well, first of all, before we even start taking the limit as w goes to negative infinity, i think we need to deal with the issue at n = 0
#help-23
delicate aspen
lean plank
Can someone explain the difference of a euler path and euler trail to me? Google seems to have conflicting info, some sites say they are the same, some say different.
Additionally, if they are the same, can it be possible for a graph to have both a trail and a circuit? I would think not, because a trail must have 2 odd vertices and a circuit must have no odd vertices, but I got this question wrong:
safe radishBOT
Can someone explain the difference of a euler path and euler trail to me? Google...
lean plank
According to the homework, this has both a trail and a circuit
safe radishBOT
@lean plank Has your question been resolved?
vocal portal
trail is the path conecting all points using all conections only once
circuit is a path connecting all points and coming back to the original
I guess the most trivial case would be single ring
formed off points
lean plank
So it is possible to be both?
vocal portal
why not
but I don't know how many sophisticated posibillites that have both there exists
junior wagon
I was able to prove this. But I was wondering for part b, what does removing the assumption gcd(a,b) =1 actually do?
safe radishBOT
I was able to prove this. But I was wondering for part b, what does removing the...
junior wagon
Intuitively, in my mind it just restricts a and b to be small to the point where theres no common factors between them other than 1
Is that way of thinking correct
eager ravine
if a, b, and c have a common prime factor
then ab will have this prime factor squared
while c only has 1 of this
thus it cannot divide
an example of this is a = 2, b = 6, c = 6
2|6, 6|6, but 12 does not divide 6
@junior wagon
junior wagon
then ab will have this prime factor squared
what do you mean prime factor squared?
eager ravine
what is prime factorization of 12
junior wagon
12 = 2x2x3
eager ravine
you don't write out the same prime multiplying you write it as a power
junior wagon
$12=2^{2}*3$
flat frigateBOT
Kurama
eager ravine
yes
and the prime 2 is squared
2 is the prime factor in a, b, and c
in fact the easiest example for this is a = b = c = 2
2^2 does not divide 2
because the power of the prime is higher
eager ravine
np
junior wagon
.close
junior wagon
For (4) and (5), why did they prove it by multiplying an element and its inverse to the identity?
safe radishBOT
For (4) and (5), why did they prove it by multiplying an element and its inverse...
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junior wagon
For part a, how would i find the identity and inverse matrices?
safe radishBOT
For part a, how would i find the identity and inverse matrices?
junior wagon
if they exist
final halo
if it were a group, it would be a subgroup of the 2x2 invertible matrices so would have the same identity
just the identity matrix
oh sorry was thinking about multiplication
junior wagon
im also not familiar with the term subgroup
final halo
a group that is a subset of another group
junior wagon
ok
final halo
okay so if we're adding matrices together what do you think the identity might be
junior wagon
yeah, the zero matrix. Since A + 0 = A = 0 + A
junior wagon
oh yeah, thats right
final halo
so we have an additive identity, great
junior wagon
Ok, so for the inverse, we can take $A^{-1} = -A$?
flat frigateBOT
Kurama
final halo
yes that will work
junior wagon
ok, linear algebra is coming back to me
final halo
one thing you need to check also, is that if we add two things in G we get back something in G
junior wagon
Right, that checks that it is a binary operation right
final halo
checking that we're our set is closed under the binary operation yeah
which is what we need for a group
junior wagon
Ok, then for part b, iirc matrix multiplication is associative
junior wagon
We can take the identity matrix where $$ \mathit{I}_2= \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}$$
flat frigateBOT
Kurama
final halo
yeah it will be
junior wagon
since regular addition is commutative
final halo
exactly, nice
so for multiplicative we have identity, and associativity
so far
what should we check next
junior wagon
Inverse, I feel like that is harder
final halo
good choice
junior wagon
So we need something such that $AA^{-1} = I_2$
flat frigateBOT
Kurama
final halo
well we already know something that does that right?
feral linden
Why do you think any element has to have an inverse for b)…
final halo
Why do you think any element has to have an inverse for b)…
we're getting there
feral linden
Sorry I see
junior wagon
Just take the inverse of A?
final halo
right... but
we need EVERY element in G to have a multiplicative inverse
if an element were to have one, it would be the usual matrix inverse
but
can we say that all our matrices in G have a matrix inverse?
junior wagon
Yeah, the zero matrix
junior wagon
I mean there are plenty, Like (1 1, 1 1)
final halo
yeah yeah good okay you see it
any where ab=c^2
so, is G a group under multiplication?
final halo
hurray
grand wolf
Does anyone know how to algebraically solve x^ln(x)=xln(x)?
I thought of, by concept, ln(x)=1 so that x^1 =(1)(x) which always hold true
So x=e is indeed the only answer (checked on the graph), but algebraically how did is x=e an answer?
safe radishBOT
Does anyone know how to algebraically solve x^ln(x)=xln(x)?
I thought of, by con...
@grand wolf Has your question been resolved?
grand wolf
I'm sorry but... <@&286206848099549185>
final halo
You would need to use the lambert W function
grand wolf
W(xe^x)=x right?
yes I need the step-by-step solution anyways
winter pivot
$\exp(\ln^2(x)) = x\ln(x)$
flat frigateBOT
EndTimes
grand wolf
x^ln(x) is e^(ln(x))^2?
Just like... someone jog down a step-by-step solution from x^ln(x) = xln(x) till x=e
final halo
,,\begin{align*} &x\ln x = x^{\ln x} \ &W(x\ln x) = W(x^{\ln x}) \ &\ln x = W(x^{\ln x}) \ &x = e^{W(x^{\ln x})}\ &x = \frac{x}{W(x^{\ln x})} \ &W(x^{\ln x}) = 1 \ &x^{\ln x} = e\ &\ln(x^{\ln x}) = 1\ &(\ln x)^2 = 1\ &\ln x = \pm 1\ &x = e^{\pm 1} \end{align*}
figured
flat frigateBOT
iCaird
final halo
@grand wolf
torpid magnet
Hey i am struggling proving convergence of the following sequence by using cauchy, i hope i phrased my questions correctly, im not a native english speaker
torpid magnet
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vagrant pasture
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vagrant pasture
I'm confused at why I'm getting the wrong answer
here are my steps:
$\frac{sin(α)}{a} = \frac{sin(β)}{b}$
flat frigateBOT
Krish
vagrant pasture
$\frac{sin(25)}{18} = \frac{sin(β)}{30}$
flat frigateBOT
Krish
grand wolf
So angle B is 44.77816685 degrees?
vagrant pasture
yeah
but imi getting 78
:/
$\frac{5 *sin(25)}{5 * 18} = \frac{3 *sin(β)}{3 *30}$
flat frigateBOT
Krish
grand wolf
grand wolf
So yeah
grand wolf
but I'm using wolfram alpha
Uhm sometimes wolfram alpha just straight up fails
Or maybe... Just maybe you typed a syntax that messed it up?
grand wolf
😳 wheeze
vagrant pasture
but i didn't know it came in decimal form
Dx
i thought it was in terms of $\frac{\pi}{6}$
flat frigateBOT
Krish
vagrant pasture
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vagrant pasture
can I assum e that the angle at the gray plane is also 30?
regal girder
I think you're meant to do smth like this?
I'm not sure what the question is though
vagrant pasture
yeah
regal girder
what are you trying to find exactly?
vagrant pasture
so I'm wondering if the angle (above one)just after the gray plane is also 30
symmetric to the one on the ground
regal girder
yeah I think it is
vagrant pasture
ok then in that case
I think I can do sin(30) = 2/hypotenuse (using km)
find the hypotenuse
and use sine law
$\frac{\frac{2}{sin(30)}}{180-(30+55)} = \frac{x}{sin(55)}$
flat frigateBOT
ウラハラキスケ
vagrant pasture
$\frac{\frac{2}{sin(30)}}{180-(30+55)}*sin(55) = x$
flat frigateBOT
ウラハラキスケ
vagrant pasture
then my answer is = $\frac{x}{\frac{1}{6}}$
flat frigateBOT
ウラハラキスケ
vagrant pasture
.close
smoky lava
In algebra, i^2 = -1. How is this possible?
safe radishBOT
In algebra, i^2 = -1. How is this possible?
smoky lava
i = imaginary unit
smoky lava
?
Seems kinda dogma-ey
I mean, this is information that has been introduced to me from khanacademy
grand wolf
It has alot of applications
But conventionally it's just i^2 = -1
smoky lava
It has alot of applications
But conventionally it's just i^2 = -1
There's not much of a reason I can deduce as to why this is the case?
peak estuary
it's literally just a definition
inner sable
because it helps resolving equation
peak estuary
which is insanely helpful. but in the end just a definition
inner sable
and it helps in physics
smoky lava
What is simply just i?
defined as
So we can't find out what i is ?
Seems so strange!
smoky lava
My misconceptions stems from the fact that an even number of negatives always results in a positive
-2^2 = 4
-2^4 = 16
peak estuary
i is neither negative nor positive
smoky lava
hmmm
peak estuary
it's in a different direction
woeful plume
i is "different"
bronze prism
you mean (-2)^2 and (-2)^4, careful about your brackets
woeful plume
It doesn't fit in with 3, 2.9, π, ...
smoky lava
it's just that based? 🤣
bronze prism
It's like this
Good luck trying to put i on the horizontal line in the middle
It's just not
woeful plume
It didn't fit in with the numbers you and I are familiar with, so we just gave it its own name
woeful plume
It's like this
Hence the axis being perpendicular here
bronze prism
Another explanation of i I saw once is that it's a number that "rotates" the numbers it's multiplied with by 90°
If you multiply 1 by i, that's i, a 90° rotation
woeful plume
Another explanation of `i` I saw once is that it's a number that "rotates" the n...
You still need to define Im(z) though
bronze prism
Multiply by it twice, you multiply by i², so a 180° rotation
But hang on, isn't reversing exactly what changing the sign does?
Indeed, to "flip" a number 180°, you multiply it by -1
Geometrically, that's why i²=-1
Both of them are 180° rotations
safe radishBOT
@smoky lava Has your question been resolved?
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@smoky lava Has your question been resolved?
smoky lava
What is a shortcut to understanding what i is
smoky lava
say i^15 or i^26
safe radishBOT
lean otter
i² = -1
i³ = i.i² = -i
i⁴ = i²×i² = -1×-1 = 1
Now just use mod 4 they all repeat in cycles of 4
i⁵ = i⁴×i = i
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lean otter
i know that when there is a midpoint the bottom line is twice that of the midpoint line but im not sure how to do it with the n-7 and 3n+12
safe radishBOT
i know that when there is a midpoint the bottom line is twice that of the midpoi...
royal kiln
i know that when there is a midpoint the bottom line is twice that of the midpoi...
Lots of typos here
Is there a way in math we can say that one quantity is equal to twice another quantity?
lean otter
uh idk
safe radishBOT
@lean otter Has your question been resolved?
thin bridge
try not to overthink this
acoustic violet
ratio of corresponding sides of a pair of similar triangles is the same
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real wyvern
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real wyvern
How do I do (ii)
fading raft
a-levels just expect u to memorize this shiz
what is the range of values of x for the normal binomial expansion
ancient scroll
$x\leq8/9$
flat frigateBOT
justinkim
real wyvern
why is it that?
vocal portal
shouldn't it work for any real number?
since numerator'll give positive number anyway
and denumerator is root of 3
so it still works for negatives
no complex numbers emerging
fading raft
$(8-9x)^{\frac{2}{3}} = 8^{\frac{2}{3}}(1-\frac{9x}{8})^{\frac{2}{3}}$
flat frigateBOT
azeem321
real wyvern
oh right so because 9*8/9 =8, |x| has to be less than 8/9 otherwise it would be 1-1
real wyvern
but then it wouldn;t be a normal binomial expansion
vocal portal
oh
real wyvern
.close
wide juniper
hello there
safe radishBOT
hello there
wide juniper
so I am currently having issues with finding the right statistical test for my data
sharp sage
ok
wide juniper
so ive tried two way anova however it only goes up to 4 variables and i have 5 including the age, weight, bmi etc
thats where im stuck because i basically dont know what test to go with
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@wide juniper Has your question been resolved?
wide juniper
<@&286206848099549185>
kind dove
any ideas ?
graph theory task:
graph G is k-edge connected => graph G is union of k-edge connected disjoint spanning trees
safe radishBOT
any ideas ?
graph theory task:
graph G is k-edge connected => graph G is unio...
@kind dove Has your question been resolved?
safe radishBOT
@kind dove Has your question been resolved?
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@kind dove Has your question been resolved?
plucky elk
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alpine crest
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alpine crest
ido kind of know the factor theorem but wouldn't you need more info to do 7a?
would u just do trial and error or is there an actual method
lean otter
ido kind of know the factor theorem but wouldn't you need more info to do 7a?
find a trivial root of f (try small values)
alpine crest
ah ok so trial and error?
lean otter
say this root is a, the factor theorem says x-a divides f
lean otter
ah ok so trial and error?
yes, but you'll quickly find the root
alpine crest
yeah makes sense ty
lean otter
np
final halo
A trick is that the root should divide the constant term
Which narrows down your choices
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turbid hawk
Hello I can’t solve this. If h(x)=ax+2 and f(x)=2(x-3)*2+1 and h(x)=f(x) then what are the two solutions of a?
plucky elk
did you try this
plucky elk
I’m finding some troubles with the a
what trouble? show your work
viral peak
its not like that
turbid hawk
plucky elk
,rotate
flat frigateBOT
plucky elk
,w expand 2 * (x-3)^2 + 1
flat frigateBOT
plucky elk
your first line's wrong
oh you correct it on the next line
-ax + 12x does not equal 12ax
-ax + 12x is just (-a+12)x
plucky elk
correct it and share again
plucky elk
method looks right, can't really read your handwriting tho
plucky elk
Hello I can’t solve this. If h(x)=ax+2 and f(x)=2(x-3)*2+1 and h(x)=f(x) then ...
what does it mean for there to be two solutions of a?
plucky elk
Yo
this channel's taken. read #❓how-to-get-help
turbid hawk
The only thing they write in the open statement is that u need to find a so that hx=fx have two different solutions
plucky elk
oh you've been doing it wrong this whole time
plucky elk
Hello I can’t solve this. If h(x)=ax+2 and f(x)=2(x-3)*2+1 and h(x)=f(x) then ...
a line can't equal a parabola
main briar
In a box there are 10 red spheres and 6 black ones . We pick randomly from the box , we get a sphere but we dont know what color it is . After we get the speheres we dont put them in the box again
Whats the probability for both spheres to be same color
plucky elk
In a box there are 10 red spheres and 6 black ones . We pick randomly from the b...
stop spamming other people's help channel and read #❓how-to-get-help for the 2nd time
plucky elk
<@&268886789983436800> thx
main briar
Can u help me with this
heavy minnow
plucky elk
Hello I can’t solve this. If h(x)=ax+2 and f(x)=2(x-3)*2+1 and h(x)=f(x) then ...
show the original question. i think you typed something wrong
turbid hawk
The original is not in English
Although there is a mistake in its not -ax, it’s ax
I m just going to ask the teacher tomorrow
Thx for helping me
.close
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Available help channel!
lean otter
I have to use u-sub here
safe radishBOT
I have to use u-sub here
lean otter
so to start it off, do I distribute?
stoic dune
You can fully distribute everything, that's a way to solve
But, uh, I wouldn't
You're looking for some expression, such that the derivative of that expression also exists somewhere
lean otter
come again?
eager ravine
clearly the (x^3 + 3x)^4 is the more nested part
so find some f(g(x)) = (x^3 + 3x)^4
for f and g
lean otter
like the deriavtive of it?
or are you mentioning another rule
nvm I got it
.close
hybrid void
i’m confused on how to solve for x,
safe radishBOT
i’m confused on how to solve for x,
hybrid void
x?*
cursive thorn
so we know this triangle is an isosceles right triangle
stray socket
cursive thorn
since it has a 90 degree angle and the missing angle is 180-90-45
is 45
so you know $y=2 \sqrt{2}$, and so $x=2 \sqrt{2} * \sqrt{2} = 4$
flat frigateBOT
Induction
worthy hemlock
so you know $y=2 \sqrt{2}$, and so $x=2 \sqrt{2} * \sqrt{2} = 4$
Don't do the work for people
cursive thorn
Don't do the work for people
oh ok. is that in the rules
worthy hemlock
oh ok. is that in the rules
cursive thorn
doesn't it just mean explain where you got it from? explaining and elaborating means giving the math behind it no?
hybrid void
oh ok. is that in the rules
i’m confused i saw my teacher do stuff like for example 4/square 2 times square2/square2 why is it just square 2 in this case
cursive thorn
if x is the side length and not the hypotenuse
by pythagorean theorem the hypotenuse is sqrt(x^2+ x^2) = xsqrt(2)
so in the case of this 45-45-90 triangle if x is the side length the hypotenuse is sqrt(2) times that length
worthy hemlock
doesn't it just mean explain where you got it from? explaining and elaborating m...
You need to give the user a chance to learn too
So you providing necessary info to help them solve it themselves is better than you doing it all yourself
hybrid void
so if we were trying to find the stuff other than the hypotenuse we would do __/square 2 times square 2/square 2 or if they already have that square thing we just times square only and don’t have to add square 2 as the denominator for the other number
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lean otter
Find all triplets of (distinct) primes such that the difference between the largest and the smallest is less than 6.
safe radishBOT
Find all triplets of (distinct) primes such that the difference between the larg...
quasi bison
first consider triplets that include 2, then consider those that don't
since this problem mentions primes directly in its statement, one might be inclined to think about the concept of divisibility, which is of course closely related to primes
safe radishBOT
@lean otter Has your question been resolved?
lean otter
first consider triplets that include 2, then consider those that don't
ok
but i don't know what to consider
don't know what to try
quasi bison
let's say a triple includes 2.
at what position can the number 2 appear in the triple? (first? second? third?)
this question is simpler than you think and definitely elementary
though i suppose that i must say explicitly that i'm treating these prime triples as being sorted in ascending order
i know for a fact that you are able to answer my question
if you have decided to do some thinking on your own then please tell me "i am currently thinking on my own without your prompts" otherwise i will get very upset
safe radishBOT
@lean otter Has your question been resolved?
lean otter
this question is simpler than you think and definitely elementary
hm
the middle number doesnt matter tho
oh does it
ah yes
so, if a triplet has 2
it has to have 8, which isnt prime
so that wont work
@quasi bison
quasi bison
are you sure?
reread the problem statement
i think you misread it.
and you also chose not to answer my questions...
lean otter
oh less than six
i am currently thinking on my own without your prompts but i still have no idea what to do
this problem feels hard
if it has a two, then it has to have 3, 5 or 7 in it
now check what if 2 is not in the triple
quasi bison
if it has a two, then it has to have 3, 5 or 7 in it
this statement could be made more precise.
lean otter
now check what if 2 is not in the triple
how?
quasi bison
you would have done much better to say "if the triple includes 2, then the only other numbers that could go in the triple are 3, 5 and 7" and then verified that (2,3,5), (2,3,7) and (2,5,7) are all valid triples and there are no others that include 2.
quasi bison
how?
if your triple doesn't contain 2, what can you say about all three numbers in it?
i mean, one thing you could say is "i am once again choosing not to follow any of your prompts" of course
lean otter
if your triple doesn't contain 2, what can you say about all three numbers in it...
they are greater than two
lean otter
but what is special about **primes** that are greater than 2?
they are odd
quasi bison
correct
so given that (p,q,r) is a triple of odd primes with p < q < r which satisfies the requirement of the problem, what can you say about r - p?
lean otter
so given that (p,q,r) is a triple of odd primes with p < q < r which satisfies t...
r-p<6
lean otter
but in light of what you said above, it can be refined
r-p=2, 3, or 5
quasi bison
i would like you to think about what you just said
lean otter
sorry, i mean 2 or 4
unique bison
r and p are odd
quasi bison
think about what the difference between two distinct odd numbers can be
think about why i would've mentioned q
lean otter
think about what the difference between two _distinct odd numbers_ can be
ah i see
if the difference were two, than q would have to be in between
making it even
which is not possible
quasi bison
yes
so in fact all triples that don't include 2 are necessarily of the form (p, p+2, p+4)
lean otter
so in fact all triples that don't include 2 are necessarily of the form (p, p+2,...
now we have to find the ones that would work?
quasi bison
there is an argument to be made here involving divisibility by three
it will help cut down the search space significantly
lean otter
there is an argument to be made here involving divisibility by three
the 3k, 3k+1, 3k+2 thing?
quasi bison
i don't know what you mean by that
lean otter
i don't know what you mean by that
is this just a bunch of trial and error now
quasi bison
no
not trial and error by any means
you need only notice that among the numbers p, p+2 and p+4, no matter what p is, there will always be exactly one number that is a multiple of 3.
lean otter
you need only notice that among the numbers p, p+2 and p+4, no matter what p is,...
ah right
thanks
.close
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runic crag
\frac{x\left(4x+1\right)}{x+4}\le1
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\frac{x\left(4x+1\right)}{x+4}\le1
runic crag
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✅
lean otter
what is the domain of x
runic crag
It doesn't matter, the questions asks how many real whole number solutions there are
I'm getting either -1 and +1 or everything below -1 depending on how I do it
lean otter
well it can't be because everything below -1 contains -4, which is impossible
runic crag
This is what I did first
Then I was told that I should set up a inequality sign thingy, so I watched some video about that
So I did this
lean otter
it's not safe to do cross product when manipulating inequalities, if x is negative, then multiplying by x in the LHS inverses the inequality
runic crag
it's not safe to do cross product when manipulating inequalities, if x is negati...
I noticed this when I put it Into Desmos, mutliplying by x completely changes the answer
lean otter
I noticed this when I put it Into Desmos, mutliplying by x completely changes th...
yes, this makes sense
in the end when you get a/b < 0
this means a and b have opposite signs
runic crag
yes, this makes sense
So is it still the way you’re supposed to do it?
lean otter
So I did this
I think you should avoid cross product, just do like you did here
runic crag
Oh, I should just do -1/x on both sides?
lean otter
Oh, I should just do -1/x on both sides?
yes
runic crag
Ahh, very smart
lean otter
this way you can't make a mistake
runic crag
Ok, should be able to solve this ez now
lean otter
runic crag
Haha noice
Slightly unrelated, is there a way to remove x terms from a denominator if it says x+4 for example?
safe radishBOT
@runic crag Has your question been resolved?
runic crag
Ok nevermind I did not in fact solve it easily
I still get the same thing
X^2 = 1
lean otter
how do you get from the fourth to the fifth line ?
lean otter
do you want to solve an inequality or an equality
runic crag
inequality
lean otter
then you can't set it =0
runic crag
runic crag
Well I thought I could find those intervalls by finding were x=0
lean otter
Well I thought I could find those intervalls by finding were x=0
nope that gives you individuals values, not intervals. here you have a ratio <=0, this means the numerator and the denominator have opposite signs
either $4x^2-4 \leq 0$ and $x(x+4) \geq 0$ or the opposite
runic crag
nope that gives you individuals values, not intervals. here you have a ratio <=0...
Yes but those individual values should be the borders of the intervalls, that's what the kind khan academy dude taught me
flat frigateBOT
AimaneSN
lean otter
Yes but those individual values should be the borders of the intervalls, that's ...
not true in general I think
lean otter
So do I have to test both cases?
yep, the would give separate interval solutions
the final solution would be the union of these two cases
runic crag
Damn, is there no way to get rid of the x in the denominator
runic crag
That cat doesn't look very happy
lean otter
haha
runic crag
but working with inequalities isn't hard
Yeah, just haven't been taught much of it in school unfortunately
Just some very basic stuff
lean otter
Yeah, just haven't been taught much of it in school unfortunately
I see, it's better to start with easier inequalities
this one is a bit messy
runic crag
I did easier ones first, in the khan video
lean otter
like try to solve $x(x+1) \geq 0$ for example
flat frigateBOT
AimaneSN
lean otter
I did easier ones first, in the khan video
Ah I see
anyway you're free to ask about anything you're stuck at
runic crag
**AimaneSN**
x can be anything except between -1 and 0 right
lean otter
x can be anything except between -1 and 0 right
yep, which interval is this ?
I mean union of intervals
runic crag
(-infinity, -1) U (0, infinity)
Umm ok, I still did something wrong but it’s looking promising
Oh it can't be -4 either I guess, since at the beginning the denominator was (x+4)
Think I got it right this time
.close
grim kraken
Hello
grim kraken
What is this formula?
This is the entire thing
but i want to know this formula
grim kraken
here
The first part Ty = T . sen 45
I understand
but the rest
2,5 is the distance
ok
but what is the formula? anyone knows the name?
$T.sen45.2,5-64.1,25=0$
flat frigateBOT
Bioleve
grim kraken
64 is kilograms
2,5 is distance
what is 1,25?
1,25 is half of 2,5
please
hello
marsh walrus
i wish i could read this 
safe radishBOT
@grim kraken Has your question been resolved?
grim kraken
sorry
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@grim kraken Has your question been resolved?
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@grim kraken Has your question been resolved?
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tawdry salmon
What is the latex for Summation?
safe radishBOT
What is the latex for Summation?
sour quiver
$\sum_{n=1}^{\infty}\frac{1}{2^n}=1$
flat frigateBOT
Dex
sour quiver
idk bro
tropic nymph
It is a GP
tawdry salmon
What’s that
tawdry salmon
No
tropic nymph
Could you show me your work?
tawdry salmon
I’m learning series in algebra 2
trying to put it into latex in my notes
@tropic nymph
tropic nymph
So, I guess you only want to expand the summation notion right?
tawdry salmon
I think (I just want to put the notation with the three numbers around it)
tropic nymph
You can expand it like $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}$..... = 1
flat frigateBOT
Trystnest
tawdry salmon
do u think theres a problem with my latex application because they aren’t in the correct place?
tropic nymph
You forgot the \frac in your latex expression
tropic nymph
This looks good to me
tawdry salmon
This looks good to me
why isn’t the infinity at the top?
tropic nymph
It is a different notation, but it means the same thing
tawdry salmon
Is there a way I can get the other notation or is it up to the app’s latex?
tropic nymph
It is mostly up to the app
broken nimbus
in 3d coordinates [x,y,z] does x represent the distance or y?
safe radishBOT
in 3d coordinates [x,y,z] does x represent the distance or y?
nova creek
All three represent a distance from the origin in a certain direction
broken nimbus
All three represent a distance from the origin in a certain direction
right but if im talking about horizontal distance what would that be?
spice grove
Conventionally x. Then again it depends on what you assume it to be.
nova creek
Dunno. It depends on the orientation of the coordinate system
Plus, couldn't both x and y be horizontal? If we're orienting it so that the z axis points straight up
broken nimbus
for e.g a person throws a frisbee at a force of [3,9,8] and at a distance of [4,8,4] and i want to find the work in the horizontal distance
so that would be [3,0,0] · [4,0,0]?
nova creek
Horizontal is too general, I think. I would use more precise wording, such as "in the x direction"
broken nimbus
Horizontal is too general, I think. I would use more precise wording, such as "i...
true, so would it still be what i showed?
[3,0,0] · [4,0,0]
bcuz on the actaul plane, the y takes the x position
nova creek
I believe so, if it's in the x direction
broken nimbus
so idk if its that or [0,9,0]·[0,8,0]
nova creek
Post the original question
broken nimbus
a person throws a frisbee at a force of [3,9,8] and at a distance of [4,8,4] and i want to find the work in the horizontal distance
nova creek
Perhaps it's [3, 9, 0] • [4, 8, 0], because both the x and the y will be horizontal if the z axis points up
broken nimbus
here it looks like y is x
nova creek
Nah, x is just jutting out from the screen. It's just hard to depict that in 2d
nova creek
The force and distance in the horizontal direction is just the force and distance without the vertical direction, aka, without z
broken nimbus
but if im only asking for the distance than what would it be?
nova creek
Well, the distance vector would be [3, 9, 0]
Because thats the distance vector projected on the xy plane
broken nimbus
but when im asking "determine the work in the horizontal axis" or "x direction" woudnt that be just one of the axis and not two?
nova creek
There is no "horizontal axis". There's an x, y, and z axes. If it's oriented with z vertical, then both x and y are horizontal
If it's asking for "x direction" that's a different story, but your question was asking for horizontal direction
broken nimbus
There is no "horizontal axis". There's an x, y, and z axes. If it's oriented wit...
so basically i made that question and i also have to answer it so idk what type of question would be best to ask if im trying to find the work only for the distance
how would i word it better
nova creek
Specify an axis. Either x, y, or z
broken nimbus
well, i saw another question where it was asking the work done by gravity
which would be the negative z axis
so is there another word for the x axis
nova creek
Gravity points downward, which is negative z
Upward and downward is pretty set in stone, most people can agree
But horizontal axes? Left and right? That's relative to the person and their orientation
broken nimbus
what if i draw a diagram so its clear for all
nova creek
I still see no reason why you wouldn't just say "along the x axis" or "along the y axis" if your set on just one axis. Saying anything else seems like it'll just lead to unnecessary confusion
broken nimbus
I still see no reason why you wouldn't just say "along the x axis" or "along the...
ok lets say i change it to along the x axis, than what would the equation look like
broken nimbus
Perhaps it's [3, 9, 0] • [4, 8, 0], because both the x and the y will be horizon...
this? or the one i showed
nova creek
The vectors along the x axis are [3, 0, 0] and [4, 0, 0]
So it'd be the dot product of those
broken nimbus
right and if i say the horizontal than it would be [3, 9, 0] • [4, 8, 0],
broken nimbus
The vectors along the x axis are [3, 0, 0] and [4, 0, 0]
which one do u think would be more efficient
broken nimbus
Using just one axis has easier computation, I guess
in terms of how the student reads the question. im making the question as if it was a test question
nova creek
Idk. Saying "horizontally" might still be ambiguous
broken nimbus
true
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Available help channel!
buoyant cipher
"express each expression without the use of radicals"
safe radishBOT
"express each expression without the use of radicals"
plucky elk
Yes
buoyant cipher
am i right then?
buoyant shadow
looks great
kind dove
safe radishBOT
kind dove
any idea how to start integrating term by term ?
obtuse jackal
it's almost the second derivative of x^(n+2)
obtuse jackal
I don't know if this can work well to compute the series, but if you then factor one x out, you have n x^n which is an easy integral. Then you have a function instead of a sum
obtuse jackal
can you send an image ? It'd be much easier to understand if I can see
obtuse jackal
they do end up integrating twice, it just looks slightly different due to the order in which it's done
obtuse jackal
just integrate f. Then factor x² out. Then call the other factor g(x)
a much more natural order of operations for the same result
kind dove
ok, thanks i will try it
kind dove
just integrate f. Then factor x² out. Then call the other factor g(x)
got it, thanks
.close
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Available help channel!
vernal gorge
safe radishBOT
vernal gorge
ive got till (x+5)^2 +(y-7)^2 +k^2-64=0
lean otter
r²>0
safe radishBOT
@vernal gorge Has your question been resolved?
forest iris
How do I decompose this into prime fractions
safe radishBOT
How do I decompose this into prime fractions
forest iris
The whole question looked like this:
answer was apparently
"False"
not sure how to prove it
safe radishBOT
@forest iris Has your question been resolved?
safe radishBOT
safe radishBOT
Available help channel!
uncut tiger
Discs of radius 4cm are cut from a rectangular plastic sheet of length 84 cm and width 24 cm. How many complete discs can be cut out?
Find :- The total area of the discs cut
and The area of the sheet wasted
safe radishBOT
Discs of radius 4cm are cut from a rectangular plastic sheet of length 84 cm and...
tribal musk
Pls help...
worthy hemlock
Pls help...
You were a bit late
tribal musk
dude my wifi is working so slow
worthy hemlock
Discs of radius 4cm are cut from a rectangular plastic sheet of length 84 cm and...
Find the area of the circle and the area of the rectangular sheet
maiden mural
Pls help...
make another channel lol, and this time maybe select the second one from the available channels
uncut tiger
Find the area of the circle and the area of the rectangular sheet
rectangular sheet = 2016
circle = 16 pi
worthy hemlock
Dividing the two values, you can determine how many discs you can make
uncut tiger
Dividing the two values, you can determine how many discs you can make
answer comes 40.10
but book answer says the answer should be 30
worthy hemlock
but book answer says the answer should be 30
Maybe the book could be wrong?
uncut tiger
Maybe the book could be wrong?
i highly doubt that
I found a solution online but it makes no sense whatsoever
worthy hemlock
Books, can have tons of typos
Ask your teacher, maybe your teacher can help you out
That was the proper process
uncut tiger
true, i'll ask him tmrw
primal lark
true, i'll ask him tmrw
Well you have to lay the discs down in a grid pattern to maximize the number of discs
You'll never get 100% packing efficiency
So that's why your answer does not match
uncut tiger
Well you have to lay the discs down in a grid pattern to maximize the number of ...
i figured it out
no worries
.close
wicked night
safe radishBOT
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@wicked night Has your question been resolved?
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@wicked night Has your question been resolved?
wicked night
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muted wolf
safe radishBOT
muted wolf
i dont know how to do this
sonic basalt
okay so
thats a transformation
mapping from ABCD to A'B'C'D
what do u notice happens with the x and y values
like B goes from (1,6) to (0, 3) and C goes from (8, 6) to (4,3)
any scale factors?
primal lark
like B goes from (1,6) to (0, 3) and C goes from (8, 6) to (4,3)
(1/2,3)*
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rustic knoll
Can anyone help me learn how Khan's Academy solved this?
safe radishBOT
Can anyone help me learn how Khan's Academy solved this?
rustic knoll
I understand that 7p^4 is the greatest common factor of 49p^8 and 42p^4 but I don't understand how they got to (7p^4 +3)^2
regal girder
$a = 7p^4, b = 3$
then we have
$$a^2 + 2ab + b^2 = (a + b)^2$$
flat frigateBOT
Doggo
rustic knoll
so the formula for this is a^2 + 2ab + b^2?
flat frigateBOT
Doggo
rustic knoll
how to factor a trinomial
regal girder
it really depends
distant trench
grouping works most times
regal girder
$$x^2 - x - 2$$ is also a trinomial but is factored as $(x - 2)(x + 1)$ instead
flat frigateBOT
Doggo
regal girder
so it doesn't have to be $a^2 + 2ab + b^2$
flat frigateBOT
Doggo
distant trench
that's only for perfect squares
rustic knoll
how do you know which way to do it?
regal girder
depends on question I guess? sometimes you try a few ways and end up finding the right one
sometimes it's easier to see
young osprey
how to prove that
safe radishBOT
how to prove that
young osprey
$\gcd(4^n,5^n)=1$
flat frigateBOT
megi-san
light shoal
$\gcd(4^n,5^n)=1$
look at the prime factorizations of 4^n and 5^n and see if they have any prime factors in common?
regal girder
Not rigorous but
$$4^n = 2^{2n}$$ which has no factor of 5
flat frigateBOT
Doggo
young osprey
look at the prime factorizations of 4^n and 5^n and see if they have any prime f...
i have to do it for every n
regal girder
i have to do it for every n
$$n \in \mathbb N$$?
flat frigateBOT
Doggo
young osprey
yes
regal girder
**Doggo**
then you can just do this to get the prime factors
there's only 2's
for the other one, there's only 5's
light shoal
i know but how to prove it with math
induction?
light shoal
example argument:
young osprey
that $\gcd(4^n,5^n)=d$
flat frigateBOT
megi-san
young osprey
the prove that d=1
light shoal
now assume it's true for a given n
then 4^(n+1) = 4 * 4^n = (2*2) * 4^n
and since 4^n only has 2 as a prime factor (because of the induction hypothesis), therefore so does 4^(n+1)
then do the same thing for 5^n, show that it only has 5 as a prime factor
young osprey
**megi-san**
man the teacher want us to do it this way
safe radishBOT
@young osprey Has your question been resolved?
safe radishBOT
young osprey
.reopen
safe radishBOT
✅
young osprey
$\gcd(4^n,5^n)=d$
flat frigateBOT
megi-san
young osprey
prove that d =1
safe radishBOT
@young osprey Has your question been resolved?
teal cape
if 2x^2-4x-5=0, solve x1^3+x2^3
safe radishBOT
if 2x^2-4x-5=0, solve x1^3+x2^3
teal cape
i used viète but it just gives me left over
fading raft
So
I am assuming again you cannot just find the roots
and then cube them?
You have to use vieta?
teal cape
i don’t see another way
fading raft
Hint: Consider $(x_1+x_2)^3$
flat frigateBOT
azeem321
teal cape
oh, so it would be (x1^3+x3^3)^3= 0
i’m wrong, that doesn’t seem right
it should be
(3x^2*3 + (-4^3))^3 - 5 = 0 but this seems wrong as well
lean otter
expand it and see what it results
lean otter
what did u get
teal cape
-2.702056333
lean otter
no when u expand
lean otter
u should get something like $x_1^3+$ ... $+x_2^3$
flat frigateBOT
Jester
teal cape
dunno, i don’t think we are meant to expand
fading raft
You remember binomial expanasion for $(a+b)^3$?
flat frigateBOT
azeem321
fading raft
Now let $a=x_1$ and $b=x_2$
flat frigateBOT
azeem321
teal cape
that’s what i did
teal cape
it should be
(3x^2*3 + (-4^3))^3 - 5 = 0 but this seems wrong as well
and this should be it
however that is wrong
flat frigateBOT
azeem321
teal cape
(2x^2 + (-4x)) ^3
there’s nothing to multiply in front of the brackets
i just replaced a with 2x and b with -4x
fading raft
bruh
we keep giving u instructions
but u keep ignoring
expand out (a+b)^3 in terms of a and b
fading raft
$(a+b)^2=a^2+2ab+b^2$
flat frigateBOT
azeem321
fading raft
Now find $(a+b)^3$
flat frigateBOT
azeem321
teal cape
oh, should be
$(2x^2)^2 + 2(2x^2 * (-4x)) + -4x^2$
flat frigateBOT
brenny
teal cape
is that right so far
lean otter
where are you getting 2x, -4x?
fading raft
🤦♂️
teal cape
if 2x^2-4x-5=0, solve x1^3+x2^3
.
fading raft
i give up
teal cape
read the first bit
lean otter
what is $(x_1+x_2)^3$
flat frigateBOT
Jester
teal cape
sorry but like x1^2 + x2^3 isn’t a + b
what am i even meant to do with x1^2 + x2^3 if they don’t have values
i just replace those with the value and then solve it
teal cape
Can you see what I have done here
can you see that these have values
what am i meant to do other than assign them values
and then solve the problem
fading raft
can you see that these have values
we will get to the that stage
but for now
lets expend out
to make sure you've done it properly
lean otter
$(x_1+x_2)^3 = x_1^3 + 3x_1^2 x_2 + 3x_1 x_2^2 +x_2^3 = x_1^3+x_2^3+3x_1 x_2(x_1+x_2)$
flat frigateBOT
Jester
teal cape
to make sure you've done it properly
i’m lost. if expand then what do i do
you expanded this already though
fading raft
do u understand what jester wrote above
teal cape
yeah of course
but right now i don’t see how that fits into the problem
the way we did it in class is different
fading raft
So your goal is to find $x_1^3+x_2^3$
flat frigateBOT
azeem321
fading raft
But from what Jester wrote above, we can rewrite it as \$x_1^3+x_2^3=(x_1+x_2)^3-3x_1x_2(x_1+x_2)$
flat frigateBOT
azeem321
fading raft
Do you get this?
fading raft
From vieta, what is $x_1+x_2$ and $x_1x_2$
flat frigateBOT
azeem321
fading raft
2x^2-4x-5=0
teal cape
yep
fading raft
what is the value of $x_1+x_2$ and $x_1x_2$
flat frigateBOT
azeem321
lean otter
$x_1$ and $x_2$ are the roots of your equation
flat frigateBOT
Jester
teal cape
sorry but i need to go over it with my prof, can i just get the answer for the question?
teal cape
i just don’t get what you’re saying
you’re talking about some expansion when you’ve already expanded
teal cape
**Jester**
and then this guy tells me the exact same thing i’ve been doing for 10 minutes
if it’s the root then x1 and x2 can be found in the equation
fading raft
You clearly don't understand this topic. Or perhaps there might some significant gaps in your knowledge. My recommendation to you is watch some videos on youtube.
try this
teal cape
does this perhaps tie into the quad formula?
i’m so tired to understand anything
i’ll figure it out in the morning
.close
dense hedge
