Available help channel!
#help-17
vocal sleetBOT
Available help channel!
uneven wave
vocal sleetBOT
silent heart
Suppose that we have N cars parked in a line occupying spaces 1 to N. Spaces N+1 to 2N are empty. Every minute a car is considered eligible to move forward one square if a) the space in front of them is empty and b) their number is less than 2N. Suppose that every minute, if a car is eligible to move it has a 1/2 chance of moving forward by one spot. What is the expected amount of minutes before the cars occupy spots N+1 to 2N?
vocal sleetBOT
Suppose that we have N cars parked in a line occupying spaces 1 to N. Spaces N+1...
glad python
whats the expected number of minutes for the car at N to get to 2N? what about the car at N-1 to get to 2N-1, or more generally whats the expected number of minutes for a car to move N spaces forward?
silent heart
whats the expected number of minutes for the car at N to get to 2N? what about t...
Easier said than done with each car is dependent on the one(s) in front of it
Finding the first car is easy but not the others
glad python
it doesnt matter by linearity of expectatio
silent heart
Thats not true
theyre dependent on each other
A car cannot move up if the spot ahead is taken
vocal sleetBOT
@silent heart Has your question been resolved?
vocal sleetBOT
@silent heart Has your question been resolved?
digital shell
vocal sleetBOT
digital shell
can someone please explain why i can't take out a 16
and factor this out
they're pulling out j an h even though 16 is a gcf of all 3 coefficients in this problem
fossil ginkgo
You can
digital shell
Doesn't that j make it harder to cancel out the h's
When evaluating derivatives
And shit
fossil ginkgo
Ok
fossil ginkgo
What are you confused about
digital shell
Or any of that
fossil ginkgo
Sure
digital shell
I'm using difference quotient right
So
How would I pull a 16h out
Bc don't I end up having to cancel out the h's
fossil ginkgo
Huh?
digital shell
And leaving the 16 in the probme
Like
When you have a DQ you need to cancel out h's num and denom
fossil ginkgo
Yes
digital shell
If you pull out a 16 doesn't it make it undefined
fossil ginkgo
Make what undefined?
digital shell
the porblem
vocal sleetBOT
@digital shell Has your question been resolved?
rugged shore
How did they get 0.0003 and 0.7109
vocal sleetBOT
How did they get 0.0003 and 0.7109
frozen bobcat
it says use technology, so either a calculator or whatever program comes with your homework
,w P(z<-3.444)
twin meteorBOT
rugged shore
it says use technology, so either a calculator or whatever program comes with yo...
The calculator I have is BA II Plus
And I don't think it has special keys
Would it be impossible to get that answer without a special program
frozen bobcat
you could also use a z-table (probably)
rugged shore
How could I use a z table for this
frozen bobcat
Go the the table and find -3.4 on the left column,
Then follow that row to the .04 column
The value there is probability for the z<-3.44
rugged shore
Go the the table and find -3.4 on the left column,
Then follow that row to the ....
What about the 0.7109
frozen bobcat
that z-score is too close to the center to get a very accurate answer using the table, you'd need a calculator to do it.
look up your calculator and see if it has stat functions
rugged shore
I have STAT on my calculator, not sure if that's what you're looking for
rugged shore
look up your calculator and see if it has stat functions
If I were to use the Alpha Wolf, how would I search for it
<@&286206848099549185>
karmic imp
If I were to use the Alpha Wolf, how would I search for it
They meant look up the manual for your calculator
You're not going to have wolfram alpha on your exam
rugged shore
You're not going to have wolfram alpha on your exam
You're right.. How would I get 0.7109 using the calculator without Alpha Wolf
karmic imp
The z table
karmic imp
How did they get 0.0003 and 0.7109
You're given z = 0.556
rugged shore
Yes
karmic imp
So use the table
karmic imp
The site rounds differently
Do you notice how your answer and the site are super close?
It's fine
frozen bobcat
You're right.. How would I get 0.7109 using the calculator without Alpha Wolf
google your calculator and look up if it has stat functions
karmic imp
Yeah but it still tells me that it is incorrect
That website has a weird tolerance. The process you are doing is fine
rugged shore
I hope that if I do the test, my prof. will give me marks for it LOL
vocal sleetBOT
@rugged shore Has your question been resolved?
karmic imp
I used the z table and didn't get -2.054
You're suppose to reverse engineer it, find the z score that has teh z value of 0.02
rugged shore
You're suppose to reverse engineer it, find the z score that has teh z value of ...
I tried finding 0.02 different ways and couldn't find it on the z table
karmic imp
You use the value that's closest to 0.02 but not more
karmic imp
Located -2.05
-2.05 does not have a z value of 0.01970
karmic imp
Yes that's the closest to 0.02, not sure what the tolerence of the site is
rugged shore
How would I search on Alpha Wolf to get that last digit 4
karmic imp
You understand that when it says "use technology", it doesn't mean wolfram alpha
It means use a calculator
rugged shore
I don't think my BA II Plus has a function for it
flat whale
I don't think my BA II Plus has a function for it
Did you Google it to be sure
rugged shore
Did you Google it to be sure
Yes. I had a STAT function but it doesn't solve for what I need
Is it possible to do it using Excel?
karmic imp
Google that
rugged shore
Did you Google it to be sure
What would I have to type in the search engine of Alpha Wolf if I were looking for the score z when the left tail is 0.02
.close
manic drum
How does sin work
vocal sleetBOT
How does sin work
twin meteorBOT
mybacon
manic drum
but how does sin work to get the answer
hidden kelp
but how does sin work to get the answer
You can't really get a nice answer except for some specific values of the input
manic drum
hm
hidden kelp
The way sine works is that it's the ratio of the opposite over the hypothenuse in a right triangle
I'll draw a quick sketch for you if you want
hidden kelp
<@&268886789983436800>
trim walrus
Bruh
karmic imp
<@&268886789983436800>
trim walrus
hidden kelp
Get a load of this idiot
hidden kelp
Idiot annihilated woot
quasi oxide
bro got destoryed by mods
lost yarrow
manic drum
:0
okay
@tulip forge
bro didnt even get banned
or kicked
lets carry on how ever 🤓🤓
manic drum
hmm
okay
well
thank you
also
last question
can @twin meteor solve equations?
like if u put 1+1 in it will it tell u 2
manic drum
,w 1+1
twin meteorBOT
worthy citrus
,w solve x + 1 = 70
twin meteorBOT
karmic imp
Yes but tbh just use a calculator or wolfram alpha itself
twin meteorBOT
Result:
69
,calc 6969696969696969696969696969696969696996966996696969*696969696969696
twin meteorBOT
Result:
2.3596943598448e+66
,w \frac{3}{6}
twin meteorBOT
errant pine
My friend said If i complete this problem i will recieve a prize, the problem is i didnt learn it yet in School so i dont know what to do
The question: Ryuu Iisou in the first round
Knowing that in a game of mahjong there are 144 pieces, and that only 24 of them can be arranged to make a ryuu iisou (combination that uses only green pieces), and considering that these 24 pieces are composed of 6 squares of pieces, respecting the rules that the ryuu iisou must be done forming 4 three of a kind or sequences and 1 pair of these pieces, what is the probability that in your first draw of the game you will be able to complete the hands of 14 pieces?
Given the addendum that, of these 24 pieces, there are 4 pieces of each type that must be combined to form with the numbers or doubles, trios or fours or sequences of three consecutive numbers, the numbers involved being (2, 3, 4, 6 and 8) and 4 colored pieces that can only form pairs or triplets, just like the 6 and 8, there cannot be sequences of 2 numbers or single pieces.
vocal sleetBOT
My friend said If i complete this problem i will recieve a prize, the problem is...
@errant pine Has your question been resolved?
dark kiln
too much work
vocal sleetBOT
@errant pine Has your question been resolved?
vast shale
On Jerry's text messaging phone plan, she paid $0.15 for every 15 messages she sent. What is the her cost per message?
vocal sleetBOT
On Jerry's text messaging phone plan, she paid $0.15 for every 15 messages she s...
vast shale
How do I create a function with this information or graph or smth
safe sable
You don't actually need that stuff, though you could make it if you want. The hint here is in the unit: "cost per message". In units, "per" typically means "divided by"
dreamy viper
yeah
so the current rate is 0.15/15
$0.15/15 text messages
just need to make that rate be for every text message instead of 15
lapis hollow
is maths hard to learn?
lapis hollow
i am having trouble understanding Expanstion Formula
dreamy viper
lapis hollow
oh sorry my bad im new here
vast shale
just need to make that rate be for every text message instead of 15
to create a function for this it would be f(x)=(0.15/15)x right?
mild flower
sure
dreamy viper
You don't actually need that stuff, though you could make it if you want. The hi...
id refer to this too
vast shale
but really, not necessary
well im studying for my placement test at my college and they provided a bunch of questions on some site to practice on and this was the question
safe sable
oh in that case then yeah, that "rate" is the slope
dreamy viper
you can eliminate any graphs whose y intercept isnt 0
safe sable
in the context of real life things, the slope of a line is usually some sort of rate
vast shale
you can eliminate any graphs whose y intercept isnt 0
ok so what about hte leftovers?
dreamy viper
do you have the slope yet
vast shale
nope
dreamy viper
find it then
dreamy viper
its just the rate
vast shale
its just the rate
and how do you get that?
safe sable
the cost per message is the rate
dreamy viper
that function is in the form of mx+b where b is 0
m is the slope or rate
or some people call it gradient
vast shale
oh so the slope/rate is 0.01
vast shale
do you have the slope yet
so yes, i have it now, how does that help me determine which one it is?
dreamy viper
whats the slope
oh
yeah 0.01
since its something nice like 0.01 we can multiply that for certain values of x and see if they match where their output should be
look at the first one
plug in 50
do yo get 0.5 as an output?
vast shale
yes
vast shale
yes
dreamy viper
no
dreamy viper
looks like maybe 0.2 or 0.3
dreamy viper
do the same for the other two
vast shale
nope, 50 for the second graph is more like 1.5 and 50 for the bottom graph is like 0.5
dreamy viper
so you have your answer
dry forge
how do i find the factors of this equation, i dont have access to a calculator
dry forge
i feel like im missing something, its too tedious to use formula, and im not sure how to split middle term
flat whale
,w factor x^2-101x+488*5
twin meteorBOT
flat whale
Ah yes, trivial
mild flower
lol
flat whale
I think you're supposed to factor 488 first
mild flower
if you put it into prime factorization you can "reasonably" quickly try all of them
flat whale
Then find all pairs of products that multiply to 488*5 that sums to 101
brisk moss
what’s so tedious about quadratic(?) formula
dry forge
ig formula would be faster at that point yea
flat whale
You know quadratic formula?!
dry forge
oh this is an advanced question, this was just a minor part of the full problem lol
mild flower
oh yes i love calculating checks notes the square root of 101^2 + 4 * 488 * 5
dry forge
this was the actual problem, but it led to the formation of that quadratic equation
and we cant use calculators, so not sure if there is a way to avoid the quadratic
flat whale
oh this is an advanced question, this was just a minor part of the full problem ...
Ah yes advanced
dry forge
i meant more advanced than just a quadratic problem 😭
dreamy viper
isnt there an equation you can use to instantly find the nth term
flat whale
Oh true
dreamy viper
or what are you supposed to find
mild flower
isnt there an equation you can use to instantly find the nth term
uh, it's probably the quadratic that we started with...
i'm assuming we're supposed to find n here
dreamy viper
uh, it's probably the quadratic that we started with...
i didnt think it would turn into that
gritty ivy
isnt there an equation you can use to instantly find the nth term
probably one of the summation ones?
Am I wrong?
dry forge
we're supposed to find n
dry forge
Am I wrong?
yea thats what i did
mild flower
,calc 101^2 + 4 * 488 * 5
twin meteorBOT
Result:
19961
,w sum n=0 to 39 of (100-2n)/5
twin meteorBOT
mild flower
why does 61 as well
flat whale
Goes up and then back down
mild flower
oh duh
twin meteorBOT
flat whale
Jk what do I know
civic drift
Factor out 1/5?
dreamy viper
what
dry forge
this was the actual problem, but it led to the formation of that quadratic equat...
answer is only 61 yes?
it says nth term is negative
mild flower
oh that would have been a way to do this without factoring a big quadratic
add up all the positive terms first
merry python
it says nth term is negative
nth term is negative should imply that its the 61th term only
since 40th term would be positive
dry forge
add up all the positive terms first
...what like manually? isnt that also extremely long
dry forge
nth term is negative should imply that its the 61th term only
yep 61 is the answer
dry forge
...what like manually? isnt that also extremely long
@mild flower
mild flower
...what like manually? isnt that also extremely long
kinda? it's "easy" to see which term is 0, so you know there are 50 positive terms and finding the sum of the first 50 is probably easier than solving a quadratic
but like overall this might be harder
dry forge
hmm ok ok
anyways yea no shortcut to this question it seems, just a tedious question, not that difficult
thanks everyone
.close
vocal sleetBOT
vocal sleetBOT
Available help channel!
elfin moon
In a triangle ABC DE parallel BC and AD/DB =4/5 and DE=12cm then Bc?
elfin moon
If we see congruence
ADE and ABC
AD/DB=DE/BC
By this property we see 4/5 =12/x
So x = 15
But this is incorrect
I don't know why
Answer is 27
tidal umbra
outer warren
did yuo draw a diagram
outer warren
the issue is that 5 isn't the side length of theb ig triangle
outer warren
you want
AD/AB = DE/BC
vocal sleetBOT
vocal sleetBOT
Available help channel!
wary mantle
Determine the set of all $x \in \mathbb R$ so that exactly two of the following statements are true: \begin{enumerate} \item $x > 1$ \item $x \leq 2$ or $x^2 < 9$ \item $x$ is a whole number\end{enumerate}
vocal sleetBOT
Determine the set of all $x \in \mathbb R$ so that exactly two of the following ...
rotund junco
Yes thats 2 differenet sets
wary mantle
rotund junco
You need to find alls sets of x values
wary mantle
Yes thats 2 differenet sets
We just need to find one set though, don't we? What do you mean?
rotund junco
We just need to find one set though, don't we? What do you mean?
Sorry i misread the question
vocal sleetBOT
@wary mantle Has your question been resolved?
wary mantle
<@&286206848099549185>
proven garden
We could let 1 and 3 or 2 and 3 be true, no?
we check all cases
wary mantle
we check all cases
What do you mean?
proven garden
find 1,2 are true and 3 is false,then 2,3 are true and 1 is false, then 3,1 are true and 2 is false
there might be some smarter way though
vast shale
Yes?
wary mantle
find 1,2 are true and 3 is false,then 2,3 are true and 1 is false, then 3,1 are ...
We can create sets and then take their union.
- A = (1, infty)
- B = (-infty, 3)
- C = x in Z
dull maple
Just find all sets and then use set theory stuff. Venn diagram must be easiest.
wary mantle
So $A \cap B = (1, 3)$
3 wouldn't be fulfilled if we took that set
so this would be a posible solution, woudn't it?
wary mantle
Just find all sets and then use set theory stuff. Venn diagram must be easiest.
Wouldn't we have multiple possibilities?
For example the one I described above would be one, no?
worthy citrus
Yeah this is just going to be pairwise intersections of the conditions
wary mantle
Yeah this is just going to be pairwise intersections of the conditions
Is it really correct to say that for 2., we have the set $B = (-\infty, 2] \cup (0, 3) = (-\infty, 3)$?
worthy citrus
Yes everything in there is either less than 2 or squares to less than 9
wary mantle
Yes everything in there is either less than 2 or squares to less than 9
Thanks. So we'd have three possible solutions, right? \begin{enumerate} \item $A \cap B = (1, 3)$
\item $A \cap C = {x \in (1, \infty) \ : \ x \in \mathbb{Z}}$ \item $B \cap C ={x \in (-\infty, 3) \ : \ x \in \mathbb{Z}} $\end{enumerate}
worthy citrus
You don't mean Z^+
wary mantle
You don't mean Z^+
These three would all be correct to answer here, right? We can just pick one out of them
worthy citrus
Right but you want the set of all x such that exactly 2 are true
Naively you would just take the union, but you have to now deal with the overlaps where all 3 might be true
Basically when we made a set such that say 1 and 2 were true, we didn't insist that 3 should not be true
Hence our overlap problem now
wary mantle
Naively you would just take the union, but you have to now deal with the overlap...
Do we need to make sure that there are always exactly two statements out of them that are true?
worthy citrus
Do we need to make sure that there are always exactly two statements out of them...
According to the question, yes
wary mantle
Then for 1., we'd need to exclude the integers for example
worthy citrus
Exactly, e.g. 2 would satisfy all 3
worthy citrus
I think that checks out
exotic mason
I used power reduction on 20(sin^4)x, is this the correct answer?
exotic mason
viral copper
,w integrate 20sin^4 x
twin meteorBOT
viral copper
Hmm
exotic mason
should I try plugging in a number into 20sin^4x
viral copper
Well the derivative of this should be your answer actually
exotic mason
and plugging in the same one in my answer
viral copper
Because Wolfram uses power reduction to integrate
So the derivative of the answer should be equal to your expression
Which it is
exotic mason
ok
so i don't know how to do derivatives
so how can I verify these answers in the future
viral copper
plug in values as you said I suppose
vast shale
vocal sleetBOT
vast shale
This would be false, correct?
scenic ravine
Wait, is this a test?
vast shale
nope, its an exit ticket
scenic ravine
what's that? If I help you I won't be helping you cheat, right?
vast shale
no you wont
elfin moon
False
scenic ravine
yeah, you're right
regal slate
an exit ticket is basically a test
vast shale
it isn't
regal slate
how come? it’s a measure of your abilities and understanding at the end of a class?
vast shale
it's not for any marks tho
its basically just hw/review questions that i wanted to confirm my answer
.close
full hatch
How is the euclidian norm defined?
proven knot
What is the norm defined on these vector spaces
i.e. when you take the magnitude of the vectors
How do you calculate here
There are different ways to do that depending on how they define the norm
Ok so then can i assume when they say
|| <2, 3> ||
You would do sqrt(4+9) = sqrt(13)
Is that correct
Yes they are
|| ******* ||
Means take the magnitude, or norm, of the vector
You have to evaluathe norm/magnitude of both vectors and see if they are equal
Magnitude is a number (scalar), not a vector
Magnitude tells me how long a vector is basically
So you can have many vectors that point in different directions with the same length
Well there are different ways of defining magnitude
The "usual" way is called the Euclidean Norm
This is akin to using distance formula with the components
Yes exactly
So whether you transpose the vector or not
The norm should be the same for the euclidean Norm
The real question is are you actually using the Euclidean norm for these problems?
To find norm of the above you'd do
Sqrt([-2]^2 + 0^2 + (2)^2]
= sqrt(8)
Yes
In fact, no non zero.vector should have zero magnitude
Because if any component is non zero then the vector would have some.length to.it
Assuming you are using the typical norm/magnitude yes
Magnitude/euclidean norm is
Square Root of the Sum of the Components Squares
They would define another norm for you and how to find it
Which if they havent mentioned that at all its probably the usual norm
b is b
You leave them in terms of b and a
You dont get it exactly
But maybe you can show
Sqrt(Of stuff on left) = Sqrt(of stuff on right)
It doesnt matter what b or a actuall equal
Real valued vectors can have any number in their components
A, b, c...
These are just names
There is some number called b in the first problem
That number is the third component of the vector on the left
b doesnt mean the second one or choice b etc
It has no significance beyond being a name
Well whether you call it 2 or b
To square it
I call
2^2. or b^2
You cant simplify the root of this butttt
Maybe you can simplify inside the root
If i can say that i have
Sqrt(x) = sqrt(c)
x = c
Regarldess of whether i can evaluate or simplify the root
That T means "transpose"
Its an operation that takes matrices and vectors and turns their columns into rows or vice versa
You can think of it rotating the numbers of the vector by 90 degrees so it makes a column
Yea its irrelevant for this one
To take the magnitude take each component and square it
Then add them under the root
Lets start on the left
(8a)^2 the whole.thing is squared
We get
64a^2
[2a]^2 = 4a^2
b^2 = b^2
Add them together we get
68a^2 + b^2
So magnitude is
Sqrt(68a^2 + b^2)
For the left
Ok great
Now carefully do the same on the right
If a component is squared it is the ENTIRE component
b^2 ofc = b^2
But what about
Sqrt(68)*a
What happens when you square this component?
Sqrt(68)
Square that number
Sqrt(2)^2 =/= 4
Sqrt(2)^2 = 2
When you square a square root
The root and the exponent cancel
Yessss
So when you add it all
Are they equal
Yesss
Nice indeed
There are 2 components with a
8a
And
2a
Square each one
64a^2
4a^2
Yes
They are like terms, both a^2
So when we add we can combine them
Yw!
Im glad to hear!
hoary frost
Are the eigenvalues of a linear operator f and its matrix representation A always the same?
vocal sleetBOT
Are the eigenvalues of a linear operator f and its matrix representation A alway...
hard atlas
yes
hoary frost
i thought so too but why?
hoary frost
modulo coordinates
what are those
hard atlas
the thing with respect you make matrix representations?
hoary frost
oh yeah
hard atlas
x vs [x]_B or however you denote it
hoary frost
usually as phi
hoary frost
.close
vast shale
is my logic sound here?
vocal sleetBOT
is my logic sound here?
vast shale
the standard matrix should be [T(e_1) T(e_2)] i believe
vast shale
yeah i guess its better to call T(x) the transformed vector or something
worthy citrus
the standard matrix should be [T(e_1) T(e_2)] i believe
this here is the matrix yes
opal obsidian
vocal sleetBOT
opal obsidian
Just wanted to verify if my reasoning is correct
We have considered a, 2a, 3a, ... (p-1)a elements
the least residues modulo p are from 0 to (p-1)
as by def p doesn't divided a, so we only have 1 to p-1 residues to choose
so if {a, 2a, 3a, ... (p-1)a} has same number of elements as to the least residues of {1,2,3,....,p-1}
so if i can show none of the integers in {a, 2a, 3a, ... (p-1)a} are congruent to each other or zero then both are have same set of residues
is that correct reasoning?
vocal sleetBOT
@opal obsidian Has your question been resolved?
frozen bobcat
yes that is correct reasoning
opal obsidian
I'm trying to extend this to understand euler's generalisation
Why does showing (ari, n) and no two elements are congruent to each other
necessarily prove that they share same set of residues as {r,r2,r3,.... , rphi(m)?
i understand proving no two congruent to each other makes sure the set of residues of ar1, ar2,..arphi(n) have same in number as r1,r2,...rphi(n)
frozen bobcat
if (ari,m) = 1 then they share no common factors. So no prime factor of m divides ari. So you can apply part 1 of the proof above. Then multiply all those primes and their residues together and you have the residue of ari mod m is nonzero
opal obsidian
i understand showing (ari, m) =1
i was wondering how does showing that necessarily show they share same residues
i think im missing something here
opal obsidian
so if {a, 2a, 3a, ... (p-1)a} has same number of elements as to the least residu...
here i can guarantee because there are only 0 to p-1 choices and residue zero is avoided
so any number that I consider thas to one of 1 to p-1
how do i see that if for any k, if (k,m)=1, then least residue of k modulo m is also relatively prime to m
@frozen bobcat
frozen bobcat
ok, take m = pq for primes p and q.
and assume (k,m) = 1
opal obsidian
sure
frozen bobcat
then k mod p is non-zero, otherwise p divides k and (k,m) is not 1
similarly, k mod q is non-zero.
so k mod pq = k mod m cannot be 0.
opal obsidian
also denote r as the least residue of k modulo m?
frozen bobcat
otherwise k is a multiple of pq, and you'd have had k mod p or k mod q are 0.
opal obsidian
i understand taking modulo m doesn't have residue ==0
frozen bobcat
then the logic is basically the same as your Lemma 7.2, none of the residues can be 0 and (with some more work) none can be equal, so the sets must be the same size.
opal obsidian
does showing none of residues can be zero and (k,m)=1 same?
frozen bobcat
basically, yes.
by showing (k,m) = 1 you are showing the residues are non-zero.
opal obsidian
thank you
vocal sleetBOT
@opal obsidian Has your question been resolved?
vocal sleetBOT
vocal sleetBOT
Available help channel!
void stone
can someone explain proof of heron's formula to me (without using trig, just good ol algebra)
vocal sleetBOT
can someone explain proof of heron's formula to me (without using trig, just goo...
urban edge
https://www.youtube.com/watch?v=9qbpmYqr4so heres one from blackpenredpen
vocal sleetBOT
@void stone Has your question been resolved?
void stone
https://www.youtube.com/watch?v=9qbpmYqr4so heres one from blackpenredpen
tysm!! everything falls together so nicely :O
vast shale
vocal sleetBOT
vast shale
Where do I start simplifying this?
I know to multiply the bottom by sqrt 18 and the top by sqrt 18
but
what do I do after I have 21*sqrt18/18
idk how to simplify further
safe sable
from there you need to simplify the sqrt(18) on the top. Do you remember how we can simplify square roots like that?
twin meteorBOT
AAces
vast shale
wait why does that matter?
karmic imp
Two different methods
safe sable
because we can then say $\sqrt{9\cdot2}=\sqrt{9}\cdot\sqrt{2}=3\sqrt{2}$
twin meteorBOT
AAces
safe sable
don't forget the 21
vast shale
63 sqrt 2 / 18?
safe sable
yeah, tho the 63 and 18 share a common factor so you could simplify the fraction
vast shale
so 7 sqrt 2 / 2
safe sable
yup!
vocal sleetBOT
Available help channel!
fossil dune
Prove the ff. reduction formula using the sum & difference identities
mild flower
it's a bit problematic because technically $\tan\f\pi2$ isn't defined
twin meteorBOT
Hayley
fossil dune
it's a bit problematic because technically $\tan\f\pi2$ isn't defined
Wdym?
mild flower
,w tan(pi/2)
twin meteorBOT
mild flower
oh it totally is
you just can't refer to tan(90º) directly
but you can use any other angle you want to talk about
well, not 270º either
fossil dune
you just can't refer to tan(90º) directly
So that means I can't use sum and difference identities?
mild flower
you can't use sum/diff identities with 90º as an argument
fossil dune
I see
mild flower
but you're welcome to use it with 45º, 30º, 60º, or 180º
or just about any other angle
or.... you can probably use limits if you know how to use those
but that might be harder
fossil dune
Is it possible to separate a two terms being added into smaller terms?
Like tan [(30+60) + x]
?
And then use sum and difference identity to prove it?
mild flower
yeah!
vocal sleetBOT
vocal sleetBOT
Available help channel!
vast shale
vocal sleetBOT
vast shale
Where did I make a mistake with the negative?
proven garden
you wrote $z^5+5z^3+10z-10z^{-1}-5z^{-3}-z^{-5}$ but it should be $z^5-5z^3+10z-10z^{-1}+5z^{-1}-z^{-5}$
twin meteorBOT
WhereWolf
woven sluice
How do I integrate this?
thin root
Determine the area of the region enclosed by $y=x^2$, $y= 2+x$, and $y =2-x$
LeGM
thin root
\begin{align*}
2\int_{0}^{4} \int_{0}^{\sqrt{y}} 1 dxdy
\end{align*}
Did I do this correctly?
split wind
on first thought i don't think so
twin meteorBOT
LeGM
split wind
it seems that you'll have to spilt it into 2 regions
split wind
hmmm
raw hollow
maybe work out the area underneath the 2 straight lines?]
split wind
is there anymore restrictions on x and y?
thin root
Determine the area of the region enclosed by $y=x^2$, $y= 2+x$, and $y =2-x$
twin meteorBOT
LeGM
thin root
this is the full question
split wind
so weird
civic otter
\begin{align*}
2\int_{0}^{1} \int_{x^2}^{2 - x} 1 dydx
\end{align*}
Alberto Z.
split wind
i was thinking it would be something like "in first Quadrant"
hasty cypress
civic otter
**Alberto Z.**
That's how I would do it
civic otter
The x is btw 0 and 1, not the y
raw hollow
idk how to explain it but the upper parts are irrelevant in your final answer?
civic otter
The y range btw the lower function and the upper one
thin root
civic otter
I don't think they ask for that region
thin root
idk how to explain it but the upper parts are irrelevant in your final answer?
it's still enclosed though
hasty cypress
You could find all just to show dominance?
thin root
You could find all just to show dominance?
?
hasty cypress
im joking sorry
civic otter
This is the region they are asking, in my opinion
obtuse sierra
why is this not enclosed?
these regions are enclosed by x^2 and one of the two lines
thin root
Determine the area of the region enclosed by $y=x^2$, $y= 2+x$, and $y =2-x$
twin meteorBOT
LeGM
civic otter
It's actually a 1-D integral, or a dummy double integral
civic otter
It could be that as well actually
But I think they ask you, implicitly, for a connected region
hasty cypress
it reality it dosen't matter just note your interpretation and as long as your solution is correct you should be fine 
civic otter
Yeah I agree
blissful needle
I need help with qn 12 thanks
blissful needle
then do substitution
Or is there another way
Cos for my learning syllabus we're not expected to find out what we need to sub by ourselves
blissful needle
Owait nvm I think I found out how to do
U can split them up right
Into 2 separate fractions
vocal sleetBOT
@blissful needle Has your question been resolved?
alpine mist
if possible, factor each of the denominators, then reduce to
same denominator “as small as possible”
vocal sleetBOT
if possible, factor each of the denominators, then reduce to
same denominator “a...
alpine mist
I think I did most of the job
But Idk what to do with the 2 and x which both factor the expressions
vocal sleetBOT
@alpine mist Has your question been resolved?
alpine mist
<@&286206848099549185>
nova plover
,rotate
twin meteorBOT
shrewd wolf
I like how you factored the denominators 👍
spare blaze
@alpine mist multiply 2nd term with x(2x-3) and the last term with 2(x-1)
That should give the same denominator
And then simplify numerator
twin meteorBOT
alpine mist
good ?
alpine mist
ah yeah my bad
spare blaze
On the numerator
alpine mist
2 instead of x
spare blaze
Yeah
alpine mist
the rest is good?
spare blaze
Yeah
alpine mist
Yeah
when i combine them all, for the denominator at the end i can either write x(2x-3)2(x-1) or 2(2x-3)x(x-1)
vast shale
Y'all down for differenciation
twin meteorBOT
obtuse sierra
no
oh wait
no, the calculation part is incorrect
whats 80 + 8?
- sorry
the expression was $-2(x^2 - 4x + 4)$ right?
twin meteorBOT
Dissrupt
obtuse sierra
which becomes $-2x^2 + 8x - 8$
twin meteorBOT
Dissrupt
obtuse sierra
now you can continue
twin meteorBOT
obtuse sierra
yep
vocal sleetBOT
vocal sleetBOT
hybrid fossil
its just written in a different form
vocal sleetBOT
Available help channel!
acoustic root
vocal sleetBOT
acoustic root
I thought of breaking it up into sets, but I don't think that'll work. Plus I'm not exactly sure how to do that anyway.
acoustic root
Yes, but I don't fully understand it
They just say the goal cannot be reached and I don't see why
hard atlas
do you know what invariant means
acoustic root
Unchanging
hard atlas
do you see why it doesnt change?
acoustic root
Oh, I was picking a middle number then increasing neighbors by 1. So it just meant like increase a_2 and a_3 by 1.
So, yh unchanging bc the neighboring term will just cancel the other's change.
It will forever remain I=2
acoustic root
0
It would have worked also if 1,1,0,0,0,0. And similar patterns
1,1,2,2,3,3
Ok
hard atlas
need to write it again
acoustic root
.close
vocal sleetBOT
vocal sleetBOT
Available help channel!
brisk patrol
I'm struggling to find a way to set up this question. Can someone help?
brisk patrol
I thought about manually counting the number of combinations where the twins are next to each other, and the other sister is at least 1 seat away. It works, but is there a better way?
So I put the twins together in each of the chairs, chairs 1 & 2, 2 & 3, 3 & 4 etc. And the sister elsewhere. When the twins are in chairs 1 & 2, the sister can be in 4, 5, 6, 7 or 8.
I counted 30 possibilities and there are 5! ways to seat the 5 boys in the other chairs.
so 30*5! = 3600.
<@&286206848099549185>
vocal sleetBOT
@brisk patrol Has your question been resolved?
dull maple
I thought about manually counting the number of combinations where the twins are...
There are other ways. They aren't as tedious but at the same time, they might not be as obvious to one as yours is.
If you want, i can walk you through it.
brisk patrol
If you want, i can walk you through it.
Yes please.
First of all, is my way correct?
dull maple
First of all, is my way correct?
Yeah
Okay. So, first thing that we'll make use of is what i call bagging.
brisk patrol
okay
dull maple
Wait. First, I just thought of this. In your case, you have considered twins to be same as you haven't considered ordering for them.
If they aren't same, then your answer should be 3600 * 2 = 7200
brisk patrol
If they aren't same, then your answer should be 3600 * 2 = 7200
oh yeah I missed that
thanks for pointing it out
dull maple
Okay. Good
dull maple
Okay. So, first thing that we'll make use of is what i call bagging.
Now, getting back to this.
It's basically this: whenever you have this condition of two objects/people always sitting side by side or being together, You can just sort of put them(imagine) in a bag and treat them as a single entity.
For example: A group of eight boys is to be seated in a row of 8 chairs. Two particular boys insist on sitting together:
We consider them as a single entity in a bag, so we are reduced to having 7 objects.
So, number of ways are 7! * 2!.
2! comes because those two boys can sit two ways inside the bag so this considers that ordering.
We can make use of similar thing for twins.
BRB. Getting a call.
brisk patrol
True
One of the example questions in my textbook explained that you can combine objects that must be together.
so if you have 8 seats, which have to seat 5 boys and 3 girls, and the girls insist on being together, you can bag the girls into a single entity that's 3!. Then you multiply it by 6! because there are 6 things to place (the bag and 5 boys).
dull maple
so if you have 8 seats, which have to seat 5 boys and 3 girls, and the girls ins...
Yes. That's right.
brisk patrol
Yes. That's right.
Is there anything more to add? or is this good
dull maple
Is there anything more to add? or is this good
That's good.
Just try your question now.
alpine mist
. After factorization(s), solve in ℝ the following equations
i can start by removing the ² then put everything on left side ?
outer warren
that is applicable later
first subtract (3x^2 - 4x -21)^2
from both sides of the equation
so that you'll have
stuff = 0
don't do any other simplification/manipulation at this stage
outer warren
you're missing the ^2
alpine mist
ahaha ok sorry
outer warren
$a^2 - b^2 \redneq (a-b)^2$
twin meteorBOT
ℝam()n()v
alpine mist
hmmmm
outer warren
note that the left side of your equation is a difference of two squares
there's a factorisation identify for that
outer warren
yes
obtuse sierra
yes
outer warren
How'd you get that +2 at the very end
outer warren
don't forget you started with an equation
alpine mist
=0
outer warren
from here you can apply zero product property
The stuff you posted earlier would be applicable to the quadratic factor
you should have no issue finding the root for the linear factor
alpine mist
from here you can apply zero product property
a*b=0 then a or b = 0 ?
outer warren
yes
alpine mist
yes
so i must do the stuff i sent before to find what would do zero in b
first solution is -12.5
outer warren
yes, that will be a solution
alpine mist
what do i do wrong @outer warren
i put 17 instead of 6
so the solutions are -12.5 ; (3-sqrt^111)/6 ; (3+sqrt^111)/6
vocal sleetBOT
@alpine mist Has your question been resolved?
alpine mist
anyone ?
vocal sleetBOT
@alpine mist Has your question been resolved?
cunning saddle
why dont ncel
vocal sleetBOT
why dont ncel
hard atlas
?
cunning saddle
(e^2x - e^-2x)
hard atlas
what about it
the numerator is of the form 2(a-b)^2 - 2(a+b)^2. can you simplify that?
frozen bobcat
you can't cancel only part of the numerator like that
cunning saddle
why
frozen bobcat
$\frac{a+b}{a} \neq 1+b$
twin meteorBOT
Zybikron
cunning saddle
i dont understand
frozen bobcat
you need (e^2x - e^-2x) in both terms in the numerator to cancel like you're trying to
you only have it in one of them. So you cannot cancel.
because in order to cancel you need it to be a factor of the numerator.
For example $\frac42 = \frac{2+2}{2} \neq 1+2 =3$
twin meteorBOT
Zybikron
cunning saddle
o thx
wind belfry
Hello, I have a pretty simple question I don't understand
I need to say true or false: (I hope my English make sense cuz idk English terms)
The binary operation match to every ordered pair (a,b) of natural numbers, all the numbers 'x' so x² = a+b is a binary operation on N(natural numbers)
wind belfry
Hopefully the English make sense
I really can't understand what does all the numbers "x" here means
I know that every ordered pair should have one matching result for a binary operation for a start
simple mason
I need to ask a simple question
what this question is about? cause i realy don't understand it
wind belfry
the subject is binary operations
silk kindle
Well x is not always a natural number, so it can't be a binary operation on N 🤔
by "all the numbers x" maybe they mean "both solutions of x^2 = a + b"
wind belfry
i think i understand it now, it was just writted in a way that went over my head. so i think if a = 2 and b =2 so a + b = 4 then if x is all the natural numbers(without 0) so x squared = a + b means that I have one x, x =2
But if x was all the whole numbers both positive and negative then I would have x =2 and x = -2
does that seem correct?
wind belfry
Well x is not always a natural number, so it can't be a binary operation on N 🤔
sorry that is my bad, there is 2 questions one where x can be all the natural numbers and the other where x can be also negative numbers
so I think the first is a binary operations cuz there will always be one x where x squared = a + b
and for negative numbers it isnt a binary operation cuz you can have 2 different x where x squared = a + b
silk kindle
for some values of a and b, x in x^2 = a+b is not a natural number
the first question says "binary operation on N", right?
wind belfry
i was about to say my bad again.. its all the positive numbers for one questions and then u include negatives for the second one
both of them on N
again sorry the english messed me up i got confused
does that seem right now?
silk kindle
ok dw
silk kindle
the first question says "binary operation on **N**", right?
this means a, b and x must be natural numbers
for example: x^2 = 2+3 = 5, x is not a natural number
wind belfry
this means a, b and x must be natural numbers
no x doesnt have to be a natural numbers so the operation is on N i dont think
silk kindle
mmm I think it has to be natural so it's on N 🤔