idle perch
sorry
#help-17
onyx flax
Ah that will help a bunch lol
Ok let me look again
I seem to get a volume of 1/12
BUT I did notice (with some help from chat) that if we have y=0 and z=0 (instead of x=0, z=0) THEN I do get an answer of 1/3
sharp scaffold
How would I do number 9
vocal sleetBOT
How would I do number 9
split wind
the calculations looks ok up till the last line🤔
sharp scaffold
What was wrong in the last line?
twin meteorBOT
Result:
4.295630140987
,calc sqrt(424)
twin meteorBOT
Result:
20.591260281974
if i read correctly, you wrote 20.2
sharp scaffold
Oh I forgot to square root
sharp scaffold
Lol ok ty
sharp scaffold
Not sure where I went wrong
vocal sleetBOT
Not sure where I went wrong
sharp scaffold
vocal sleetBOT
@sharp scaffold Has your question been resolved?
brazen isle
ur rainbow is wrong
sharp scaffold
Rainbow?
sharp scaffold
Why not?
split wind
did you put the squareroot properly this time?
sharp scaffold
I think so?
split wind
it's sqrt(12²-4(1)(-70))
brazen isle
oh yea why is x in ur quadratic
sharp scaffold
Wdym
brazen isle
sharp scaffold
Thats a bracket
brazen isle
bruh LOL
sharp scaffold
Just my handwriting is ass
split wind
careful
brazen isle
ur steps seem correct
lucid bane
,calc (-12+sqrt(12^2+(4*70)))/2
twin meteorBOT
Result:
4.295630140987
yep
split wind
remember not to round off so earily
brazen isle
the sqrt should be 20.59.....
sharp scaffold
Ah ok
lucid bane
???
sharp scaffold
But it says to the nearest tenth so
split wind
yea
sharp scaffold
I thought but mb
brazen isle
ur ans should be to nearest ten
lucid bane
oh the sqrt
brazen isle
not ur numbers used in the calculation
sharp scaffold
Ok got it
lucid bane
,calc sqrt(12^2+(4*70))
twin meteorBOT
Result:
20.591260281974
be precise until the ans
twin meteorBOT
Result:
424
okay rather uhm
which number do u look at when rounding 3.15 to the nearest whole num
sharp scaffold
5
brazen isle
❌
brazen isle
yep haha
split wind
brazen isle
okay so
sharp scaffold
I thought I said 4.3 is that wrong?
sharp scaffold
;-;
brazen isle
sharp scaffold
I think imma go jump off a cliff my test is tmrw
brazen isle
vocal sleetBOT
vocal sleetBOT
Available help channel!
lunar dew
vocal sleetBOT
vocal sleetBOT
vocal sleetBOT
✅
lunar dew
49th and 51th
lavish sluice
49:
GP : a,ar,ar²
AP : a , 2ar , ar²
4ar = a + ar²
4r = 1 + r²
r = 4+-2root3/2 = 2 +-root 3
lunar dew
@lavish sluice please explain your 3rd step
lavish sluice
In any AP, if we take any 3 consecutive terms,the double middle term is the sum of its preceeding and succeeding term
Try to prove this also
lunar dew
But in the question we were asked to double the middle term of gp but you doubled ap
lavish sluice
After doubling middle term
We get the sequence
a, 2ar , ar²
Right?
Now this sequence is an AP
lavish sluice
In any AP, if we take any 3 consecutive terms,the double middle term is the sum ...
So in AP.....
vocal sleetBOT
@lunar dew Has your question been resolved?
vocal sleetBOT
@lunar dew Has your question been resolved?
lunar dew
.close
narrow hamlet
I'm trynna prove it's a subspace but I tried my best to prove it but it seems like it's wrong, i need help please
vocal sleetBOT
I'm trynna prove it's a subspace but I tried my best to prove it but it seems li...
@narrow hamlet Has your question been resolved?
vocal sleetBOT
@narrow hamlet Has your question been resolved?
vocal sleetBOT
@narrow hamlet Has your question been resolved?
narrow hamlet
.close
candid nova
hi im currently struggling on finding the particular solution to
candid nova
this is as far as i have gotten, i have checked the answer and i just dont see what they are doing
it is in danish but maybe you can see through the math
i dont understand how they get to 34a-17=0 and -6a+34b-14=0
shadow raft
on the left hand side you have (34a-17)t and on the right hand side you have 0
the only way to remove the t term is to get 34a-17=0
candid nova
i mean i see that but how does(34a-17)t make the right hand side equal 0 when we have 6a-34b+14 there
thats where im getting lost as surely that would also depend on what b is etc
shadow raft
because this relation works for all t
and our solution for a and b are independent of t
so if 34a-17 is not 0, then the t term will never be 0 and can never satisfy the relation
vocal sleetBOT
@candid nova Has your question been resolved?
vocal sleetBOT
vocal sleetBOT
Available help channel!
elfin cypress
In the below formula 1.34 - are those matrix operation? or just an algorithm to follow to compute the matrix - for example 1.35 is valid matrix operations: scalar x matrix(transpose) x matrix
elfin cypress
similarly: if I have a matrix X, is it considered a valid matrix operation to subtract a row vector u from all rows in the matrix? (ie despite u not being a matrix of equal size)
cyan talon
no 1.34 everything is just a number
so yeah an algorithm to compute all the entries of T bar if you want
elfin cypress
ok - so I have X and I can compute u as a Row Vector
is there a such an operation for row vector subtraction from a matrix?
cyan talon
ok - so I have X and I can compute u as a Row Vector
your u is µ right?
elfin cypress
yeah
cyan talon
is there a such an operation for row vector subtraction from a matrix?
it's a thing in some matrix libraries for example
elfin cypress
like if i expanded the row vector to a matrix to match X... then i can do valid matrix subtraction
cyan talon
it's a thing in some matrix libraries for example
but it's really just a convenient notation
behind the scenes it just substraction on all rows/columns
elfin cypress
OK - so no formal operation/operator exists... it just convenience
cyan talon
you could define one yourself if you really use such operations all the time
elfin cypress
got it - was just curious.. cause like you can multiply Row Vector x Matrix or Matrix x Column Vector.. (with the correct dimensions) wasn't sure if such a rule existed for addition - but you answered
.close
frail jungle
y = | 4 - | 3- |2- |1-x| | | |...; If the arrangements of such series continues till 100 modulus then find the number of points of non differentiability
vocal sleetBOT
y = | 4 - | 3- |2- |1-x| | | |...; If the arrangements of such series continues ...
frail jungle
So i know, y is non differentiable at x = 1
and then |1-x| = 2 so we get
1-x = +-2
x = 3 and -1
and then the next one which is |2 - |1- x| | = 3
so we get, - |1-x| = 1 and - |1-x| = 5
solving which will get us 4 points
and then the next one
and so on
how can i solve this
vocal sleetBOT
@frail jungle Has your question been resolved?
tribal moss
you have to solve following euqations:
| 1 - x | = 0, |1 - x | = 2, |2 - | 1 - x | | = 3, | 3 - | 2 - | 1 - x | | | = 4
these equations show zeros of "modules"
and "module" is not differentiable in its zero
vocal sleetBOT
@frail jungle Has your question been resolved?
river parcel
Plss can someone help
flat whale
!1c
vale ermine
What would the approach be in these types of probs? Is graphing complex equations possible like z +1/z gives a feel it should be a hyperbola or ellipse but how do i graph it
vocal sleetBOT
What would the approach be in these types of probs? Is graphing complex equation...
half imp
vale ermine
Sorry i didnt send the rest part they r asking for if c1 and c2 intersect or not
vocal sleetBOT
Available help channel!
wary mantle
Regarding the Lambert W function over R
vocal sleetBOT
Regarding the Lambert W function over R
wary mantle
We have that it is the inverse of f: x -> x * e^x
So its domain will be the range of x * e^x, thus (-1/e; infinity)
And its range will be the domain of x * e^x, thus R.
But looking at its graph, the latter is not true
hushed pewter
,w plot y=x*e^x
twin meteorBOT
wary mantle
,w plot y=x*e^x
Yeah, that's x * e^x
hushed pewter
But looking at its graph, the latter is not true
What graph are you looking at?
oh you know what?
duh
I believe it's only defined for x>=0, where it is one-to-one
wary mantle
The graph of the Lambert W function.
wary mantle
I believe it's only defined for x>=0, where it is one-to-one
No
It is defined on the range of x * e^x
x * e^x has its minimum at -1/e
wary mantle
What is W_-1?
hushed pewter
What is W_-1?
dull bear
And its range will be the domain of x * e^x, thus R.
You restrict the domain of f so that f is actually invertible - so, say, x >= -1
wary mantle
You restrict the domain of f so that f is actually invertible - so, say, x >= -1
Is it standard to pick x >= -1 for the Lambert W function?
We could also pick x <= -1
dull bear
You usually do - that's where these branches come in here, if you chose x <= -1, you get the W_{-1}
wary mantle
Oh
dull bear
For the most part, when people use the Lambert W function, they use the branch W_0, though some exceptions sometimes
wary mantle
For the most part, when people use the Lambert W function, they use the branch W...
I read there is also W_n
What does that mean?
nth branch? Why?
dull bear
When you get to complex numbers, you have infinitely many branches, kinda like how complex logs have many branches
wary mantle
Oh
wary mantle
When you get to complex numbers, you have infinitely many branches, kinda like h...
So over the reals, only W_0 and W_{-1} exist, right?
dull bear
They're pretty much the only ones you'll care about - I mean in theory you could restrict the domain of f further such that you're still injective, but then those would be restrictions of W_0 and W_{-1} anyway
wary mantle
They're pretty much the only ones you'll care about - I mean in theory you could...
Oh, and also, if we want to cover the entire range of x * e^x with W, then we need to define two functions with edges at -1/e, since x * e^x has a minimum there
So it only makes sense to look at those two
We can't really define any other two and still cover the entire range
Thanks
.close
vocal sleetBOT
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rancid kestrel
Hi why did this equation change into that? I understand 64 turned into 4 because 4x4x4 is 64 but why the 2 and x change ???
vocal sleetBOT
Hi why did this equation change into that? I understand 64 turned into 4 because...
strange crater
they didn't change, looks like
rancid kestrel
But 1/3 is gone
strange crater
what do you mean "fine"?
rancid kestrel
what do you mean "fine"?
Gone *
outer warren
2x = (2)(x)
outer warren
64^(1/3) is 4
rancid kestrel
How do I do 64 times itself 1/3 times
outer warren
view fractional exponents like this as the nth root
64^(1/3) is the cube root of 64
the number when cubed gives 64
rancid kestrel
3 radical 64 ?
outer warren
$\cbrt{64}$
twin meteorBOT
ℝαμΩℕωⅤ
rancid kestrel
Yeah that
outer warren
which is 4
since 4 * 4 * 4 = 64
outer warren
$64^{\frac 14} = \sqrt[4]{64}$
twin meteorBOT
ℝαμΩℕωⅤ
rancid kestrel
Okay
outer warren
$64^{\frac 1n} = \sqrt[n]{64}$
twin meteorBOT
ℝαμΩℕωⅤ
rancid kestrel
$64^{\frac 1n} = \sqrt[n]{64}$
Wait there’s more
How did the stuff on the right happen
2/5 (4^2)^x why is x not with the 2 exponent
gaunt sparrow
Because $x^{ab} = (x^a)^b$.
twin meteorBOT
Azyrashacorki
rancid kestrel
Oh ok thx
thin vale
vocal sleetBOT
Oh ok thx
If you are done with this channel, please mark your problem as solved by typing .close
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Available help channel!
dire rose
not sure how to do this ^^
vocal sleetBOT
not sure how to do this ^^
gaunt sparrow
Do synthetic division to get the other factor.
Or you can always multiply (x-3) by a general quadratic term (ax^2 +bx + c) and compare the coefficients with P(x) to get the values of a, b and c.
sage ermine
cos^4 A - sin^4 A how do I prove this
vocal sleetBOT
cos^4 A - sin^4 A how do I prove this
empty frigate
...what's the original question?
empty frigate
...ok so the original question isn't to prove cos^4 A - sin^4 A, it's to select the best answer
sage ermine
i mean same thing?
empty frigate
no that's completely different
"prove cos^4 A - sin^4 A" doesn't even make sense, because cos^4 A - sin^4 A isn't a statement
it's like asking to prove my hair
empty frigate
did they give you options to choose from or are you just supposed to write something?
empty frigate
alright, what are the options
empty frigate
alright
hmm
...well cos^4 A - sin^4 A is the same as (cos^2 A)^2 - (sin^2 A)^2
so you can apply difference of two squares to that
crude swan
this question
vocal sleetBOT
this question
@crude swan Has your question been resolved?
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 was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
Any part in particular?
crude swan
how does the thing turn out to be one or zero
crude swan
Any part in particular?
is this correct for g_k(t) when n = 2?
dull bear
is this correct for g_k(t) when n = 2?
Not quite the best form - note that you’d want to consider what k is
crude swan
my thought process is
k can be either 1 or 2 since n goes up to 2 only
and since i can't be equal to k
then we would have to possibilites
one where k=1 and i=2 and vice versa
dull bear
one where k=1 and i=2 and vice versa
If k=1, you’d only be thinking about (t - x2)/(x1 - x2)
crude swan
hmm
so we would have a single term for n=2?
multiplied by noithinG?
and for n=3 we would have 2 terms then?
dull bear
Yep - remember you’re taking the product of everything between 1 and n - except the k term
crude swan
except when k and i are equal
all other terms are included
so that would mean that we have multiple possibilites
for different values of k?
we would have 2 terms for n=3 and k =1 and two more for n = 3 and k = 2 and 2 more for when n = 3 and k = 3
each being a different multiplication
dull bear
Well you choose your k in advance, anything between 1 and n, and then once you have that, you decide you don’t want i=k in the product - but everything else between 1 and n is fine
(Worth bearing that in mind!)
crude swan
Well you choose your k in advance, anything between 1 and n, and then once you h...
okkk
so k you choose
then you go off of that
makes sense
so now I understand this
the question is asking to show that when t = x_i g_k(t) = 0
and g_k(x_k)=1
that seems pretty straightforward
we just have to plug in the x_k and x_i for t and see what happens
it's that simple?
or am I missing something
@dull bear u still there homie?
dull bear
For $g_k(x_k)$, remember that it's the whole product
[
g_k(x_k) = \prod_{i = 1, i \neq k}^{n} \frac{ x_k - x_i }{x_k - x_i}
]
but yep, it becomes 1
twin meteorBOT
@dull bear
dull bear
None of the terms in that product will ever be zero, so you're happy 
crude swan
so i need imply that there are more terms
dull bear
Similarly the idea is that in the product, you shall find this for the other one, hence there's a zero in your product, hence zero
crude swan
hmm
so i'd just write dot dot dot
then the term that is zero
and dot dot dot again since it's a product
the entire thing will be zero
dull bear
Yep, that would do 
crude swan
ok cool
and for part b
i don't quite understand the notation
f here is a function right?
and it is part of a set of function on the fiel K
dull bear
Yep - a polynomial, which you want to be of degree n-1, that passes through each point (xi, yi)
crude swan
wait wait wait
cobalt crypt
you're gonna need to write out the full product here, not just those terms
cobalt crypt
Yep - a polynomial, which you want to be of degree n-1, that passes through each...
also, a polynomial is distinct from a function K -> K, as emphasised by (c)
crude swan
so the thing with the big P is like a polynomial generator
and we want to show that one of these polynomials f that has a degree with that condition has a point at (xi, yi)
cobalt crypt
part (b) can be done independently of part (a)
vocal sleetBOT
@crude swan Has your question been resolved?
unreal sparrow
Is there a video covering the topic of this problem?
vocal sleetBOT
Is there a video covering the topic of this problem?
strange crater
looks like it's using the Inverse Function Theorem, you could search for that
vocal sleetBOT
@unreal sparrow Has your question been resolved?
vast shale
If y=300sin(pi/2.2(x-4))+500, when would it equal 600?
vocal sleetBOT
If y=300sin(pi/2.2(x-4))+500, when would it equal 600?
vast shale
I need desprate help, I would really appreciate it!
I got 1.56, 4.24, 5.96, and 8.64
I need the x values from 0 to 10
<@&286206848099549185>
When I put it in desmos I get 1.56 and 5.96 also equal 600, but when I solve the equation, I only get 4.24 and 8.64
vocal sleetBOT
@vast shale Has your question been resolved?
crude swan
Yep - a polynomial, which you want to be of degree n-1, that passes through each...
hey u still there?
vocal sleetBOT
dull bear
hey u still there?
Sure, but open a new channel, this one might lock
dull bear
Cause you deleted the original message, often the channels sometimes lock without warning
dull bear
It's already closed!
vocal sleetBOT
Available help channel!
lilac panther
I need help with 9a)
vocal sleetBOT
I need help with 9a)
silk osprey
what’s 5cos(3x)-cos(3x)
vocal sleetBOT
vocal sleetBOT
Available help channel!
celest forge
vocal sleetBOT
celest forge
What do I do in d)
livid horizon
If s is perpendicular to r+s, then the dot product between the two vectors will be 0
Since cos90⁰= 0
celest forge
there is no dot product in my syllabus
livid horizon
Damn
lime basin
If s is perpendicular to r and r+s then it must be the zero vector
This question is nonsensical
Assuming s is perpendicular to r
Oh whoops wrong assumption
Yeah I’m not sure how you do this without dot product
Actually hold on it’s quite simple
It’s just a right triangle
celest forge
then guess I'll have to learn the dot product.
lime basin
r is the vector correspond going along the hypotenuse
livid horizon
$| \vec{s} + {(\vec{r} + \vec{s})}|= \sqrt{{|s|}^2 + {|{(r+s)}|}^2}$
twin meteorBOT
Lorentz
lime basin
$| \vec{s} + {(\vec{r} + \vec{s})}|= \sqrt{{|s|}^2 + {|{(r+s)}|}^2}$
Consider the difference of the vectors instead
lime basin
Just Pythagoras
livid horizon
there is no dot product in my syllabus
You can find the difference of two vectors and then it's magnitude, there is more than one way to solve this question
vocal sleetBOT
@celest forge Has your question been resolved?
vocal sleetBOT
vocal sleetBOT
Available help channel!
quartz iron
how to write in interval notation that x can be any real number, except for +2 and -2?
vocal sleetBOT
how to write in interval notation that x can be any real number, except for +2 a...
quartz iron
should i just write X ≠ +2, x ≠-2
lapis marten
What about x = 1 btw
desert hornet
that doesn't seem right.
quartz iron
why can't x be 1?
pale perch
negative argument
frank swan
how to write in interval notation that x can be any real number, except for +2 a...
All (
Like this
quartz iron
so x in r
[2, inf)
(-inf, -2]?
that's for this?
frank swan
Yes
quartz iron
okay thanks
quartz iron
yeah
quartz iron
oh okay
quartz iron
how to add that x is in the set of real numbers
jsut add r? in front of the 1 pair of brackets?
frank swan
(-inf, inf)
frank swan
jsut add r? in front of the 1 pair of brackets?
That’s not interval notation tho
quartz iron
no sorry, i meant like the symbol R
frank swan
Some teachers don’t allow it
frank swan
no sorry, i meant like the symbol R
You can do that
quartz iron
no worries then
frank swan
But some teachers don’t allow
vocal sleetBOT
quartz iron
.reopen
vocal sleetBOT
✅
frank swan
Lemme see
quartz iron
frank swan
Well
It kinda stops at -2
But doesn’t continue
But -1 is included
up to 0
So no domains wrong
frank swan
That’s the domain
We have to remember
-1 and 0 is included
In the middle
If u put the graph into desmos
quartz iron
oh yeah
frank swan
I mean visually
We can see from the graph
That the graph has values of x within -1 and 0
Domain is all values that x can be
and -1 and 0 is there
so 😊
frank swan
Alg
vocal sleetBOT
frank swan
It’s a neither function
For the second question
Btw
You can tell looking at the graph
quartz iron
yeah cause it's not symmetrical right
vocal sleetBOT
Available help channel!
frank swan
Yeah not symmetrical
quartz iron
i still need to show working though but i'll get back that later
vocal sleetBOT
quartz iron
lol
vocal sleetBOT
Available help channel!
unborn wadi
vocal sleetBOT
unborn wadi
if my matrix looks like so
D = 64 and f_xx is negative
so it is a local minimum
right
paper depot
so it is a local minimum
maximum
paper depot
yes
unborn wadi
if this is my f(x,y) = x^2y^2 - 4xy + 6x - 6*y
is there a quick way instantly spot wheter it increases or decreaes as x and y increase
vocal sleetBOT
@unborn wadi Has your question been resolved?
celest forge
You can find the difference of two vectors and then it's magnitude, there is mor...
what do you mean by the difference of vectors here
vocal sleetBOT
what do you mean by the difference of vectors here
livid horizon
The difference in vectors r+s and s
celest forge
I still don't get it 😭
livid horizon
$|{(\vec{r} + \vec{s})} - \vec{s}| = \sqrt {{|(r+s)|}^2 + {(|-s|)}^2} = |\vec{r}|$
twin meteorBOT
Lorentz
livid horizon
The reason the 2ab cos theta term didn't come is coz angle is 90⁰
And cos 90⁰ is 0
@celest forge
celest forge
ohh
celest forge
The reason the 2ab cos theta term didn't come is coz angle is 90⁰
And cos 90⁰ is...
these cos theta things aren't in my syllabus for vectors
so I'm still confused on how you got that equation
and how will I use it to find |(r+s)|
livid horizon
$|\vec{a} + \vec{b}|= \sqrt {|\vec{a}|^2 + |\vec{b}|^2 + 2|\vec{a}||\vec{b}| \cos{\theta}}$, $\theta$ is the angle between $\vec{a}$ and $\vec{b}$
twin meteorBOT
Lorentz
celest forge
Have you seen anything like this?
nope not in syllabus
celest forge
I am in igcse
I figured 4 comes with sqrt(5^2-3^2)
its just simple Pythagoras😭
I'm so dumb
livid horizon
It is
celest forge
tysm
livid horizon
**Lorentz**
If the angle is 90⁰(which it is), it becomes the Pythagorean
celest forge
I see
livid horizon
I was just specifying the general case
celest forge
thank you very much
celest forge
.close
vocal sleetBOT
celest forge
I see
vocal sleetBOT
Available help channel!
manic lagoon
can someone walk me through on how to factor this?
vocal sleetBOT
can someone walk me through on how to factor this?
gritty sage
OK, so you look at the powers.
You have (3x - 1)^5 and (3x - 1)^4, right?
So, what's the lowest exponent?
manic lagoon
4 it seems
twin meteorBOT
Chai T. Rex
gritty sage
Then, the ys. What's the smallest exponent?
manic lagoon
hang on, still trying to make sense of that step.
gritty sage
OK.
manic lagoon
so the (3x-1)^5 disappeared because we factored out (3x-1)^4, leaving inside (3x-1)^5-4 right?
gritty sage
Right.
manic lagoon
ok 😄 so now, smallest exponent, is the one on (3x-1)? which is 1
gritty sage
Oh, you already did that part.
You'd say 0 because there's (3x - 1)^0 in the 7y^6 part.
So, if the lowest power is 0, you can't factor any further.
Back when it was 5 and 4, you could factor out the 4.
gritty sage
so the (3x-1)^5 disappeared because we factored out (3x-1)^4, leaving inside (3x...
Like what you said here, it would be (3x - 1)^(4 - 4) = (3x - 1)^0.
And anything to the 0 power is 1, so we can just get rid of it.
manic lagoon
ok
gritty sage
So, we're done with the (3x - 1) factoring part.
gritty sage
Oh, when your factoring, you don't want to distribute.
Factoring is kind of the opposite of distributing.
What we do is we look at the terms you have (terms are the things you add or subtract).
twin meteorBOT
Chai T. Rex
gritty sage
Another common factor is both terms have ys.
gritty sage
Right, but what exponent of it?
manic lagoon
6?
gritty sage
Right, that's the smallest exponent.
twin meteorBOT
Chai T. Rex
gritty sage
Does that make sense?
manic lagoon
yes absolutely
manic lagoon
1?
gritty sage
Right, so we would factor that out, but factoring out 1 doesn't do anything.
If it was like 5, we'd put 5 in front and then divide both coefficients by 5.
manic lagoon
yes
gritty sage
OK, so we're done because there's no more common factors.
manic lagoon
are you sure? solution says otherwise 😄 one sec
gritty sage
Well, they might do the distribution.
twin meteorBOT
Chai T. Rex
gritty sage
You only distribute if the factor you're inside has a possible distribution, but you can only do it on some of the terms instead of all of them.
Like the (3x - 1) is only on the first term inside the last factor, so you can distribute that.
gritty sage
I shouldn't have forgotten that.
You need to do the distributions to see if what remains afterward can be factored further.
You're welcome.
manic lagoon
okay! will remember that. thanks again 😄
gritty sage
No problem.
manic lagoon
.close
scenic ravine
Just wanted someone to verify if this proof is right for : Let P (x) be an odd degree polynomial in x with real coefficients. Show that the equation P(P(x)) = 0 has at least as many distinct real roots as the equation P (x) = 0.
vocal sleetBOT
Just wanted someone to verify if this proof is right for : Let P (x) be an odd d...
scenic ravine
So as P(x) is an odd degree polynomial, it has atleast one real root say a_1, a_2...a_n, where a_n is the last real root it has, and let P(x) be of degree m> or equal to n
lyric relic
then?
scenic ravine
so P(x)=(x-a_1)(x-a_2).....(x-a_n)(q(x) where q(x) is some non constant function is m is not n
then we have P(P(x))= (P(x)- a_1))(P(x)-a_2)....q(P(x))=0
scenic ravine
oops
lyric relic
but is not important
scenic ravine
so as every odd degree polynomial is a map from R to R , we'll have atleast n roots
QED
Is the proof right? or is it missing anything?
lyric relic
mh.. i think you have meant that every odd degree polynomial is surjective
because EVERY polynomial is a map from R to R ,
scenic ravine
because EVERY polynomial is a map from R to R ,
Yeah, my bad
But barring that, is the proof alright?
lyric relic
yes...but i would add an extra sentence (if not, your previous work seems quite useless)
because P is surjective
for every a_k
there is b_k such that P(b_k)=a_k
so P(P(b_k))=P(a_k)=0
and the b_k are as many as the a_k's
just like your proof ...but with some extra details at the end
pallid zenith
I'm not sure i understand what youre doing
for one you are ruling out some of this domain, right?
oh, yea, on the right
youve used a bit of shorthand here
,w Plot sqrt( (x^2-3x+2)/(x-2) )
twin meteorBOT
pallid zenith
is there a reason you chose to not simplify the function?
vocal sleetBOT
@stuck maple Has your question been resolved?
vocal sleetBOT
vocal sleetBOT
Available help channel!
celest forge
When the angle may not be 90⁰
also i tried drawing a vector triangle diagram for thi
vocal sleetBOT
also i tried drawing a vector triangle diagram for thi
celest forge
bc for sqrt(5^2-3^2)
r needs to be hypotenuse
is this right?
i feel like it's wrong as (r+s) and s should be connected head-tail
but someone told me that the head-tail thing only works for component vectors and we have (r+s) here which is a resultant vector
what does that mean?
vocal sleetBOT
@celest forge Has your question been resolved?
low swan
I’ve got a question that might involve advanced algebra or calculus, I’m not sure. The problem arose from the idea of putting 10% of your paycheck into savings each week. I then thought, “what if I put 10% of (paycheck + current checking balance) into savings each week. Eventually checking would reach a limit and all of the paycheck would go into savings each week. What I need to figure out is how to find the limit, given the paycheck amount & the %left in checking
vocal sleetBOT
I’ve got a question that might involve advanced algebra or calculus, I’m not sur...
low swan
Y = checking balance
R = %left in checking
N = # of paychecks
X = paycheck value
In the end, I want to be able to say “here’s what I want my checking limit to be, and here’s my paycheck value, now I plug it into the formula and it tells me what % of (paycheck + balance) to put into savings, and ~ how many paychecks it’ll take to get to the limit”
<@&286206848099549185>
Did I do this wrong? Or are people just busy lol
vocal sleetBOT
@low swan Has your question been resolved?
low swan
Nope
What do I do if nobody can help me mr. Bot?
I’m learning now from looking around that it may take a while to get help, and that’s okay
vocal sleetBOT
@low swan Has your question been resolved?
lilac bronze
the value of money going into each account after an infinite amount of paychecks assuming you put (1-R)% of checking balance and check for each check, will eventually converge. The balance of the checking account, if defined as $Y_n = R(Y_{n-1} + X), Y_1 = RX$, $Y_2 = R^2X + RX, Y_n = X(R^n + ... + R + 1), Y_n = X\sum_1^n{R^n}$ $lim_{n\to\infty}X\sum_1^n{R^n}{X\sum_1^n{R^n}} = \frac{X}{1-R}$ if |R| < 0 (which I assume, since it's a percent), meaning it converges to a specific value. Same thing happens for the balance account, since |(1-R)| < 0.
twin meteorBOT
fish
the value of money going into each account after an infinite amount of paychecks assuming you put (1-R)% of checking balance and check for each check, will eventually converge. The balance of the checking account, if defined as $Y_n = R(Y_{n-1} + X), Y_1 = RX$, $Y_2 = R^2X + RX, Y_n = X(R^n + ... + R + 1), Y_n = X\sum_1^n{R^n}$ $lim_{n\to\infty}X\sum_1^n{R^n}{X\sum_1^n{R^n}} = \frac{X}{1-R}$ if |R| < 0 (which I assume, since it's a percent), meaning it converges to a specific value. Same thing happens for the balance account, since |(1-R)| < 0.
lilac bronze
$Yn = R(Y{n-1} + X), Y_1 = RX$, $Y_2 = R^2X + RX, Y_n = X(R^n + ... + R + 1), Y_n = X\sum_{i=1}^{n}{R^n}$ $lim_{n\to\infty}X\sum_1^n{R^n}$ ${X\sum_1^n{R^n}} = \frac{X}{1-R}$ if |R| < 0
twin meteorBOT
fish
low swan
Okay I’ve gotta figure out how to translate this lol
I understood everything up until Yn= X\sum{i=1}…etc.
But also, since R is a percentage, it would be written as a decimal greater than zero and less than 1
So wouldn’t that mean that |(1-R)| > 0?
In my “plug and chug” testing, I found that the two lines meet, and then the savings quickly overtakes the checking.
vocal sleetBOT
@low swan Has your question been resolved?
vocal sleetBOT
vocal sleetBOT
Available help channel!
scenic ravine
What does f(x) even mean here?
vocal sleetBOT
What does f(x) even mean here?
wraith venture
see if the squeeze theorem can yield anything
scenic ravine
see if the squeeze theorem can yield anything
for g(x) that makes sense, but I don't know what f(x) even means here
wraith venture
if the lower and upper bound agree, we're good
cyan talon
that's what f looks like
wraith venture
why do we always meet at the unlikeliest times platypus
cyan talon
we love the night
scenic ravine
that's what f looks like
hmm, how? Say n=4, then f(x) is 2 only when x is between (1/3,1/2), right?
wraith venture
no, n=4 gives (1/5, 1/4)
scenic ravine
my bad, yes
wraith venture
on which f = sqrt(4)
wraith venture
defining f
for all x, x is in exactly one of these intervals
So that defines n, and therefore f(x)
wraith venture
and g to zero
looking at the upper bound of g, since f(x) ~ 1/sqrt(x)
We have lim fg <= 2
But does the lower bound also give 2 ?
scenic ravine
looking at the upper bound of g, since f(x) ~ 1/sqrt(x)
We have lim fg <= 2
But ...
won't the upper bound be 0 at 0
scenic ravine
near 0 it would still tend to 0 though , right?(The upper limt)
wraith venture
depends on g
if g(x) = 2sqrt(x)
the limit is 2
but if g(x)'s lower bound turns out to equal x, then you could have a case where g beats f and the limit is 0
for example
scenic ravine
how? This for instance is the graph of the upper bound(2sqrtx)
wraith venture
it's above x
so x is a possible lower bound
though probably not
depends on what that integral evaluates to
though you don't need to evaluate it
if you find a sufficiently tight bound
depends
scenic ravine
Is there any way to do this without the squeeze theorm, I'm not really familiar with it, I know what i means, but that's about it
wraith venture
did you see it in the lecture ?
scenic ravine
did you see it in the lecture ?
I've never had a lecture on this, it's from a competitive exam for 12th graders in my country
wraith venture
ok
Having had a look at this integral and evaluating its asymptotic behavior
This is definitely just the squeeze theorem
I don't think there's much of a way around it
wraith venture
I've never had a lecture on this, it's from a competitive exam for 12th graders ...
and is it in the syllabus for 12th graders ?
scenic ravine
and is it in the syllabus for 12th graders ?
in the standard reference material, no, but they expect students to know a lot out of the book too.(This is from JEE adv , you may have heard of it )
wraith venture
indians graduating HS ?
scenic ravine
yes
wraith venture
it's got to be known
surely JEE adv would have anything the french curriculum dares include, as its most fundamental tools
scenic ravine
it's got to be known
nevermind , I was wrong, it was just not there when I did 11th a couple of years ago. Would MIT opencoureware be a good reference ?
sorry, once again
wraith venture
I don't know references
buoyant notch
a+b+c=35
abc=1000
Geometric series
vocal sleetBOT
a+b+c=35
abc=1000
Geometric series
buoyant notch
I started with changing into terms of a and r
a(1+r+r^2)=35
a^3r^3=1000
I'm unsure how to continue
hushed pewter
a^3r^3=1000
I would solve for r here
buoyant notch
How do I do that
flat whale
How do I do that
xy = a implies y = a/x
buoyant notch
a^3=1000/r^3?
twin meteorBOT
riemann
flat whale
Use those to help
buoyant notch
oh
r^3=1000/a^3?
But how do I get rid of ^3
Do I take the third root
Wait can I do (ar)^3=1000
Wait nvm
I'm stuck
<@&286206848099549185>
Oh is it
third root of (ar)^3= third root of 1000
ar =10
r=10/a?
a(1+r+r^2)=35
a(1+10/a + 100/a^2)=35
a+10+100/a=35
a+100/a=35-10
a+100/a=25
a^2+100=25a
a^2-25a+100=0
(a-20)(a-5)=0
a=20 or a=5
First scenario
r=10/20
r=0.5
20(1+0.5+0.5^2)=35
Wait what am I doing
a=20
ar = (20)(0.5) = 10
ar^2= (20)(0.5^2)= 5
Same thing other way
So it's 5 and 10 and 20
Can someone confirm
azure harness
<@&286206848099549185>
what ru trying to solve?
buoyant notch
what ru trying to solve?
Oh I have to find the first three terms of geometric series
So I need a b and c
which is a, ar and ar^2
buoyant notch
No they only give two equation
buoyant notch
a+b+c=35
abc=1000
Geometric series
These
Then I bring to a and r and do some math stuff
buoyant notch
Yea
azure harness
hmm
azure harness
So it's 5 and 10 and 20
if u wanna verify the answer u can do this,,
5+10+20 should be 35 yea
and then 5 x10 x 20
is also 1000 so i think its aight
haughty cairn
need some help
vocal sleetBOT
need some help
haughty cairn
at a price of 9000$ per box of oranges the supply is 320000 boxes and the deman is 200000boxes. at a price o 8.50$ per box the supply is 270000 boxes and the demand is 300000 boxes
-find a price supply equation in the form p= mx+b where p is the price in dollard and x coresponding supply in thousand of boxes
-find a price demand equation of the form p=mx+b where p is the price in dollars and x is the corresponding demand in thousands of boxes
vocal sleetBOT
@haughty cairn Has your question been resolved?
flat whale
just close it
.close
celest forge
but someone told me that the head-tail thing only works for component vectors an...
?
vocal sleetBOT
twin meteorBOT
Mycobacterium
lucid bane
$\sum_{n=1}^{\infty} \f{n+1}{n}$ does NOT converge!!!
twin meteorBOT
🫎 Mοοsey🫎
lucid bane
$\sum_{n=1}^{\infty} \left( 1+\f{1}{n}\right)$
twin meteorBOT
🫎 Mοοsey🫎
lucid bane
and this definitely does NOT converge (by divergence test)
lost warren
yeah cause harmonic series
brisk moss
simpler: it's a sum of numbers all of which are greater than 1
vocal sleetBOT
vocal sleetBOT
Available help channel!
tough valve
How many 3 digit nos divisible by 3 can be formed by using digits 0,1,2,3,4,5 with repetition
vocal sleetBOT
How many 3 digit nos divisible by 3 can be formed by using digits 0,1,2,3,4,5 wi...
lime basin
tedious, show your working
tough valve
@lime basin
tough valve
what check did you make for that?
Sum of nos is divisible by 3
So it doesn't matter if the sum of first 2 digit is 3k,3k±1, there is 2 choices for all
wraith mist
Is the answer 60?
60 seems like low for an answer
tough valve
Can you do it and try?
wraith mist
Intuitively the amount of numbers not divisible by 3 should be about 2/3 of the total mount of given numbers
wraith mist
Can you do it and try?
The answer is 120
tough valve
Intuitively the amount of numbers not divisible by 3 should be about 2/3 of the ...
But qstn is divisible by 3
lime basin
Sum of nos is divisible by 3
ok, then what about the number 445?
wraith mist
The question was to find the amount of numbers not divisible by 3, right?
lime basin
it is included in your permutation
deft sedge
The question was to find the amount of numbers not divisible by 3, right?
nos being numbers not a misspelling of not
wraith mist
Oh so "nos" wasn't a typo
sorry english isn't my first language, I didn't know this slang
ok then the answer is 60
tough valve
ok, then what about the number 445?
Its not
lime basin
you shouldn't do stuff like this intuitively
tough valve
4+4=8
lime basin
I'm aware
tough valve
Either 1 or 4 will be at last digit
lime basin
I'm just not clear about your method
tough valve
I'm just not clear about your method
Its simple method to use when you have symmetry
.close
delicate void
vocal sleetBOT
vocal sleetBOT
@delicate void Has your question been resolved?
delicate void
^
vocal sleetBOT
@delicate void Has your question been resolved?
vocal sleetBOT
@delicate void Has your question been resolved?
torn abyss
This is my working but im pretty sure its wrong
.
shadow raft
https://cdn.discordapp.com/attachments/903463353417072641/1187353757101596732/im...
!occupied
vocal sleetBOT
https://cdn.discordapp.com/attachments/903463353417072641/1187353757101596732/im...
Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).
shadow raft
yeah sorey i was trying to help him
I know, but its channel
The person vreating the channel is responsible
sweet flower
ive just gotten help in another channel, Thanks anyway tho
Then close this channel
delicate void
.close
ember tendon
\begin{gather*}
F_1(a,b) = a \oplus b\
F_2(a,b) = a'b
\end{gather*}
twin meteorBOT
metnal
ember tendon
for as far as i can tell, this isnt a functionally complete system
a second pair of eyes would help
.close
gritty widget
If i have A, where $e^{det(A)}det(e^{A}) = e$, find a
vocal sleetBOT
If i have A, where $e^{det(A)}det(e^{A}) = e$, find a
twin meteorBOT
ZmijaZ
gritty widget
how do i get to the $det(e^A)$
twin meteorBOT
ZmijaZ
paper depot
$\det(e^A) = e^{\tr(A)}$ says wikipedia
twin meteorBOT
Ann
paper depot
(and in fact it is not hard to prove/justify)