help but at the moment im ignorant
google: Subtracting two vectors involves putting their feet together and drawing the resultant vector, which is the difference of the two vectors, from the head of the vector you're subtracting to the head of the vector you're subtracting it from.
haha, its 1 dimensional
@carmine whale
or not hm
topaz sinewBOT
@carmine whale Has your question been resolved?
carmine whale
.close
carmine whale
topaz sinewBOT
carmine whale
Is this right?
topaz sinewBOT
@carmine whale Has your question been resolved?
sharp pier
Pls help my script doesn’t work
local myTable {1, 2 ,3 name=“John”}
return 1 or 2
end
golden mesa
!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).
topaz sinewBOT
@carmine whale Has your question been resolved?
golden mesa
yes
topaz sinewBOT
@carmine whale Has your question been resolved?
golden mesa
ok
carmine whale
yes
it says to find the direction in the question also do i have to do soemthing for that?
topaz sinewBOT
topaz sinewBOT
Available help channel!
wanton zinc
I need help with combinatorial math, if you had 16 oak and 16 pine trees in how many ways could you place them so that everytime they are next to each other. (Example one long row of pine oak pine oak pine oak etc)
topaz sinewBOT
I need help with combinatorial math, if you had 16 oak and 16 pine trees in how ...
@wanton zinc Has your question been resolved?
gritty trout
placing them in a row so that every oak tree has an adjacent pine and every pine tree has an adjacent oak? should we consider the oak and pine trees indistinguishable?
like is oak_1 pine_1 oak_2 pine_2 different than oak_2 pine_1 oak_1 pine_2
topaz sinewBOT
@wanton zinc Has your question been resolved?
wanton zinc
like is oak_1 pine_1 oak_2 pine_2 different than oak_2 pine_1 oak_1 pine_2
So every oak tree is considered same
gritty trout
okay, so you want to know how many ways you can alternate oaks and pines in a row of length 32
topaz sinewBOT
@wanton zinc Has your question been resolved?
wanton zinc
okay, so you want to know how many ways you can alternate oaks and pines in a ro...
Doesnt have to be a row
gritty trout
Doesnt have to be a row
If you are placing them into an unspecified body of land, what does it mean for them to be next to each other?
wanton zinc
Then I wasnt told it properly
gusty estuary
i need a hint
topaz sinewBOT
i need a hint
gusty estuary
i have written down all definitions, and im trying to make the distance d(x_m, x_n_k) < epsilon, so i can apply the triangle inequality
im stuck here
odd pagoda
well x_m is close to x_n_k by cauchy sequence. and x_n_k is close to x by convergence. and you have to show that x_m is close to x
gusty estuary
x_n is close to x_m by cauchy, x_n_k is close to x_m by cauchy aswell and x_n_k is close to x by convergence. can i now apply the triangle inequality to these three cases to show that x_n is close to x?
why would i have to show that x_m is close to x?
odd pagoda
where is the difference between x_n and x_m except for the choice of letter?
you only need one of x_n and x_m
gusty estuary
oh i see, my bad. yeah i just let m,n >= N for some positive integer N
i dont see how x_m is close to x_n_k by cauchy. do i have to let m and k be larger than some number?
odd pagoda
if k is large enough then n_k is also large
yes you'll need to choose m and k large enough
gusty estuary
if i in the cauchy case let m,n >= N and in the converging case let k>M, can i chose min{N,M}?
odd pagoda
well you have to check that n_k>=M
odd pagoda
yes
gusty estuary
amazing, thanks
tender girder
topaz sinewBOT
tender girder
what properties would I use to expand and simplify?
bold nebula
open your own channel, #❓how-to-get-help
next fjord
Determine the orthogonal projection of the vector (0.4) in the direction (1,1).
topaz sinewBOT
Determine the orthogonal projection of the vector (0.4) in the direction (1,1).
@next fjord Has your question been resolved?
vital vector
Can someone help with part b)? It asks to find the Derivative and this is the answer key, when I went to check my work and the the x on the 4 is squared. And I’ve tried everything idk :( can someone explain why it’s squared (my work is after this message)
topaz sinewBOT
Can someone help with part b)? It asks to find the Derivative and this is the an...
vital vector
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vital vector
<@&286206848099549185>
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fluid belfry
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fluid belfry
help help, i did find an eq by using product of slopes of perpendicual line is -1
but i want one more si that i can solve this
acoustic pecan
could you show exactly what youve done so far?
fluid belfry
sure
acoustic pecan
im not sure how youre getting to 2-m=l-1
that implies the gradient =1, which you dont actually know
fluid belfry
oh
formula for slope when cordinates of the line is given, any two actually is y2-y1/x2-x1
and if two line are perpendicular to each other then prod. of thier slopes are equal to -1
thats what i did here azo
fluid belfry
that implies the gradient =1, which you dont actually know
well gradient of QR is actually 1
like this, so for QR line it is, 4-2/3-1=2/2=1
acoustic pecan
oh thats what you meant, yeah okay then
fluid belfry
hm, what should i next then, we have 2 variables but not 2 eq so that we can solve them
in order to find the final ques
fluid belfry
but, like ans is given 11, ie we have to take this second case
why not the first one
fluid belfry
oh ye
thanks you so much, i was on this server for like 3 hours, thanksyou so much
!!
.close
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verbal gulch
How could I prove that the Dirichlet Function (or a modification of it) is continuous at a point where x=0?
I am taking a calculus course and I have learned thus far that a function is considered continuous at point c if the limit as the function approaches c is equal to the value of the function at c (i.e., limx->c f(x) = f(c) )
topaz sinewBOT
How could I prove that the Dirichlet Function (or a modification of it) is conti...
sweet shard
do you know eps delta definition of continuous
verbal gulch
it has been covered in class but i dont know it by heart yet; i have learned the eps delta definition of limits
sweet shard
can you show exactly what you think "eps delta definition of limits"means
like a screenshot
verbal gulch
sorry for the late reply
sweet shard
that's equivalent to f(x) continuous at x=c
oh shit it's not
you need L = f(c)
that's the only difference
verbal gulch
hold on ill try formulating a proof with that and ill come back
topaz sinewBOT
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fossil coyote
The ranges are in degrees, so your final answer should also be in degrees
And yeah think of your quadrant signs
golden mesa
And yeah think of your quadrant signs
you don't have to tag 3 times
there's other angles that give cos=0, etc.
shrewd horizon
$\frac{-5\pi}{6}$?
thorny flameBOT
bee [it/its]
Jeremy
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)
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golden mesa
$\sin\theta=-\frac12$ for example?
thorny flameBOT
chlamydia
golden mesa
check where the y coordinate is -1/2
because sin corresponds to y
so that gives -30 or 330 as the main answer you might think of
and then you can also get 210 because sin(180-x)=sinx
yes
so there, sin0=0, cos0=1
pastel tide
I need help solving for a value that makes a linear system consistent.
pastel tide
golden mesa
!status
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
Solved it 2x but both answers wrong
1 attempt left
<@&286206848099549185>
<@&286206848099549185>
hazy pumice
<@&286206848099549185>
it would be lovely if you would post a screenshot of the probblem statement
pastel tide
hazy pumice
you need to open a fresh channel.
pastel tide
set up an augmented matrix and row-reduce it
I keep getting the wrong answer
pastel tide
how far did you get doing that?
Got 2 values both wrong
Its prolly a reduction error but I cant see it
hazy pumice
Its prolly a reduction error but I cant see it
prolly
are you starting by getting a 1 in (1,1)?
then a 1 (2,2)?
and so on?
(and zeros elsewhere, naturally)
oh, scuze me
pastel tide
are you starting by getting a 1 in (1,1)?
I am trying to get the bottom row to 0s
hazy pumice
the lower-left trianglular matrix should be
pastel tide
then simplify for B
hazy pumice
I am trying to get the bottom row to 0s
ok, that would demonstrate an inconsistent system
hazy pumice
let's set up this one
because I can't make up a 4-equation system that is consistent
pastel tide
Alright
hazy pumice
*3
pastel tide
Yeah
hazy pumice
1 -3 -6 | -5
4 -5 -4 | 4
-8 3 -12 | b
Yep got the same
hazy pumice
ok so we have our 1 in (1,1)
now we need zeros below it
so we do row operations on R2 and R3 until that happens
then we can move on to (2,2)
pastel tide
R2-4R1?
hazy pumice
||```
1 -3 -6 | -5
0 7 20 | 24
0 -21 -60 | b - 40
Got the same
hazy pumice
ok, so now how do you get (2,2) to become 1?
hazy pumice
no.
because you have an element in column 1 in row 1, but you don't have (and don't want) a nonzero element in column 1 for row 2
pastel tide
Ohh makes sense
hazy pumice
divide row 2 through by 7 instead
||```
1 -3 -6 | -5
0 1 20/7 | 24/7
0 -21 -60 | b - 40
now, to get the zero below (2,2), what must you do?
pastel tide
Add R2 To R3?
hazy pumice
Add R2 To R3?
no.
pastel tide
Then?
pastel tide
I j did R3 + 21R2
hazy pumice
what multiple of R1 will cause (2,2) to cancel with (3,2)?
hazy pumice
I j did R3 + 21R2
yes
pastel tide
Yeah that’s what i originally did
Then that makes the bottom row 0 0 0 b+41
Then u solve for B which gives u -41 but that’s wrong
hazy pumice
||```
1 -3 -6 | -5
0 1 20/7 | 24/7
0 0 0 | b - 40 + 72
Wait how did u get 480/7?
pastel tide
Yeah so u get B+32
hazy pumice
so now we have an inconsistent system
pastel tide
Yes
hazy pumice
there is no value of b that will make this system consistent
hazy pumice
I take that back
pastel tide
Nvm@
pastel tide
It’s -32 no?
hazy pumice
if we just realize that R3 = -21R2
then we can compare RHS of R2 and R3
so that
hmm
lemme check
hazy pumice
b - 40 = -21(24/7)
pastel tide
Yeah
hazy pumice
b - 40 = -72
pastel tide
B = -32?
hazy pumice
b = - 32
see, I told you
but you
just
didn't
listen!
(jk)
I know you got it
and now I am sure too :3
pastel tide
Yeah i made a simple mistake
I calculated it wrong
I did it 2x lol same mistake
U j helped me see it and then i got it
hazy pumice
ah ok
pastel tide
Thanks!
hazy pumice
np
pastel tide
The thing about matrices is that there’s so many places for a tiny error
that make the entire thing wrong
same thing happened here
Have a good one
hazy pumice
you too
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Available help channel!
mossy garnet
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sweet shard
What's the question
sweet shard
What's the question
Do you see this
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
1
sweet shard
Here
mossy garnet
how to
sweet shard
How to what
frail dove
Don't worry I just showed ten mintues ago
Expanding probably will work
Wait never mind I don't think it will
mossy garnet
Im worried about the cubic terms
mossy garnet
I mean (x-m)^3 is alr, we can expand that
mossy garnet
Maybe graphing it will give us insight
mmm maybe
(x-m)^3 = x^3 -3*(x^2)*m + 3xm^2 -m^3
frail dove
So there're 3 cases: 1. there's 1 solution; 2. there's 2 solutions; 3. there's 3 solutions
frail dove
Somehow I got this from my CAS
mossy garnet
frail dove
Can you give some context, that might help
topaz sinewBOT
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torn knoll
yo
torn knoll
option b is singular
cuz sin(x) / x is not analytic at x = 0
it should be singular point
but then why its ordinary point
@odd pagoda can u help
plz
🥹
this is so confusing
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molten igloo
<@&286206848099549185>
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north sparrow
according to this, x=0 for b) is a singular point
nvm
its conditional that A2(x) =/= 0, but it is
north sparrow
it should be singular point
cant be singular or ordinary right?
urban thicket
Hello.
urban thicket
I'm working on a project related to the game 999: Nine Hours, Nine Persons, Nine Doors. In particular, I want to make a tabletop game based on it for some friends to play, with myself as the DM.
This means I have to do math, something I am, quite frankly, terrible at; the game uses digital roots as a central mechanic, and while they're easy to understand in the game, the task I'm working on is less easy to understand.
Basically, here's the thing I'm trying to do:
Imagine there are 11 people; each of these 11 people has a numbered bracelet; these bracelets correspond to a certain number:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
Imagine there are also 11 doors; 10 of these doors are numbered; 1 of them is labelled with a letter:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, A
The characters can use their numbered bracelets, in groups of at least 3 people at a time, to open the doors; they do this by adding the numbers of their bracelets together, then calculating the digital roots from the total.
For example, suppose that a group formed between the persons with the 0, 1, and 9 bracelets ( 0 + 1 + 9 = 10 ).
The digital root of 10 is 1 ( 10 = 1 + 0 = [1] ).
( 0 + 1 + 9 = 10 = 1 + 0 = [1] )
As a result, they can open Door 1.
The only exception are the doors labelled 10 and A. As 10 cannot be the digital root (what with digital roots being single-digit inherently), Door 10 is opened via reaching a combined total of 10 before the digital root calculation (as the door does not apply it). Door A follows similar rules, but for a combined total of 55 (the sum of all the numbered bracelets).
I want to figure out every possible combination of 3 or more people (with each number being unique) that could open each door.
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urban thicket
<@&286206848099549185>
topaz sinewBOT
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urban thicket
<@&286206848099549185>
north sparrow
Imagine there are 11 people; each of these 11 people has a numbered bracelet; th...
with these rules, wouldnt 0, 1, 9 open door 10
urban thicket
with these rules, wouldnt 0, 1, 9 open door 10
It opens both!
north sparrow
ohh okay
urban thicket
See, Door 1 and Door 10 work on different systems.
Door 1 opens when a digital root of 1 is reached; Door 10 opens when a total of 10 is reached.
Since 0 + 1 + 9 = 10, and the digital root of 10 is 1 + 0 = 1, it's a valid solution for both doors.
north sparrow
this would be pretty complicated since number like 10, their digits dont have individual single values, but represent tens places. ill try sm tho but not guaranteeing anything,
urban thicket
Kind of an "not all rectangles are squares, but all squares are rectangles" situation.
Since every combination that would open Door 10 will also open Door 1, but not vice-versa.
north sparrow
Since 0 + 1 + 9 = 10, and the digital root of 10 is 1 + 0 = 1, it's a valid solu...
ah okay i get it
urban thicket
You can also ignore Door A since it only has one valid solution.
north sparrow
so when door A is opened, door 10 and 1 also open
urban thicket
No, they're all seperate.
north sparrow
so if 1, 9, 0 go how do we determine if door 1 or 10 opens?
since 10 is reached and the root is 1
urban thicket
The door itself!
Basically, in the actual scenario, they'd be in different locations.
With different places to input the numbers.
urban thicket
(In the 999 game, you scan the bracelets on a machine on the wall next to the door, then pull a lever to confirm the scan.)
So it's best to conceptualize all the doors as seperate entities from one another.
urban thicket
You can also ignore Door A since it only has one valid solution.
(that being 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55)
You can also ignore the numbers 0 and 9 as they don't actually change the digital roots at all.
So they only change things in terms of meeting the requirement of three people minimum.
ie. 8 + 3 = 11 = 1 + 1 = 2 is invalid, since you need three people.
But 0 + 8 + 3 = 11 is fine.
As is 8 + 3 + 9 = 20 (which reaches the same end result of 2)
So 0 and 9 effectively 'act' the same in terms of the calculations of the digital root, and only change the structure (adding an additional component to render an equation valid).
Since adding nine to any digital root's preceding total will just result in a total with the same digital root.
north sparrow
honestly im not sure how id generalise this
it might take some time to figure out
if its even possible to without just brute forcing it
north sparrow
@urban thicket brute forcing: 😭
got this so far
top part is all the combinations where the digital root is calculated on the first sum
besides for door 1 because thats the same as 10
on the left of the boxes is the number of possible ways
at the bottom half is all the total sums that will result in the specific root
for example if the sum of character numbers is 24, 51 or 33, the 6th door will open
now we need to find the number of different ways each first sum can be reached from 3 or more numbers 1-8, then add 3 for each of them and youll get it
which might take a little more time
sorry if you expected an algorithm or a general formula of some sort, couldnt find out how to do it
digit sum formulas are specifically defined
urban thicket
sorry if you expected an algorithm or a general formula of some sort, couldnt fi...
Oh, no worries. I'm just glad to have help.
north sparrow
might take more time tho
its like 11:30 so i cant keep working on this but ill try finish it tommorow
it i found out all the possible combinations for each door is that good?
also again no promises, but i think i can figure it out
urban thicket
Oh, that'd be spectacular!
urban thicket
It's for a TTRPG game I'm working on for some friends.
Basically, I'm trying to account for all of the different paths that can be taken at a given time.
ie. "What if this set of people wants to enter this door?"
north sparrow
so far ive found 60 combinations for 5 different sums (first sum roots)
theres 45 more sums to consider, and the larger the value the more combinations (or maybe not cuz itll limit out)
found this pattern tho:
not sure how viable brute forcing is
theres 219 different combinations of character bracelets
north sparrow
It's for a TTRPG game I'm working on for some friends.
ooh thats cool
north sparrow
Basically, I'm trying to account for all of the different paths that can be take...
well theres a bit over 200
urban thicket
well theres a bit over 200
Yeah. It's just so I can keep track of it for if a player does something.
north sparrow
do you know coding?
urban thicket
No, unfortunately.
This'd be run as a fairly standard tabletop game (so just manually keeping track via a document).
north sparrow
ah i see
topaz sinewBOT
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plush copper
Hi
plush copper
I need help
solemn jay
Send ur question :)
plush copper
I dont really understand the n part
solemn jay
i'm pretty sure u take the same bases and the. just use the powers only to find n
plush copper
It doesnt want the n the lesson here is
Complex numbers
The( i) is the most important part
topaz sinewBOT
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neon iron
I have no clue where im supposed to go from here
topaz sinewBOT
I have no clue where im supposed to go from here
neon iron
epsilon-delta proof
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neon iron
<@&286206848099549185> any help please?
barren lion
<@&286206848099549185> any help please?
your denominator is wrong
try reducing the denominator (thereby making the fraction larger)
neon iron
your denominator is wrong
whats wrong with it
neon iron
try reducing the denominator (thereby making the fraction larger)
oh I was working on paper but copied over so I could ss
barren lion
whats wrong with it
you are multipling the numerator by 2 when subtracting the fractions
barren lion
what do you have
barren lion
so you want to choose an N (which depends on epsilon) such that if x > N, then that inequality is satisfied
neon iron
wdym
barren lion
that is just the epsilon-delta definition
neon iron
so you want to choose an N (which depends on epsilon) such that if x > N, then t...
yes
Im trying to find the N
barren lion
start by noting that $\frac{10x^2 + 29}{4x^3+2} < \frac{10x^2 + 29}{4x^3}$
neon iron
but im not sure what kind of manipulation im supposed to be doing here
thorny flameBOT
tushar
thorny flameBOT
Bandera
neon iron
right
barren lion
**Bandera**
so you want ot answer the question: x must be at least _ such that this fraction is less than epsilon
you can intuitively see that if x is large enough, then the fraction will be small enough (because it tends to 0)
neon iron
you can intuitively see that if x is large enough, then the fraction will be sma...
this does make sense intuitively but how do I write this down formally
or would it better to continue reducing the fraction?
how does it lead to finding an N
barren lion
or would it better to continue reducing the fraction?
yes
you must make it formal
you can do this in many ways
e.g. splitting the fraction
neon iron
is it fine to say that $\frac{10x^2}{x^3} < \epsilon$
thorny flameBOT
Bandera
barren lion
$\frac{10}{x} + \frac{29}{x^3} < \frac{10}{x} + \frac{1000}{x^3}$, so if $\frac{10}{x} < \varepsilon$, then $\frac{10}{x} + \frac{1000}{x^3} < \varepsilon + \varepsilon^3$
neon iron
woah
neon iron
**Bandera**
is this fine though
thorny flameBOT
tushar
neon iron
since (10x^2+29)>10x^2
barren lion
since (10x^2+29)>10x^2
but reducing the numerator would make the fraction smaller
neon iron
i feel like the $\epsilon + \epsilon ^2$ is complicating it a bit
thorny flameBOT
Bandera
neon iron
but reducing the numerator would make the fraction smaller
oh right
neon iron
but reducing the numerator would make the fraction smaller
wait isnt that fine
since itll still be less than epsilon
neon iron
you want an upper bound on the fraction
I thought its fine as long as with have epsilon greater
barren lion
let me think of the cleanest way of doing this
barren lion
**tushar**
this works, but it's not very simple
neon iron
i can imagine
barren lion
**Bandera**
I mean I'd probably go with this
neon iron
Am i allowed to add a conditon like this?
barren lion
well again, just because 10x^2/(4x^3+2) is < epsilon, it doesn't mean (10x^2+29)/(4x^3+2) is
neon iron
Man I think im going to switch degrees if this is supposed to be the easy stuff
neon iron
well again, just because 10x^2/(4x^3+2) is < epsilon, it doesn't mean (10x^2+29)...
is less than epsilon?
barren lion
yeah
neon iron
but 10x^2+29 is greater than 10x^2
wouldnt that mean that 10x^2/4x^3+2 is less than 10x^2+29/4x^3+2
neon iron
what direction do I want
barren lion
you want to make sure 10x^2+29/4x^3+2 is less than epsilon
neon iron
but isnt the whole point of this step working backwards and finding an N
neon iron
you want to make sure 10x^2+29/4x^3+2 is less than epsilon
and then showing this after
neon iron
oh actually
I think that part is fine
but the condition I put on the bottom wont hold
or I cant use that
neon iron
yh but you might need to explain it
barren lion
taking N = 10/epsilon
neon iron
wait
oh maybe it might work
I might need to wrap my head first but looks promising
@barren lion minor problem
$\frac{10x^2+29}{4x^3} >\frac{10x^2+29}{4x^3+2}\text{ but that doesnt imply that }\frac{10x^2+29}{4x^3} <\epsilon$
thorny flameBOT
Bandera
barren lion
you want to start with (10x^2+29)/(4x^3+2) and bound it above by epsilon
neon iron
mb
neon iron
you want to start with (10x^2+29)/(4x^3+2) and bound it above by epsilon
you wanna make it greater than epsilon?
barren lion
no
you want to make x large enough so that you can derive a string of inequalities with < leading to epsilon
try taking N = 10/epsilon and making x > N
neon iron
try taking N = 10/epsilon and making x > N
my problem would be how would I even come up with this
barren lion
then (10x^2+29)/(4x^3+2) < (10x^2+1000)/x^3 < 10/x + (10/x)^3 < epsilon + epsilon^3
neon iron
on like a different example
i thought we're supposed to be finding an N using |f(x)-L| < epsilon
barren lion
if epsilon < 1, then epsilon^3 < epsilon