@vague raptor Has your question been resolved?
#help-41
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@vague raptor Has your question been resolved?
oak quartz
a real-valued sequence $x_n$ is bounded if $\forall x_n\exists M$ such that $M\geq|x_n|$. Otherwise, it's unbounded.
grizzled pagodaBOT
00100000
vague raptor
I still don’t know if I understand the examples I’ve seen of this
winged quest
a real-valued sequence $x_n$ is bounded if $\forall x_n\exists M$ such that $M\g...
Wait, that's not right. The choice of M can't depend on x_n.
amber waspBOT
@vague raptor Has your question been resolved?
vague raptor
Wait, that's not right. The choice of M can't depend on x_n.
Hm?
winged quest
As written, the sequence x_n = n would be bounded because for every x_n you can choose M = x_n + 1, and then M > x_n.
grizzled portal
a real-valued sequence $x_n$ is bounded if $\forall x_n\exists M$ such that $M\g...
a real-valued sequence $x_n$ is bounded if $\exists M\forall x_n$ such that $M\geq|x_n|$. Otherwise, it's unbounded.
grizzled pagodaBOT
Supyovalk
grizzled portal
Pretty sure it's this.
Wait no this
a real-valued sequence $x_n$ is bounded if $\exists M$ such that $\forall x_n M\geq|x_n|$. Otherwise, it's unbounded.
grizzled pagodaBOT
Supyovalk
dark sierra
As written, the sequence x_n = n would be bounded because for every x_n you can...
According to the definition, a series x_n is bounded if there exists M for all x_n such that M>=|x_n|
so if we have a series x_n = n
as n goes to infinity, we will find certain x_n that cannot be bounded for any choice of M
(For any choice of M, there exist x_n>M if we choose n>M)
this is not true for a series such as -1,1,-1,1-1,1-1,1,-1,1 ..
where say M=1 works for all x_n
amber waspBOT
@vague raptor Has your question been resolved?
vague raptor
I think I’m starting to get it but the series types
Geometric series for example
What defines a geometric series and how do you identify it
gray nebula
cloud scarab
What defines a geometric series and how do you identify it
Let $x_0, q \in \mbb{R}$ or $\mbb{C}$. Define $x_n=x_0 q^n$. This is also known as geometric progression.
Series $\sum_{n=0}^\infty x_n$ converge if and only if $|q| < 1$ or $x_0=0$. In that case we have
$$\sum_{n=0}^\infty x_n = \frac{x_0}{1-q}.$$
grizzled pagodaBOT
EQUENOS
vague raptor
I think I’m starting to get this I think
I’ll see if I can summarize my current thinking and maybe you can tell me if it’s correct
If the absolute value of our q value is less than 1
Then we reformat into the fraction x0 / 1 - q and find the value it converges at
If no such value exists, it diverges
Is this correct?
oak quartz
Wait, that's not right. The choice of M can't depend on x_n.
oops, I messed up my quantified order 😳
amber waspBOT
@vague raptor Has your question been resolved?
earnest minnow
Then we reformat into the fraction x0 / 1 - q and find the value it converges at
you dont reformat, thats what the limit is. when |q| < 1 the series converges to that fraction x0/ 1-q
amber waspBOT
@vague raptor Has your question been resolved?
vague raptor
you dont reformat, thats what the limit is. when |q| < 1 the series converges to...
Okay so that’s just the baseline rule for it
What about for things like p series
earnest minnow
by p series do you mean $\sum_{n=0}^\infty \frac{1}{n^{p}}$
grizzled pagodaBOT
alephzer0
earnest minnow
this converges iff p > 1
vague raptor
Yes
Okay
So putting it together
A sequence is the numbers given and the pattern
To test what kind of sequence it is, I have to do a test on it
Like a geometric test
A p series test
Etc.
And it has to follow a certain form
(I hope im understanding correctly)
So do I just plug in each value given into one of these tests to see if it fits? If so what does that look like and how would I do that
earnest minnow
Heres how I would correct what youve said:
a sequence is a just a collection of numbers x1, x2, x3, ... labelled by the naturals (the counting numbers)
a series is when you take a sequence and look at what happens when you sum each term, so a series looks like x1 + x2 + x3 ... (this is an infinite sum).
a series converges if this infinite sum actually evaluates to some finite number. We say the series converges to this number. Otherwise, it 'diverges' which means the infinite sum adds up to (+-)∞
There are standard 'tests' for checking whether a series converges or diverges which is probably what you are referring to.
vague raptor
Yes that is what I’m referring to.
I think I’m trying to understand how to put it together though
Like I understand everything you’ve said
But for example
Let’s say I’m given a sequence
And I’m told I need to do a geometric series test
Let’s go with the idea that we’re working in the naturals
A geometric series test (afaik) has a certain format
Hold on I’m trying to explain what I want to say but it’s probably easier if I just attach a photo
So in this instance
Let’s say we’re looking at numbers 2 and 3
I need to identify the common ratio between them right
Which would be 1
Sorry not one
I don’t think that’s correct
But I need to identify a common ratio between them yes?
earnest minnow
Let’s say we’re looking at numbers 2 and 3
this doesnt really make sense to me 'looking at 2 and 3'
Could you clarify
vague raptor
Just an example
Like
Let’s say we’re given a series
Idk how to use the teXit hot so bare with me
S = {1,2,3,4,5….}
Or sorry a sequence
Now a series as you said is when I sum each term
In a geometric series test
That would look like (for example with this sequence)
Let’s say we’re starting at 2
earnest minnow
i think a problem here is that the sequence youve written is not a geometric sequence so the geometric test wouldnt apply here
vague raptor
Right okay
So this is what I’m confused about
How do I identify what kind of sequence I am looking at and what the appropriate test to perform on it is
earnest minnow
I see. So a geometric sequence would be something like {1,2,4,8,16,...} because you are multiplying by the same thing 'the common ratio' (=2 in this case) each step.
vague raptor
This makes sense yeah
earnest minnow
if you can identify such a sequence, then it is easy to see whether the series would converge
vague raptor
Okay so maybe it’s me who’s over complicating this then
When do the tests come into play
earnest minnow
bc as someone said the series converges if and only if |r| <1
vague raptor
Is it just to see if it converges or diverges?
earnest minnow
yeah p much
the tests just check if we converge or diverge. not all of them are applicable to each sequence
thats why there are a few different tests
vague raptor
How would I use the integral tests for series
earnest minnow
heres the integral test as it is in my analysis notes from first year
so for example say you wanted to see if the series $\sum_{n=1}^\infty \frac{1}{n^{2}}$ converges
grizzled pagodaBOT
alephzer0
earnest minnow
in this case ur f(x) = 1/x^2 which is nonnegative and decreasing so we can apply the integral test
then $\int_{1}^{k} \frac{1}{x^{2}} = 1 - k^{-1}$
grizzled pagodaBOT
alephzer0
earnest minnow
as a sequence in k , this converges, so we can conclude via the integral test that $\sum_{n=1}^\infty \frac{1}{n^{2}}$ converges
grizzled pagodaBOT
alephzer0
vague raptor
Hm
So I take the integral of that rational function
And use that to tell me if it will converge
earnest minnow
pretty much
spring marsh
Hello
Can someone help me with " les suites geometrique " (idk how to say it in english) ?
Pls
I have a test tomorrow
weak breach
Can someone help me with " les suites geometrique " (idk how to say it in englis...
girl i just had an exam yesterday
but i only understand the arethmetique un peu
j ai un exercise des suite geometrique
ils sont vraimon facile d apres l exercices donner par mon prof
weak breach
bah malheureusement je n ais les comprend pas trop bien
pallid canopy
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fervent oasis
Hi, can I get help with this problem that involves conditional probability and LOTP law
fervent oasis
here is what I got:
A = double-headed coin, B = 7 heads in a row ---- P(A) = 1/100, P(B) = 1/128
I need P(A | B)
by bayes theorem: P(B | A) P(A) / P(B) ===> (1 * 1/100) / (1/128) ===> 128/100 which is impossible
oh, I just had to use LOTP in P(B) 😄
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graceful badger
amber waspBOT
graceful badger
Can someone please explain how to arrange expression A into expression B?
grizzled pagodaBOT
Azyrashacorki
full elk
And then the second ||d|| just brought in as well
graceful badger
I don't understand how |d|^2 gets split up and placed below both d
amber waspBOT
@graceful badger Has your question been resolved?
graceful badger
reading it...
In the second one, do you divide (c1 c2) / c2, and then also divide A by that?
Sorry, algebraic manipulation is my weakest area.
barren furnace
lol im dumb
i meant (c1c2)A = (c1) (c2A)
the order u use to multiply the constants doesn't matter
you can view it as c1 =(v dot d)/d and c2 = 1/d, that's why the manipulation is valid
graceful badger
I think this is illuminating: " c1 =(v dot d)/d and c2 = 1/d"
I'll have to take my time to parse it out. Like I said, this is my weakest area. I'll go over it offline to understand it from the steps you showed.
I appreciate the help.
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keen pawn
I'm trying to determine the relationship between the eigenvalues of T, and T-cI
sage chasm
cI ?
keen pawn
$cI$
sage chasm
I identity matrix ?
grizzled pagodaBOT
A dense set(Ping when reply)
sage chasm
c here is the eigenvalue of T ?
sage chasm
well
Tx = ax | (T-cI)x = bx
Tx - cIx = bx
ax - cx = bx
a-c = b
🤔
maybe i dont understand the task
i did a lot of assuming here too bruh
indigo cloud
I already know the relationship between their eigenvalues
wdym. you just said you want to determine that relationship
sage chasm
well maybe he has the answer and want to find it himself
sage chasm
ah
indigo cloud
how do you know the relationship between the eigenvalues but not between the eigenvectors
keen pawn
I compared the chacteristic equations
indigo cloud
well that works. frankly I don't know a hint which doesnt give it away so I'll just let you experiment
keen pawn
Welll, the eigenvectors of $T$ are the solutions to $(T - \lambda I) (v)=0$
\
Those of T-cI are the solutions to $(A- ( \lambda -c))(v)=0$
grizzled pagodaBOT
A dense set(Ping when reply)
keen pawn
so the $v_{\aleph} - cv$ , where $v_{\aleph}$ is a eigenvector of $T$
grizzled pagodaBOT
A dense set(Ping when reply)
keen pawn
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split sail
Q. Prove that x^2 congruent to 1 (mod p) implies that x is congruent to +-1 (mod p)
split sail
Solution:
x ≡ y(mod p)
so,
x² ≡ y²(mod p)
so, y² =1
=> y = +1 or -1
=> x ≡ -1 or +1 (mod p)
Is this correct?
amber waspBOT
@split sail Has your question been resolved?
indigo cloud
so the $v_{\aleph} - cv$ , where $v_{\aleph}$ is a eigenvector of $T$
this very much misses the point of the relationship between the eigenvectors. think again
split sail
Nah
indigo cloud
Solution:
x ≡ y(mod p)
so,
x² ≡ y²(mod p)
so, y² =1
=> y = +1 or -1
=> x ≡ -1 o...
you used the fact you want to prove, namely that y^2=1 mod p implies y=+-1 mod p
as a hint, x^2=1 mod p means x^2-1=0 mod p
sweet drift
amber waspBOT
sweet drift
Unsure if whether the three non-negative squares are to be natural or not. If they aren't then I believe I have solved it, though I highly doubt this is the case
If they are supposed to be natural then I have no clue how to prove so
amber waspBOT
@sweet drift Has your question been resolved?
wet breach
Unsure if whether the three non-negative squares are to be natural or not. If th...
It's highly likely they're natural numbers because every real is the square of a real number. The closest would be rationals, but they probably don't want those either.
Oh, every positive real, sorry.
Nonnegative.
sweet drift
Yeah I thought so too. If these could be non-naturals I can just call x y z respectively sqrt(2/3 a^2), sqrt(2/3 ab) and sqrt(2/3 b^2) and that would be that
Nvm I got it
Thanks chat
.clopse
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wet breach
No problem.
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acoustic lodge
this is just part of whole exercise, but anyway, they showed this step. so how do i exract this? like exponential or
sage chasm
$\rho (e^{-i\rho} - e^{i 0})$
grizzled pagodaBOT
Herels
sage chasm
?
acoustic lodge
yes
like they said its trick to use e^-rho/2i
i dont kina get it why? is like that exponential function
sage chasm
$\rho (e^{-i\rho} - e^{i 0}) = \rho e^{-\frac{i\rho}{2}} \left(e^{-\frac{i\rho}{2}} - e^{i 0 + \frac{i\rho}{2}} \right)$
grizzled pagodaBOT
Herels
acoustic lodge
$\rho (e^{-i\rho} - e^{i 0}) = \rho e^{-\frac{i\rho}{2}} \left(e^{-\frac{i\rho}{...
i just dont understand how to get it
sage chasm
just exponent laws tho
acoustic lodge
sage chasm
what is this
acoustic lodge
is like taht or
sage chasm
$e^{-i\rho} = e^{-i\left(\frac{\rho}{2} + \frac{\rho}{2}\right)} = e^{-i\frac{\rho}{2}} e^{-i\frac{\rho}{2}}$
grizzled pagodaBOT
Herels
acoustic lodge
$e^{-i\rho} = e^{-i\left(\frac{\rho}{2} + \frac{\rho}{2}\right)} = e^{-i\frac{\r...
ok ok
got it
thanks
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prime knoll
Is it possible to parametrize the curve of intersection of y = x^2 and z = x
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still stump
help me pls, i don’t know what to do
amber waspBOT
help me pls, i don’t know what to do
split sail
help me pls, i don’t know what to do
factor out 5^x
rocky vapor
amber waspBOT
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.
boreal imp
Sorry
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untold shoal
Find all functions f(x), that satisfy ,tex [
f'(x) = \tan\left( \arccos\left( \frac{x}{\sqrt{x^2 + f(x)^2}} \right) \right)
]
grizzled pagodaBOT
nikolipo
slow atlas
**nikolipo**
You can convert arccos to arctan
Then you'll get a sweet ODE, you can solve it by separation of variables
untold shoal
Can you elaborate pls, i dont understand
slow atlas
Can you elaborate pls, i dont understand
You know how to convert $\arccos(x)$ to $\arctan$
?
grizzled pagodaBOT
@slow atlas
untold shoal
No
slow atlas
No
Alr
See whats inside an inverse function
Ratio of sides
So in a arccos function you have ratio of Base to Hypotenuse
Using the Pythagoras theorem, you can find the adjacent sides value
Now because you know all the sides of a the triangle
You can find arctan
Which will have the ratios of the adjacent to Base
Basically
$\arccos{(x)} = \arctan{\frac{\sqrt{1-x^2}}{x}}$
grizzled pagodaBOT
@slow atlas
untold shoal
So, is it [
f'(x) = \frac{f(x)^2}{x\sqrt{x^2 + f(x)^2}}
]
grizzled pagodaBOT
nikolipo
untold shoal
,ask sqrt(1-(a/sqrt(a^2+b^2))^2)/(a/sqrt(a^2+b^2))
grizzled pagodaBOT
untold shoal
,ask [
f'(x) = \frac{f(x)^2}{x\sqrt{x^2 + f(x)^2}}
], for f(x) = 2x
grizzled pagodaBOT
untold shoal
<@947094726513942578>
I might have miscalculated that
grizzled pagodaBOT
untold shoal
,ask separation of variables
grizzled pagodaBOT
amber waspBOT
@untold shoal Has your question been resolved?
untold shoal
Find all functions f(x), that satisfy ,tex [
f'(x) = \tan\left( \arccos\left( \frac{x}{\sqrt{x^2 + f(x)^2}} \right) \right)
]
grizzled pagodaBOT
nikolipo
untold shoal
Find all functions f(x), that satisfy ,tex \[
f'(x) = \tan\left( \arccos\left( \...
.
deft swift
maybe draw a triangle to start
amber waspBOT
@untold shoal Has your question been resolved?
honest dagger
Find all functions f(x), that satisfy ,tex \[
f'(x) = \tan\left( \arccos\left( \...
Try simplifying f'(x)
reef kraken
Try to bring arccos to arctan
amber waspBOT
@untold shoal Has your question been resolved?
light musk
How did they jump from the blue to the green
amber waspBOT
How did they jump from the blue to the green
light musk
and how did the + C turn into a + K?
high sparrow
$\ln(\frac{A}{B}) = \ln(A) - \ln(B)$
grizzled pagodaBOT
ColdTe²
high sparrow
-ln(7) will be a constant
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neon kettle
Topology:
If two continuous function f and g are cont, is the function h(x,y) = f(x)g(y) (R^2 -> R) continuous? Using this can I then prove the function h(x,x) (R -> R) is continuous proving that the product of two cont. functions is continuous? (definition of continuity is f^-1(open) is open, not the epsilon delta thing)
amber waspBOT
Topology:
If two continuous function f and g are cont, is the function h(x,y) = ...
@neon kettle Has your question been resolved?
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@neon kettle Has your question been resolved?
fervent oasis
amber waspBOT
fervent oasis
Here is what I got>
D = Disease, Dc = Not disease. P(D) = p, P(Dc) = q = 1-p
Tj = result in jth test
a = P(Tj | D), b = P(Tj | Dc)
I need P(D | k of the n test are positive)
so by bayes theorem: P(D | K +tests) = P( K+tests | D ) P(D) / P(k of the n test are positive
but im stuck there
amber waspBOT
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rigid swallow
question: if i purchase something for £3.99 GBP and get 2,500 of said item how much value does 1 of the item come for in pound/pence?
indigo cloud
can you answer it if you get 2 of said item?
amber waspBOT
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@rigid swallow Has your question been resolved?
amber waspBOT
@rigid swallow Has your question been resolved?
trim tundra
a) Let P, Q, R be sets with |P | = s, |Q| = t, and |R| = u. Determine exactly or state a
range of values:
i. |P ∪ (Q ∩ R)|
ii. |(P ∪ Q) \ R|
trim tundra
our prof did not go over operations with only cardinality so i'm a bit lost
for i. i started with $|P \cup (Q \cap R)| = |(P\cup Q)\cap(P\cup R)|$
grizzled pagodaBOT
mads
trim tundra
but idk how to find it with cardinality as i said 💀
little widget
this might be useful
trim tundra
but what's $n(A\cap B)$?
grizzled pagodaBOT
mads
hearty inlet
it's just that
trim tundra
oh, no need to do anything more?
hearty inlet
this might be useful
you can rearrange this to write it in terms of n(A U B)
hearty inlet
this might be useful
do u know why this is true?
trim tundra
principle of inclusion exclusion
little widget
trim tundra
yeah
hearty inlet
ermmm sure but that's a way more general version of this
doesn't tell me u get why it's true
trim tundra
the diagram is helpful to see why it can be applied to this
little widget
that one has weird notation
hearty inlet
a) Let P, Q, R be sets with |P | = s, |Q| = t, and |R| = u. Determine exactly or...
i feel like this question can get messy tho
like even if u just ask what's the range of |P n Q|
can u say what that is? it's not that obvious
hearty inlet
yeah u can't so u need to give a range
it's possible that P n Q is empty
or that one is contained in the other
trim tundra
ohh
hearty inlet
it's possible that P n Q is empty
what's n(P n Q) in this case?
hearty inlet
this might be useful
after rearrangeing this for AnB
trim tundra
ugh this is gonna suck
hearty inlet
is this for hw or smth? from my judgement of ur understanding it's prolly a better use of ur time to think of easier questions for the purpose of just learning, cuz this one might get tricky
trim tundra
is this for hw or smth? from my judgement of ur understanding it's prolly a bett...
no, it's an assignment
or unfortunately i wouldn't have been doing this
hearty inlet
same thing
even for solving this maybe u wanna go to basics first. you know what they say, rather sharp ur axe for 2 hours cut the tree in 1 hour than cut the tree for 5 hours.
trim tundra
do you know where i can do questions like this? our textbook doesn't have any involving cardinality
hence why i'm here 
hearty inlet
maybe chatgpt ngl
fleet pilot
Can someone explain why the 9t is negative
amber waspBOT
Can someone explain why the 9t is negative
fleet pilot
Its not reflected in my answer
compact yarrow
you should get a -ln bit on the lefthand side here
fleet pilot
ohows that
compact yarrow
well when doing the antiderivative, there's an implicit step where you do a substitution like u = 8-y
so you get a negative
compact yarrow
basically
we can also work backward from your final answer. What's dy/dt for what you got?
fleet pilot
wdym
fleet pilot
its displayed in the image i sent
fleet pilot
would it be -9e^-9t +8t?
compact yarrow
that's not your answer
fleet pilot
based on mine or the correct answer
fleet pilot
9e^9t
fleet pilot
no
compact yarrow
y = e^9t + 8. What is y-8?
fleet pilot
but im not sure i could convince myself with the correct answer's derivative either
compact yarrow
we'll get to that
fleet pilot
e^9t
compact yarrow
just direct substitution
fleet pilot
is this just to check
compact yarrow
it's showing that your original answer can't be correct, so yeah basically checking
usually a good practice for these questions is to differentiate your final answer to show it indeed does satisfy the original diffeq
compact yarrow
same deal, what is y-8?
fleet pilot
e^-9t
compact yarrow
so dy/dt = ...?
fleet pilot
would that not just be the derivative i stated
compact yarrow
sure but we're looking to transform it
indeed the derivative is -9e^-9t
but what is e^-9t? and thus what is an equivalent way of expressing dy/dt?
fleet pilot
im not sure
compact yarrow
you just told me that y-8 = e^-9t
so how can we rewrite -9e^-9t to be in terms of y (like the original differential equation) and not in terms of t?
fleet pilot
you just told me that y-8 = e^-9t
can u explain why its important to establish this
to write the deriv in terms of y
compact yarrow
because that's the form the original question is in
if you're able to properly check if your answer is correct, you'll save yourself from answering incorrectly and not having any way to find out before you submit it
fleet pilot
so how can we rewrite -9e^-9t to be in terms of y (like the original differentia...
idk tbh
fleet pilot
-9(y-8)
compact yarrow
quite so
so we have that dy/dt = -9(y-8), however the original question says dy/dt = 9(8-y). Can we do some algebra to rectify this discrepancy?
fleet pilot
dont we just distribute the negative
compact yarrow
indeed
keen pawn
$\lim_{x \to \infty} \frac{x}{\floor{x}}= \lim_{x \to \infty} \frac{x}{x- {x}}= \lim_{x \to \infty}\frac{1}{1- \frac{{x}}{x}}=1$
keen pawn
<@&268886789983436800>
grizzled pagodaBOT
A dense set(Ping when reply)
keen pawn
How does this look?
brisk leaf
It's correct
keen pawn
But are the steps detailed enough
brisk leaf
I mean you could try and explain why (fractional part of x / x) is 0
If you deem necessary
But other then that it's fine
brisk leaf
I mean you could try and explain why (fractional part of x / x) is 0
The explanation being that fractional part of infinity will be some number between 0 and 1 and numbers from 0 to 1 on being divided by infinity yield 0
keen pawn
Thnaks
brisk leaf
Anything else I can try and help with?
keen pawn
Yea
This
so my idea was first determine the value of a via differential calc
so m=-2
so ax=-1
and $(b-2x)=ax^2$
grizzled pagodaBOT
A dense set(Ping when reply)
grizzled pagodaBOT
A dense set(Ping when reply)
keen pawn
so $y_1'=2x_1$; $y_2'=2x_2-2$. so $x_2-x_1=1$. So $\frac{y-y_1}{x-x_1}=. \frac{y_2-y_1}{x_2-x_1}$
grizzled pagodaBOT
A dense set(Ping when reply)
keen pawn
Yeah, probably a bad idea they day before my final exam
keen pawn
wait
I know x_2-x_1=1
Well, the equation of the first tangent line is
I'm lost tbh
sorry
brisk leaf
Their is only 1 common tangent tho
keen pawn
so we have $\frac{y-x_1^2}{x-x_1} = \frac{(1+x_1)^2-x_1^2}{1}$
grizzled pagodaBOT
A dense set(Ping when reply)
keen pawn
so I equate the discriminant to zero to find x_1
brisk leaf
Isn't y2 = (x2-1)^2 + 1 ?
grizzled pagodaBOT
A dense set(Ping when reply)
keen pawn
ah
keen pawn
x2-1 = x1 - 2
this feels wrong
brisk leaf
x2 - x1 = -1
x2 = x1 -1
x2 -1 = x1-2
keen pawn
oops
the 12 stages of grief
so $\frac{y-x_1^2}{x-x_1}=. \frac{(x_1-1)^2+1-x_1^2}{-1}$?
grizzled pagodaBOT
A dense set(Ping when reply)
brisk leaf
$\frac{y-x_1^2}{x-x_1}= \frac{(x_1-2)^2+1-x_1^2}{-1}$
grizzled pagodaBOT
Lex
keen pawn
mhm
and I now just equate the discriminant to zero
right
so $y-x_1^2 =( 4x_1-5)(x-x_1)$
okay, I can solve for x_1 now
thanks!
brisk leaf
Yup
grizzled pagodaBOT
A dense set(Ping when reply)
brisk leaf
Usually we can go a simpler path
We can just assume a common tangent
Of the format y = mx + b
keen pawn
mhm
brisk leaf
So,
For 1st parabola
x^2 = mx +b
Here the discriminant has to be 0
So m^2 -4b =0
This is our condition of tangency
brisk leaf
and then compare
Yup
These 2 equations
Must have a common solution
For the tangent to exist
Find that solution
keen pawn
Cool
brisk leaf
And you are done
grizzled pagodaBOT
A dense set(Ping when reply)
brisk leaf
Wait I just recalled you in the helpers lounge aswell right?
keen pawn
yes
keen pawn
**A dense set(Ping when reply)**
so I have to find the discriminant of $y=-3x_1^2 +x_1(5+4x)-5x$ where the quadratic is in$ x_1$
grizzled pagodaBOT
A dense set(Ping when reply)
grizzled pagodaBOT
A dense set(Ping when reply)
keen pawn
$16x^2-20x+15=0
brisk leaf
Hmm
keen pawn
That means no tangent exists?
brisk leaf
It has to exist
keen pawn
hmm
keen pawn
yeah
brisk leaf
1t make sense for a tangent to exist
grizzled pagodaBOT
Lex
brisk leaf
y_2 = (x_2-1)^2+1
keen pawn
yes
keen pawn
that means $y=x_1^2+x-x_1$
grizzled pagodaBOT
A dense set(Ping when reply)
brisk leaf
Yup
keen pawn
x=1/2
*1/4
Cool
let me graph it
so the equation is
$y-1/16/x-1/4= 1/2
That isn't a common tangent
brisk leaf
-_-
brisk leaf
It's (x-1)^2 +1
brisk leaf
Issok
keen pawn
thanks
brisk leaf
Got it all?
brisk leaf
Yeah sure
keen pawn
.close
amber waspBOT
brisk leaf
Ur welcome
amber waspBOT
Available help channel!
lime jasper
lime jasper
How would i have found theta is also 270 degrees
fresh ocean
You correctly realised that cos theta needs to be 0. But does theta need to be 90? Think of when cos theta is 0, and use that
lime jasper
So i just have to memorize cos 270 is 0
lime jasper
I haven’t learned those yet
crystal cloak
or sohcahtoh as some call it ¯\_(ツ)_/¯
ASTC and sohcahtoh are two different things
lime jasper
Whats astc
fresh ocean
ASTC and sohcahtoh are two different things
Oh is it? idk, never heard of it, and some helpee told me they are the same
fresh ocean
Whats astc
Tells you how the order in which sin cos tan are +ve in each quadrant
all are + in 1st Q. Only sin in 2nd, tan in 3rd and cos in 4th
amber waspBOT
Available help channel!
worn stump
How to get help
amber waspBOT
How to get help
outer hull
brisk leaf
Check #❓how-to-get-help
amber waspBOT
@worn stump Has your question been resolved?
amber waspBOT
amber waspBOT
Available help channel!
spring crater
amber waspBOT
grizzled pagodaBOT
turbid notch
I think you’re more likely to get help if you translate it
amber waspBOT
@spring crater Has your question been resolved?
split sail
(complex numbers)
why is i^5 different from (i^4)^(5/4), i being the square root of negative one,
split sail
if it was a real number instead of i, they represent the same quantity, whats creating the problem here
but i can still write i^12 = (i^4)^3
whats with fractional powers and complex numbers not going together
shadow stump
you can use fractional powers with complex numbers, but you run into the issue that are 4 different numbers which satisfy z^4 = i, so there is not a clear way to choose which one should be i^(1/4)
amber waspBOT
proven vapor
<@&268886789983436800>
amber waspBOT
gusty rampart
.close
amber waspBOT
Available help channel!
brisk nexus
Assume that the chances of the patient having a heart attack are 40%. It is also assumed that a meditation and yoga course reduce the risk of heart attack by 30% and prescription of certain drug reduces its chances by 25%. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options the patient selected at random suffers a heart attack.
Find the probability that the patient followed a course of meditation and yoga?
Here if I want to take out probability of the patient having a heart attack provided he followed the yoga and meditation course what would it be? The question says this course reduces the risk by 30% so does it mean theres still a risk of 70%? Does that make any sense?
brisk nexus
Can someone help me with the interpretation of this numerical
sinful sapphire
it could be either
can't tell for sure
40−30 = 10 or
40×0.7 = 28
it can't be what you said, it's not 100 at the beginning
brisk nexus
ah
merry rivet
It's asking you to find the conditional probability
brisk nexus
yes
merry rivet
but the 40*0.7 is the expected way of calculating it
maiden pebble
∫ (cosh^3(x) * sinh^2(x)) dx
amber waspBOT
∫ (cosh^3(x) * sinh^2(x)) dx
split sail
nvm
dull pike
Use trigonometric identities
amber waspBOT
@maiden pebble Has your question been resolved?
maiden pebble
I need explanation too
dull pike
About identities?
maiden pebble
Yes
dull pike
Lets start with the derivative of sinh
maiden pebble
Yes
dull pike
What it is
maiden pebble
The derivative of the hyperbolic sine function, sinh(x), is the hyperbolic cosine function, cosh(x). In mathematical notation, this is expressed as:
d/dx(sinh(x)) = cosh(x)
Im just want to help my brother
To solve it
dull pike
Dont use anything to ans my ques
Cosh was enough
If cosh is derivative of sinh
Then u need to make a sub for sinh but
When u do
U have this
u = sinh
du = cosh dx
dx=du/cosh
But when u sub in the integral
U have cosh^3 not cosh
So u cant really eliminate it
Then u need to let only cosh
How to do that?
Before substituting
Replace cosh^3 with cosh * cosh^2
And now use identities to replace cosh^2
In terms of sinh
amber waspBOT
@maiden pebble Has your question been resolved?
mint nacelle
yes it's by parts
glad pulsar
mint nacelle
<@&268886789983436800> spamming same msg in multiple channels
edgy sky
Iirc, you should take out the a to make it a^-3((1+(b/a)x)^-3), then use the binomial series to find a and b
glad pulsar
Iirc, you should take out the a to make it a^-3((1+(b/a)x)^-3), then use the bin...
so u make it less than 1?
edgy sky
so u make it less than 1?
Make what less than 1?
glad pulsar
bcz b/a<1
mint nacelle
yeah it's actually literally just $a^{-3} = 1/27$
and $(a+bx)^{-3} = a^{-3} (1 +(b/a)x)^{-3} = a^{-3} (1 - (3b/a)x + \cdots)$
edgy sky
Thats just a condition of the binomial series
glad pulsar
why do u not solve it directly
edgy sky
yeah it's actually literally just $a^{-3} = 1/27$
and $(a+bx)^{-3} = a^{-3} (1...
oh yeah how did i miss that 
edgy sky
why do u not solve it directly
what do you mean by solve it directly
glad pulsar
what do you mean by solve it directly
nvm
edgy sky
Any parts you don't understand?
glad pulsar
yeah it's actually literally just $a^{-3} = 1/27$
and $(a+bx)^{-3} = a^{-3} (1...
how a^-3=1/27?
mint nacelle
how a^-3=1/27?
compare the constant terms
glad pulsar
oh
grizzled pagodaBOT
southlander!
mint nacelle
but then this split form where you take a^(-3) out is better for comparing all the terms
glad pulsar
do i expand (1+b/a)^n like (1+x)^n?
mint nacelle
not just the constant term
grizzled pagodaBOT
southlander!
ruby dome
$\cos(2x)=0$
amber waspBOT
$\cos(2x)=0$
grizzled pagodaBOT
crazytime
ruby dome
are the solutions in [0,2π] just x=π/4 and x =3π/6?
ruby dome
But so I don't go outside the interval [0,2π]?
elder harbor
U do that first and then u restrict x to the interval later
ruby dome
I only need the ones in [0,2π]
elder harbor
U do that first and then u restrict x to the interval later
.
ruby dome
Okay
ruby dome
2x = pi/2 + pik
The ones I found come out
elder harbor
There should be 4 solutions in the interval
Cuz ull get pi/4 + pik/2
In which case the x are pi/4, 3pi/4, 5pi/4, 7pi/4, 9pi/4, …
Then u intersect this with [0, 2pi]
It should agree with this
elder harbor
In which case the x are pi/4, 3pi/4, 5pi/4, 7pi/4, 9pi/4, …
.
ruby dome
In which case the x are pi/4, 3pi/4, 5pi/4, 7pi/4, 9pi/4, …
The solutions are the first four, i.e. up to k=3
elder harbor
Yeah
Basically u restrict the interval too fast
U have to do that at the end
That’s why u lost some answers
amber waspBOT
amber waspBOT
Available help channel!
fierce edge
dumb question but for |-4x + 8| <= 2
amber waspBOT
dumb question but for |-4x + 8| <= 2
grizzled pagodaBOT
mmmm7
fierce edge
why can't we plug this into the definition and make caseworks like:
round sierra
I think it is because this will going to make 2 y values for x =2 which is not acceptable for the function
grizzled pagodaBOT
mmmm7
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
round sierra
Imagine you draw a graph and then like revolve x axis
shadow stump
the definition has us make cases based on when the inside is positive or negative
fierce edge
Imagine you draw a graph and then like revolve x axis
no i know how to solve the inequality
that is not my question
fierce edge
the definition has us make cases based on when the inside is positive or negativ...
mhm what about it?
shadow stump
**mmmm7**
Compile Error! Click the <:errors:504266556054962196> reaction for mor...
so where is this coming from?
round sierra
**mmmm7**
Compile Error! Click the <:errors:504266556054962196> reaction for mor...
It is compiled error for me.
fierce edge
I mean we have two cases:
- x < 2
- x >= 2
we know that |-4x + 8| is equivalent to either -4x + 8 or 4x - 8 depending on the interval
so why can't i just plug those in
shadow stump
so why can't i just plug those in
plug what in to what?
fierce edge
consider x < 2
then |-4x + 8| is equivalent to -4x + 8
so the first inequality is -4x + 8 <= 2
that part is good
what about now considering the case where x >= 2
|-4x + 8| is equivalent to 4x - 8
shadow stump
where does this inequality between |-4x + 8| and 2 come from?
fierce edge
so shouldn't it be 4x - 8 <= 2
fierce edge
where does this inequality between |-4x + 8| and 2 come from?
that's the "main question"
if i had an inequality like that to solve
why can't i use the absolute value definition and casework to solve the inequality
shadow stump
you could, you just have to make sure you combine the results you get with the intervals of x on which each case is valid
fierce edge
so basically put:
Answer (but wrong for obvious reasons):
1) consider x < 2
then |-4x + 8| is equivalent to -4x + 8
so the first inequality is -4x + 8 <= 2
2) Now consider the case where x >= 2
|-4x + 8| is equivalent to 4x - 8 and so 4x - 8 <= 2```so you could consider the system of inequalities given by
-4x + 8 ≤ 2
x < 2
and similarly for the other case
fierce edge
so you could consider the system of inequalities given by
-4x + 8 ≤ 2
x < 2
u mean x >= 3/2 for 1)
and for 2) we get x <= 5/2
round sierra
Second one is wrong
fierce edge
yep
latent nest
Yeah but I think it’s easier if you just solve it the other way
round sierra
4x -8 should be bigger then 0 for x larger (only) 2 , as we consider sign change of the function inside the absolute when the function is below zero. However this case the the Turing point is x larger then 2.
fierce edge
now what? do we have to consider the union of the solution space?
fierce edge
yeah
oh i guess this always holds
okay so uh
where does the "trick" come from?
the trick being that
you do something like:
|-4x + 8| <= 2 is equivalent to solving -4x + 8 <= 2 or 4x - 8 >= 2
round sierra
Both
shadow stump
you do something like:
|-4x + 8| <= 2 is equivalent to solving -4x + 8 <= 2 or ...
just from the piecewise definition of absolute value
cunning birch
you do something like:
|-4x + 8| <= 2 is equivalent to solving -4x + 8 <= 2 or ...
and this is kinda off
you're actually solving
-4x + 8 <= 2 AND 4x - 8 <= 2
both need to be true at the same time
fierce edge
oh yeah my bad
fierce edge
|x| = max(x,-x)
is this for my "trick"?
cunning birch
this is literally the property being used
fierce edge
just from the piecewise definition of absolute value
but isn't the piecewise definition exactly what i did and not the "trick"
cunning birch
We have shown two ways to do it
either use piecewise def of absolute value
or use max definition
cunning birch
no
if both a and b are under some value c
then no matter which one of a or b is greater
fierce edge
oh yeah okay
cunning birch
it's gonna be under c
fierce edge
yes
cunning birch
and vice versa
if the biggest of the two is under c
then the other one is also gonna be under c
fierce edge
yes fair enough
fierce edge
round sierra
I am having confusion isn't it x <= 2 not the y value
cunning birch
?
fierce edge
4x-8 <= 2 AND -4x+8 <= 2
okay i get it now
wait is this a justification of the "trick"
or is this an independent way
fierce edge
just experimented and yeah okay |x| = max(x,-x) looks like it holds for all R
cunning birch
it's just about using the max def of absolute value
cunning birch
just experimented and yeah okay |x| = max(x,-x) looks like it holds for all R
if you don't understand why it's true I suggest you prove it
prove |x| = max(x,-x) when x>= 0
and |x| = max(x,-x) when x< 0
fierce edge
if you don't understand why it's true I suggest you prove it
yeah no i get it i think
idk how to formally prove it but i'm sure it holds
for x >= 0 max(x,-x) is x
|x| = x because |x| >= 0
same argument for |x| = max(x,-x) when x < 0
right?
fierce edge
if you don't understand why it's true I suggest you prove it
thanks!
i learnt smth new about the max thing ig
makes sense now
round sierra
Um
Like I am high school math person
I dont understand what you actually want to know
fierce edge
I dont understand what you actually want to know
when solving |-4x + 8| <= 2
people instantly go that this is equivalent to solving: -4x + 8 <=2 AND 4x - 8 >= 2
i wanted to ask why
round sierra
So for me test trick is u think time -1 inside the absolute which will allow cause the inequality filps
round sierra
So can you draw this function? When you draw it and then flip as we apply absolute on the drawing we get two different function and still only y<=2 is valid so formula form like that
Like we multiply-1 to negative value to make it positive
fierce edge
Like we multiply-1 to negative value to make it positive
wot
u just multiply by -1 to one side lol
anyway yes i'm aware
i wanted to make sense of why we do that
anyway that is solved
round sierra
Cool
coral wigeon
anyway that is solved
fierce edge
<a:waves:880516410948853840>
severe scroll
wait is this a justification of the "trick"
it’s sort of a justification. whenever we have |f(x)| < c, then we can go from first principles of abs, either f(x) > 0 or f(x) <= 0. so we can do it in cases.
if f(x) > 0, then |f(x)| < c if and only if f(x) < c. if f(x) <= 0, then |f(x)| < c if and only if -f(x) < c. and now, we are gonna use the fact that c is positive to show that if one of these inequalities don’t hold at x, then we must also be in the case where f(x) was (positive, negative, whichever one lets us conclude what we want)
i tried to word the last part better but there isn’t a nice wording.
if the first inequality doesn’t hold, then f(x) > c and therefore f(x) is also positive because c was
and if -f(x) > c, then f(x) < -c so f(x) must be negative at x
one can also justify it by noting that |f(x)|=max(f(x),-f(x))
amber waspBOT
@fierce edge Has your question been resolved?
fierce edge
it’s sort of a justification. whenever we have |f(x)| < c, then we can go from f...
yeah that makes sense too
thank you!!