so that negates it
#help-17
upbeat fractal
false
how do u show that something is true for ALL x tho?
i understand how to negate it now but i dont get the opposite
iron parrot
Well, that is slightly more difficult
upbeat fractal
the answer to 5 is true
iron parrot
But in most cases it should be slightly obvious, or with some manipulation, you can do it
upbeat fractal
but i believe thats the case where u need to show that its true for all
could u help me w 5?
in terms of y again
?
∃ x ∈ R ∀ y ∈ R : xy^2 + 2^x = 1
iron parrot
For this question, we see that there is a variable exponent
upbeat fractal
yes, the 2^x right
upbeat fractal
ys
iron parrot
So start out by trying some small values for x, like 0,1, and 2
upbeat fractal
okay
upbeat fractal
So start out by trying some small values for x, like 0,1, and 2
wait 1 for x doesnt work
shouldnt i be plugging those in for y instead?
iron parrot
shouldnt i be plugging those in for y instead?
You could, but the x as an exponent means that it is not very straightforward to solve for x when you assume y
Ok
You tried x=1, right?
upbeat fractal
yes
iron parrot
Which term caused it to become negative?
iron parrot
or rather in the equation, it would be the $2^x$ term, right?
twin meteorBOT
Dork9399
upbeat fractal
yes
upbeat fractal
2^x must be > 1 ?
iron parrot
and what does that mean for x
upbeat fractal
it can only be 0
iron parrot
it has to be $\leq 0$
iron parrot
sqrt0 is still real
upbeat fractal
o ok
twin meteorBOT
Dork9399
iron parrot
So trying 0, what do you get?
upbeat fractal
0 for x?
iron parrot
yes
upbeat fractal
y can be anything
ohh
since its just 0*y + 1 = 1
and anything times 0 is 0
?
upbeat fractal
yes
iron parrot
I really don't like this question, because it deals more with if you can see that 0 will work, and deals less with your mathematical skills than your guessing ability
upbeat fractal
yeahh
iron parrot
But ig thats some questions
upbeat fractal
do i not have to account for when x<0?
iron parrot
y cant take all reals if its fixed
upbeat fractal
since, for any real y, it works as long as x=0 ?
iron parrot
wait what
upbeat fractal
wait
iron parrot
explain your question
upbeat fractal
**Dork9399**
so here we found that the bound for x is that x<=0
and for x=0, we know that all real y's will have a corresponding real x which is 0
but do we not have to do anything for when x is <0
iron parrot
Yes, because of the "∃" symbol, we only need to find one value of x
iron parrot
Rather, we have to find if there exists an x such that xy^2 + 2^x = 1 for all real y.
upbeat fractal
okk!
i understand it a lot better now
if the question was, for all real x and for all real y, then it would be false the moment i found the bound for x right
iron parrot
yes
upbeat fractal
since we figured out that x must be <=0 for there to be any real solution for y, and that means that it wont work for all real x
iron parrot
yes
iron parrot
If you ever get stuck on a question like this, an easy way to start is by plugging in things like 0, -1, sqrt2, or numbers of that nature
upbeat fractal
If you ever get stuck on a question like this, an easy way to start is by pluggi...
to chooser whether i plug into x or y, do i just go by the there exists/for all
like for this problem we plugged into x because its there exists an x
iron parrot
o wait no
iron parrot
for this problem we just chose the more convenient one
For question 4, x = 1/(y^2 + 1)
We could have gone by y
iron parrot
an alternate solution would be looking at all x's for the values of y, and figuring out that x has a bound
main light
yo do i just put the question here?
upbeat fractal
yo do i just put the question here?
upbeat fractal
an alternate solution would be looking at all x's for the values of y, and figur...
for all x, there exists a y: find a y that bounds x to prove that its not for all x
?
upbeat fractal
whats the alternative method?
iron parrot
We find a real that is impossible to make, like x = -1
upbeat fractal
TO NEGATE
for all real x, there exists a real y: find a y that bounds x to prove that its not for all x
OR
find a x with no real y
upbeat fractal
We find a real that is impossible to make, like x = -1
is this to prove true or false?
iron parrot
False
upbeat fractal
oh ok so
upbeat fractal
**TO NEGATE**
for all real x, there exists a real y: find a y that bounds x to p...
does this look right
wait uhh
upbeat fractal
and then to prove true, u need to find a y that works for any real x?
iron parrot
Yes
upbeat fractal
is there any alternative way to do that or no
iron parrot
Not that I can think of, no
An easier way would be to ask your teacher or to reference your textbook lol
iron parrot
np
upbeat fractal
.close
dapper pine
How would I do this one
vocal sleetBOT
How would I do this one
iron parrot
Just send a screenshot of the first part
iron parrot
im assuming that tn is the nth term and Sn is the sum of the first nth terms, right?
dapper pine
mhm
silk osprey
do u know the formula for sum of arithmetic sequence
dapper pine
If this helps
dapper pine
do u know the formula for sum of arithmetic sequence
Yes
dapper pine
Yeah
silk osprey
so we use the common difference
dapper pine
Got it
silk osprey
so u did part a?
dapper pine
Mhm
dapper pine
Okay
silk osprey
is S=n(a+an)/2
we know S, n, and an
and 2 is obviously a constant so this is a single variable equation we can solve
dapper pine
Wait what is an
N?
Oh last term
How do we know that 50 is the last term tho?
OH
NVM GOT IT
vocal sleetBOT
fervent geyser
ah
vocal sleetBOT
Available help channel!
scenic ravine
$\int \frac{ln(1+x^4)}{x^3}$
vocal sleetBOT
$\int \frac{ln(1+x^4)}{x^3}$
twin meteorBOT
Why am. I here
scenic ravine
!status
vocal sleetBOT
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
1
what sub would I start with?
or do I go straight to IBP
ok, just realised IBP works, but is there any clever sub I can use instead?
inner osprey
ibp is best
scenic ravine
OK, thanks
inner osprey
🎉
scenic ravine
.close
scenic ravine
how would I do this?
vocal sleetBOT
how would I do this?
scenic ravine
The problem would be easy if not for the sin(x) term
inner osprey
how would I do this?
rewrite tan(x)sin(x) !
well a bit boring if you give the exact solution but ok
unique sonnet
lol, that's funny
scenic ravine
rewrite tan(x)sin(x) !
Yeah I tried that, but never thought of moving the sin term outside the root. Thanks!
thanks!
.close
vocal sleetBOT
vocal sleetBOT
Available help channel!
slender brook
How do I solve this?
vocal sleetBOT
@slender brook Has your question been resolved?
slender brook
<@&286206848099549185>
vocal sleetBOT
@slender brook Has your question been resolved?
vocal sleetBOT
@slender brook Has your question been resolved?
gloomy wadi
How do I solve this?
I believe with conditional probability
vocal sleetBOT
@slender brook Has your question been resolved?
blissful sentinel
How do I solve this?
What have you tried so far?
vast shale
How would I solve number 5? (Ignore the work) I don’t know where to begin
vast shale
,rotate
twin meteorBOT
elder scaffold
yo
paper depot
@vast shale do you know in general how to find the area of a sector using its angle and radius?
elder scaffold
i mean they have the right idea just executed it wrong
paper depot
op said to ignore their work.
elder scaffold
fair, just assumed it was their attempt
paper depot
and imo they should be able to state the formula i asked them for.
vast shale
<@456226577798135808> do you know in general how to find the area of a sector us...
Yes
vast shale
It’s part of circle x/360 * circle area
paper depot
ok right... that's if your angle is given in degrees
vast shale
Yes
paper depot
but if it is in radians it's actually simpler
vast shale
Oh ok
paper depot
$A = \frac{1}{2} \theta r^2$
twin meteorBOT
Ann
paper depot
theta is the angle, so yes in your case it is 5pi/9.
vast shale
Ok thx
elder scaffold
I actually have never see that before but is theta 5pi/9
x/360 * πr^2 is the normal formula, but 360 degrees is equal to 2π so instead we have x/2π * πr^2 and from there we can just cancel the π to get x/2 * r^2
vast shale
Am I doing this right
vast shale
x/360 * πr^2 is the normal formula, but 360 degrees is equal to 2π so instead we...
Oh okay thanks
elder scaffold
yeah you did it right
vast shale
x/360 * πr^2 is the normal formula, but 360 degrees is equal to 2π so instead we...
Would all the pi(s) cancel each other out
Then it would be 5/9 * 18^2
vast shale
Oh ok
vast shale
just from the original fraction
Am I doing something wrong
vast shale
Am I doing this right
It’s different from this
elder scaffold
5π/(9*2π) = 5/(9**2)
vast shale
.close
dreamy lark
There are 64 handball teams who play single elimination tournament, hence 6 rounds,
and you have to predict all the winners in all 63 games. You gain 32 points for correctly
predicting the final winner, 16 points for each correct finalist, 8 points for each correct
semifinalist and so on, down to 1 point for every correctly predicted winner for the first
round. Knowing nothing about the teams, you decide every one of your bets by tossing
a coin. For a given match, when you get Heads you bet on the first team and when you
get Tails you bet on the second team. Which team is first and which is second is chosen
at random. The coin is biased towards heads with a probability 0.6.
\1. What is the maximum number of points you can get? (5 marks)
\2. What is the expected number of points you will get? (15 marks)
vocal sleetBOT
There are 64 handball teams who play single elimination tournament, hence 6 roun...
dreamy lark
I did question 1 i got 192 points
i believe that is correct
but for question 2
i feel like my approach is all wrong
and i keep getting answers that dont seem right
Correct first round: 32 predictions, probability of correct prediction = 0.632, so
expected points = 32 ∗ 1 ∗ (0.6^32) = 0.0000025472
Correct second round: 16 predictions, probability of correct prediction = 0.616,
so expected points =16 ∗ 2 ∗ (0.6^16) = 0.018055104
Correct third round: 8 predictions, probability of correct prediction = 0.68, so
expected points = 8 ∗ 4 ∗ (0.6^8) = 0.53747712
Correct fourth round: 4 predictions, probability of correct prediction = 0.64, so
expected points = 4 ∗ 8 ∗ (0.6^4) = 4.1472
Correct fifth round: 2 predictions, probability of correct prediction = 0.62, so
expected points = 2 ∗ 16 ∗ (0.6^2) = 11.52
Correct final winner: 1 prediction, probability of correct prediction = 0.61, so
expected points = 1 ∗ 32 ∗ (0.6^1) = 19.2
Adding all the expected points together, the expected number of points you will
get is 35.4227347712
this was my first approach
i think its wrong
then i tried without the powers
and you get 6x19.2
115.2
but i dont feel like its right either and im unsure how to go about it
and the probability of getting a correct prediction isnt 50/50 since the coin is 60% wieghted but its not 60% either
spiral turtle
Consider what you know about the teams, which is nothing, so the fact that the coin is biased towards heads doesn't actually matter, because under this assumption, there is a 50/50 probability that either team wins, and so the ordering of the teams in any specific pairing also doesn't matter. Let's show this explicitly: Let the teams be named A and B. There are four possibilities. Team A corresponds to Heads, and you throw a Heads. Team A corresponds to Tails, and you throw a Heads. Team A corresponds to Heads, and you throw a Tails. Team A corresponds to Tails, and you throw a Tails. We don't know how we select Team A to be Heads vs Team B, so let's let P(A is Heads) = x. P(A is Heads and Heads is thrown) = 0.6x, P(A is Tails and Heads is thrown) = 0.6(1-x), P(A is Heads and Tails is thrown) = 0.4x, and P(A is Tails and Tails is thrown) = 0.4(1-x). We can add up the probabilities that you are correct, as these are disjoint, and we get 0.6x + 0.4(1-x) = 0.4 + 0.2x. So, now we have to make an assumption. And our assumption is that these teams are seeded at random, that is, there is no correlation between how we choose Team A to be Heads vs Team B and how well they will do. We can ensure this assumption, if we go out of our way to randomize which of Team A and Team B is Heads, instead of picking the same each time. If that's the case then x = 0.5, so our winning probability is 0.4 + 0.2 * 0.5 = 0.5 for any given round.
@dreamy lark
dreamy lark
I see so you're assuming that since nothing is know abt them that the chance of picking the correct team would still be 50%
spiral turtle
yes.
dreamy lark
I see ok then assuming 50%
Round one has 32 rounds
With the ability to win 1 point per guess
Mutliplied by the 50% chance of getting it correct
So its 16 points in round 1
Then 16 rounds 2 points 50%
So its always 16
So 16 * 6
So 96 points?
I guess that makes sense
Since max points is 192
Which is double 96
And since the chance of a correct answer is 50%
Then it would follow that the estimated points would be half of the maximum
Thanks that makes a lot of sense :)
Now just gotta write it in latek
.close
vocal sleetBOT
spiral turtle
.reopen
vocal sleetBOT
✅
spiral turtle
@dreamy lark
dreamy lark
hi
spiral turtle
Sorry, one second
dreamy lark
ye?
spiral turtle
ok so question about the process
dreamy lark
mhm
spiral turtle
do you have to place all of the bets before any games are played?
dreamy lark
no from the wording i assumed that right before a game you simply flip a coin
and you assign head to team a tails to team b
and the coin is wieghted to heads 60% of the time
dreamy lark
Thanks i appreciate it :)
well tbf
it does say predict all the winner
but i assume he means before each game
spiral turtle
rather than before the tournament
dreamy lark
it would still be the same if you predicted before all the games tho right?
dreamy lark
youd still be flipping the same coin
for team a or b
not knowing who team a or b are
spiral turtle
yes, but I assume you'd be entering the bracket in like Team 1-64
not Team A and B
dreamy lark
no you flip for each match
spiral turtle
like you know that in the first match you have Team 3 vs Team 9
spiral turtle
So you'd flip a coin for that match
dreamy lark
ye
spiral turtle
then write down that (for instance) Team 9 won
dreamy lark
yes
spiral turtle
then in the next round of 16 you'd have Team 9 vs Team someone else
dreamy lark
yes
spiral turtle
but if Team 3 won the first round and advanced to the round of 16
then there'd be a 0% chance of team 9 winning
dreamy lark
it wouldnt matter who won since you never cared who wins
yes but youre not betting on teams
youre betting on team a or b in that match in that round
spiral turtle
well, that is a matter of semantics, it depends on how the betting is structured in the tournament
and that's not clear
dreamy lark
i guess so ill just write assuming you take the bets before each match
wait no it doesnt matter still
spiral turtle
if you had to fill out the entire bracket before the tournament started, then there's only a 25% chance to correctly guess the winner of the round of 16 (because you'd be guessing 1 team from 4)
dreamy lark
because youre not betting on team 3 or team 9
dreamy lark
in this situation youre betting on an arbitrary team a or team b
each match will have a team a or b
and you dont care who advanced from before
becuase the coin chooses
spiral turtle
well, here's the thing
let's say that Team 64 wins
like overall
and Team 64 participates in the following matches:
dreamy lark
if you had to fill out the entire bracket before the tournament started, then th...
no you wouldnt youd geuss team a or team b from each match not if its a specific team
spiral turtle
64 vs 32, 64 vs 16, 64 vs 8, 64 vs 4, 64 vs 2, and 64 vs 1 in the final.
dreamy lark
ye
spiral turtle
with our coin, we predict at the beginning that team 64 will win against team 32, and team 16, but lose against team 8.
So once we have predicted a loss, no bet involving team 64 will be won.
spiral turtle
no you wouldnt youd geuss team a or team b from each match not if its a specific...
This is just your interpretation of the prompt, it's not the only interpretation of the prompt, is what I'm getting at.
dreamy lark
thats where i think youre wrong
youre not betting on a specific team 64 for example
youre betting on an arbitary team a or b in match 1
of round 1
spiral turtle
like, I totally get what you're driving at.
spiral turtle
I understand how you're interpreting the betting
dreamy lark
This is just your interpretation of the prompt, it's not the only interpretation...
i see
spiral turtle
I'm only pointing out, that your interpretation is not the only interpretation that makes sense here.
It's an ambiguous question
dreamy lark
so you're saying you could interpret it that you have to bet on exactly who the team winning would be
and not who wins the match arbitrarily
but specifically who
i guess you're right that would change it a lot
since if they lose they wouldnt do the next matches
dreamy lark
For a given match, when you get Heads you bet on the first team and when you
get Tails you bet on the second team. Which team is first and which is second is chosen
at random. The coin is biased towards heads with a probability 0.6.
spiral turtle
I think the purpose of the question was to see if you could see through the tricky bit with the weighted coin
dreamy lark
this part of the question i think clarifies it
that its an arbitrary team
cuz he says heads is the first team and tails is the second
spiral turtle
well, there's still two ways to think about this.
spiral turtle
"For a given match" could mean that you are currently considering a match that has already been paired by the actual tournament, or a match that has been paired by your hypothetical tournament that you've made from coin flips.
dreamy lark
i see
inner osprey
irrespective of the intended interpretation, the first interpretation does make a lot more sense language-wise here
spiral turtle
It all hinges on whether all of the bets are required to be placed before the tournament begins, or before each round begins.
well, seeing as the value for each round doubles, it makes a lot of sense that the bets are required before the tournament begins.
dreamy lark
i wrote "Assuming you bet on an arbitary team A and team B before each match in the tournament with the weighted coin."to clear any misinterpretation
inner osprey
well, seeing as the value for each round doubles, it makes a lot of sense that t...
yes that's the discrepancy 👀
dreamy lark
well, seeing as the value for each round doubles, it makes a lot of sense that t...
thats true
spiral turtle
because that would make the expected value of winning any particular bet the same.
dreamy lark
but youre not betting on a specific team
youre betting for each match
on multiple different teams
like in round 1 you have to choose 32 different teams to win each match
spiral turtle
I think you're probably in the clear so long as you stated your assumption.
dreamy lark
ye lets hope he agrees with me
spiral turtle
I don't think it's a matter of him agreeing or not with you. He's probably also aware of the multiple different ways that this question can be interpreted.
dreamy lark
thats possible but its also possible he didnt realize the ambiguity
english isnt his first language he is greek tbf
and his english isnt great
spiral turtle
Unless this is the first time he's teaching this course or giving this problem, he's definitely has run into this ambiguity before.
spiral turtle
neat, well, it will be a learning experience for the both of you
dreamy lark
haha i guess so
spiral turtle
might as well do the problem both ways
dreamy lark
the second way seems.. a lot more complicated ngl
spiral turtle
One way you get 96 points (half of the total points) and the other way you get 31.5 points (half of the total matches)
dreamy lark
half of the total matches?
spiral turtle
yeah, there's 63 matches right?
spiral turtle
because the expected value for each match is 0.5
for the first round, you get 1 point if you win, and 0 points if you lose.
dreamy lark
but the point amount increases with each round ? ooooh
i see
cuz its a 1/4th chance in the 2nd round
spiral turtle
yes
dreamy lark
then 1/8 in the 3rd
spiral turtle
exactly
dreamy lark
i see
spiral turtle
so both ways it's easy
Just have to explain the assumption and give both answers. If Assumption A then 96, if Assumption B then 31.5
That would certainly demonstrate that you understand how the problem works.
spiral turtle
👍
spiral turtle
I'm going to go ahead and close this again.
spiral turtle
.close
dapper pine
Can someone check what I’m doing wrong
vocal sleetBOT
Can someone check what I’m doing wrong
strong grove
seems correct
strong grove
maybe they are wrong
languid dune
How do I do the two examples
vocal sleetBOT
How do I do the two examples
desert hornet
what do you think a non-decreasing sequence of sets is?
rough pine
Hi
rough pine
Say u have ab + bcos(u)
And a > b >0
And u is from 0 to 2pi
Can the above expression never equal 0?
tidal dock
Can the above expression never equal 0?
yes
silk comet
a=1 b=anything u=pi?
astral shadow
a > b
rough pine
Yh
astral shadow
Can the above expression never equal 0?
What do you think?
silk comet
oops
rough pine
What do you think?
I don’t think it can = 0 but I just wanted some clarity
rough pine
Just to confirm
desert hornet
a<1?
tidal dock
I don’t think it can = 0 but I just wanted some clarity
i mean
astral shadow
You do not.
astral shadow
yes.
tidal dock
ab+bcos(u)=0
is equivalent to
a+cos(u)=0
if 0<a<1, then no, it will have at least one zero
desert hornet
it will have infinite zeroes*
astral shadow
0 to 2pi
astral shadow
So only 2.
desert hornet
yes
tidal dock
yeah exactly 2
astral shadow
Hence it can never equal 0
cos(u) = -a
-a < 0
But cos(u) can also be < 0
rough pine
It can’t right
astral shadow
It can.
tidal dock
Can it equal 0 then?
yes
astral shadow
That's the condition for the solution.
silk comet
a=1 b=anything u=pi?
this?
desert hornet
$ab+b\cos(u)=0$
twin meteorBOT
kheerii
desert hornet
$\cos(u)=-a$
twin meteorBOT
kheerii
desert hornet
this will have solutions when a<1
you can have 1>a>b>0 for that expression to be able to be 0
sinful sky
can someone give me an intuition on how the Ring Q[X,Y] looks like?
silk comet
the elements of $\Q[X,Y]$ are sums of monomials of the form $aX^iY^j$ for $a \in \Q$ and $i,j$ nonnegative integers
twin meteorBOT
Tushar
silk comet
e.g. X^3 + (1/2)X^2Y + (1/2)XY^2 + Y^3 + (2/3)XY
sinful sky
so like how can I find out which of them are the same:
silk comet
you can ask whether elements of one are in the others
Q[X^2, X^3] are polynomials in X^2 and X^3
is X in Q[X^2, X^3]?
sinful sky
no
silk comet
so the first two are not the same
then you can compare whether the third one is equal to either of the first two
sinful sky
in your definition of monomials, can i and j be 0?
trim relic
The population of a suburb of Los Angeles increases exponentially over time for a period. Ten years ago, the suburb had 85,000 inhabitants. Now 102,000 people live there.
After how many years, counting from today, will the population increase by 3,000 people per year, if we assume that the population continues to increase in the same way?
vocal sleetBOT
The population of a suburb of Los Angeles increases exponentially over time for ...
trim relic
yep
vast shale
You need to get alpha
trim relic
i got something along the lines of 0.183 not sure if thats correct
vast shale
For alpha?
trim relic
yes
twin meteorBOT
Palahoo
trim relic
<@&286206848099549185> yall wouldnt mind
lucid bane
i got something along the lines of 0.183 not sure if thats correct
!showwork
trim relic
yeah i dont even have much to show got stuck here
mild flower
well i mean how did you get 0.183
lucid bane
oh did you just approximate?
paper depot
<@&268886789983436800> ?
unique silo
Def
unique silo
I know I have to find the area of the dirt path and the total area of the thing combined and subtract the dirt from it to find the area of the garden
My teacher said I was on the right track and stuff but idk what to do after this
paper depot
,rccw
twin meteorBOT
unique silo
what does that do
unique silo
wdym notational hiccups
paper depot
minor mistakes in notation that can be fixed quickly and don't wreck you
but you've got the thing down to 2x^2 - 36x + 144
unique silo
yeah
paper depot
that... is it, really?
unique silo
what too you mean
paper depot
btw
unique silo
What do you mean
paper depot
these are the symbols you missed
aside from that, your work is correct and finished
unique silo
It is?
unique silo
I have to find the area tho
unique silo
I thought I had to find the area of the dirt path, and the total area of the garden and path combined
Then subtract it
paper depot
no
you just found the green area directly
cause it's a rectangle with width 2x-12 and height x-12
you did it correctly...
unique silo
It says, find the area in square feet. So would 2x^2 be the answer
unique silo
oh
thank you
I just didn’t realize it was that simple
I guess I was over complicating things
Thank you Ann
Hey Mrs Ann do you study? If so how would I study and be smart like you
I just don’t want to disappoint my parents
paper depot
uh
😂 sorry i really do not have any anti-parental-disappointment advice lmao
nor any real study advice
like yeah i am a uni student but i have no routine to speak of
unique silo
you just answered it really fast and I just want to know how to know how to solve it
Because tbh I didn’t even know how to set this up and solve it
Until my friend spouted some random things abt it
paper depot
i mean like. idk
i answered it really fast because this is things ive had... like a decade of experience with, one way or another
i really have no idea what to say as far as advice. honest.
tidal dock
Because tbh I didn’t even know how to set this up and solve it
practice and experience really are the only things
pearl ingot
Stuck on the inverse of e to an exponent...
At this point, I'm not sure how to pull the exponent off of e in order to manipulate y
Yeah, it'll be the natural log, but against what? (y-7) or... ?
Sidenote: I KNOW WE'RE SOLVING FOR Y... THAT'S THE ENTIRE POINT OF AN INVERSE FUNCTION
This is my original problem and how far I've gotten:
lyric relic
hey here we go again
pearl ingot
But because people want to answer 15 questions that are NOT my question, let's break it down to something simpler
y = e^x
x = e^y
pearl ingot
hey here we go again
You don't like it, leave
visual oracle
ln x = ln e^y
pearl ingot
I don't need your smartass comments
pearl ingot
ln x = ln e^y
Wait.. why does taking the ln of e eliminate e?
pearl ingot
why is there animosity
Not important
visual oracle
ln is defined like that
lyric relic
because ln is the inverse function of e^x
tidal dock
Wait.. why does taking the ln of e eliminate e?
because that's literally the definition of ln
inner osprey
visual oracle
ln x asks e to the power of what = x
so ln(e^y) asks e to the power of what = e^y
which is of course just y
does that make sense?
pearl ingot
Ok... so ln e^(y-7) is just y-7?
visual oracle
correct
tidal dock
Ok... so ln e^(y-7) is just y-7?
yep
pearl ingot
Holy shit, why couldn't people just say that to start with... thank you, ed
visual oracle
no problem
pearl ingot
.close
vocal sleetBOT
tidal dock
Holy shit, why couldn't people just say that to start with... thank you, ed
that's what i was trying to tell you but you closed it lol
pearl ingot
.reopen
vocal sleetBOT
✅
pearl ingot
that's what i was trying to tell you but you closed it lol
Nah.. you tried to tell me that taking the inverse means solving for y
Thanks but no, hasta
lyric relic
that's what i was trying to tell you but you closed it lol
yeah its like wtf ...just let me to help you
pearl ingot
.close
vocal sleetBOT
inner osprey
😂
vocal sleetBOT
Available help channel!
slow ice
for this https://i.imgur.com/idBrl4D.png
vocal sleetBOT
for this https://i.imgur.com/idBrl4D.png
slow ice
i want to say:
1/n < x when 1 < xn
since both x and n are positive, xn will grow as n grows
the positive integers are unbounded, so no matter how small x is, there will always be n > 1/x which will satisfy our inequality
is that good enough
and how do i put it into math
tidal dock
i want to say:
1/n < x when 1 < xn
since both x and n are positive, xn will grow...
maybe n > 1/x ?
yes
slow ice
yeah wa sjust about to edit that
tidal dock
are there any theorems/axioms you can refer to?
the proof itself is correct, but idk how rigorous you want it to be
dull bear
[if not, there’s a way to find an explicit n based on your x]
slow ice
[if not, there’s a way to find an explicit n based on your x]
do you just mean n = 1/x? that's what I had it as before
remote arch
Do you have the Archimedean Property available?
slow ice
but I wasn't sure if that is always an int
slow ice
Do you have the Archimedean Property available?
I do
dull bear
but I wasn't sure if that is always an int
It may not be: you can make it into one though!
remote arch
I do
Cool. So given real positive numbers x and y, you can always find an integer n > 0, such that nx > y.
In particular, you can choose y = 1.
slow ice
Cool. So given real positive numbers x and y, you can always find an integer n >...
yeah, I think this is what they wanted
@dull bear still would like to hear what you meant by a way to find n
dull bear
<@788085606483361802> still would like to hear what you meant by a way to find n
Round 1/x up, then add one
slow ice
oh sure, that's a little code-ish
dull bear
Choose $n = \ceil{1/x} + 1$, then $n$ is a positive integer and greater than $1/x$
twin meteorBOT
@dull bear
dull bear
Do you have the Archimedean Property available?
That was more if you didn’t have any theorems to work with though, as per above, Archimedean property is probs easier
slow ice
ok thx guys
hushed pewter
Even though ceil is just archimedean property in application 
slow ice
.close
brave zenith
need help with this. I went in a loop and ended up with 2Pr(Q) + 2Pr(R) - 2 but that doesn't seem close to what I need
vocal sleetBOT
need help with this. I went in a loop and ended up with 2Pr(Q) + 2Pr(R) - 2 but ...
hushed pewter
What is Pr? Probability?
brave zenith
yeah sorry. I need to use set theory to boil it down tho
So the Pr(QUR) = Pr(Q) + Pr(R)
or something... the identity used is if the sets are mutually exclusive tho :/
remote arch
or something... the identity used is if the sets are mutually exclusive tho :/
If you want to brute-force it, you can use the full identity, instead of the one for mutually exclusive sets.
Pr(Q u R) = Pr(Q) + Pr(R) - Pr(Q n R).
Also $Pr\left(\bar{Q}\cup\bar{R}\right) = Pr\left(\overline{Q\cap R}\right) = 1 - Pr\left(Q\cap R\right)$
twin meteorBOT
OneTrackPony
remote arch
So you can pop this in on your left-hand-side, so it becomes
Pr(Q u R) - (1 - Pr(Q n R))
Then use the identity Pr(Q u R) = Pr(Q) + Pr(R) - Pr(Q n R) on the first term to get
Pr(Q) + Pr(R) - 1
Rewrite this as Pr(R) - (1 - Pr(Q)), which is what they are asking for.
vocal sleetBOT
@brave zenith Has your question been resolved?
brave zenith
Rewrite this as Pr(R) - (1 - Pr(Q)), which is what they are asking for.
ahh thank you very much
ember flame
what would be the best way to solve this? like graphically or with substitution or elimination
vocal sleetBOT
what would be the best way to solve this? like graphically or with substitution ...
ember flame
wait for subsitiution do u need to multiply both of them or just one
cos i multipled the top by 2 yk and it gave 6x + 10y = 20 but then if i do thst then both Y and x r eliminated
😓😓
inner osprey
yeah that's right
inner osprey
you've proven this system has no solutions
ember flame
oh
inner osprey
if x and y are both eliminated, they cannot be found
tidal dock
cos i multipled the top by 2 yk and it gave 6x + 10y = 20 but then if i do thst ...
if both are eliminated you are getting something like 0=20
ember flame
ohh ok
tidal dock
which isn't true hence no solutions
vocal sleetBOT
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Available help channel!
cosmic void
vocal sleetBOT
cosmic void
having trouble with this
i keep getting the wrong answer
i got 107 degrees and 0 degrees
vocal sleetBOT
@cosmic void Has your question been resolved?
gloomy wadi
Hi! So the vectors are u=(-1,8,1) and v=(1,-4,0)
I found that $\vec{u} \times \vec{v}=(4,1,-4)$
twin meteorBOT
cristorenzo99
gloomy wadi
Given that $|\vec{u} \times \vec{v}|=|\vec{u}| \cdot |\vec{v}| \cdot \sin(\alpha)$, where $\alpha$ is the angle between $\vec{u}|$ and $|\vec{v}|$, we have: $|\vec{u} \times \vec{v}|=\sqrt{4^2+1^2+4^2}=\sqrt{16+1+16}=\sqrt{33}$;
$|\vec{u}|=\sqrt{(-1)^2+8^2+1^2}=\sqrt{1+64+1}=\sqrt{66}$
$|\vec{v}|=\sqrt{1^2+(-4)^2+0^2}=\sqrt{1+16}=\sqrt{17}$, you can say that:
\begin{equation}
\sin{\alpha}=\frac{|\vec{u} \times \vec{v}|}{|\vec{u}|\cdot |\vec{v}|}=\frac{\sqrt{33}}{\sqrt{17}\cdot\sqrt{66}}=\frac{1}{\sqrt{34}}
\end{equation}
So $\alpha = \asin{\frac{1}{\sqrt{34}}}=9.874°$. The additional obtuse angle is $180°-9.974°=170° ,7' ,30.14'' $
vocal sleetBOT
twin meteorBOT
cristorenzo99
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Available help channel!
echo plaza
i simplified the equation to 2xy=p(x+y) but im not sure how to continue
echo plaza
anyone?
inner osprey
my way of doing this after reaching your current point would be first ||multiply by both sides by 2||, then ||rearrange||, then ||apply simon's favorite factoring trick||
hybrid flicker
Get rid of the fractions first
inner osprey
my way of doing this after reaching your current point would be first ||multiply...
and the solution directly follows
inner osprey
Get rid of the fractions first
they already did that
hybrid flicker
i simplified the equation to 2xy=p(x+y) but im not sure how to continue
Ah here didn't see that
The trick is to write 4xy + ...x + ...y = 0
echo plaza
my way of doing this after reaching your current point would be first ||multiply...
oh ok ill try that
4xy=2p(x+y)
hybrid flicker
And then indeed notice that the left side is the beginning of a factor
echo plaza
And then indeed notice that the left side is the beginning of a factor
i see
hybrid flicker
4xy-2px-2py=0
Exactly
Add something (no x or y factor) on both sides so that the left side is (...)(...)
Additional hint : ||(2x+a)(2y+b) = ... and identify terms and what's missing||
vocal sleetBOT
@echo plaza Has your question been resolved?
echo plaza
Additional hint : ||(2x+a)(2y+b) = ... and identify terms and what's missing||
this is what i get (4y-2p)(x-2p)=4p^2
how does this show that there is only one integer solution
you there?
@hybrid flicker
inner osprey
(you factored it incorrectly)
echo plaza
(you factored it incorrectly)
actually?
this is what i did
ohhh i see what i did wrong
should have been 4x(y-2p)-2px
wait
2x(2y-p)
@inner osprey this is correct i believe (2x-p)(2y-p)=p^2
both 2x-p and 2y-p equal p i believe
.close
main mesa
vocal sleetBOT
https://cdn.discordapp.com/attachments/903481102591750184/1198106684128768060/Sc...
blissful sentinel
<@&268886789983436800> <@&286206848099549185>
(Please don't ping Moderators for math help, only for moderation issues)
vocal sleetBOT
<@&268886789983436800> <@&286206848099549185>
Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
main mesa
(Please don't ping Moderators for math help, only for moderation issues)
Oh my
Bad
Can I ping helpers for help?
blissful sentinel
Only after 15 minutes of no response, but since you already pinged once, don't ping again
severe current
https://cdn.discordapp.com/attachments/903481102591750184/1198106684128768060/Sc...
You know that the center of a side of the square will be touching the sphere or it isn’t optimal
main mesa
Yeah
main mesa
You know that the center of a side of the square will be touching the sphere or ...
ohh do you think it would make it easier if I took a cross section so I don’t have to worry about 3d coordinates
severe current
ohh do you think it would make it easier if I took a cross section so I don’t ha...
Idk. I have no paper rn and i cant work out stuff online or in my head lol
main mesa
Idk. I have no paper rn and i cant work out stuff online or in my head lol
oh yeah ill do the work i was just wondering if takinga. cross section would be a good idea
severe current
Idk sorry
inner osprey
https://cdn.discordapp.com/attachments/903481102591750184/1198106684128768060/Sc...
are you familiar with the three dimensional pythagorean theorem?
inner osprey
are you familiar with the three dimensional pythagorean theorem?
(or equivalently relating the length of a diagonal in a rectangular prism to the side lengths)
main mesa
(or equivalently relating the length of a diagonal in a rectangular prism to the...
I think it’s like each of the 3 sides squared then added
Then square rooted right
vocal sleetBOT
@main mesa Has your question been resolved?
vocal sleetBOT
@main mesa Has your question been resolved?
inner osprey
Lemme try that out
how has that gone?
vocal sleetBOT
@main mesa Has your question been resolved?
sturdy sleet
Can any one help me with this
lucid cloak
vocal sleetBOT
@lucid cloak Has your question been resolved?
lucid cloak
<@&286206848099549185>
lucid cloak
Am I doing the working corrently? (pencil parts)
charred shell
,rotate
twin meteorBOT
charred shell
@lucid cloak did you work out why ABC = 90?
cursive turret
what is 1 and then what is 2?
twilit stratus
vocal sleetBOT
As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.
lucid cloak
<@216616675666165763> did you work out why ABC = 90?
no
lucid cloak
what is 1 and then what is 2?
ohh the Z, so 180-168 = 12°, so 1 and 2 and 12°
so that's part A right?
was I attempting part b) correctly
cursive turret
so that's part A right?
yes
cursive turret
was I attempting part b) correctly
with the sam method as in a) whats 3?
lucid cloak
with the sam method as in a) whats 3?
ohhhh okay
i got it so it'd be 12° as well
if it's a 90° at B then those two at A and C are 168° right
i wasn't sure if C would be 168°
cursive turret
i got it so it'd be 12° as well
no, 3 is 78.
lucid cloak
OHH RIGHT
lucid cloak
no, 3 is 78.
can i send one more question here or do i have to make new channel
cursive turret
you can, its your channel as long as it isnt closed.
lucid cloak
if two lines are parallel this angles sum up to 180.
ohh alright thank you
so like do that and proceed like how i was doing it?
180-80 = 100, so A from B would be..
360-100-115= 145
145+115= 260?
cursive turret
100 or 260, depends on how you defined "bearing A from B".
lucid cloak
100 or 260, depends on how you defined "bearing A from B".
okk thank youu
what would the other C from B be?
will it be
145
and for b) and c) i have to use cosine rule right
cursive turret
what would the other C from B be?
use this:
cursive turret
and for b) and c) i have to use cosine rule right
for b) yes. if you know all length in a triangle there are different ways to calc the angles. cosine rule is one of them. sine rule would be another one.
or you can calculate the angle without sine/cosine as you did in the previous example.
vocal sleetBOT
@lucid cloak Has your question been resolved?
lime oriole
I got an exam back today and I'm a bit curious to one of the solutions to a trig question that I couldn't figure out. The question is as such:
You are already given the identity $cossec x - sin x \identity cos x cot x, x \ne (180x)$. Hence, or otherwise, solve for $0 < x < 180, cosec x - sin x = cos x cot(3x-50)$
vocal sleetBOT
I got an exam back today and I'm a bit curious to one of the solutions to a trig...
twin meteorBOT
james
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
lime oriole
can someone help
can you get a different channel
vocal sleetBOT
To ask for mathematics help on this server, please open your own help channel or help thread. See #❓how-to-get-help for instructions.
full pumice
yes no one’s answering
queen skiff
Where
full pumice
in my channel
full pumice
i did
lime oriole
I got an exam back today and I'm a bit curious to one of the solutions to a trig...
idk how to do an identity in latex but you probably get the idea
the solution was the substitute the left-hand side for the identity so you get cos x cot x = cos x cot(3x - 50)
then rearrange to cos x cot x - cos x cot(3x - 50) = 0
factorise out the cos x
cos x [cot x - cot(3x-50)] = 0
then you come to two equations (1): cos x = 0
and (2): cot x = cot(3x-50)
then for equation 2 you can simplify to x = 3x-50
thats the part I don't understand
Is there some rule of function I am missing where if you have some function, say f in the equation f(x) = f(y)
does this mean you can rearrange this to x = y
spring tapir
∋ how is this different than ∈
vocal sleetBOT
∋ how is this different than ∈
twin meteorBOT
Ann
paper depot
some books apparently use $\ni$ to mean ``such that'', especially mixed in with quantifiers. but i personally think that usage is dumb.
twin meteorBOT
Ann
spring tapir
some books apparently use $\ni$ to mean ``such that'', especially mixed in with ...
how would you interpert this? the first way or the such that way?
paper depot
$\mathcal{A} \ni A \mapsto \mu(A \cap F)$?
twin meteorBOT
Ann
paper depot
this?
spring tapir
okay thats what i wrote it as
paper depot
what?
spring tapir
sorry i misunderstood
yes this bit here
i just wrote the map is an element of A
or interperted it
paper depot
no
this is meant to be a somewhat terse way of defining a function from \mathcal{A} to R without giving it a name
something like... $\mu' : \mathcal{A} \to \bR$ given by $\mu'(A) = \mu(A \cap F)$ maybe.
twin meteorBOT
Ann
paper depot
that is how i would write it
jade vale
Can someone help with activity 1? I don’t know how to solve it
vocal sleetBOT
Can someone help with activity 1? I don’t know how to solve it
jade vale
@ me when someone answers
vast shale
Can someone help with activity 1? I don’t know how to solve it
do you know the definition of continuity
jade vale
Yes
vast shale
state it
jade vale
The limit of a has to equal f(a)?
twin meteorBOT
vast shale
what is f(1/2)
jade vale
1
jade vale
1
vast shale
so is it continuous
jade vale
Yes
jade vale
Don’t we have to figure out the right side limit and left side or is that not necessary
jade vale
The second one isn’t continuous
vast shale
state why
jade vale
Because as x approaches 0.5^ is 1 while f(0.5) is 1.5
vast shale
Because as x approaches 0.5^ is 1 while f(0.5) is 1.5
yes you could also say that a limit doesnt even exist at 0,5
as the left and right sided limits dont match
jade vale
@spring tapir thanksksk
gaunt forge
vocal sleetBOT
vocal sleetBOT
@gaunt forge Has your question been resolved?
spring tapir
youre just doing the chain rule here right?
gaunt forge
youre just doing the chain rule here right?
yes
but is that how we are supposed to be doing
spring tapir
It looks right to me
gaunt forge
It looks right to me
Oh okay thank you and how to do the 2.2 I’m not sure
Can u help explain please
vocal sleetBOT
@gaunt forge Has your question been resolved?
gaunt forge
@spring tapir
vocal sleetBOT
@gaunt forge Has your question been resolved?
vocal sleetBOT
@gaunt forge Has your question been resolved?