$P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}$
#help-27
wispy oyster
woven radishBOT
mia
wispy oyster
seems like it
im guessing the notation (18, 0.14) means 18 trials and 0.14 probability of success
pearl grotto
im guessing the notation (18, 0.14) means 18 trials and 0.14 probability of succ...
yes
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peak wave
why did they not use tabular integration here?
devout snowBOT
why did they not use tabular integration here?
unreal thunder
what's tabular integration ?
i'm unaware sorry
Perhaps the shortcut method for integration by parts?
peak wave
yep exactly
unreal thunder
did you try using it ?
peak wave
i didn't 😅 but i automaticially assumed that when we have smthing like x^4 or a x^5 etc then we use tabular int.
unreal thunder
That's not true, tabular method just cover up a lot of algebra
otherwise you have to do it by hand
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granite bough
hi
granite bough
i need help solving a first order differential equation
y' + p(x)y = q(x)
my p(x) is -x/(1+x)
and my q(x) = 1
I(x) = e^-(integral(x/(1+x))
i dont know how to continue
honestly i forgot how to do basic integrals too
but when i figure that out
i dont know how the hell i am supposed to use e^(whatever this is) for
mystic scarab
You multiply by it to the whole diff eq
granite bough
isnt there another method?
y = 1/(I(x) integral(I(x)q(x)dx + c)
wait
wait
x/(1+x)
mystic scarab
You multiply by it to the whole diff eq
Also,
$$\int{\frac{x}{x + 1}\dd{x}} = \int{\frac{x {\color{red}{+ 1 - 1}}}{x + 1}\dd{x}} = \int{\frac{x+1}{x + 1}\dd{x}} - \int{\frac{1}{x + 1}\dd{x}} = $$
woven radishBOT
Alberto Z.
mystic scarab
I think you know how to continue from here
granite bough
i get some kind of log
so it cancels the e
i need to freshen up on integrals before i try problems like this
i am drawing a complete blank
ty for the help
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granite bough
**Alberto Z.**
this is actually really useful thanks for this
mystic scarab
Yw
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kindred mauve
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kindred mauve
can someone check this please (as in the working out as there is an error somewhere
thanks
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@kindred mauve Has your question been resolved?
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@kindred mauve Has your question been resolved?
potent tusk
okay
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@kindred mauve Has your question been resolved?
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@kindred mauve Has your question been resolved?
humble dirge
having trouble finding a way to begin tackling this
devout snowBOT
having trouble finding a way to begin tackling this
sharp kayak
having trouble finding a way to begin tackling this
well you have a known angle and side length for one right triangle
potent tusk
having trouble finding a way to begin tackling this
question is a bit cut off
sharp kayak
that means that you're able to solve the rest of the side lengths
humble dirge
that's the full question past that little cutoff
sharp kayak
(and angles if you want)
humble dirge
well i am solving for angle theta so yeah i'll need to do that too
sharp kayak
well yes in the end you want to do that
but youre unable to solve things in triangle ABD since you only have 1 known side length
but you are able to solve values in triangle BCD
humble dirge
yeah i just realized i could get C w/o even needing to do anything mildly complex
not sure if i even need it though
sharp kayak
also notice how line BD is in both triangles ABD and BCD
humble dirge
yea i noticed that, if i get that part that's pretty helpful
sharp kayak
hint trig
humble dirge
although i dont know if ABD is a right angle triangle since its not on the diagram
sharp kayak
although i dont know if ABD is a right angle triangle since its not on the diagr...
angle DBC is a right angle
humble dirge
i know that
potent tusk
although i dont know if ABD is a right angle triangle since its not on the diagr...
is AC a straight line?
humble dirge
and i'm racking my brain on what to do w/ it
humble dirge
is AC a straight line?
🤷
humble dirge
yeah thats how i got C
41.3 degrees
am i missing a law
like i'm thinking about it and i dont think either sine or cosine law applies here
or SOH CAH TOA i think
humble dirge
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vale lava
hi
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short jasper
How to solve this
Multiple options can be correct
devout snowBOT
How to solve this
Multiple options can be correct
supple knot
set the two y equations equal to each other and solve for x
strong igloo
Can you show your work?
supple knot
i see yes this is inefficient
try solving for x in the second equation and plug that x into the first equation
should have a quadratic in y i believe
short jasper
Still it becomes 4 degree equation
short jasper
Yeah
supple knot
i guess you can "argue" that the graphs are convex and concave, respectively, so if they intersect for varying a, then they only intersect once for a single value otherwise they don't intersect or intersect at two places
so only one correct choice
short jasper
Bro the equation are straight up parabolla in x and y
supple knot
what does that have to do with what i said
short jasper
In the equation where their is a
1/a^3 is coefficient of x which differ the width of the parabolla and at different widths their might be different points of intersection and some of them will lead to two points and some of them will lead to only one common point
So what you are saying cant be true for all cases
The answer given in the book says that option b and c both are correct
short jasper
Yeah
supple knot
that's when the parabola turns convex
supple knot
i guess you can "argue" that the graphs are convex and concave, respectively, so...
so you have to append that case to the arguemnt here
short jasper
???
supple knot
the first equation is only convex when a > 0
short jasper
K
short jasper
How
supple knot
i guess you can "argue" that the graphs are convex and concave, respectively, so...
a > 0, you do this argument and get a=1/3
short jasper
How do you know that all negative cases only one have one common point
supple knot
a < 0 you make a different one
short jasper
How do you know that all negative cases only one have one common point
All negative a will lead to one common point
supple knot
do you know derivatives
short jasper
Yeah
supple knot
you can use invermediate value theorem or something similar to show there's at least one intersection point
short jasper
Both must have a common tangent ??
supple knot
then show that one function is increasing and the other is decreasing after the intersection point so they won't intersect again
supple knot
short jasper
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vale lava
hi
supple knot
just post it so someone can see it
small nest
Seems to me that you didn't integrate the inner expression, just replaced the x^5 immediately into it
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supple knot
,w int 1 to 3 of (-x^2 - 5x + 2)
woven radishBOT
supple knot
both are right
vale lava
and correct work?
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cinder bay
help
cinder bay
I need to prepare for my algebra test tommorow, made this question based on ai on some areas I lack
basically stuff on vertex form and other stuff, because I don’t understand how the ai got the answers I searched up
dense jay
what are your answers to it
cinder bay
what are your answers to it
I did a different question
I js asked ai again to give me some problems based on that same question so I can understand ig
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vast ledge
how to do 53
vast ledge
<@&286206848099549185>
lilac moat
how to do 53
which graphing utility are you using
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forest bolt
I think my answer times 5 and 8 is supposed to equal k, but it doesn’t, and i’m not sure where i went wrong
devout snowBOT
I think my answer times 5 and 8 is supposed to equal k, but it doesn’t, and i’m ...
thin fern
I think my answer times 5 and 8 is supposed to equal k, but it doesn’t, and i’m ...
how did you get k=36
forest bolt
i did 3 times 6 times 2
thin fern
that doesn't really match your equation
Plug in x=3, y=6, z=2
$2 = \frac{k \cdot 3^2}{6}$
woven radishBOT
Ari
forest bolt
Plug in x=3, y=6, z=2
okay thanks i got it!! also before that i had looked up how to get k and google just said do xyz, do you know why it said that? like is there a circumstance where that would work
tribal jungle
devout snowBOT
https://i.imgur.com/MSfFp79.png
tribal jungle
what can possibly be wrong about this?
supple knot
Maybe 1/tan
supple knot
Same, 1/tan
tribal jungle
lmao
supple knot
This question is a joke
tribal jungle
I'm going to ask my professor but I can't believe I lost over 1 point for this
it's a quiz
hopefully he can change my grade on it
supple knot
Oh then you shouldn't be cheating on a quiz
tribal jungle
Oh then you shouldn't be cheating on a quiz
I am not cheating. It's already permanently graded.
that's how the quizzes work in my class
I'm not in the zoom class, this is homework.
supple knot
cot from the beginning then
hollow ice
I am not cheating. It's already permanently graded.
did you not just change the answer in an attempt to change your grade?
supple knot
If it's none of those three then this question is awful
tribal jungle
did you not just change the answer in an attempt to change your grade?
you can't change the grade even if you get it right
hollow ice
Ah icic
supple knot
This question is a joke
.
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pseudo basin
but I don't know how this is teadching me anything
it's teaching you bureaucracy 
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icy burrow
How can i find the total number of 7-digit numbers that have a digit sum of 36?
icy burrow
I'm attempting to use bar-stars but im struggling to figure out how to exclude the case when the first digit is 0
solid oriole
I'm attempting to use bar-stars but im struggling to figure out how to exclude t...
Well, why not just subtract the number of cases where the first digit is 0?
icy burrow
i've been trying that but i keep getting the wrong answer >_<
like subtracting 6-digit numbers that have a digit sum of 36 (inc those with 0 to starT)
solid oriole
Yeah that's what you should be trying to do
icy burrow
hmm ill guess i just try again
hollow ice
like subtracting 6-digit numbers that have a digit sum of 36 (inc those with 0 t...
Do your stars and bars include the restriction of decimal system? That the single digit cant be more than 9?
icy burrow
Do your stars and bars include the restriction of decimal system? That the singl...
yh
i do like $(9-d_1)+(9-d_2)+...(9-d_7)=36$
woven radishBOT
nacho
icy burrow
$$\begin{align*}
(9-d_1)+(9-d_2)+...(9-d_7)&=36 \
d_1+d_2+...+d_7&=63-36=27\
&= C(27+(7-1), 7-1)\
&= C(33,6)\
\text{Now the cases when the first digit is 0:} \
(9-d_2)+...(9-d_7)&=36 \
d_2+...+d_7&=54-36=18 \
&= C(18+(6-1), 6-1)\
&= C(23,5) \
\text{Subtracting, we obtain:} \
C(33,6) - C(23,5)
\end{align*}$$
wait
i cant do math
undone chasm
9-0≠0?
icy burrow
oh i meant i did the subtraction wrong lol
icy burrow
i was considering the ways 6 numbers <=9 can make 36 since the first digit would be 0
undone chasm
im not real carry on
woven radishBOT
nacho
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icy burrow
yh this is still wrong i think im being silyl
icy burrow
9-0≠0?
u might be right?
i kinda get what ur saying idk
Wait is it because the total is 27 and 18 which is outside [0,9]
do i need to inclusion-exclusion
YIPPEEE I GOT IT'
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restive river
Hey, can someone please help with this?
A matrix $A$ is given. Find, if possible, an invertible matrix $P$ and a diagonal matrix $D$ such that
[
A = P D P^{-1}.
]
Otherwise, explain why $A$ is not diagonalizable.
[
A = \begin{bmatrix}
-1 & 2 & -1 \
0 & -3 & 1 \
0 & 0 & 2
\end{bmatrix}
]
woven radishBOT
Cizy
restive river
Do you know how to do that
The matrix $A$ is upper triangular, so its eigenvalues are the diagonal entries: $-1, -3, 2$
The matrix $A_{3 \times 3}$ has $3$ distinct eigenvalues, so it is diagonalizable
sand quarry
Yeah
woven radishBOT
Cizy
sand quarry
So D is the diagonal matrix with those eigenvalues so you're good to go with that
P is the matrix of the eigenvectors
restive river
So D is the diagonal matrix with those eigenvalues so you're good to go with tha...
So $D = \begin{bmatrix} -1 & 0 & 0 \ 0 & -3 & 0 \ 0 & 0 & 2 \end{bmatrix}$, right?
sand quarry
Ye
woven radishBOT
Cizy
restive river
P is the matrix of the eigenvectors
The basis of each eigenspace?
I don't understand the theory behind this
Like—how come this even works?
sand quarry
Like—how come this even works?
If you collect enough linearly independent eigenvectors to form a basis, then in that eigenvector basis, matrix multiplication becomes very simple. Its just basically scaling each eigenvector by its eigenvalue
restive river
If you collect enough linearly independent eigenvectors to form a basis, then in...
Okay, still confused
How come $A = P D P^{-1}$ magically holds?
woven radishBOT
Cizy
restive river
By the way
If I compute the basis of each eigenspace—in this case, $E_{-1}$, $E_{-3}$, and $E_2$—are they always linearly independent?
woven radishBOT
Cizy
sand quarry
How come $A = P D P^{-1}$ magically holds?
It comes from realising that AP = PD holds
Try to multiply out AP symbolically
restive river
It comes from realising that AP = PD holds
This is the matrix version of $A\vec{v} = \lambda\vec{v}$?
woven radishBOT
Cizy
restive river
Seems like it
sand quarry
If I compute the basis of each eigenspace—in this case, $E_{-1}$, $E_{-3}$, and ...
Yes as long as they come from unique eigenvalues
sand quarry
This is the *matrix version* of $A\vec{v} = \lambda\vec{v}$?
I'm not going to draw analogies like that that might mislead you, but I guess it's symbolically similar
restive river
I'm not going to draw analogies like that that might mislead you, but I guess it...
Made it here
sand quarry
Are you sure you row reduced the second and third correctly
restive river
Are you sure you row reduced the second and third correctly
I didn't row-reduce actually
Maybe I should've shown more steps, oops
sand quarry
Nevermind they seem fine, I just didn't calculate them on my own
sand quarry
Yeah
restive river
Why are they linearly independent?
sand quarry
Why are they linearly independent?
I mean suppose for the sake of contradiction that they're in fact linearly dependent
Could you try to figure out why this would not work
Use the definition of linear dependence
restive river
Could you try to figure out why this would not work
I'm guessing that $E_{\lambda_1} \cap E_{\lambda_2} = \emptyset$ for $\lambda_1 \neq \lambda_2$, by definition?
woven radishBOT
Cizy
restive river
Thus making any two eigenvectors, respectively in $E_{\lambda_1}$ and $E_{\lambda_2}$, linearly independent?
woven radishBOT
Cizy
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finite briar
gm
woven radishBOT
rak³en
finite briar
we can choose any sign for alpha beta doesnt matter
pseudo basin
0.25 
finite briar
0.25 <:pain:849218577633640449>
too lazy to type it out \frac{1}{4}
finite briar
that works?
pseudo basin
$3^{1/4}$
woven radishBOT
Ann
acoustic leaf
inline fractions are generally preferable for exponents
finite briar
so $\alpha = \frac{3^{1/4}}{\sqrt{2}} e^{i \cdot {3 \pi/4}}$
woven radishBOT
rak³en
pseudo basin
ok so far so good
finite briar
and $\beta = \frac{3^{1/4}}{\sqrt{2}} e^{i \cdot {5 \pi/4}}$
woven radishBOT
rak³en
pseudo basin
strange, im not getting any of these answer options...
hold on
i think your alpha and beta are wrong
lemme recalc
finite briar
i did get the right answer just i think i am blundering something in this approach
the other approach was noticing that alpha^12 = 3^3 = beta^12
pseudo basin
yes they are wrong
finite briar
yes they are wrong
i see where did i blunder
pseudo basin
finite briar
okay shit
goddayum
such a rookie mistake
ugh this is getting messy
maybe i shouldnt do it this way and just accept what the other approach did
hollow ice
ugh this is getting messy
why is it getting messy? its just a multiple of 30deg now
finite briar
why is it getting messy? its just a multiple of 30deg now
its -1+/-27i * stuff
my theta is terrible
hollow ice
more rookie mistakes
finite briar
oh wait is it root 3
pseudo basin
finite briar
-1+/-root3 i
hollow ice
> Solving JEE ahh questions
> Numbers dont simplify to nice things
You must realize you fucked up
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opal flare
Hello, can someone help me on this question? I have no idea where to start. Please and thank you
devout snowBOT
Hello, can someone help me on this question? I have no idea where to start. Plea...
opal flare
dapper fable
primal ferry
might help to look at it like this
dapper fable
these should be the same size. how would you find the value of x?
runic pelican
the key step in most problems of this form is drawing the correct right triangle
the hypotenuse will often be given by the radius of a circle, and the two legs will be horizontal and vertical distances of some form (perhaps think in terms of coordinates)
primal ferry
the hypotenuse will often be given by the radius of a circle, and the two legs w...
this is what i was suggesting
runic pelican
if you can find all three side lengths of the triangle in terms of a single parameter, you can set up an equation to solve for the parameter
primal ferry
that look at the circles as 0,5 and 10,5 with r=5
devout snowBOT
@opal flare Has your question been resolved?
opal flare
do u assume the botom length is 2x + a?
dapper fable
do u assume the botom length is 2x + a?
yeah, if we let the distance from the tangent to the square be x
opal flare
what does x represent again?
opal flare
so we know that the center point of the right circle to the top right corner of the square is 5cm, how do we find the length of the other legs?
opal flare
I see, so what is the next step to finding it?
opal flare
r=5 in this case
yes thats right
thin inlet
thin inlet
solve for x in the first equation
opal flare
so x^2 + (r-a)^2 = 5?
opal flare
ok right i see
thin inlet
so you can solve for x in 2x+a=10, then subsitute it into the x^2
then its just algebra to solve for a
thin inlet
so x = 5 - a/5
not a/5
thin inlet
x^2+(5-a)^2=5^2
yes, x=5-a/2, then subsitute it into this
devout snowBOT
@opal flare Has your question been resolved?
opal flare
so a is equal to 2
next wadi
Hi there, is your question solved?
The general idea is to form 2 equations from the drawing and solve it
Given r = 5, a = length of the area of the square, we can let x = distance from the center of the circle base to the square base
next wadi
so a is equal to 2
Oh u got the answer already ahhh
opal flare
Oh u got the answer already ahhh
Thank you for the help guys
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finite briar
devout snowBOT
finite briar
so i was doing this one
and i have 4 pairs
and i understand the solution which gives 6
but i dont understand where the 2 extra pairs in my approach got missed
i think my approach itself is flawed seeing the soln
because i started with $x^2+ax+b=0$ having the roots $\alpha$ and $\alpha^2 - 2$
woven radishBOT
rak³en
finite briar
i never considered $\alpha = \alpha^2 -2 = \beta^2-2$ but $\alpha \neq \beta$
woven radishBOT
rak³en
finite briar
that is the mistake right
restive river
have you considered the case (a^2 - 2)^2 - 2 = (a^2 - 2)
finite briar
have you considered the case (a^2 - 2)^2 - 2 = (a^2 - 2)
uhhhhhhhh?
wwwwhat
i think i have
restive river
its either that or (a^2 - 2)^2 - 2 = a
worthy heath
do you know vieta's formulas?
finite briar
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devout snowBOT
worthy heath
why?
restive river
u dont need it in this question
worthy heath
or wait
$(x- \alpha) \cdot (x- \alpha ^2 + 2)=0$
and compare it with original eqn
to get pairs
restive river
dawg
worthy heath
wait
restive river
just stop talking
woven radishBOT
∮Ē.dĀ = Qₑₙ꜀/ε₀
restive river
please
worthy heath
?
restive river
that doesnt do shit
worthy heath
how? please explain
restive river
whats ur plan after that
worthy heath
can we not get pairs of a and b with values of alpha?
restive river
hell nah
restive river
i already suggested the approach scroll up
worthy heath
ok i misinterpreted the question
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pure kelp
Question: Solve for $x$. \ $|x-2|^{10x^2-1}=|x-2|^{3x}$ \
I got two solutuions by equating the exponents and another two by setting the base to 1. but i have a doubt. which is: can we also set the base to be equal to 0 to get another solution? because $0^x=0, \forall x \in \mathbb{R}-{\s 0 }\s$
devout snowBOT
Question: Solve for $x$. \\ $|x-2|^{10x^2-1}=|x-2|^{3x}$ \\
I got two solutuions...
faint zinc
You can, this should get you one solution. If you want to explicitly write { in latex you should escape it with \s.
pure kelp
okii
pure kelp
You can, this should get you one solution. If you want to explicitly write { in ...
but my teacher said that setting the base to 0 is not applicable in this case. so like i really dont know what to do
woven radishBOT
T&C
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faint zinc
Well, try it and see if it causes an issue, the base = 0 implies that x = 2
pure kelp
Well, try it and see if it causes an issue, the base = 0 implies that x = 2
yes. it works out. we dont get $0^0$ form
faint zinc
Not "\s" instead "\" plural
woven radishBOT
T&C
faint zinc
yes. it works out. we dont get $0^0$ form
Yes but, if x = 2 does 10x^2 - 1 = 3x ?
pure kelp
Yes but, if x = 2 does 10x^2 - 1 = 3x ?
oh we need to check that too?
pure kelp
But only to make sure it's not 0
yes on one side we get 0^39 and on the other we get 0^6 but both sides ultimately yield 0=0
so technicaly x=2 should be in our solution set right?
faint zinc
Yes exactly
pure kelp
the complete solution set so far is {-1/5,1/2,3,1}
pure kelp
Yes exactly
okayy. thank you for clarifying. my teacher got me really confused.
pure kelp
yess 2 is there!!
thank you!! i totally forgot i could check the graphs too
god bless u @faint zinc
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restive river
Can someone explain Tensors to me? I dont get it.
-What do upper and lower indicies mean and whats the difference?
restive river
Or how do i start best herausbeuget understanding then
devout snowBOT
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graceful cosmos
What's the context? Are you learning differential geometry?
restive river
What's the context? Are you learning differential geometry?
not really im just curious about them
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lament imp
Hellooo, can i get help with this integral?
devout snowBOT
Hellooo, can i get help with this integral?
lament imp
runic grove
which one
runic grove
did you solve the first one/
lament imp
yes i did
runic grove
good
lament imp
this is the result i got
runic grove
can't you just multiply out x(x+4)^2
like x(x^2 + 8x + 16)
x^3 + 8x^2 + 16x
integrate that
@lament imp
lament imp
omg thank u so much it was an exam and i got it right
have an excelente day
😁
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solemn jewel
I need help with geometry
devout snowBOT
I need help with geometry
solemn jewel
,im stuck on the exercises, ive drawn the points where this are located but I don't know how to connect them together and how to form.
in geogebra it came out as an Cube but how should I know, I don't understand how to draw in it
raven egret
Can you send the translated question?
solemn jewel
of couse
. Draw the solid (3D shape) in a spatial coordinate system (1 unit = 2 squares) with the vertices
while taking visibility into account.
raven egret
I think you messed up in taking your z-axis units
solemn jewel
z is the green one in my paper
i just have it 0.5 going up
and every 2 boxes a 1
no a 2
tame palm
Label your axes on your work.
devout snowBOT
@solemn jewel Has your question been resolved?
solemn jewel
Label your axes on your work.
i will do it but the one going high is y, right is x and the last one going down Z
im going to redraw it but without the y axe, how deep should z go, 0,5 or just 1
okay so I draw it without the y and it came a long rectangle, so now i surely know its a cube, do i just remake it with y and connect the same angles?
should i in an exam do the same, before without an axe just to see and then again?
but because i did the z axe going 1x and not 0.5x its wider, should i redoit with 0.5?
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worn flax
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worn flax
So this is 10C5 * 5C2 * 3C3
if they asked: What is the coeff of x5y2z3 in (2x+y+z)^10
Would be be 2^5 * (10C5 * 5C2 * 3C3)?
supple knot
yea. x^5 becomes (2x)^5 and that's equal to 2^5 * x^5
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restive river
im confused on d's work
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im confused on d's work
misty crest
lilac moat
isn't it supposed to be equal to -1/6
no
lilac moat
instead of like -1/6 <= x+2 <=1/6
why would you think this is true?
restive river
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wind mason
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Did you get it? Or what?
You should notice the limit is infinity and that |x+2| is a constant, so it “takes priority” over the limit when it comes to assigning a value to it
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valid blaze
Is this solution correct
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Is this solution correct
valid blaze
Also my teacher said he prefers parallelogram method but o have no clue what that means
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rain flint
@strange nimbus
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velvet urchin
Could you please help with with couple questions
Send
rain flint
Send
Could you guys please help?
@velvet urchin@strange nimbus @misty crest@dire spruce
<@&286206848099549185>
small nest
Help with whtat?
rain flint
Help with whtat?
the question above
small nest
What question specifically?
rain flint
What question specifically?
question number 3
small nest
OK,. what have you tried?
rain flint
OK,. what have you tried?
idk nothing, idk what to do
small nest
Ok, look at the graphic
g(x) = -2 means get the point in the line that matches -2 in the vertical axis
as a first step
Got it?
rain flint
We're not gonna do your homework
bro Idk how to do it 😭
small nest
I'm explaining you
exotic geyser
gng is playing fortnite while doing his homework 🙏
rain flint
Otherwise it's just handing you the answer
???????
small nest
<@&268886789983436800> Can we clear the channel?
small nest
This person is expecting to be handed the answer
faint zinc
Hi
rain flint
<@&268886789983436800> bro this max hetfield guy is being rude, I actually don't know how to do the problem
faint zinc
I'm already here
crisp stratus
Let's start with a)
rain flint
I'm already here
oh mb
rain flint
Let's start with a)
👍
crisp stratus
What does g(4) mean?
rain flint
What does g(4) mean?
g of 4, idk
small nest
See?
crisp stratus
As a function of x
faint zinc
!noans @rain flint
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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.
crisp stratus
rain flint
!noans <@1007753182744682636>
I didn't ask him for the answer though
faint zinc
I didn't ask him for the answer though
Regardless, you are acting as if you just want the answers and it doesn't seem like you're trying very hard to work with the people attempting to guide you to that answer.
rain flint
horizontal
rain flint
Regardless, you are acting as if you just want the answers and it doesn't seem l...
I don't really get it and my teacher doesn't even teach us that much😭
exotic geyser
I don't really get it and my teacher doesn't even teach us that much😭
if the function shown is g(x), what does g(4) mean
crisp stratus
horizontal
Cool now g is a function of x. Meaning in this graph you are looking at the x axis. So far so good?
exotic geyser
what is "4"
small nest
@rain flint Follow @crisp stratus's lead
rain flint
Cool now g is a function of x. Meaning in this graph you are looking at the x ax...
yes
crisp stratus
Ok now to find the value of g(x) for some arbitrary x, you just going to match it with the y axis (vertical one)
crisp stratus
Ok so what is g(4)
faint zinc
To be clear, when you look at a point on the graph, it will be a point that we normally notate as (x, y)
In this case y = g(x), so your point is going to be (x, g(x)) or more specifically (4, g(4))
(hopefully you recall plotting points on the Cartesian plane!)
rain flint
-5?
small nest
Yes
crisp stratus
See, great
Now before doing b), what does the uncolored circle mean?
As opposed to the colored one
rain flint
Now before doing b), what does the uncolored circle mean?
not including
small nest
not including
Also technically named a "discontinuity" in a function. Or open interval, if it's at one of the extremes of a range of values
crisp stratus
So what is g(-3)?
rain flint
Now we can answer b) confidently
I don't think there is an answer for that
rain flint
?
crisp stratus
There's a discontinuity there as Max said
rain flint
btw
small nest
Just to confirm, this isn't a class about limits, is it?
rain flint
one question
crisp stratus
If you're not comfortable with this term don't worry about it unless it is a calculus class
rain flint
When do we use point slope form
cuz it is going to be on my exam tmr
I am kinda cooked
crisp stratus
We haven't finished the questions yet
Unless you did c) already
c) is asking the opposite now. What value of x gives you -2
crisp stratus
Yeah this is correct
rain flint
Yeah this is correct
Deltoid can u pls answer my question now
crisp stratus
Ok now we can go to your question. I'm not sure what you mean by this
Do you have a sample question
Or the "form"
rain flint
Ok now we can go to your question. I'm not sure what you mean by this
Why do we use point slope form
/ what do we use it for
small nest
To represent a line when you only got 1 point and the slope of the line
crisp stratus
Are you referring to y-y_0 = m(x-x_0)?
When the question asks you to find the equation of the line, you use it
rain flint
On my exam tmr, one part is going to be on graphing lines: domain range, parrellel perpendicular and 3 forms of a line
rain flint
To represent a line when you only got 1 point and the slope of the line
Can you gimme an example pls
crisp stratus
Alright I'll leave it to Max I guess. Good luck with your exam
small nest
Can you gimme an example pls
You got a line with slope = 2 and and point (0, 1)
small nest
Are you referring to y-y_0 = m(x-x_0)?
Find the equation of the line with this equation
rain flint
Alright I'll leave it to Max I guess. Good luck with your exam
Alright, ty Deltoid
small nest
In this case, m=2, x_0 = 0, y_0 = 1
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kind ridge
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full cloak
I suppose you want like a clever way to do this problem?
kind ridge
I suppose you want like a clever way to do this problem?
yes😭
i keep getting 2root 7
x root7 + root x/x
so can x be canceled by x
full cloak
do you know how to turn 7+4root3 into a square?
full cloak
alright what do you get?
soft umbra
Hint: 4*3 = 12; 4+3 = 7
full cloak
yes
he did say he can turn it into a square
alright dont worry
stuff like this requires you to know your perfect square formulas
a^2+2ab+b^2=(a+b)^2
kind ridge
stuff like this requires you to know your perfect square formulas
a+b whole square?
yes
kind ridge
Hint: 4*3 = 12; 4+3 = 7
this?
full cloak
one more thing to remember: roota*rootb=root(ab)
I am just asking you where you get 2ab
would it be the 7 or 4root3?
kind ridge
I am just asking you where you get 2ab
we can plus 2ab-2ab
kind ridge
would it be the 7 or 4root3?
4 root 3?
kind ridge
how do you turn 4root3 into 2 times something
2 2 root 3
kind ridge
what is your a and what is your b?
a=root 3
b = 2
cause
root 3 sqaure
=3
and b swarue = 4
and 4+3= 7
which is given
there
full cloak
yes very nice
kind ridge
:0
full cloak
so what is 7+4root3 equal to?
kind ridge
so what is 7+4root3 equal to?
root 3 + 2 whole swaure
full cloak
yes very good
kind ridge
yes very good
full cloak
...
full cloak
if x is (root3+2) squared
kind ridge
shud be
full cloak
what is square root of x
kind ridge
5
kind ridge
so therefore on the right you end up with:
that shud be easy to rationalize now
alr
mme cook
full cloak
yeah you should rationalize the denominator
kind ridge
how would there be 2 answers?
look
we have root 3+2
in denomintator
if we take 2+root 3
then root 3 would be negative in numberator
and root 3 would cnacle negative root 3
and then 4 would be answer
kind ridge
we have root 3+2
but if i take
this
then
2 would cancel
and 2 root 3 would be answer
full cloak
alright show me your steps
full cloak
also please talk in full sentences, it is kinda painful reading yopur seperate words
kind ridge
also please talk in full sentences, it is kinda painful reading yopur seperate w...
sorry
my bad
full cloak
alright I can show you why what you said is false
so to rationalize 1/(2+root3)
if you do 2+root3
you get (2-root3)/(2+root3)(2-root3)
and that is equal to (2-root3)/1
if you do root3+2
kind ridge
so to rationalize 1/(2+root3)
where did 1/2 came
full cloak
where did 1/2 came
looks better?
full cloak
if you do root3+2
anyways continue
you get (root3-2)/(root3+2)(root3-2)
and that is equal to (root3-2)/-1
which is equal to 2-root3
full cloak
also I just found like a quicker way to solve this question, you probably dont need to know this, but it is fun to know
do you want to hear about it
kind ridge
also I just found like a quicker way to solve this question, you probably dont n...
YES
full cloak
first of all, what do you get as final answer
I need to make sure you got the answer before i tell you the secret method
shhhh dont answer for him lol
welp if he wants to play the act of disappearing so shall I
random pike
xD
kind ridge
first of all, what do you get as final answer
4
full cloak
finally you are back
kind ridge
welp if he wants to play the act of disappearing so shall I
i went to eat fod💔
mom kill if no me eat food
full cloak
alright are you ready for the fast method
kind ridge
alright are you ready for the fast method
yes
full cloak
this one not gonna make you think through because you probably never going to use
so on the left:
x=7+4sqrt3
btw sqrt means square root
kind ridge
yesir
full cloak
therefore:
x-7=4sqrt3
square them
x^2-14x+49=48
therefore:
x^2+1=14x
on the right:
kind ridge
therefore:
x-7=4sqrt3
dem
full cloak
sqrtx+1/sqrtx
=(x+1)/sqrtx
if you expand the square root:
sqrt[(x^2+2x+1)/x]
but x^2+1=14x
therefore that is equal to:
sqrt(16x/x)
which is equal to sqrt16=4
full cloak
dw you probably don't need to know this method lol
full cloak
this method HEAVILY relies on number sense
full cloak
number sense is like how good you are with numbers
kind ridge
practice math
oh boi i ahve been doing that sinc ei was bor
full cloak
like how quickly you can realize to turn an expression into something you can work with
for example:
turning 7+4sqrt3 into (a+b)^2
that requires number sense
kind ridge
sqrtx+1/sqrtx
i didnt understand thi
full cloak
that is all I am going to do, try figuring it out yourself, if you cannot, stick with the first method
full cloak
I have a bad feeling about that question
feral agate
why does uniform convergence let me swap the differentiation and summation operations?
devout snowBOT
why does uniform convergence let me swap the differentiation and summation opera...
feral agate
meaning if the series $\sum_{n=1}^\infty f_n(x)$ converges uniformly to $f(x)$ on some domain then why is $f'(x)=\sum_{n=1}^\infty f_n'(x)$?
woven radishBOT
kheer257
feral agate
and are there more conditions required for this to be true?
pseudo basin
hmm... 1/n sin(n^3 x)?
jaunty mantle
Oh
feral agate
oh
sharp crow
if there are 6 colours of labubu's with equal chances and a special edition labubu that has a probability of 1/72
and you pull 5 of the normal colours + the special edition in 6 tries
whats the probability that you pull the last normal colour on the 7th try
My answer:
Here's what we know:
- There are 6 regular Labubu colors, each with an equal probability.
- There's a special edition Labubu with a probability of 1/72.
- You pulled 5 of the regular colors + the special edition in 6 tries.
- We want to find the probability of pulling the last regular color on these
- Probability of each regular color:
Since there are 6 regular colors with equal chances, the probability of pulling any specific regular color is 1/6.
- Probability of NOT pulling a specific regular color:
This would be 1 - (1/6) = 5/6
- Probability of pulling the last regular color on the 7th try:
To pull the last regular color on the 7th try, it means you haven't pulled it in the first 6 tries. So, the probability is 5/6.
Therefore, the probability that you pull the last normal colour on the 7th try is 5/6.
<@&286206848099549185>
wispy oyster
!occupied
jaunty mantle
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pseudo basin
if there are 6 colours of labubu's with equal chances and a special edition labu...
please open your own channel
feral agate
oh
so we need at least one point where the original sequence of functions converges
and the derivatives need to converge uniformly?
jaunty mantle
Mhmm theorem 7.17 of rudin
Also this says on closed intervals as well so infinities might not work either
feral agate
https://math.stackexchange.com/questions/484250/is-there-a-simple-proof-that-a-power-series-can-be-differentiated-term-by-term
I was asking in the context of this answer here, particularly the second fact they mentioned
I just don't really understand where it came from
faint hearth
In practice the application you will come across most is:\
$f_n , f_n'\rightarrow f, f'$ uniformly on $[a,b]$, then $\lim_{n\rightarrow \infty} f_n' = f'$ on $[a,b]$
woven radishBOT
Asteroid
faint hearth
The theorem stated in books is slightly stronger than this
feral agate
yeah that's the one they used in the answer
feral agate
**Asteroid**
I think you mean f_n' -> g, then f' = g
faint hearth
Uh yeah
feral agate
is there some way to see why this would be true?
analysis seems kinda random at times
jaunty mantle
There’s a proof in rudin for it but it also says that its easier if f_n’ can be assumed to be continuous
Which I guess you have because you’re talking about power series
feral agate
hmm
faint hearth
uniform convergence allows you to "control" the function in a neighbourhood of a point, so it behaves nicely with limits which also use neighbourhoods. So it makes sense why uniform convergence is the condition that a lot of these limit switching theorems ask for
Maybe there is a more intuitive reason
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restive river
so we need at least one point where the original sequence of functions converges
.This is the minimal requirement because if $F'(x) = \sum_{n=1}^{\infty} f'n(x)$ then by FTC, $\int{x_0}^{x}F'(x) = F(x) - F(x_0)$ considering $x_0,x \in [a,b]$
So we need the existence of $F(x_0)$ and hence this minimal requirement $\sum_{n=1}^{\infty}f_n(x_0) < \infty$ is needed. The stronger conditions on this like $F(x)$ being continuous, or uniformly continuous in $[a,b]$ (mentioned in stack exchange) essentially fulfil this requirement
To compute $\int_{x_0}^{x}F'(x) = \int_{x_0}^{x} \lim_{N \rightarrow \infty} \sum_{n=1}^{N}f'n(x)$, the requirement of uniform convergence of $\sum{n=1}^{\infty} f'_n(x)$ is needed to interchange the integral and limit, and the reason is nicely explained in the "To integrability" part in the screenshot sent here (although the sequence used there is not a sequence of partial sums, but it is any sequence of Riemann Integrable functions)
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à뜜
eager stone
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eager stone
In the seconds question I dont get what its doing with z-f(1,2)
fx(P0)(x − x0) + fy(P0)(y − y0) + fz(P0)(z − z0)=0
I know this is the equation
also pls ping me if you answer
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eager stone
<@&286206848099549185> 
midnight echo
In the seconds question I dont get what its doing with z-f(1,2)
Looks correct
eager stone
Looks correct
it is correct but I dont know why
midnight echo
fx(P0)(x − x0) + fy(P0)(y − y0) + fz(P0)(z − z0)=0
I know this is the equation
It's just rearranging that equation
I mean the equation of a tangent plane is z=f(a,b) + (x-a)f_x(a,b) * (y-b)f_y(a,b)
The answer just moved the f(a,b) term to the lhs
where P0 = (a,b)
eager stone
I mean the equation of a tangent plane is z=f(a,b) + (x-a)f_x(a,b) * (y-b)f_y(a,...
isnt that the equation of linear approx
does tangent plane also have the f(a,b) at the start?
eager stone
fx(P0)(x − x0) + fy(P0)(y − y0) + fz(P0)(z − z0)=0
I know this is the equation
i know tangent plane as this
midnight echo
how do you define fz then?
It is equivilant, going from implicit to explicit form.
You start with your equation, but fz = -1, then move z-z0 to the rhs, which matches the formula I gave.
eager stone
You start with your equation, but fz = -1, then move z-z0 to the rhs, which matc...
oh hm i kinda get it
.close
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chrome isle
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chrome isle
Can someone tell if my attempt is correct
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chrome isle
<@&286206848099549185>
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worthy heath
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