No my original equation was
#help-0
hallow verge
I forgot them
wind cloak
Can you add 10 to both sides of that equation
hallow verge
Yes
wind cloak
What would you have
hallow verge
The original equation?
wind cloak
4x + y-10 = 0
This one yes
wind cloak
Yeah
hallow verge
4x + y =10
hallow verge
Yea
wind cloak
What would you have
hallow verge
Y=-4x+10
wind cloak
bingo that is the slope intercept form
hallow verge
How do I turn this into a sketch graph
hallow verge
10
wind cloak
yes mark it on your paper
lone heartBOT
y = -4x + 10
y= -4(1) + 10
y= -4 + 10
y= 6
As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.
wind cloak
Do you know what y = -4x + 10 represents
hallow verge
No
I don't get how not that I have y=6 this help my sketch a graph
Of the original equation
wind cloak
what
the goal here is to get two points that lie on the line
You can then connect them and you will have your line
hallow verge
Yea
hallow verge
Oh ok
Can I keep drawing the points
Using rise over run?
To have more than 2 points
?
wind cloak
Yes 10 is on the y axis
hallow verge
You have the points (0, 10), and (1, 6)
But
wind cloak
That's what (10, 0) means, the x coordinate is 0
wind cloak
I'm sorry for that again
wind cloak
God nacht
hallow verge
.close
undone ledge
can we do mod x derivative by defination
lone heartBOT
can we do mod x derivative by defination
wind cloak
sure
paper mango
step two?
undone ledge
like just formed the limit , and done know how to proceed
like i wrote it in the form of limit
oo wait i can try to use conjuate
undone ledge
are you trying to find the derivative at x=0 or just in general
general
vale wigeon
right
granite badger
can we do mod x derivative by defination
|x| is differentiable at all points except 0 because there is a cusp point
vale wigeon
consider three cases separately.
granite badger
it just does not exist
vale wigeon
for x > 0, |x| = x and its derivative is 1 as you will quickly find out
for x < 0, |x| = -x so its derivative is -1
undone ledge
oh just like we do integral of it
vale wigeon
and yeah, |x| is nondifferentiable at 0. its derivative at 0 doesn't exist.
undone ledge
in integral of mod x we also consider casses right-
undone ledge
and yeah, |x| is _nondifferentiable_ at 0. its derivative at 0 _doesn't exist_.
noted thanks
vale wigeon
in integral of mod x we also consider casses right-
the consideration of cases doesn't have anything to do with integration or differentiation
it is simply a fact about the function itself
no it isn't
granite badger
nvm
undone ledge
the consideration of cases doesn't have anything to do with integration or diffe...
right got it .
sorry about that
this question.
winter light
this question.
You'll have to consider separately the cases for $x \to 0^-$ and for $x \to 0^+$
ocean sealBOT
Alberto Z.
zealous lichen
|x| is not differentiable at x=0 but it doesn't mean all functions with |x| is not differentiable
winter light
|x| is not differentiable at x=0 but it doesn't mean all functions with |x| is n...
Yep, example y = |x|² since it is equivalent to x²
undone ledge
Yep, example y = |x|² since it is equivalent to x²
right yeah thats understandable
can i just do this perhaps?
is this ok
undone ledge
oo wait i can try to use conjuate
took the conjuate
zealous lichen
it's ok
you still need to take $x\to 0^-$ and $x\to 0^+$ to solve for $\frac{x}{|x|}$ though
ocean sealBOT
WhereWolf
alpine sable
Let (f :\mathbb{Z} \rightarrow \mathbb{Z}) where (f(x) : x^2 + x) and let (g : \mathbb{Z} \rightarrow \mathbb{Z} ) where (g(x) = 4x + 3). \
(a) Write down ((f \circ g (x))) and (g \circ f(x)) as expanded polynomials. \
(b) Determine the ranges of ((f \circ g(x))) and ((g \circ f(x))).
ocean sealBOT
scholablade
alpine sable
For both functions we have the signature (\mathbb{Z} \rightarrow \mathbb{Z}) so we can do function composition we will start with (f \circ g), let (f \circ g : \mathbb{Z} \rightarrow \mathbb{Z}) where (g \circ f = (4x + 3)^{x^2 + x}) and let ((g \circ f : \mathbb{Z} \rightarrow \mathbb{Z}) where (f \circ g = (x^2 + x)^{4x + 3}). \
Since we have (\mathbb{Z} \rightarrow \mathbb{Z}) for both function means that without looking at the definitions we have at maximum (\mathbb{Z}) and via the definitions we can deduce that the range of the functions are (]-\infty, +\infty [).
ocean sealBOT
scholablade
rigid smelt
That is not what function composition is. f\circ g, means that you will replace every x in f(x) with g(x)
what you are doing instead is raising f to the power of g, i.e f(x)^g(x)
alpine sable
Okay, am gonna rewrite
What do you think now?
For both functions we have the signature (\mathbb{Z} \rightarrow \mathbb{Z}) so we can do function composition we will start with (f \circ g), let (f \circ g : \mathbb{Z} \rightarrow \mathbb{Z}) where (g \circ f = 4(x^2 + x) + 3) and let ((g \circ f : \mathbb{Z} \rightarrow \mathbb{Z}) where (f \circ g = (4x + 3)^2 + (4x + 3)). \
Since we have (\mathbb{Z} \rightarrow \mathbb{Z}) for both function means that without looking at the definitions we have at maximum (\mathbb{Z}) and via the definitions we can deduce that the range of the functions are (]-\infty, +\infty [).
ocean sealBOT
scholablade
alpine sable
@rigid smelt
rigid smelt
ok much better. But they require both to be expressed as expanded polynomials so you will need to do a few more work.
Anyway, moving to the next issue. You claim that the range of both functions are (-infty, infty). Can you find an x such that f(g(x))=sqrt(2)?
I know this is probably what you are not saying, but what you are implying is that f(g(x)) outputs every real numbers
alpine sable
True, will fix it
rigid smelt
ok, and beyond that issue, if you are trying to say that the range is x\in Z then this is also wrong. Because there is no x such that f(g(x))=-1
alpine sable
For polynomial
For both functions we have the signature (\mathbb{Z} \rightarrow \mathbb{Z}) so we can do function composition we will start with (f \circ g), let (f \circ g : \mathbb{Z} \rightarrow \mathbb{Z}) where (g \circ f = 4(x^2 + x) + 3) or as expanded polynomials (4x^2 + 4x + 3) and let ((g \circ f : \mathbb{Z} \rightarrow \mathbb{Z}) where (f \circ g = (4x+ 3)^2 + (4x + 3)) or as expanded polynomial (16x^2 + 28x + 12) \
ocean sealBOT
scholablade
rigid smelt
seems good. You also should just expand 4(x^2+x) as well. Either way, this is correct
oh nvm, didn't see the bit after "or as expanded polynomials"
alpine sable
What is the process of finding the range?
Aam really having trouble with this concept.
rigid smelt
Are you having trouble with finding the range of these compositions specifically or you just don't know how to find the range of f and g at all
Since the range of f and g is very important when determining the range of their compositions
alpine sable
Are you having trouble with finding the range of these compositions specifically...
Range at all without using the signature
By signature i mean (f : \mathbb{N} \rightarrow \mathbb{N})
rigid smelt
Let's just start with the easy one first, can you determine the range of g(x)?
ocean sealBOT
scholablade
alpine sable
**scholablade**
Something like this
rigid smelt
Let's just start with the easy one first, can you determine the range of g(x)?
basically, what I'm asking you to do is to solve for 4x+3=y
rigid smelt
right, now can you describe in words what the range would be?
meaning what are all values of y such that x is an integer?
rigid smelt
sure
alpine sable
I'l look for others
rigid smelt
don't, just notice here that y-3 has to be divisible by 4
and hence that implies y belong to a set of some odd integers
rigid smelt
ok, but that would not help us too much, here is how you can describe it in detail
since we know y is some odd integers, we know that y=2k+1, for k in Z. So that means we will have that x=(k-1)/2
now we can say that y=2k+1, for odd k
meaning, for any odd k, this will give us a value y that will be in our range of g(x)
this is usually how it goes for determining range of functions over N or Z. However, you have to remember that not all functions can be described explicitly, meaning not all functions can be described using just notations
anyway, can you do the same thing for f(x), and find a way to describe its range?
alpine sable
Okay will do that
lone heartBOT
@alpine sable Has your question been resolved?
rigid smelt
yes, but this is something you should notice, you have factored it into y=(x+1)x
what this is saying is that y is always even
can you tell why?
azure frigate
A point is such that (abscissa of the point, other than zero) that it equals to the ordinate of the point. In which quadrants can the point lie?
rigid smelt
alpine sable
This room is occupied by me currently, there is a guide in #❓how-to-get-help on how to claim your own room :) @azure frigate
azure frigate
ah I am new
alpine sable
It's fine
azure frigate
btw thanks
alpine sable
No problem
I see it now
It's because if x is odd we add 1 to it and it becomes even multiplied by another even number
And if x is even x adding one becomes odd but by multiplying an odd number with another, it becomes even
Example
if x is even
2(2 + 1)
2 * 3 = 6
if x is odd
1(1 + 1)
if x is odd
1(1 + 1)
2
rigid smelt
yes pretty much, product of two consecutive intgers has to be even
however, one more conclusion that we can pull out of this is that our range does not contain any even perfect square
so numbers like 4, 16, and so on
anyway, so our range for f is some positive even integer that can be expressed as a product of two consecutive integers.
now, by knowing these two ranges, can you determine something about g(f(x)) first?
alpine sable
however, one more conclusion that we can pull out of this is that our range does...
How did you conclude that?
rigid smelt
because like I have have said, product of two consecutive integers
perfect squares aren't products of two consecutive integers
alpine sable
Ah true
alpine sable
now, by knowing these two ranges, can you determine something about g(f(x)) firs...
Okay, will do that
rigid smelt
I'm sorry, I have to go now. Anyway, for g(f(x)), you should be able to find that the range are some positive odd integers, this is the best that you can describe the range. For f(g(x)), you will see that it will be some even positive integers, again, this is the best that you can describe the range.
You can perhaps prove that the range of g(f(x)) is either prime or is divisible by 3, but I haven't think of a way to prove this formally, nor disprove it either.
if you can't find what I have found, you can ask for others' help
late drum
can someone pls verify that the function in the image is this heaviside function: t* (u(t-2) - u(t-4))
lone heartBOT
can someone pls verify that the function in the image is this heaviside function...
@late drum Has your question been resolved?
steady mantle
lone heartBOT
@late drum Has your question been resolved?
candid swan
lone heartBOT
lone heartBOT
Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
lone heartBOT
<@&286206848099549185>
Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
@candid swan Has your question been resolved?
lone heartBOT
@candid swan Has your question been resolved?
patent swallow
in point 1b, how can I find the intersection of A and C?
lone heartBOT
in point 1b, how can I find the intersection of A and C?
tardy stag
quizás podés dibujar algo para ayudarte
patent swallow
quizás podés dibujar algo para ayudarte
But dont I need to find an equation for both of them
tardy stag
yeah
patent swallow
like one that represents both
patent swallow
yup
tardy stag
what does A look like, graphically?
patent swallow
I cannot plot all those infinite points
tardy stag
if you were to take the coordinate plane and shade in the area that A represents though
lone heartBOT
Available help channel!
ivory fjord
Can anyone explain to me how to do #18 and #19?
lone heartBOT
Can anyone explain to me how to do #18 and #19?
@ivory fjord Has your question been resolved?
Available help channel!
alpine sable
Help
alpine sable
How do I find the number
vale wigeon
well you might want to find the fraction first, and then adjust it so that it has the right number at the top or bottom (wherever it is known)
for example, 1/4 + what = 1?
dull dove
There's no way this dude's over the age of 13
paper mango
@alpine sable are you still present
vale wigeon
There's no way this dude's over the age of 13
benefit of the doubt.
also she has a she/her role icon.
dull dove
My bad I don't really check people's profiles
alpine sable
There's no way this dude's over the age of 13
Im 16 but This is what we learn in sweden😭
alpine sable
for example, 1/4 + what = 1?
Ok I kinda get it now ill try to solve it
vale wigeon
My bad I don't really check people's profiles
it's the role icon right next to the name
alpine sable
Wait is a 5?
dull dove
Ok I kinda get it now ill try to solve it
For you to add a fraction to 1/4 to make 1/1, what fraction do you need?
dull dove
it's the role icon right next to the name
I'm colorblind
vale wigeon
Wait is a 5?
no
vale wigeon
I'm colorblind
you can mouse over or tap it
alpine sable
I dont get it
dull dove
I dont get it
For you to add a fraction to 1/4 to make 1/1, what fraction do you need?
dull dove
you can mouse over or tap it
I didnt even see it in the first place but I'll take note for future circumstances
alpine sable
For you to add a fraction to 1/4 to make 1/1, what fraction do you need?
1/8??
vale wigeon
no
do you know how fractions work?
do you know how to add and subtract fractions?
1/4 + 1/8 isn't anywhere near 1
vale wigeon
@dull dove do you mind taking this over so i can sleep
dull dove
No problem
@alpine sable you have this
Three of the parts aregone
And there's only 1 part
alpine sable
2/8?
dull dove
How many do you need to complete it
How many parts do you need to complete the circle if there's only 1 par
dull dove
Okay yes you need 3 parts
That's just the basic part
So if you were to write this ot
dull dove
We are not talking about that for now
Just focus on the circle
It's 3/4
We will get to your question later
alpine sable
Yes but I dont know in This case
dull dove
Okay so what fraction will you have that simplifies into 3/4 and has 8 as the numerator
Denominator*
alpine sable
Multiplication?
lone heartBOT
@alpine sable Has your question been resolved?
shell nimbus
hello all! would like to request your help on my research report regarding quadratic fractions:
paper quartz
Im gonna fucking fail my math exam 😭
Can someone come in call and teach me smth ppeaeeeeee
shell nimbus
hello all! would like to request your help on my research report regarding quadr...
was wondering if the numerator would also be multiplied by the Qa or only the denominator would be affected for this case?
finite flax
was wondering if the numerator would also be multiplied by the Qa or only the de...
what happened to your denominators
and why don't you have quadratic terms in your working
* a quadratic term
shell nimbus
ah its economics but let me ss the full portion
but was confused if the calculation / working is correct or na
finite flax
$500Q_A - \frac{{Q_A}^2}{200} - \frac{Q_A X}{200} - 200Q_A$
ocean sealBOT
Disorganized
finite flax
$300Q_A - \frac{{Q_A}^2}{200} - \frac{Q_A X}{200}$
ocean sealBOT
Disorganized
finite flax
$-\frac{1}{200}{Q_A}^2 + \left(300 - \frac{X}{200}\right)Q_A$
ocean sealBOT
Disorganized
shell nimbus
ahhh okay i got it now thank you so much though!
finite flax
are you differentiating with respect to X or Q_A?
shell nimbus
X 🙂
finite flax
$\pi_A = -\frac{Q_A}{200}X +\left( 300Q_A - \frac{{Q_A}^2}{200}\right)$
ocean sealBOT
Disorganized
safe tartan
For definite integrals, for let’s say x^5 from 0 to 1 equal the definite integral from -1 to 0 for x^5+1
safe tartan
Isn’t it an odd function
fresh parcel
yes but not all functions are odd
wait wait wait hold up i think im doing something wrong
forget what i said about the odd function stuff
safe tartan
Ok
fresh parcel
but for odd functions
fresh parcel
yeah
safe tartan
Not vertical translations
safe tartan
Yeah
fresh parcel
in that case it works for any function
safe tartan
Right
fresh parcel
because you're essentially just shifting the entire graph
ocean sealBOT
safe tartan
Oh I thought it didn’t exist
What I’m tryna ask is that can we make use of this to find the definite integrals of integrals which don’t exist
fresh parcel
what do you mean by integrals that don't exist
safe tartan
Ones that can’t be written with standard notation
fresh parcel
non elementary integarls
safe tartan
Yeah I think that’s what it’s called
Or even the quintic one I sent above which looks very long
fresh parcel
probably not, because its still the same portion youre concerned with
so therefore the integral is the exact same
safe tartan
For example integral from 0 to 1 of 1/x^5 is equal to the integral from -1 to 0 in 1/(x^5-1)?
Same area but x^5 is way easier
1/x^5
Oh wait from 1 to 2
I’m trippin so hard rn
lone heartBOT
@safe tartan Has your question been resolved?
safe tartan
.close
wheat lion
how can i find the value of tan(β)? Im studying for a test and i'm completely lost on this. I tried to use the Trigonometry Table but i wasnt able to find the answer
icy lodge
Use the rule tan(90 - A) = cot(A)
lone heartBOT
lone heartBOT
Available help channel!
vapid steppe
$\int{\frac{x²}{x²+1}}dx$
lone heartBOT
$\int{\frac{x²}{x²+1}}dx$
ocean sealBOT
Jash
vapid steppe
is it legal to like add and subtract 1/(x²+1)
proven leaf
is it legal to like add and subtract 1/(x²+1)
$\frac{x^2}{x^2+1}=\frac{x^2+1-1}{x^2+1}$ may be easier :)
coarse skiff
thats fine yeah
ocean sealBOT
MrFancy
proven leaf
oh wait same thingy
coarse skiff
good job
vapid steppe
so the integrand would be 1- 1/(x²+1)
limpid spade
Yes
vapid steppe
how would you do it normally
vapid steppe
so the integrand would be 1- 1/(x²+1)
wait isn’t this just what u get after long division
proven leaf
how would you do it normally
wdym by "normally"?
vapid steppe
$\int{\frac{x²-1}{x²+1}}dx$
coarse skiff
x=tantheta might help but not sure
ocean sealBOT
Jash
limpid spade
No
vapid steppe
this is a different problem
coarse skiff
this is a different problem
its the same integral as before
the first one
x^2/1+x^2
vapid steppe
oh wait yea
vapid steppe
so x-arctan(x)-arctan(x)+c
coarse skiff
yes
vapid steppe
or x-2arctan(x)+C
coarse skiff
x-2arctanx + C
vapid steppe
how would you find ∫1/x dx without knowing about logs
limpid spade
Ibp idk
vapid steppe
so differentiate 1/x integrate 1
coarse skiff
this is ho e arrive at antiderivatives for functions e can difffferentiate
but dont kno ho to integrate
yes
lone heartBOT
tardy stag
@coarse skiff something wrong with your w key?
lone heartBOT
Available help channel!
fathom wasp
Okay so
fathom wasp
I got a 200mg Star shaped edible
how many mgs are in a corner / tip /whatever you wanna call it
fathom grove
how big is a corner?
fathom wasp
I got you 1 sec
like regular stars
i've always been really bad at math but can you tell how much of the stars total % is a corner?
fathom grove
i’d guess 32 just based on size
lone heartBOT
remote heron
I'll go with that
no dont!
cant you weigh it?
200mg is too much to be guessing
if you cant weigh it just cut in towards the center from each spike
or between each spike
then you can create 5 40mg edibles
fathom wasp
That's a really good idea
remote heron
yea
fathom wasp
thanks mate
remote heron
you too take it easy
lone heartBOT
Available help channel!
vale storm
vale storm
ok
tacit arch
.close
clever dust
Given the set X = {x;x∉X}
Can i say that X is its one complement? Would it implie that X∩X = ∅?
Is so, would X∪X = U? U being the universe set
sour dove
I believe that's a form of the definition of the empty set. You're saying that the set X is defined as elements that are not in X. Thus, you wouldn't have any elements in it, so it's the empty set. Someone else correct me if I'm wrong but it would just imply that X = ∅
clever dust
You get the self reference problem, it is fun to think about
maiden glen
X = {x;x∉X} isn't a well-formed set under ZFC
sour dove
ah right
clever dust
Make sense
maiden glen
clever dust
Thats cool to learn
maiden glen
yes, set theory is very fun
clever dust
Do you know of any set of axioms that this is allowed?
maiden glen
not any standard ones at least
you really need some kind of restriction on what kind of predicates you allow, or you get russell's paradox
there are some logical systems that are paraconsistent and allow some contradictions, but I don't know of any off the top of my head
clever dust
Don't know by name, ill check it out
lone heartBOT
@clever dust Has your question been resolved?
primal galleon
How do I find the determinant of \begin{pmatrix}-1&2\ 2&-3\ -1&3\end{pmatrix}
ocean sealBOT
dgh
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
primal galleon
I've tried laplace expansion but I don't get the answer 11
sonic ridge
im pretty sure non square matrices dont have determinants?
primal galleon
,w determinant {{-1, 2}, {2, -3}, {-1, 3}}
ocean sealBOT
primal galleon
I've encountered a challenging problem while working on a matrix equation, and I'm seeking some guidance to proceed further. The problem involves finding the minimum least squares solution to the inconsistent system (Ax = b), where (A) is given by:
[ A = \begin{pmatrix}
-1 & 2 \
2 & -3 \
-1 & 3
\end{pmatrix} ]
and (b = \begin{pmatrix}
4 \
1 \
2
\end{pmatrix}). The goal is to solve the normal equation (\hat{x} = (A^T A)^{-1} A^T b).
I've been attempting to calculate (\hat{x}) but seem to have hit a roadblock. The matrices (A^T A) and (A^T b) are as follows:
[ A^T A = \begin{pmatrix}
-1 & 2 & -1 \
2 & -3 & 3
\end{pmatrix} \begin{pmatrix}
-1 & 2 \
2 & -3 \
-1 & 3
\end{pmatrix} = \begin{pmatrix}
6 & -11 \
-11 & 22
\end{pmatrix} ]
[ A^T b = \begin{pmatrix}
-1 & 2 & -1 \
2 & -3 & 3
\end{pmatrix} \begin{pmatrix}
4 \
1 \
2
\end{pmatrix} = \begin{pmatrix}
-4 \
11
\end{pmatrix} ]
primal galleon
How do they get 1/11 there?
sonic ridge
thats the 1/determinant of the matrix
wary stream
The shortcut to inverse of 2 x 2
sonic ridge
sicne the inverse of a matrix is 1/det(A) adj(A)
primal galleon
so how do i get 1/11
wary stream
The det of
primal galleon
that's the matrix A?
royal socket
It's the det of A^T A
lapis crow
yo i need help
lone heartBOT
yo i need help
lone heartBOT
i need help on 4
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
sonic ridge
do you know what the $\sqrt{,}$ is
ocean sealBOT
light
lapis crow
yes
sonic ridge
yup
sonic ridge
there is no question 5, do you mean 7?
sonic ridge
well, what do you think the operations are?
lapis crow
i wrote it is being subtracted by 9 💀
sonic ridge
yes thats a good start
lapis crow
yayy
sonic ridge
so x subtracted by 9 would get you x-9 right
lapis crow
alr bet
sonic ridge
so the operation is subtract 9, square the expression
lapis crow
how would I solve?
sonic ridge
altho at that point idk if thats necessary an operation on x. im not sure how pedantic ur teacher is
lapis crow
naw she checks for completion but i wanna undertand it
sonic ridge
the order in which you undo the operation is the opposite of PEDMAS (or BEDMAS/BODMAS if thats how you learned it)
lapis crow
PEMDAS
sonic ridge
square root both sides?
yup thats the first step
so once you square root both sides, what do you get?
lapis crow
x-9=7
sonic ridge
yup
lapis crow
then add 9
sonic ridge
correct!
sonic ridge
x-9=7
TECHNICALLY this is one of the 2 expressions you get
when youre describing how you would solve it, i suppose saying square rooting both sides would be correct
but note that
$$(x-9)^2=49$$
$$(x-9)=\pm7$$
ocean sealBOT
light
lapis crow
😭
sonic ridge
so saying square root both sides would i think be the best solution
sonic ridge
**light**
but just a note for yourself or for your teacher to show that you do understand that there are 2 values that the RHS can take on when you go from these two steps
lapis crow
so saying square root both sides would i think be the best solution
yeah
lapis crow
but just a note for yourself or for your teacher to show that you do understand ...
yep
sonic ridge
but yeah it seems like u got it
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river shore
I just started a pre calculus class, it’s still only like the first week of school. I absorbed absolutely no information from the way my teacher taught it, I think it’s summation notation?
river shore
i can send pictures of previous pages we did friday , I just don’t know how to set it up mostly
river shore
19
sour verge
How many do you add each row?
river shore
4 each row
sour verge
Ok so if it's the third row from the top how many times did you add DVDs?
river shore
twice?
river shore
okay, how do i put it into the equation though?
sour verge
Well n is the row
So in general, if I'm on row n how many times did I have to add 4 dvd.s
river shore
okay
sour verge
Second part is adding all the rows together. So 15 + (15 + 4) + (15 + 2(4)) + (15 + 3(4)) + ... + (15 + (n-1)*4)
Maybe you've seen something about an expression for adding the first k integers?
river shore
i dont think i have. this is the first day we did summation notation
i might know it if you could give me the general idea?
sour verge
Well it's just a formula for the first k integers : the sum of the first k integers is k(k+1)/2
river shore
oh, yeah i dont think ive learned that yet
sour verge
ok then maybe you can just leave it as an expression using summation notation
river shore
alright
so what would the expression look like? I have a few different formulas on my other papers
sour verge
A particular row has 15+4k DVDs (starting with the top row being k = 0)
So you add all n rows
from 0 to n
river shore
okay
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Available help channel!
icy violet
lone heartBOT
icy violet
What am I actually supposed to find in this
In question d
???
Guys I swear this question should be easy
<@&286206848099549185>
sour verge
Denote A(t) the area of the ripple after t seconds and r(t) the radius of the ripple after t seconds.
Then we know A(t) = pi * (r(t))^2. So what is A'(t)?
icy violet
Denote A(t) the area of the ripple after t seconds and r(t) the radius of the ri...
Is this a kinematic and rates combination question?
sour verge
Yeah
icy violet
2pirt
icy violet
.close
remote terrace
yo
remote terrace
Why do the power become positive when (3/4)^-2 and ans is 4^2/3^2
and the numbers in the numerator and denominator get witched too
naive valley
Recall the property of exponents that $x^a x^b = x^{a+b}$.
Consider the special case $b = -a$, in which case the equation becomes $x^a x^{-a} = 1$ since $x^{a - a} = x^0 = 1$.
Now apply this with $x = 3/4$ and $a = 2$ and see what happens
ocean sealBOT
Bungo
remote terrace
Recall the property of exponents that $x^a x^b = x^{a+b}$.
Consider the special ...
Explain in simple terms gang
😭
thorn monolith
and the numbers in the numerator and denominator get witched too
For "numerator and denominator getting witched" to get rid of -2 in the exponent, multiply the expression with (3/4)² and then divide by it
naive valley
Explain in simple terms gang
what part is not simple?
thorn monolith
Recall the property of exponents that $x^a x^b = x^{a+b}$.
Consider the special ...
Then use this
remote terrace
what part is not simple?
ooo ok
remote terrace
For "numerator and denominator getting witched" to get rid of -2 in the exponent...
that makes sense
Also uh
How do u do
(1 4/4)^-2
thorn monolith
And be aware that multiplication of two numbers with the same parity never will be negative
thorn monolith
A number has same parity with itself, so when you are taking square of it, it multiplies with a number with same parity which will be non negative number
remote terrace
remote terrace
How come
remote terrace
That is same as (5/4)^-2
I dont understand how u get 5/4
so from what im seeing im guessing u did 4x1=4 and then added +1
and then 4x1
for the denominator
thorn monolith
That one represents that there is a whole 1 of that fraction, which is same as adding 1 to that fraction
Its a primary school notation
wheat isle
x^-2 = 1/(x^2)
remote terrace
remote terrace
Its a primary school notation
Im dumb asf so idk
thorn monolith
Uhh dont type equal though because it lacks the power -2 which first expression has
$$ 1 \frac{1}{4} = 1 + \frac{1}{4}$$
ocean sealBOT
Cyrenux
thorn monolith
Same thing
remote terrace
Same thing
but how do u make them just numerator and denominator
how u simplify it
wheat isle
What’s 1 equal to here
thorn monolith
Multiply 1 by 4 then divide by 4
thorn monolith
You are going reverse
You are inserting 4/4 where you 1, NOT inserting 1 where you see 4/4
Purpose of this is to get the same denominator as 1/4
So we can apply 4/4 + 1/4 = (4+1)/4
$$\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c} $$
ocean sealBOT
Cyrenux
remote terrace
thorn monolith
Yes
remote terrace
My question is
how did u even get 4/4 and 1/4
out of 1 1/4
😭 im super dumb im sorry
thorn monolith
**Cyrenux**
Reread
thorn monolith
Dunno how you got 6, but yeah you add 4 and 1 after getting same denominator like this screenshot
remote terrace
Dunno how you got 6, but yeah you add 4 and 1 after getting same denominator lik...
1+4 = 5
and 5+1 and 6
right to left left to right
thorn monolith
Also be aware for future notice: $$ 1 \frac{1}{4} \neq 1 . \frac{1}{4}$$
ocean sealBOT
Cyrenux
remote terrace
How did you add them when they dont have same denominator
Can u uh step by step show me what to do
cause im lost
thorn monolith
**Cyrenux**
I told you first step already
remote terrace
So do i times them?
remote terrace
I told you first step already
But idk what to do at that step
am i supposed to be multipling, adding, subtracting
thorn monolith
Its always adding..
thorn monolith
An example:
$$\frac{7}{4} = \frac{4}{4} + \frac{3}{4} = 1 + \frac{3}{4} = 1 \frac{3}{4}$$
ocean sealBOT
Cyrenux
lone heartBOT
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Available help channel!
vivid heath
lone heartBOT
vivid heath
how is the 0.25 solved?
tardy stag
0.25 = 1/4
tardy tapir
squaring both sides
vivid heath
so we square both sides and are left with 0.25
tardy tapir
yeah
vivid heath
so if i had 1/3 and only a square root what would i do?
tardy tapir
square both sides
vivid heath
and i would have 1/9
tardy tapir
yes
ember badge
prove that (x-a)(x-b) = x² - 2x - 1/3 and deduce a simple expression of a and b using the square function. there are 2 prior question, that i did, but i'm completely lost on that one
finite flax
prove that (x-a)(x-b) = x² - 2x - 1/3 and deduce a simple expression of a and b ...
what have you done so far
x^2 - (a+b)x +ab = x^2 - 2x - 1/3
so then
(a+b) = 2
ab = -1/3
I could verify ab
ember badge
i have the value of ab, but its not a round number, because its not supposed to be done using a calculator
and after calculating it it does give 1/3, but idk if there is any way to calculate it "by hand"
alpine sable
I don't understand the wording here
lone heartBOT
I don't understand the wording here
wild trail
I don't understand the wording here
point to the exact line please
echo socket
I think the first line
echo socket
v seems to me like it only consists out of one column, not many or even *all*
It just gives a single arbitrary column vector, yeah
But it could be any element of R^n
As long as it has n real entries
wild trail
@alpine sable the elements of R^n look like (r1, r2, ..., rn), you can write it as a column vector
lone heartBOT
@alpine sable Has your question been resolved?
alpine sable
What does the "all columns" mean?
A (vector?) "space" can also have multiple columns instead of just one like in the example?
lone heartBOT
tall talon
\begin{cases}xy+xz=a&\ xy+yz=b&\ xz+yz=c&\end{cases}::::a+b+c\ne 0
ocean sealBOT
themathboi #2137
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)
tall talon
\begin{cases}xy+xz=4&\ :xy+yz=9&\ :xz+yz=-6&\end{cases}
ocean sealBOT
themathboi #2137
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)
lone heartBOT
alpine sable
A (vector?) "space" can also have multiple columns instead of just one like in t...
Like
-- -- |v1, va| |v2, vb| |..., ...| |vn, v...| -- --
echo socket
\begin{cases}xy+xz=4&\\ \:xy+yz=9&\\ \:xz+yz=-6&\end{cases}
You need to put this inside math mode
echo socket
Like
`-- --
|v1, va|
|v2, vb|
|..., ...|
|vn, v...|
-- --`
The channel is closing, open an available one
alpine sable
The channel is closing, open an available one
My previous question is here. The bot closed it because i reacted too late
echo socket
Still, open a channel
alpine sable
Ok
lone heartBOT
Available help channel!
meager dirge
Is this right?
I know that x=1, and I know that m=9/2.
BUt y=9/2x -19/2 seems off somehow.
subtle birch
Is this right?
I know that x=1, and I know that m=9/2.
BUt y=9/2x -19/2 seems of...
why does it seem off to you?
The steps are all correct tho
meager dirge
It does too. 🤔
It does too!
I think I was staring at it for too long.
Thank you!
.close
grizzled hawk
How would i solve this differential equation
x'' = -w^2 x - g
alpine sable
what have you tried?
grizzled hawk
Not that good with differential equations but tried, x= Acos(wt) - gt^2 / 2, because Acos(wt) would work for x'' = -w^2 x, but the -g is messing me up
not sure if this can help https://math.stackexchange.com/a/4181357
lone heartBOT
@grizzled hawk Has your question been resolved?
lone heartBOT
@grizzled hawk Has your question been resolved?
glad gale
lone heartBOT
lone heartBOT
@glad gale Has your question been resolved?
thorn monolith
draw a perpendicular line to side with length 6
from the point which intersects the sides : with length 4 and length a + b
glad gale
Like
thorn monolith
then you get a rectangle which has sides 4 and 2
glad gale
Triangle?
thorn monolith
yeah you will get a right triangle
thorn monolith
yes, you want to draw one more thing
glad gale
?
thorn monolith
do you see the side which is split with 1 and 1
you want to draw a line from there, to side that includes a and b
since its split with ratio -> 1:1 , the line you will draw will also split it 1:1
glad gale
Yes
thorn monolith
then you gotta use proportion on triangles
glad gale
Ahh okay
thorn monolith
the expression i typed in blue indicate the blue side which i higlighted
ok i lied
2nd blue equation is false
glad gale
Yes
thorn monolith
from the picture, you can see a is longer than (a+b)/2 , so we extract (a-b)/2 from a to find length highlighted in blue
on other side, (a+b)/2 is greater than b, you should be able to see it from the picture
we had to extract b from (a+b)/ 2 so
(a+b)/2 - b
but that gives us same equation as the first blue equation so it doesnt matter
thorn monolith
we get that length of blue side is (a-b)/2 but im too blind to see how that helps
thorn monolith
And a+b/2 - b
also if you type a+b/2 it will be interpreted as $$ a + \frac{b}{2}$$ instead
ocean sealBOT
Cyrenux
thorn monolith
always have paranthesis if there are more than 1 variables or constants in the numerator or denominator
i see what we are missing now... lol
im so blind
we missed using the fact that area is split equally as well
thorn monolith
sorry got carried away back now
@glad gale so you should be able to see full area is 10 which makes its half 5
glad gale
Yes
thorn monolith
from earlier , you know that if we draw the line to middle of a and b it will be the middle segment which we can calculate by (4+6)/2
i mean you dont need to even use 'middle segment' you can use proportion to see that bottom length of triangle is 1 by proportion
but forget about it for now
since area of that is 9/2, we get the tiny area beneath is as 1/2 :
since their addition must be 5
ember lotus
Make auxiliary lines to a complete triangle, then calculate the proportion.
ask if any question
lone heartBOT
<#490557019623915520>
halcyon monolith
I found out the locus of Z-4, I think and it's a straight line. I believe I am asked to find the locus of Z and I don't know what that is
How can I go by solving it.
vale wigeon
<#490557019623915520>
writing the channel name in the channel itself doesn't do anything
halcyon monolith
I am sorry, I don't know what to do abt that?
vale wigeon
anyway this is still a straight line (or a ray to be more precise, but it seems not to be in the answer options) it's just emanating from the point 4 rather than 0
vale wigeon
I am sorry, I don't know what to do abt that?
nothing, just don't do it again.
don't delete it tho.
halcyon monolith
no, wait, what's the right procedure?
vale wigeon
the right procedure is to just post your question. the channel will open when any message is written in it.
random slate
i proved 1.2 and 1.3
but i dont quite understand 1.4
so
if you divide each side by (a,b)
you get a'x + b'y = n'
and (a',b') is 1
but it says due to theorem 1.2 it has integer solutions
but that only is true when n' is 1
and they havent said how its 1
ebon sparrow
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Available help channel!
flat vale
mb for opening again can someone check my notation?
lone heartBOT
mb for opening again can someone check my notation?
flat vale
Let $\varphi: \mathbb{N} \times X^2 \to X, ; \varphi(n,x,y) = x+y$, \
and $F_n$ such that $F_n = \varphi(n, F_{n-1}, F_{n-2}), ; n>1$ with $F_{0} = F_{1} = 1$.
ocean sealBOT
TheWhiteShadow
flat vale
in reference to this
actually I can
Let $\varphi: \mathbb{N}^3 \to \mathbb{N}, ; \varphi(n,x,y) = x+y$,\
and $F_n$ such that $F_n = \varphi(n, F_{n-1}, F_{n-2}), ; n>1$ with $F_{0} = F_{1} = 1$.
ocean sealBOT
TheWhiteShadow
lone heartBOT
@flat vale Has your question been resolved?
lone heartBOT
@flat vale Has your question been resolved?
lone heartBOT
@flat vale Has your question been resolved?
keen prawn
lone heartBOT
keen prawn
What formula should I use for the 10th question?
native cloud
,rccw
ocean sealBOT
cobalt dock
woah
native cloud
Chemistry..?
keen prawn
Yes
cobalt dock
im in
native cloud
I'm out