to simplify
#help-23
quiet cypress
swift palm
Actually a lot of that doesn't make sense. Where did 12^x come from in second to last line.
quiet cypress
what is 2*6?
livid sparrow
Yes
quiet cypress
Distribute
livid sparrow
what is 2*6?
2x * 6x = 12x^2? no?
swift palm
There is no powers of x in that entire question so it makes no sense that 12^x would ever be a term
What you said is right purple don't worry
livid sparrow
There is no powers of x in that entire question so it makes no sense that 12^x w...
Hmmm
livid sparrow
What you said is right purple don't worry
so 12x^2 is right?
swift palm
I'm just not sure what is going on in that working out
quiet cypress
2x * 6x = 12x^2? no?
what? that is right
swift palm
Yes that's right it's not what I'm talking about.
Just that working out at the beginning is wrong
quiet cypress
Yes that's right it's not what I'm talking about.
you're confusing
swift palm
The initial working out that you are confused about is wrong purple.
livid sparrow
Lmao guys chill you're making my head hurt lol
quiet cypress
what is the beginning of the problem? like what is starting point?
livid sparrow
The initial working out that you are confused about is wrong purple.
Which part specifically?
quiet cypress
2+2x(2(3x+2)+2)
??
livid sparrow
2+2x(2(3x+2)+2)
??
Yes
quiet cypress
or is there something before ?
livid sparrow
or is there something before ?
No this is the start
livid sparrow
so what is the goal?
Just to simplify
swift palm
Mostly in the transition from 3rd last to 2nd last line of working.
quiet cypress
this 2+(12x^2+12x)?
livid sparrow
Ahhh guys my bad
quiet cypress
Is the + supposed be = ? becasue you can't really divide 2 in this case if it was a +
swift palm
Ohh ok.
livid sparrow
But still I dont understand why this
swift palm
Do you have the textbook or something where it came from because it's still a bit hard to understand
If it's put equal to 0 then you can divide by 2
But otherwise it makes no sense
quiet cypress
Because if you divide 2 you can simplify it to the right answer
livid sparrow
Do you have the textbook or something where it came from because it's still a bi...
It's from this website https://www.storyofmathematics.com/simplifying-expressions/
example 2
livid sparrow
Because if you divide 2 you can simplify it to the right answer
But why does this work?
quiet cypress
This expression can be simplified by dividing each term by 2 as;
livid sparrow
*This expression can be simplified by dividing each term by 2 as;*
Which rule is being applied?
livid sparrow
you are missing 6x. so the answer is 6x^2+6x+1
Damn my bad I should learn how to read
livid sparrow
Its okay man.
I still don't understand why everything is being divided by 2 though
quiet cypress
I still don't understand why everything is being divided by 2 though
" In mathematics, simply or simplification is reducing the expression/fraction/problem in a simpler form. It makes the problem easy with calculations and solving."
So if your goal is to simply then you do whatever you can do make it more simpler
So in this case, dividng by 2 makes it better than 2 + 12x 2 + 12x
2 + 12x 2 + 12x becomes 6x^2+6x+1 which is easier to do if you have to use the quadratic equation
swift palm
Actually I'm going to have to somewhat disagree unfortunately. Symbolab and wolfram alpha keep a factor of 2 outside brackets.
Try it on both and yes they divide by 2 to simplify but they keep the final answer as 2(6x^2+6x+1)
livid sparrow
Try it on both and yes they divide by 2 to simplify but they keep the final answ...
So I'm guessing the divided all terms by 2 and then kept the factor of 2 there so it stays the same?
I have no clue what im talking about though
But it is becoming clearer
swift palm
That is the only way it makes sense but the website doesn't keep the factor. I am actually unsure about what the website is doing...
livid sparrow
That is the only way it makes sense but the website doesn't keep the factor. I a...
You mean the website I linked to?
Or symbolab/wolfram?
swift palm
Yeah.. maybe killer can explain it to me better but I'm working through y12 and this shouldn't be something I'm failing.
If I'm wrong
quiet cypress
So from the website you showed me it the part where it says, "Now eliminate the parentheses by multiplying any number outside it;" that is where hydro is taking about. The final answer is 2(6x^2+6x+1). The expression itself can be simplify to 6 x^2 + 6x + 1 and that is where the 2 comes from (12x^2/2 + 12x/2 + 2/2)
swift palm
Yeah, I get that but here's my confusion. For an equation equal to 0, so for example 12x^2+12x+2 = 0, you can divide both sides by 2 easily. If you have an expression though, like 12x^2+12x+2, dividing by 2 gives you a different expression totally.
livid sparrow
Yeah, I get that but here's my confusion. For an equation equal to 0, so for exa...
Thats what I thought was strange about it. It's like they were just halving it
swift palm
As you can see from desmos, 12x^2+12x+2 and 6x^2 +6x+1 are completely different expressions however they do have identical x intercepts. So the solutions are the same but substituting values of x give different values of y. What is wrong with my logic killer?
quiet cypress
Ahh, based on the graph I see. Gotcha. Yeah, I guess is not totally worth simplifying then. I guess it will depend on the teacher
swift palm
This is a really unsatisfying conclusion
I actually think the website is objectively poorly worded, where did you find it @livid sparrow
If it's official then maybe I'm just thinking about this wrong, but otherwise I think they could improve wording...
livid sparrow
I actually think the website is objectively poorly worded, where did you find it...
Just google. Do you have any other resources you would recommend?
quiet cypress
Khan academy is a good one
livid sparrow
What would I search on there to find this?
quiet cypress
swift palm
Simplification of expressions or simplification of equations
livid sparrow
Simplification of expressions or simplification of equations
Alright
Thanks for the help fellas, appreciate it
quiet cypress
swift palm
Honestly though you seem to understand it fine and just got tripped up by the website which would have happened to me too.
quiet cypress
Honestly though you seem to understand it fine and just got tripped up by the we...
It seems multiple websites have the same example tho
I will ask my mentor about this tho. It seems interesting
swift palm
Can you show me the other website with that example?
I'm guessing they have it as an equation not an expression but maybe not.
livid sparrow
Honestly though you seem to understand it fine and just got tripped up by the we...
I'm pretty new to it really. It's been 8 years since I've studied any math. I'm just self learning because I find it interesting. I didn't even know it was called a polynomial. There are huge gaps in my knowledge
swift palm
A polynomial is just an expression with 2 or more terms.
Algebraic expression to be specific
swift palm
I assume you live in the US right?
livid sparrow
Nope, New Zealand
swift palm
I use the Australian mathematical curriculum textbooks
livid sparrow
I use the Australian mathematical curriculum textbooks
I could probably find those here
quiet cypress
College Algebra by Robert F. Blitzer . You might have a good luck finding it online but its an amazing book with good explanation
livid sparrow
College Algebra by Robert F. Blitzer . You might have a good luck finding it onl...
I'll check it out, thank you
quiet cypress
Anytime!
swift palm
I could probably find those here
Haese mathematics is the author of the Australian official textbooks.
quiet cypress
If you want, I can create a group chat here. If Hydro is fine with it. There you can ask all the questions you need. Good to see people come back to math after years.
swift palm
If you go to haesemathematics.com that has yrs1-12 of Australian math curriculum including specialist pathways.
I'm fine with it 100%
https://www.haesemathematics.com/ that has the aus textbooks
quiet cypress
Just added you both
livid sparrow
https://www.haesemathematics.com/ that has the aus textbooks
Will definitely have a look at this, thanks
livid sparrow
Just added you both
š I'll try not to pester you too much!
quiet cypress
š I'll try not to pester you too much!
You're fine! You are actually motivating me to teach again. I am hoping to get my masters in math soon! I am taking a break from school now and just enjoying life. But man I do miss helping people out. I was a TA for calculus 2 in grad school for one semester but didn't do well in my classes so I had to leave.
I taught it last year so not quite a break from school.
livid sparrow
Nice man, glad to have someone knowledgeable on my contact list
Having someone explain things can be really helpful
You guys want to make a group chat?
plucky elk
W_n are usually already column vectors
That's awkward phrasing
R^n exists outside the scope of W_n
safe radishBOT
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lean otter
Is there a difference between saying perpendicular vs orthogonal or is it just semantics?
safe radishBOT
Is there a difference between saying perpendicular vs orthogonal or is it just s...
lean otter
I have seen some people divided on it online
vestal sedge
depends on the context
cosmic grove
kinda the same thing, but I use "perpendicular" for lines and orthogonal for vectors
vestal sedge
perpendicular is orthogonal in planar geometry
orthogonal isnt perpendicular necesarily in spatial geometry
cosmic grove
orthogonal is best suited for vectors
vestal sedge
yes agreed
and when reffering to orthogonal matricies you cant say perpendicular matricies
cosmic grove
perpendicular is a geometry term
vestal sedge
even in geometry it's not quite the same
perpendicular is just that perpendicular
orthogonal involves perpendiculars but isnt necesarily perpendicular on its own
cosmic grove
for example ?
vestal sedge
orthogonal projections in 3d space
the segment projected isnt perpendicular on the plane
but the ends of it that determine the projections are
vestal sedge
the segment projected isnt perpendicular on the plane
if it was it would just be a point
a contradictory example would br orthodiagonal quadrilateral which by definition have perpendicular diagonals
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@lean otter Has your question been resolved?
lean otter
Will keep this channel open for you guys to discuss about it lol
cosmic grove
the segment projected isnt perpendicular on the plane
huh, when you project a segment on a plane, you project each point of the segment on the the plane and thats the line that goes through a point on a segment and the plane which is perpendicular. Thats why its called orthogonal projection
or did I miss something
The projection of the segment isnt perpendicular to the segment itself of course
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proven zephyr
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proven zephyr
found out dĀ²y/dxĀ² but cant prove = 0
hybrid osprey
found out dĀ²y/dxĀ² but cant prove = 0
whats ur dy/dx value
proven zephyr
-sin theta / (1+ cos theta)
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proven zephyr
<@&286206848099549185>
proven zephyr
nvm i got it i forgot to multiply by dĪø/dx when taking the higher derivative
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pseudo scroll
$\int \tan^5 \theta d\theta$
safe radishBOT
$\int \tan^5 \theta d\theta$
flat frigateBOT
NEONPerseus
pseudo scroll
How do I even start man :/
vestal sedge
god what kind of integral homework do you have
pseudo scroll
Ah welcome back
vestal sedge
yeah lol
pseudo scroll
My teacher said the x^5 one was too hard and they assigned it as a joke š
They said that this was within scope though
So...
vestal sedge
hmm
pseudo scroll
I know there exists a reduction formula for tangent but can't use it at school
vestal sedge
oh sad
thought of that exactly lmao
like you cant use secĀ²x-1 = -tanĀ²x
1
sorry
vestal sedge
yeah but u said the formula is off bounds
thats what i thought you meant
you can square the ahole thing to get tanā“x
pseudo scroll
$\int \tan^m x dx = \frac{\tan^{m - 1} x}{m - 1} + \int \tan^{m - 2} x dx$
flat frigateBOT
NEONPerseus
pseudo scroll
This is the reduction formula
pseudo scroll
its fun yeah
vestal sedge
anyways i believe you can work with that identity
pseudo scroll
lets see what that does
flat frigateBOT
NEONPerseus
vestal sedge
how the heck do u use that tho i gebiuanely dont know
yeah i think thats right
like the bot
pseudo scroll
$\int (1 - \sec^2 \theta)\tan \theta d\theta$
flat frigateBOT
NEONPerseus
pseudo scroll
So secant squared is u
$$\sec^2 \theta = u$$ $$2\sec^2 \theta \tan \theta d\theta = du$$
flat frigateBOT
NEONPerseus
pseudo scroll
$d \theta = \frac{du}{2u\tan\theta}$
flat frigateBOT
NEONPerseus
vestal sedge
mhm
violet owl
hello
violet owl
the answerās (-infinity,-3)U(9,infinity)
however none of these seem to represent the correct answer
ši just need someone to confirm tht wat i did is correct
trim swan
ši just need someone to confirm tht wat i did is correct
what did you do? š¤
violet owl
divide bith sides by 2
then set two possibilities
3x-9>18 then x=9
3x-9<-18 , x=-3
violet owl
oh okay
trim swan
For example R - [-3,9] would be correct for this inequality
elfin cove
so i have to show this identity step by step but i am confused how i would write the first part in the gauĆ formula
safe radishBOT
so i have to show this identity step by step but i am confused how i would write...
elfin cove
for e.g. this is understandable:
but with n+1 and v=1 i'm confused how it becomes (n+1)(n+2)/2
safe radishBOT
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jade gust
I probably won't be much help cause 1. I'm dookie at maths 2. 12 am
Probably perform an index shift
after doing that use the commutativity and associativity property
Index Shift: \[\sum^t_{n=s}[f(n)]=\sum^{t+p}_{n=s+p}[f(n-p)]\]
Commutativity and Associativity: \[\sum^t_{n=s}[f(n)]\pm\sum^t_{n=s}[g(n)]=\sum^t_{n=s}[f(n)\pm g(n)]\]
elfin cove
sadly i never heard of the index shift š„²
and the thing is my prof skipped this part [induction] in uni whysoever
jade gust
damn don't you learn mathematical induction during highschool
but it's not like i listened anyway
Basically induction is just tryna prove some formula works for all natural numbers
safe radishBOT
jade gust
The simplest and most common form of mathematical induction infers that a statement involving a natural number \(n\) (that is \(n\in\mathbb Z\geq0\lor 1\) holds \(\forall n\). The proof consists of two steps:
\begin{enumerate}
\item The \textbf{base case} (Synonym: \textbf{initial case}): prove that the statement holds for \(0\lor 1\) or both.
\item The \textbf{induction step} (Synonyms: \textbf{inductive step}, or \textbf{step case}): prove that \(\forall n\), if the statement holds for \(n\), then it holds for \(n+1\). In other words, assume that the statement holds for some arbitrary \(n\in\mathbb N\), and prove that the statement holds for \(n+1\).
\end{enumerate}
The hypothesis in the induction step, that the statement holds for a particular \(n\), is called the \textbf{induction hypothesis} or \textbf{inductive hypothesis}. To prove the induction step, one assumes the induction hypothesis for \(n\) and then uses this assumption to prove that the statement holds for \(n+1\). \\~\\
Authors who prefer to define natural numbers to begin at \(0\) use that value in the base case; those who define natural numbers to begin at \(1\) use that value.
south skiff
What is the inclusion exclusion formula for n(A and not B
safe radishBOT
What is the inclusion exclusion formula for n(A and not B
south skiff
so n(A and B with a line on top )?
sorry how would i do n(A and not B and not C)>
?*
safe radishBOT
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orchid wave
How do I do this question ?
safe radishBOT
How do I do this question ?
orchid wave
When I try to get the area while using an integration of the equation, I get the result as 3/2
obtuse plover
What was ur integration equation
obtuse plover
Notice how part of ur curve goes under the x-axis
obtuse plover
No need to apologize. Yes it does, it adds the areas above the x-axis and subtracts from it the areas underneath the x-axis
But we just want a sum of everything shaded
orchid wave
But we just want a sum of everything shaded
Yea
obtuse plover
Try manipulating ur bounds on the integral, u will be using 3 integrals, 1 for the first area above the x axis , the next for the area under the x axis, and one finally for the last area above the x axis
orchid wave
Alright
Iāll try and let you know
obtuse plover
Sounds good
orchid wave
Well I tried it and it doesnāt work
I basically took the integral of x^2 -3x +2 (which is [x^3]/ 3 - [3x^2]/2 + 2x
Then I equated it to a function which I called f(x)
Then I did the following
f(1) -f(0) + f(2) - f(1) + f(3) - f(2)
which results in my answer being 3/2
obtuse plover
What did u get for the f(2)-f(1)
orchid wave
-1/6
obtuse plover
Thereās ur problem
orchid wave
Ahhh
obtuse plover
Thatās the area under the x axis
orchid wave
I see
obtuse plover
So take abs value of it, and add it to the other values
obtuse plover
Well I tried it and it doesnāt work
I basically took the integral of x^2 -3x +2 ...
By doing this, itās the same thing as integrating from 0 to 3 straight
Which woulda worked if there was an absolute value around the f(2) - f(1)
orchid wave
Now I get 19/6
orchid wave
f(0) = 0
f(1) = 5/6
f(2) = 2/3
f(3) = 3/2
So 5/6 + 1/6 + 3/2 - 2/3
= 11/6
orchid wave
Ohhh okay
Thanks
One more thing
Isnāt there a way to find the area of a curve for a specific range without considering whether the curve dips below the x axis or not ?
obtuse plover
Iām not sure Iām understanding
So the area under the curve above and below the x-axis added together?
All as positive ur asking?
safe radishBOT
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orchid wave
So the area under the curve above and below the x-axis added together?
Basically yea
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Available help channel!
pallid spade
Could anybody explain what I'm doing wrong? I checked wolfram alpha and synolab, and both say that it's 1/sqrt(2x-1) but I'm not quite sure how do they get to that solution as there is 1/2
safe radishBOT
Could anybody explain what I'm doing wrong? I checked wolfram alpha and synolab,...
split ether
coral meadow
help
lean rover
safe radishBOT
lean rover
The 2nd one
safe radishBOT
@lean rover Has your question been resolved?
wheat cave
The 2nd one
What exactly are you stuck on?
also, with 2nd, do you mean ii, (b) or the entire 2. task?
lean rover
Iām not even sure what question 2 is asking
wheat cave
Iām not even sure what question 2 is asking
In mathematics and particularly in algebra, a linear or nonlinear system of equations is called consistent if there is at least one set of values for the unknowns that satisfies each equation in the systemāthat is, when substituted into each of the equations, they make each equation hold true as an identity. In contrast, a linear or non linear equation system is called inconsistent if there is no set of values for the unknowns that satisfies all of the equations.
Stolen from wikipedia
tl;dr, is it possible to solve them
lean rover
Not A
wheat cave
no, cramers rule was the first task
here you can prove very fast that this is not a possible equation
lean rover
Yh and my teacher wants us to use matrices as that is what the modules is about
We unfortunately canāt use any other ways
wheat cave
Well then write it like that
i) 2 -3 8
ii) 4 -6 9
you can multiple i) with 2 to find out that its not possible
lean rover
Hold on let me do it quickly with the way my teacher told me to do it give me 1 second
wheat cave
sure, and can you send an example so I know how to explain better?
lean rover
its basically saying 18=9
What do you mean
I know the determinant has to not equal Zero
For Cramer rule to be applicable
Is that my answer
Probably not since it talk about consistency
lean rover
Yh
wheat cave
Since D=0, you know that its not solvable for any numbers, meaning its just not consistant
lean rover
Which this does not satisfy
Yh so I say not consistent because Cramer rules needs Dā 0
wheat cave
Yupp
I think you could've stopped after the first line, finding out that D=0 and Dx!=0 means its not solvable
lean rover
I think you could've stopped after the first line, finding out that D=0 and Dx!=...
Yh
I donāt know why I continued but yh
Is the same for B
wheat cave
tbh when we did cramers rule we always simplified first so our divisions in the end weren't that big, but I guess you have to do it this way
lean rover
Yh my teachers we quite adamant on us doing it this way
May I ask for help on another question?
wheat cave
sure
lean rover
A
I havenāt really done much studying on so it maybe be best I learn to do it
What is question A called so I can watching some videos on before asking you as it maybe easier on you
wheat cave
What do you mean called?
lean rover
The topic
wheat cave
Well, since you know you have to use cramers rule, its cramers rule
lean rover
3A?
wheat cave
I believe?
lean rover
Ah not an issue
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Available help channel!
tawdry hornet
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tawdry hornet
<@&286206848099549185>
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rancid lion
i need someone to help me with these 2 worksheets
safe radishBOT
i need someone to help me with these 2 worksheets
peak estuary
general note: dont expect people to download random files to help you
rancid lion
i can stream it
im not good at showing my work so i need someone to help me with that
<@&286206848099549185>
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mental patrol
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rigid inlet
What's giving you issue?
mental patrol
What's giving you issue?
The -29.29. I think that would mean there is a decrease of sometime? I just donāt know where
I was thinking B but not sure
rigid inlet
-29.29 is the y-intercept
mental patrol
Yes
rigid inlet
It's the estimated value of Y when you set X=0
mental patrol
Okay :/ I donāt really understand
rigid inlet
What don't you understand?
mental patrol
The negative does not matter then?
The negative makes me think that it will be a negative relationship
rigid inlet
I wouldn't say it doesn't matter
Does it make sense for a donation rate to be negative?
mental patrol
No but some of the answers say that the college would be expecting fewer alumni to donate, not that there would be negative donations I think?
rigid inlet
Well I'm trying to illustrate what the -29.29 represents
mental patrol
I think the Answer is A I just donāt understand what -29.29 is
I know it is Y
But like what does the number come from/what is is representing
rigid inlet
It's the estimated value of Y when you set X=0
It's this
mental patrol
That I donāt understand
rigid inlet
An important concept in regression is the idea of a relevant range
There's a relevant range on the output (it must be in [0,1]) and there's one on the input (SAT scores have bounds)
However, a line is continuous for all real numbers, and thus it has to have a y-intercept if it has finite slope
So while it may physically be nonsense (a donation proportion of -29.29% makes no sense), it needs to exist for the regression line to be an actual line
rigid inlet
Y is the donation percentage
mental patrol
Yes I thought -29.29 is Y?
rigid inlet
We're estimating Y through Yhat = -29.29 + 0.05X
mental patrol
What is -29.29 then
rigid inlet
It's the y-intercept
mental patrol
I know but what is it for, donation, score, etc?
rigid inlet
It's the value of Y when X=0
mental patrol
That is part I do not get
rigid inlet
So the donation percentage when we look at SAT scores of 0
mental patrol
If an SAT is 0, the donation is -29.29?
rigid inlet
Yes
rigid inlet
Yes
mental patrol
Thank you!
rigid inlet
Np
mental patrol
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sinful tinsel
Hi, i understand case 1 and 2, but how did they get 4C1 in case 3?
safe radishBOT
Hi, i understand case 1 and 2, but how did they get 4C1 in case 3?
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winged fern
Need help proving for n > 16, Ļ(16) < n -4 where Ļ is the eulur toitent function
safe radishBOT
Need help proving for n > 16, Ļ(16) < n -4 where Ļ is the eulur toitent function
winged fern
woops leftthe other help channel
safe radishBOT
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wise field
How do I get the equation x^5-k=0 to equal the iterative formula shown in the picture.
safe radishBOT
How do I get the equation x^5-k=0 to equal the iterative formula shown in the pi...
wise field
I've found the derivative to be 5x^4
and the Newton Raphson Formula is xn+1 = xn - f(x)/f'(x)
but i still can get the answer
<@&286206848099549185>
safe radishBOT
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@wise field Has your question been resolved?
modern pollen
Simplify: -12/25+3/-5-1/15
wise field
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safe radishBOT
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naive agate
How do you find the inverse of this exponential (and logs) function?
naive agate
Can you show me all the steps
naive agate
I thought you had to
merry sleet
like replace 2x by a
naive agate
Swap the x and y
naive agate
You are saying.. 4x = a?
naive agate
You move 4 to the other side
merry sleet
a=-4x right
but we re not done
lets replace a by its value
2^y = -4x
now we can apply log 2 on both sides
merry sleet
wdym?
merry sleet
if you have
-2/4
you only have to count opne of them as negative
because if its negative divided by negative
the result is positive
it doesnt matter which
merry sleet
2^y = -4x
like here it could be -2^y = 4x
you just have to multiply by -1 on both sides
to see its the same
naive agate
So it doesnāt matter?
merry sleet
yeah
naive agate
Wouldnāt that be log-2(4x)??
merry sleet
log -2 doesnt exist
merry sleet
like here it could be -2^y = 4x
but this is true
i'm not sure if you dont understand the algebra i did before
or the log thing
naive agate
I literally just learned logarithms like a week ago
merry sleet
yeah but for now
if you ignore the next part
which is taking a log
there shouldnt be any confusion
naive agate
You need log to inverse an exponential donāt you though?
merry sleet
yeah
ln
or log base e
is the inverse of the exponential function
and log base a
is defined as
log_a(x) = ln(x)/ln(a)
and log_a is the inverse of a^x
naive agate
Ok
merry sleet
we were at -2^y = 4x
naive agate
Right
merry sleet
you could apply log2 directly here
but to be sure about things
and not make a mistake in general
its better to have the power isolated
naive agate
If it was 2y instead of -2y, would it still be log2
merry sleet
it would
but as i said
the best practice if you are starting with logs
is to isolate 2^y completely
so multiply on both sides by -1
naive agate
4x and -2yā¦?
merry sleet
yeah
naive agate
I multiplied both
merry sleet
2^y =-4x
naive agate
Ok
naive agate
Yea thatās the right answer
merry sleet
and if you know rules about logarithm
you can simplify
what is the log of a product?
merry sleet
log(a*b) =log(a)+log(b)
naive agate
I know almost nothing on logs
naive agate
y = log2(-4x)
This is the right one
naive agate
Can you do one rq?
merry sleet
also maybe you need to be careful about the domain
naive agate
Domain??
merry sleet
since we are talkiung about a function
naive agate
Like the X axis stuff
flat frigateBOT
merry sleet
basically the domain of the inverse is the same as the rangez of the original funciton
function*
so here its non positive numbers
naive agate
Cool
dark sluice
hi can someone help me with a proof
safe radishBOT
hi can someone help me with a proof
dark sluice
.close
safe radishBOT
lean otter
.close
stray bluff
what is $x * x$
flat frigateBOT
cantprogram
safe radishBOT
@formal pumice Has your question been resolved?
fiery tree
How exactly do I factorise this expression?
safe radishBOT
How exactly do I factorise this expression?
violet sundial
to make it easier to understand let n= (y+1)
so it looks like n^2 - 4n
@fiery tree
fiery tree
Yupp makes sense
violet sundial
alg
fiery tree
uh then where do I go from there š
fiery tree
like n^2 -4n simplifies to n(n-4)
er I've got the answer here from my homework site and I'm not quite following, I'm just stuck on the method and the steps of the problem
Still a lil confused š
violet sundial
then you have to substitute (y+1) back into n
from n(n-4) you get ((y+1)(n-3)
MAKES SENSE
fiery tree
Ahhh
violet sundial
sry for caps
fiery tree
Yup I've got the hang on it now I think
molten finch
safe radishBOT
molten finch
Hello guys
I am not sure what the question is asking in this problem am I just need to take the derative of the function twice
Or do I need to find the anti deravativr#e
<@&286206848099549185>
static ledge
Or do I need to find the anti deravativr#e
for f''(x)=12+3e^x, you want to find the second antiderivative
static ledge
yup
safe radishBOT
@molten finch Has your question been resolved?
safe radishBOT
safe radishBOT
Available help channel!
austere kestrel
safe radishBOT
austere kestrel
can someone verify my answers to see if theyāre correct š
austere kestrel
i think that was a typo
itās supposed to be 4
bc in our worksheets we used 33
lean otter
yeah i mean 1-7 seems fine if u managed to figure out the formula
which seems like u did
lean otter
well it just means that
as t increases, v decreases
and as d increases, v increases
reverse is true too
so what do u think it is? @austere kestrel
lean otter
mm going over them, i am kind of doing it in my head doe so might not be completely right proofreading
not sure what 11 is asking for
austere kestrel
alright alright iāll see tomorrow
Iāve already reviewed everything
tysm š„°
lean otter
yeah i went over it
it seems correct for the most part, even if u did something wrong, it is probably something minimal i didn't notice
good job
safe radishBOT
@austere kestrel Has your question been resolved?
fiery valley
why is 1/2?
safe radishBOT
why is 1/2?
quasi bison
the permutations in which z comes before a can be matched up one to one with the ones in which z comes after a
fiery valley
Can you please do an example?
lean otter
Can you please do an example?
For a given permutation in which z preceds a, there exists a permutation with all the letters at the same position but that of z and a interchanged
fiery valley
permutation in which z preceds a = 24! ?
permutation with all the letters at the same position but that of z and a interchanged = 1?
<@&286206848099549185>
safe radishBOT
@fiery valley Has your question been resolved?
noble sail
When is the algebraic multiplicity greater than the geometric multiplicity for an eigen vector
safe radishBOT
When is the algebraic multiplicity greater than the geometric multiplicity for a...
noble sail
what's the intuition
floral hornet
Where did the 4/3 come from
safe radishBOT
Where did the 4/3 come from
vestal pagoda
volume formula
lean otter
hmmm i think if you are asking for its derivation, integrating using disk method is what i can think of
but i bet thats not what you are looking for haha
lean otter
big A š
vestal sedge
ive always wondered where the volume for a sphere is derived from
floral hornet
lean otter
can really only think of cross sections
vestal sedge
can really only think of cross sections
same tbh
vestal sedge
Ok also where did the x^2 + y^2 come from
pythagoras?
floral hornet
Bro i rly donāt remember anything from algššš
Tysm
Wow this was embroiled
Embarrassing
lean otter
calculus be like: i forgor š
vestal sedge
calculus be like: i forgor š
real
lean otter
also never said it, but cultured kel enjoyer šš» @vestal sedge
floral hornet
.close
vestal sedge
ty ty
leaden scaffold
Is this just finding the derivative of the function or is there more to it?
vestal pagoda
finding derivative and settings x to 1
pseudo scroll
yeah
vestal pagoda
f'(1) = f'(x = 1)
leaden scaffold
finding derivative and settings x to 1
So whatever I find for the derivative of the function, I set it equal to 1?
pseudo scroll
no you sub x = 1
leaden scaffold
no you sub x = 1
So I find the derivative of 1^20e^1?
pseudo scroll
So I find the derivative of 1^20e^1?
no
differentiate first
and in the derivative sub x = 1
leaden scaffold
thanks!
safe radishBOT
@leaden scaffold Has your question been resolved?
ashen gust
I have a packing problem where I want to find an optimal layout for maximizing the number of nodes in a 3D grid where the rule is that nodes should be within a taxicab distance of at least 5 of each other. Any idea how to go about it? And do you expect that for this distance it's intractable?
safe radishBOT
I have a packing problem where I want to find an optimal layout for maximizing t...
ashen gust
Yes, I feel like it should be some way to find it. In one dimension, the repeating pattern has 2 nodes, for 2D it has 4 nodes. I am assuming I need a pattern of 8 nodes.
The only way I can think of solving this problem is to maybe brute force it but... That's probably not a good idea.
(I have zero experience with packing, maybe there are better approaches.)
buoyant shadow
ashen gust
How does one even prove what the optimal packing is other than going through all permutations?
buoyant shadow
so like "the lattice generated by the vectors (2/3,1,1/3), (ā1/3,ā2/3,1), and (ā1,1/3,ā2/3)" may mean you;re right, it's a pattern of 4 nodes
i don't know
safe radishBOT
@ashen gust Has your question been resolved?
safe radishBOT
@ashen gust Has your question been resolved?
safe radishBOT
@ashen gust Has your question been resolved?
compact ledge
help pls
vestal sedge
whats the question?
safe radishBOT
@compact ledge Has your question been resolved?
compact ledge
whats the question?
Oh nvm
vestal sedge
so you figured it out yourself?
compact ledge
yes
safe radishBOT
@compact ledge Has your question been resolved?
safe radishBOT
safe radishBOT
Available help channel!
flint coyote
safe radishBOT
untold sky
factorize the insides of the cuberoots to find things you can pull out of them
untold sky
yes and yes
flint coyote
ok
then what do I do
so its 3*2
and a 2 stays on the inside
for the first one would it be 6 3sqrt2^2
untold sky
correct, what about the second term?
flint coyote
correct, what about the second term?
-15 3sqrt 4
then would i just simplify the numbers on the outside of sqrt since the numbers in the square root are the same
so final answer -9 3sqrt4
untold sky
also it's cube root, if you keep calling it square root it might lead to some issues eventually
fierce stirrup
Hi, how do I do linear extrapolation?
safe radishBOT
Hi, how do I do linear extrapolation?
fierce stirrup
I have one point which is x0=1 and I have the function f(x)=x^3
I need to find the approximation at f(x) when x = 1.01
is it just f(x0) = 3 and since x0=1 then we can have g=0.01 and then that would be 0.01*x0
1.03?
lean otter
Are u talking about linearization?
Ig that's the same name
I have a general question. Is the Jacobian matrix thought of as a linear transformation that changes the unit basis vectors at each point in space to create a new coordinate plane?
Anyways @fierce stirrup
where those unit vectors may or may not be "similar" at each point
L(x) = f(a) + f'(a)(x-a)
Take a = 1
obtuse plover
@lean otter #āhow-to-get-help
fierce stirrup
so L(1.01) = f(1) + f'(1)(1.01-1)
So I got L(1.01) = 3 + 3(0.01)
which is 3.01
3.03
lean otter
Yeah I mean
Approximate enough
Linearization is just first order Taylor series but thats a spoiler for the future 
lean otter
Oh lol
fierce stirrup
I'm going to watch the recorded lecture for this, hopefully it is helpful, if not I'll be back š
thanks for the help
.close
split fulcrum
safe radishBOT
split fulcrum
For part B itās the same steps as part A?
But how would the line y = -1/2x change this equation formula wise
safe radishBOT
@split fulcrum Has your question been resolved?
split fulcrum
.close
lean otter
Doing rational root theorem. If the problem in question has no trailing coefficient (shouldnāt be a x term) what do I do?
safe radishBOT
Doing rational root theorem. If the problem in question has no trailing coeffici...
untold sky
well, if the coefficient is 0 you can factor out x from each of the terms and obtain a smaller polynomial with a constant coefficient
lean otter
well, if the coefficient is 0 you can factor out x from each of the terms and ob...
can you express this with an example
Iām trash without an example for it usually just words donāt work for me
like in the context of above what should I try to do
untold sky
for example:
a polynomial x^3-5x^2+6x
as there are no constant coefficient, you can factorise x from each of the terms to get
x(x^2-5x+6)
notice how you now have a polynomial of degree 2 with a constant coefficient inside the brackets
lean otter
for example:
a polynomial x^3-5x^2+6x
as there are no constant coefficient, you...
final question what should I do when after synthetic division there is 4 terms
Bottom question
Past the squiggly line
safe radishBOT
@lean otter Has your question been resolved?
compact egret
I'm so
compact egret
I'm so stuck on this shit $$(P \land R) \lor [ \neg R \land (P \lor Q)]$$ $$[(P \land R) \lor (\neg R \land P) \lor (\neg R \land Q)]$$
flat frigateBOT
studying lang calculus
compact egret
<@&286206848099549185>
lean otter
Help me
safe radishBOT
@lean otter Has your question been resolved?
lean otter
I need help
safe radishBOT
@lean otter Has your question been resolved?
uneven gale
So our math teachers decieded to ruin our week by telling us to explain a picture using ratio, fractions, probability, and rates
safe radishBOT
So our math teachers decieded to ruin our week by telling us to explain a pictur...
heavy terrace
just put a picture of a piechart lol
safe radishBOT
@uneven gale Has your question been resolved?
uneven gale
i mean
safe radishBOT
safe radishBOT
Available help channel!
short sparrow
I've asked this before but I only know how to prove it for a chosen n, not all.. here it is again:\
\
Let $A \subseteq \mathbb{R}$ be nonempty and bounded. Show that there are monotone sequences $(a_n){n\in \mathbb{N}}$ and $(b_n){n\in \mathbb{N}}$ with $a_n, b_n \in A$ for all $ n \in \mathbb{N}$ with
$$
\lim \limits_{x \to \infty} a_n \text{ sup}A \quad \text{ and } \quad \lim \limits_{x \to \infty} b_n \text{ inf}A
$$
safe radishBOT
I've asked this before but I only know how to prove it for a chosen n, not all.....
flat frigateBOT
Levens
solar hazel
did you understand what i wrote before?
short sparrow
yeah not really.. i didnt quite understand how that squeeze theorem was connected to that one line youve written.. wait let me look it up
here it is
ashen kraken
short sparrow
also, idk if im even allowed to use that squeeze theorem, if we havent addressed it yet in our lectures
solar hazel
can you at least see why (a_n) converges to sup A?
short sparrow
can you at least see why (a_n) converges to sup A?
where would i see that?
solar hazel
like even if you don't have proof yet
short sparrow
you mean by looking at the question?
solar hazel
just from looking at the way i constructed the sequence
toxic stratus
basically "do you understand why it should work"
short sparrow
no actually
short sparrow
ye
solar hazel
the sequence (a_n) whatever it is will satisfy
1 - 1/1 <= a_1 <= 1
1 - 1/2 <= a_2 <= 1
1 - 1/3 <= a_3 <= 1
1 - 1/4 <= a_4 <= 1
and so on
do you see why those conditions force it to converge to 1?
short sparrow
okay i see that
solar hazel
here it is
so does it sounds promising that (a_n) converges to sup A?
short sparrow
yeah, if it's monotonically increasing
short sparrow
where $a_n < a_{n+1}$
solar hazel
yeah, if it's monotonically increasing
well actually no, (a_n) converges to sup A regardless of that
flat frigateBOT
Levens
short sparrow
well actually no, (a_n) converges to sup A regardless of that
then shouldnt it be the same for any b_n?
toxic stratus
no need for monotonic
solar hazel
the question requires them to be monotone
short sparrow
but it doesnt say whether theyre increasing or decreasing
solar hazel
yea
short sparrow
convergence doesnt require it tho
yea
short sparrow
well actually no, (a_n) converges to sup A regardless of that
so why wouldnt this apply for b_n
solar hazel
wdym?
solar hazel
here it is
you don't have this if you replace sup A with inf A
short sparrow
what do you have instead? infA + (1/n) >= b_n >= infA?
solar hazel
sure
short sparrow
how did you come up with that tho
solar hazel
idk, i think i had this exact same problem once without the monotone requirement