red helm
Smart
#help-27
stone stump
which btw we can see if we are smart just from the matrix
cause the second column is just (0,2,0)
in general, if the i'th column is just c*e_i, then c is an eigenvalue and e_i is an eigenvector
unborn geyser
is there a reason why dsqrt(x)/dx where dx = New area is the derivative of sqrt(x)? Because isnt the derivative of sqrt(x) = dsqrt(x)/slight nudge in x
devout snowBOT
is there a reason why dsqrt(x)/dx where dx = New area is the derivative of sqrt(...
unborn geyser
<@&286206848099549185>
im genuinely confused
does d(sqrt(x)) the difference in y value?
devout snowBOT
@unborn geyser Has your question been resolved?
coral wraith
Twelve volleyball teams just completed a round-robin tournament. (There are no ties in volleyball.) If no team won exactly 7 matches, prove that there exist 3 teams A, B, and C such that A beat B, B beat C, and C beat A.
my intuition is that is is possible, but im not sure how to prove it. can i get a hint please?
devout snowBOT
Twelve volleyball teams just completed a round-robin tournament. (There are no t...
velvet vale
Twelve volleyball teams just completed a round-robin tournament. (There are no t...
This is completely random and potentially not the right approach. But I’m thinking some graph coloring set up
Like the final set up feels like finding a cycle in the graph
Ah not graph coloring but directed graph
coral wraith
Ah not graph coloring but directed graph
?
velvet vale
Yea
It’s a complete graph with 12 nodes
And one node can’t have 7 edges leaving it
Which also feels like pigeon hole will be used at some point
coral wraith
Which also feels like pigeon hole will be used at some point
im not sure where tho
and im not sure how to continue with the graph
velvet vale
I'm not either lol
I'm just trying to give ideas
ok let's uh
restate the problem
Given a complete digraph with 12 vertices, show that there are no 3 member cycles
coral wraith
and no node connects to more than 7 other nodes
velvet vale
right right
coral wraith
im trying to set up a contradiction
If there doesn’t exists 3 teams A, B, and C such that A beat B, B beat C, and C beat A,
.... not sure what happens
@velvet vale
what if
A, D, E, F and G beat B
and B beat C
then did C not beat A, D, E, F and G assuming that there is no cycle of three?
coral wraith
Im really confused on how to continue with this
velvet vale
I need to go to class, but I’ll think about it. Or just ask someone else
velvet vale
What if you get rid of the win restriction and just try constructing an acyclic graph
coral wraith
or using pigeonhole and contradiction
velvet vale
Does it follow that it contradicts the win flcondiditon
coral wraith
What if you get rid of the win restriction and just try constructing an acyclic ...
idk im honestly not great with graph theary
every teams either wins or loses to another team
velvet vale
Let’s say A beats B, then A can’t lose to any team that B beats
So now A also beats the set of teams B beats
And then you can keep applying that argument maybe
And might imply that in order for it to be acyclic A must be undefeated
coral wraith
interesting
velvet vale
Hmmm
If B loses to C then A can either beat or lose against C
Maybe you can then pivot to C or something
coral wraith
considering the restrictions, A can only be beat by 11-(the number of teams B beat)
at most 5 teams
im honestly really not sure
:despair:
ill come back to it later
.close
ruby fjord
stuck on how to do proof by induction with inequalities
one specific question would be
prove 2^k > k^2 for k >= 5
i can get up to the point of expanding the k+1 part into 2(2^k) > k^2 + 2k + 1, but im not sure where to go from there
if i take out the assumed 2^k > k^2 part, im just left with 2 > 2k + 1 but thats always false for k >= 5
devout snowBOT
stuck on how to do proof by induction with inequalities
one specific question wo...
magic thicket
2(2^k) = 2^k + 2^k
So since 2^k > k², you get
2^k > 2k+1
Does that hold ?
ruby fjord
2(2^k) = 2^k + 2^k
So since 2^k > k², you get
2^k > 2k+1
Does that hold ?
i guess so
magic thicket
can you prove it ?
magic thicket
could it be that 2^k > k² >= 2k+1 ? If so we're pretty much done
ruby fjord
could it be that 2^k > k² >= 2k+1 ? If so we're pretty much done
how do you know if k^2 >= 2k+1
magic thicket
or you can try and prove this result by induction too, but that's slower (though no analysis involved, depends on what you prefer)
In general a polynomial of degree n will be (at least in absolute value) bigger than any polynomial of degree < n when x gets big enough. Here we can expect x >= 5 to work according to the question
ruby fjord
In general a polynomial of degree n will be (at least in absolute value) bigger...
what
magic thicket
for any constant k > 0 and integer n, x^(n+1) / (kx^n) -> +inf as x -> +inf
so you can expect x² > 2x+1 when x is large enough
,w plot x² - 2x - 1
woven radishBOT
magic thicket
you can see that for x >= 5 (and even x >= 2.4) it is positive
ruby fjord
for any constant k > 0 and integer n, x^(n+1) / (kx^n) -> +inf as x -> +inf
i dont understand this either, is / meant as division symbol or smth else
and how does that mean that k^2 > 2k + 1
magic thicket
of course it is
prime egret
and how does that mean that k^2 > 2k + 1
Try solving the inequality for k
magic thicket
yes that's more explicit
ruby fjord
Try solving the inequality for k
i cant get k on its own tho
magic thicket
you never studied quadratics ?
ruby fjord
wait
sorry my brain has been fried from being forced to look at this type of question for an hour straight today
and no one else in the room knew wtf was going on either
if you get (k+1)^2 > 0 then k = -1 to make that true right
woven radishBOT
ruby fjord
shouldn't it be (k-1)² - 2 > 0 ?
why would it be
magic thicket
,w plot x² - 2x - 1
it's this
magic thicket
k² > 2k+1
k² - 2k - 1 > 0
k² - 2k + 1 - 2 > 0
(k-1)² - 2 > 0
ruby fjord
makes sense
ruby fjord
could it be that 2^k > k² >= 2k+1 ? If so we're pretty much done
ok so how exactly did you think up of putting k^2 into the middle of it tho
its solved now cause A > B B >= C then A > C but is that applicable to other questions in the exact same way
magic thicket
I know it's bigger than 2k+1 so I directly know it will work. So it's much easier to find it compared to you who has to spend several minutes to check it
I intuitively notice immediately that the LHS is <= 2k² and therefore we conclude from the induction hypothesis
with enough experience anything becomes immediately noticeable I guess
"How can I simplify this expression to something that will be simple enough and still correct ?"
ruby fjord
lemme get the main example that broke my brain cause this was just smth i thought up to get the basic idea of it
“prove (1 + x)^k >= 1 + kx for k >= 1”
this was an example for note-taking actually and i still couldnt understand the explanation so i just have a picture of the explanation that i dont get
magic thicket
a^x - a >= 0 for every x >= 1 and a >= 0
The equality holds for x = 0
ruby fjord
i understand up to expanding the k+1 part into (1+x)(1+x)^k >= 1 + kx + x
ruby fjord
a^x - a >= 0 for every x >= 1 and a >= 0
The equality holds for x = 0
a is said to be a real number
magic thicket
you only want it for integer values of k ?
magic thicket
ok k integer
I thought we were doing the more general version, but then no induction so it was weird
ruby fjord
or not integer actually, natural number
magic thicket
we said k >= 1 so that was obvious
magic thicket
i understand up to expanding the k+1 part into (1+x)(1+x)^k >= 1 + kx + x
you want to do it by induction ?
ruby fjord
you want to do it by induction ?
im not sure what you mean cause i thought the entire process im doing rn is induction
prove for 1, assume for k, prove for k+1
magic thicket
yeah but you could prove it different ways, I was just confirming that's the method you want to use
induction is very doable yes
ruby fjord
so yeah induction
magic thicket
i understand up to expanding the k+1 part into (1+x)(1+x)^k >= 1 + kx + x
it's actually a bit of a similar idea of breaking up the inequality into parts that are each easy to handle
magic thicket
so it works by comparing the two functions
and so just for your knowledge, the inequality also holds for non integer values of k >= 1
ruby fjord
it's actually a bit of a similar idea of breaking up the inequality into parts t...
actually, wouldnt you be able to just remove the assumed part and be left with x + 1 > x
magic thicket
it's actually a bit of a similar idea of breaking up the inequality into parts t...
still ends up applying that same idea
real grail
-
we will solve it by reasoning by recurrence
-
let's prove that
2 ^ K > K²whenK >= 5:for
K = 5:
2^5 = 32 > 5^2 = 25
so2 ^ K > K²is true for K = 5 -
we will suppose that : 2^k > K² and we will prove that
2 ^ (K + 1) > (K + 1)²:- we know that : 1. `2 ^ (K + 1) = 2 ^ K + 2 ^ K` 2. `(K + 1)² = K² + 2K + 1`
so we have to prove that 2 ^ K + 2 ^ K > K² + 2K + 1
- we already supposed that `2^K > K²` so need just to prove that `2^K > 2K + 1`
and since `2 ^ K > k²` then `2^K - 2K - 1 > K² - 2K - 1`
so if we prove that `K² - 2K - 1 > 0` then `2 ^ K - 2K - 1 > 0 ` is also true
- for
f(K) = K² - 2K - 1
the derivation :f'(K) = 2K - 2 > 0when K >= 5
so the function K² - 2K - 1 is converging and f(K) >= ( f(5) = 14)
so K² - 2K - 1 > 14 > 0
soK² > 2K - 1
and since 2 ^ K > K² then 2 ^ K > 2K - 1
- so we can conclude that :
2 ^ (K + 1) > (K + 1)²
and by the principle of reasoning by recurrence we can say that
2 ^ K > K ^ 2
devout snowBOT
@ruby fjord Has your question been resolved?
velvet vale
considering the restrictions, A can only be beat by 11-(the number of teams B be...
Ok I figured it out
devout snowBOT
Ok I figured it out
velvet vale
You can make a contradiction trying to say it’s acyclic
Team A either has to win 11 games or lose 11 games
If it wins 11 then it’s a contradiction
If it loses 11 then you can just remove it because it won’t impact our cyclic case
So we reduce it to 11 teams and 6 max wins
You can do the same argument and eventually it leads to another contradiction
coral wraith
Team A either has to win 11 games or lose 11 games
why not anything else?
velvet vale
Because then it’s cyclic
coral wraith
right
coral wraith
If it wins 11 then it’s a contradiction
not really
velvet vale
you can only have max 7 wins right?
velvet vale
oh wtf
coral wraith
sorry
velvet vale
hmmm
velvet vale
ah ok
velvet vale
no worries
coral wraith
@velvet vale you had the right idea
but you just had to apply it to team B, C and so on
based on the quora post
pulsar ermine
devout snowBOT
pulsar ermine
I’m confused as to how setting X_i to have all zeros apart from one row with entry 1 gives the following expansion?
How can (x_i)^T(…)=(…)
real grail
A^T is a matrix that if you multiply it by the matrix A will give you In
A^T * A = I
example :
[\begin{bmatrix*}[r]1 2 \ 2 1 \end{bmatrix*}] * [\begin{bmatrix*}[r]a b \ c d \end{bmatrix*}] = [
\begin{bmatrix*}[r]
1 0 \ 0 1
\end{bmatrix*}
]
if you find a , b , c and d then its A^T
pulsar ermine
Thanks, but I don’t think that’s what I’m asking?
woven radishBOT
Mehdi_Moulati
pulsar ermine
I’m referring to when they set x_i =(0,0,0,…1…,0,0,0)
torn vessel
I’m confused as to how setting X_i to have all zeros apart from one row with ent...
because (P^TP-I)y is a vector. x^T is a row, So basically your taking the dot product of x and (P^TP-I)y
Since that product is 0 for all x,y in R^n
You can pick x = x_i.
Then the dot product will give you the i'th entry of (P^TP-I)y = 0, for each entry. So you can conclude that (P^TP-I)y is the 0 vector
pulsar ermine
I see thanks
So when they’re referring to (P^TP-I)y they’re referring to each row in regards to x_i right
torn vessel
i'm not sure I understand your question
pulsar ermine
So when they write (P^TP-I)y they’re referring to the ith row of (P^TP-I)y
devout snowBOT
@pulsar ermine Has your question been resolved?
devout snowBOT
devout snowBOT
Available help channel!
tulip summit
Karl reads that on average there is a win on every 3rd scratchcard. He claims that if he buys
3 scratch calendars, he is sure to win a prize.
Explain why Karl is not right in his claim.
devout snowBOT
Karl reads that on average there is a win on every 3rd scratchcard. He claims th...
meager blaze
maybe the answer is in the word "on average"
tulip summit
Can i just type that in my answer box
hold up
Karl buys a scratch calendar for Christmas. If he scratches and gets 10 Santas, he wins
1 million DKK. There are 2,520,000 scratch calendars printed and there are 5 scratch calendars with 10
Santas on it.
they also give this info
devout snowBOT
@tulip summit Has your question been resolved?
heavy crow
What is the motivation behind orientation of isometry
devout snowBOT
What is the motivation behind orientation of isometry
@heavy crow Has your question been resolved?
devout snowBOT
@heavy crow Has your question been resolved?
dusky adder
devout snowBOT
wooden wraith
Pretty much that
Since it's an odd power of sin, if you factor out sin^2 (or in general the highest even power you can) and change it to 1-cos^2, then you have a nice u-sub you can do
dusky adder
odd power of cos?
wooden wraith
pythag identity
wooden wraith
It's basically the same idea as the image you posted
wooden wraith
since cos and sin are derivatives of each other (up to a negative sign), if you can leave one sin alone and change everything else to cos, then you're golden
or vice versa
graceful lava
$y''=\frac{d^{2}y}{dx^{2}}$
devout snowBOT
$y''=\frac{d^{2}y}{dx^{2}}$
woven radishBOT
cookie2
graceful lava
is this just taking the implicit derivative twice?
jade vessel
Z=(-3+3i)^2021, Find the Re part and Im part
I solved it and i got
(-1+i)*54^1010
Is this right?
zenith jacinth
is this just taking the implicit derivative twice?
wdym ?
jade vessel
So Re would be -54^1010 and Im would be 54i^1010
zenith jacinth
y is a function, so d^2 y/dy^2 is his second derivative
zenith jacinth
So Re would be -54^1010 and Im would be 54i^1010
woven radishBOT
cookie2
graceful lava
If we have this fun
Then we solve for the thing i did up there
it's just taking the derivative twice
(2nd derivative)?
zenith jacinth
what do you want to get
graceful lava
i dont understand the problem
100%
is it saying i have to do dy/dx on both sides twice?
zenith jacinth
can you send the whole exercise
graceful lava
can you send the whole exercise
$Consider\ the\ equation\ y^{3}-\left(\frac{3}{2}x^{2}\right)=1.\ Solve\ \operatorname{for}\ y''=\frac{d^{2}y}{dx^{2}}.$
woven radishBOT
cookie2
zenith jacinth
$Consider\ the\ equation\ y^{3}-\left(\frac{3}{2}x^{2}\right)=1.\ Solve\ \operat...
then differentiate two times on both sides and find y''
graceful lava
then differentiate two times on both sides and find y''
do i have to isolate dy/dx before doing it a second time?
zenith jacinth
No
graceful lava
Idk how to do this, would it just be multiplying 2 dy/dx together
$6y\left(\frac{dy}{\left(dx\right)}\right)^{2}-3=0$
woven radishBOT
cookie2
graceful lava
like this?
zenith jacinth
First derivative :
$$3y'y^2 - 3x = 0$$
woven radishBOT
Ｈｅｒｅｌｓ
graceful lava
First derivative :
$$3y'y^2 - 3x = 0$$
Is this equivalent to $$3y^{2}\left(\frac{dy}{dx}\right)-3x=0$$
woven radishBOT
cookie2
zenith jacinth
Its the same thing
graceful lava
ok
So my logic is that I just take derivative of 3y^2 again and -3x again
And I end up with $$$$6y\left(\frac{dy}{\left(dx\right)}\right)^{2}-3=0$$$$
woven radishBOT
cookie2
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
graceful lava
Its the same thing
Is this correct?
zenith jacinth
false
real grail
So Re would be -54^1010 and Im would be 54i^1010
nope because :
(-3+3i) ^ 2021 = (-3 + 3i) ^ 2020 * (-3 + 3i)
= ((-3 + 3i)²)^1010 * (-3 + 3i)
= (-18i)^1010 * (-3 + 3i)
= -(18)^1010 * (-3 + 3i)
Why not?
zenith jacinth
nope because :
```
(-3+3i) ^ 2021 = (-3 + 3i) ^ 2020 * (-3 + 3i)
...
go somewhere else my dude
graceful lava
Product rule?
zenith jacinth
yes
graceful lava
What the heck is dy/dx of dy/dx
woven radishBOT
Ｈｅｒｅｌｓ
zenith jacinth
the gradient
dude, not now
fresh elm
wat
zenith jacinth
thats not the main topic
graceful lava
$6y\left(\frac{dy}{dx}\right)+3y^{2}\left(\frac{d^{2}y}{dx^{2}}\right)-3=0$
woven radishBOT
cookie2
zenith jacinth
$6y'^2 y + 3y''y^2 -3 = 0$
woven radishBOT
Ｈｅｒｅｌｓ
graceful lava
Why y prime is squared on the left
zenith jacinth
recheck your calculations
graceful lava
i have no idea
i used product rule
idk where the extra dy/dx came from
im only differentiating dy/dx in the right side
the rightmost one of the product rule*
zenith jacinth
$(3y'y^2 )' = 3(y'' y^2 + 2y'yy') = 3y''y^2 +6y'^2 y$
woven radishBOT
Ｈｅｒｅｌｓ
graceful lava
$(3y'y^2 )' = 3(y'' y^2 + 2y'yy') = 3y''y^2 +6y'^2 y$
$6y\left(y''\right)+y^{2}y^{''}-3=0$
woven radishBOT
cookie2
zenith jacinth
what are you doing
graceful lava
what are you doing
6y\left(\frac{dy}{dx}\right)\left(\frac{dy}{dx}\right)+3y^{2}\left(\frac{d^{2}y}{dx^{2}}\right)-3=0
$6y\left(\frac{dy}{dx}\right)\left(\frac{dy}{dx}\right)+3y^{2}\left(\frac{d^{2}y}{dx^{2}}\right)-3=0$
woven radishBOT
cookie2
graceful lava
Is this right?
zenith jacinth
false
graceful lava
WHAT
graceful lava
LES GO
zenith jacinth
No its not
graceful lava
no
I just added the extra dy/dx
multiplied
when taking derivative of 3y^2
for the leftside of product rule
zenith jacinth
$3y'y^2 = 3(y'' y^2 + 2y'y \times y') = 6y'^2 y + 3y'' y^2$
woven radishBOT
Ｈｅｒｅｌｓ
zenith jacinth
$y'^2 = \left(\frac{dy}{dx}\right)^2$
woven radishBOT
Ｈｅｒｅｌｓ
graceful lava
oh well, you were right, I saw it as d^2 y/dx^2
should i switch to y'' instead of dy/dx
notation wise
but then what would i do for other oens like d/dx
zenith jacinth
$y'' = \frac{d^2 y}{dx^2}$
both notations are valid
woven radishBOT
Ｈｅｒｅｌｓ
woven radishBOT
Ｈｅｒｅｌｓ
zenith jacinth
its not the same as y''
graceful lava
its not the same as y''
oh so it is $y'^{2}$
woven radishBOT
cookie2
zenith jacinth
yes
graceful lava
yes
ok now i have this result, the question is how do i simplify it
$6y^{'^{2}}y+3y^{2}y^{''}=0$
woven radishBOT
cookie2
zenith jacinth
you want y'', so its ez
graceful lava
but isn't it bad if i have y' on the other side?
zenith jacinth
idk, its about implicit differentiation so its fine
graceful lava
idk, its about implicit differentiation so its fine
$y''=\frac{-6y\left(y`\right)^{2}y}{3y^{2}}$
woven radishBOT
cookie2
zenith jacinth
$y'' = -\frac{6y'^2 y}{3y^2} = -\frac{2y'^2}{y}$
woven radishBOT
Ｈｅｒｅｌｓ
devout snowBOT
@graceful lava Has your question been resolved?
quartz gale
how to find max or min ?
devout snowBOT
how to find max or min ?
quartz gale
i know for quadratic we take differtiation and stuff
zenith jacinth
differentiate
scarlet sequoia
and check boundary values
zenith jacinth
yes
quartz gale
then what
quartz gale
oh so we are finding critical points?
zenith jacinth
thats what they mean by finding maximum values tho
quartz gale
dont we have to differentiate again for finding max or mi
zenith jacinth
?
quartz gale
we differntiate again then if answer is less than 0 its max if more than 0 then min
zenith jacinth
you are talking about convexity and concavity
quartz gale
uh
im just saying local max or min
how do i solve for the function
its sin and cos
@zenith jacinth
zenith jacinth
you dont need to differentiate two times to have the max
zenith jacinth
its look kinda impossible
zenith jacinth
,w tan(2x) = -x/3
woven radishBOT
zenith jacinth
,w -12sin(2x) -4xcos(2x)=0
woven radishBOT
devout snowBOT
@quartz gale Has your question been resolved?
wide hemlock
how to find max or min ?
differentiate and restrict your function to the interval. After differentiating, the maximum value is either at the critical points or at the edges of the interval
devout snowBOT
differentiate and restrict your function to the interval. After differentiating,...
wide hemlock
you test to see if it's a maximum or minimum based on the second derivative's value at the critical points
devout snowBOT
@wide hemlock Has your question been resolved?
upbeat karma
help
shadow bobcat
with what
upbeat karma
.close
restive river
if x=3 and y=7 how much times has my grandpa been constipated this past 2 weeks ?
devout snowBOT
if x=3 and y=7 how much times has my grandpa been constipated this past 2 weeks ...
restive river
please help this came in my maths test and i dont know if its right
winter patrol
don't troll
restive river
im not trolling 😡😡😡
main gull
im not trolling 😡😡😡
Definitely trolling
winter patrol
.close
restive river
devout snowBOT
winter hound
how do u multiply both sides
For the first equation one of the term is a fraction
The fraction has a denominator of 5
We don’t want to have fractions in quadratics, so we multiply each term in that equation with the value of the denominator
restive river
ok so i multiply each term in the whole equation by the denominator of the fraction to get rid of the fraction]
winter hound
yes
restive river
ty
winter hound
@restive river type .close when you’re done
restive river
.close
devout snowBOT
restive river
ok ty
cloud ether
Ah no- wait.
devout snowBOT
shadow bobcat
thats ez
cloud ether
Help me out please? 😭
shadow bobcat
why did they have to make it BB lol
just make one equation equal BB
and one z
and then substitute
cloud ether
why did they have to make it BB lol
Nvm- issokay. Someone is helping out <3
shadow bobcat
okayy
devout snowBOT
Available help channel!
verbal lintel
Quick inequality question, is this correct? I always thought the intersection part to be the answer, hence x< 3/5 to be the answer.
devout snowBOT
Quick inequality question, is this correct? I always thought the intersection pa...
supple knot
Quick inequality question, is this correct? I always thought the intersection pa...
You're right
restive river
how do i do quadratic inequalities
devout snowBOT
how do i do quadratic inequalities
trim vine
Do you know how to find the roots of this quadratic?
clear pond
,rotate
woven radishBOT
clear pond
for cosx - 1 series how do we know to subtract 1 from the exponent of x?
isnt it just subtracting the first term which is 1?
wooden wraith
No, starting the sum at n=1 instead of n=0 already takes care of that
That's not right, it should be 2n
wooden wraith
Yeah, all you did was subtract a constant, 1, which happens to be equal to the first term of the series
So all you have to do is remove that term
Which you did by changing the lower bound
wooden wraith
No problem 👍
devout snowBOT
@clear pond Has your question been resolved?
short orchid
Hello I don't know how to solve for x
7x - 3 = 2x + 12 + 2x
devout snowBOT
Hello I don't know how to solve for x
7x - 3 = 2x + 12 + 2x
rigid sandal
hey mate
short orchid
hei
rigid sandal
are u familiar w the concept of like terms
short orchid
yes, i have to move the x
rigid sandal
so
short orchid
ok, 11x
short orchid
yes i know i have to
short orchid
balane
rigid sandal
good
rigid sandal
nah mate
stop for a sec
its the same thing w the x's
u need to add 3 to both sides
to make there no 3 on the left
which gives 3x = 12 + 3
short orchid
so 3x=15?
rigid sandal
yes
rigid sandal
yes
short orchid
k thanks
rigid sandal
all good
rigid sandal
all g mate
short orchid
omg it wont let me get the question hold up
rigid sandal
holding
short orchid
ok
rigid sandal
type that
short orchid
.close
restive river
for b) what does it mean by stabilize
devout snowBOT
for b) what does it mean by stabilize
brazen parcel
I think it means the as t goes to infinity what does f(t) go to
devout snowBOT
@restive river Has your question been resolved?
restive river
I think it means the as t goes to infinity what does f(t) go to
?????
restive river
.close
devout snowBOT
west palm
I don't know aii
a(ii)
I let x=pi-u
But I don't know how to make pi-u become u here
No wait
brave vapor
But I don't know how to make pi-u become u here
f(pi-x) = f(x)
restive river
Crickets
devout snowBOT
Available help channel!
unborn geyser
hmm i dont get it. The video said that if you invest your money, the rate at which it grows is proportional to the amount of money at any time
devout snowBOT
hmm i dont get it. The video said that if you invest your money, the rate at whi...
unborn geyser
i dont understand what the sentence said by the video mean
pseudo basin
do you understand proportionality in general
sounds like you may have trouble with that more than with the actual calculus
unborn geyser
do you understand proportionality in general
well i understand around 50% of it
i havent done proportionality in quite some time
warm fulcrum
Basically two things are proportional if you can multiply the first with some constant to get the latter and vice versa
warm fulcrum
hmm i dont get it. The video said that if you invest your money, the rate at whi...
In here it says the rate that the money grows (dM/dt) is proportional to the money (M). Here the proportionality constant is (1 + r) since to get dM/dt you can take your M at a given time and multiply it with this constant to get dM/dt
warm fulcrum
by dividing (dM/dt) by M?
Usually the constant is given when solving a differential equation, but yeah if it's unknown and you know your amount of money and the rate your money changes at the same time, you can find the proportionality constant by what you said here.
unborn geyser
Usually the constant is given when solving a differential equation, but yeah if ...
hmm actually what does the proportionality constant tell us?
warm fulcrum
Say that you are told your money grows by 1.05 times your current amount of money. This would be a differential equation where the change is proportional to the original function, which can be modelled by dM/dt = 1.05M
unborn geyser
Say that you are told your money grows by 1.05 times your current amount of mone...
take this for example
how does the proportionality constant relate to the two ratios
warm fulcrum
It isn't really, since when the proportionality constant is 1, the two are just equal
warm fulcrum
how does the proportionality constant relate to the two ratios
It just tells you that 1/2 = 1 ∙ 2/4
warm fulcrum
No, it tells you that 3/5 = 33/35 ∙ 7/11
unborn geyser
oh
warm fulcrum
So 3/5 is proportional to 7/11 by the constant 33/35, or said in another way 3/5 is 33/35 times as big as 7/11
unborn geyser
warm fulcrum
In general, we say y is proportional to x, if there exists some constant value k such that y = kx
unborn geyser
theres always 2 proportionality constant for 2 not proportional ratios?
unborn geyser
In general, we say y is proportional to x, if there exists some constant value k...
can we also say that x is proportional to y?
warm fulcrum
Well, kind of. Saying 3/5 is proportional to 7/11 is equivalent to say 7/11 is proportional to 3/5, but the different wordings have different implied meanings. The first usually mean we want it on the form 3/5 = k ∙ 7/11, while the other wants it on the form 7/11 = c ∙ 3/5
But if one is proportional to the other, the other is proportional to the one, yeah
unborn geyser
oh ok
unborn geyser
hmm i dont get it. The video said that if you invest your money, the rate at whi...
this still doesnt make sense
unborn geyser
Well, kind of. Saying 3/5 is proportional to 7/11 is equivalent to say 7/11 is p...
i dont how to relate this to my first question
warm fulcrum
Which part doesn't make sense?
unborn geyser
Which part doesn't make sense?
the proportionality equation
dM/dt = (1+r)M
rate of change = k * M
M is the amount of money right?
warm fulcrum
At a given point in time, yes
unborn geyser
so we can rewrite M as y on the graph
warm fulcrum
Yeah
unborn geyser
dy/dt = k * y
warm fulcrum
Yes
unborn geyser
dy/dt = k * y
so from this we can say that the rate of change is proportional to the amount of money at a given point in time
warm fulcrum
Yeah, that's what it said in the beginning
unborn geyser
oo
but how can we know that its proportional?
by finding if a constant exist such as dy/dt can be express by k * y?
warm fulcrum
It's just the information given
unborn geyser
oh ok
warm fulcrum
It says it's proportional, and modelled by dM/dt = (1 + r)M, for some constant value r
So we can directly read from this that the proportionality constant is 1 + r
umbral niche
devout snowBOT
umbral niche
m^294=(m^18)^16*m^6 and I found 21->41->61->81->01
so (m^18)^16's last two number is 61
and then i don't know how to solve this
long sundial
$m^{18} \equiv 21 \mod{100}$, trying to find $m^{294}\mod{100}$
woven radishBOT
Max..
umbral niche
aha..
long sundial
$m^{294}\equiv {m^{18}}^{16}m^6 \mod{100}$
woven radishBOT
Max..
long sundial
now you have found $(m^{18})^{16} \mod{100}$ so we need to find $m^6\mod{100}$
woven radishBOT
Max..
long sundial
let $y=m^6$
woven radishBOT
Max..
long sundial
then you are trying to solve $y^3\equiv 21 \mod{100}$
woven radishBOT
Max..
midnight laurel
guys i need help
devout snowBOT
guys i need help
midnight laurel
got a lot of topics to cover
starting with finding length using pythagoras theorum
i mean length of a segment*
ill go grab screenies of the other problems while i wait
btw we can just do one or 2 of the equations
and then move on to the next topic
after i do a few practice equations
<@&286206848099549185> pls help this is not homework btw
just revision for exam
manic coral
im just asking is that like finding the distance between the points?
midnight laurel
im just asking is that like finding the distance between the points?
no its length
formula is ACpower 2 + BC power2
=ABpower 2
manic coral
ye i wont be able to help 💀
wanton pumice
i mean length of a segment*
so distance between 2 points?
u can use pythagoras theorem using by treating the length, the horizontal distance and vertical distance as a right angled triangle
wanton pumice
u can find the coordinates of each point and find distance same way u did for question 1
midnight laurel
i dont understand
can u explain pls
@wanton pumice
😐
<@&286206848099549185>
AAAAAAAAA
CAN ANYONE HELP PLS
wanton pumice
once u know the coordinates of two points, u can find the distance between them
In qeustion 1, they directely gave u the 2 points so u can go ahead and apply the formula directly
midnight laurel
that's with a right angle
wanton pumice
In question 2, they didnt give points but you can find the coordinates using the graph
midnight laurel
ya
midnight laurel
In question 2, they didnt give points but you can find the coordinates using the...
do one example pls
wanton pumice
devout snowBOT
devout snowBOT
Available help channel!
left trench
devout snowBOT
unborn geyser
how is the difference in y value after increasing dx for the f(x) graph $df/dx\text{ }(a)\text{ }dx?$ shouldn't it be just $df$?
devout snowBOT
how is the difference in y value after increasing dx for the f(x) graph $df/dx\t...
woven radishBOT
LichBaw_GT
unborn geyser
<@&286206848099549185>
warm fulcrum
df/dx(a) is the slope for f at your point a, multiplying this with dx gives you df if you view it as a triangle
That's just the interpretation of slope for a straight line
unborn geyser
df/dx(a) is the slope for f at your point a, multiplying this with dx gives you ...
oh ok but how did df becomes df/dx(a) dx?
warm fulcrum
When you have a straight line on the form y = ax + b, a is the slope meaning how much you go up/down in y everytime you go 1 to the right on your x-axis. If you take this slope a and multiply it with 2, you get how much the y-value changes when you go 2 to the right on the x-axis and etc. with 3, 4, 5 and so on. This can also be done with non-integers like ½. If you multiply the slope of your line with ½ this is the change in y-value as you go ½ to the right on your x-axis.
Looking at your triangle, the slope of the line that is your hypothenuse, will be the slope of f at point a, or said in another way df/dx(a). Multiplying this slope with dx will give you how much your y-value changes, dy = df, when you go dx to the right on your x-axis. So therefore df = df/dx(a) ∙ dx
unborn geyser
actually i noticed something
df/dx (a) * dx = df(a)
df(a) is the df near the point a after increasing dx
unborn geyser
df/dx (a) * dx = df(a)
the dx cancels out
unborn geyser
When you have a straight line on the form y = ax + b, a is the slope meaning how...
i dont understand the 2nd section
unborn geyser
When you have a straight line on the form y = ax + b, a is the slope meaning how...
is there another explanation for this?
unborn geyser
thanks for the explanation
devout snowBOT
Available help channel!
eternal sluice
i need the solution for this differential equation, not just the final answer but the process to get there. I have been banging my head around this for 2 days. I know it is a bernoulli equation and i know the steps to solve it, but something is not working so i need help.
devout snowBOT
i need the solution for this differential equation, not just the final answer bu...
bright burrow
i need the solution for this differential equation, not just the final answer bu...
i think x=yt substituion could work(?)
eternal sluice
i am thinking of solving it as a homogenous equation
do you think that could make it easier?
bright burrow
i think x=yt substituion could work(?)
idk what homogenous equations mean, but my teacher said if sub this and y vanish then is homogenous equation however idk if thats the actual way to define them lol
bright burrow
i need the solution for this differential equation, not just the final answer bu...
also i couldnt find the form to see it an a bernoulli equation
eternal sluice
also i couldnt find the form to see it an a bernoulli equation
divide the whole thing by 2xy
pass the 3y/2x to the other side
assume x/2y as x2y^-1
that's it
eternal sluice
i think x=yt substituion could work(?)
but either way, you think that using this would be the best way to go about it?
bright burrow
but either way, you think that using this would be the best way to go about it?
maybe, ive not done enough questions on this topic to tell you that for certain
devout snowBOT
Available help channel!
robust mason
can someone help me understand absolute value pls
devout snowBOT
can someone help me understand absolute value pls
wooden veldt
If the number is positive, it does nothing
If the number x is negative, it gives you -x
|3|=3
|-5|=5
|0|=0
robust mason
wait that's all
zenith jacinth
yes
robust mason
but
prime egret
Yeah all for reals but in general it describes the distance of the point from the origin
robust mason
Yeah all for reals but in general it describes the distance of the point from th...
what does that mean pls elaborate
restive river
Absolute value is the distance from 0 of that number
restive river
R u sure
robust mason
yes
restive river
Don’t lie
robust mason
why would I lie
restive river
What does ur name say
robust mason
why do u ask
restive river
Cuz I need to know who I’m talking to
ashen bear
where can i ask for help i mean which channel shud i ask?
prime egret
restive river
?
robust mason
you can call me yassin
robust mason
yes
restive river
R u Muslim yassin
robust mason
not really
robust mason
Well I'm not though
robust mason
I suppose so
restive river
Ok great
robust mason
but that doesn't matter does it
restive river
So u stole Muslim name and aren’t Muslim
robust mason
I live in a muslin country but I'm not nessecarily Muslim though besides wdym by stealing
robust mason
no but why does it matter
restive river
Wow ok
robust mason
alright?
restive river
Doesn’t matter
robust mason
ofcourse
restive river
Yep
robust mason
what did I do
robust mason
yes
restive river
And show ur teacher ur a good student
robust mason
thanks for helping
restive river
Yw
devout snowBOT
@robust mason Has your question been resolved?
stark shuttle
Can anyone help me with this question?
devout snowBOT
Can anyone help me with this question?
left robin
we can pull in the 4
4log(x)=log(x^4)
then move that to the other side
apply 3^() to both sides
hybrid snow
Ah but make sure that x > 0 for the solutions
left robin
the minus should not be there
we have something like a-b=0 so a=b
log3(16)-4log3(x)=0
log3(16)=4log3(x)
log3(16)=log3(x^4)
left robin
almost
i would wait with eliminating the 16
so that we get
4log3(x)=log3(16)
log3(x^4)=log3(16)
apply 3^() to both sides
x^4=16
x^2=+-sqrt(16)=+-4
x=+-sqrt(+-4)
x=+-2, x=+-2i
devout snowBOT
@stark shuttle Has your question been resolved?
devout snowBOT
@stark shuttle Has your question been resolved?
high badge
devout snowBOT
high badge
havent seen this type of graph before
restive river
Well I guess whenever cos(2theta) is 0 would give u r = B
high badge
when theta is 0?
fossil bay
Yes, you can also count it clearly when theta is pi/2 yes?
high badge
yep
restive river
Ahh I see now
high badge
-2?
restive river
Hmm I figured the answer to be -4 ig?
fossil bay
yeah but idk why i keep getting +4 hm
lofty crater
i cannot use a comparison test on this can i?
devout snowBOT
i cannot use a comparison test on this can i?
restive river
Alternating series test may work here nicely if u can prove that series is completely monotonic
lofty crater
but the series doesnt alternate right
lofty crater
all good
i just dont want to do a integral test on this but i cant figure out how to use the comparison test
restive river
Anyways @lofty crater
Direct comparison would work nicely here I think
Start with 6+ (-1)^n <= 7
lofty crater
but how because if we turn it into a pseries it would converge but the new series would not be less than the orignal series
restive river
Start with 6+ (-1)^n <= 7
Divide both sides by n sqrt(n)
You get the comparison with 7/nsqrt(n) which converges
lofty crater
oh i thought you mean it points to 7
okay yeah
so what would i compare this to a pseries or a geometric series
restive river
Wdym
lofty crater
well like would would i turn the b_n into
lofty crater
so 1/nsqrt(n)
restive river
Don't forget 7
lofty crater
why do we need the 7
restive river
Because that's how we defined our bounds to be earlier
lofty crater
yeah but how do i solve a pseries with 7 in the numerator
restive river
It does not matter it is just a constant
lofty crater
wait
so youre telling me this whole time i can solve pseries problems if the constant in the numerator is any number
restive river
Yeah
Because dun dun, p series are just integrals
And what can u do with integrals? Take out constants
lofty crater
that makes sense by why does every example use 1 i thought this whole time it could only be 1
devout snowBOT
@lofty crater Has your question been resolved?
devout snowBOT
devout snowBOT
Available help channel!
unborn geyser
what does the video mean by there is no limiting output where everything is within a distance of epsilon of that output?
devout snowBOT
what does the video mean by there is no limiting output where everything is with...
radiant drift
it's just a form of discontinuity?
A jump discontinuity occurs when the curve “breaks” at a particular place and starts somewhere else.
ive no clue why u need epsilon to explain it
but it's essentially limit from the left and right both exist at f(x)
but they don't equate
unborn geyser
ok
unborn geyser
ive no clue why u need epsilon to explain it
the video was showing the definition of a limit in terms of epsilon and delta
radiant drift
just use intuition
like however small the epsilon is
there exists a positive integer N such that whenever n > N, then |an – L| < ε.
epsilon>0
supple knot
what does the video mean by there is no limiting output where everything is with...
No matter how close x gets to 0, the function left of zero and the function right of zero will always be greater than 0.4
unborn geyser
No matter how close x gets to 0, the function left of zero and the function righ...
so if the function left of zero and the function right of zero is always decreasing as it approaches 0 then the limit exist at 0?
radiant drift
and the opposite is true: A sequence {sn} is said to diverge to ∞, if, for any positive number M, however large, there exists a positive integer N such that whenever n > N, then |sn| > M.
unborn geyser
ok
radiant drift
this applies to limit of functions too. Cause the concept of limit of function is derived of sequence, a thing called heine's theorem
unborn geyser
oh okay
uncut crow
what's heine's theorem?
radiant drift
what's heine's theorem?
uncut crow
interesting, never heard it called that before
supple knot
so if the function left of zero and the function right of zero is always decreas...
No. What I said applies to the image only
devout snowBOT
@unborn geyser Has your question been resolved?
lyric folio
A box of 60 colored marbles contain 3 times as many as blue as yellow and 12 white marbles. What is the probability of choosing not a blue marble from the box?
devout snowBOT
A box of 60 colored marbles contain 3 times as many as blue as yellow and 12 whi...
rotund heron
lilac portal
blue has 3 times more as the number of yellow marbles
lyric folio
3 + x
lilac portal
It just mean equals
lyric folio
ohh
lilac portal
blue has 3 times more than yellow
lilac portal
Yeah that would be much more appropriate
lilac portal
3x=y right? x is yellow while blue is y
lyric folio
Yep