but there are values greater than the local max
#help-38
rustic nest
tepid hamlet
try to come up with some other examples
rustic nest
tardy hemlock
This is the graph of 1/x
It has no absolute maximum or absolute minimum because it keeps getting bigger or smaller for x -> 0
Now you just need to get a local min/max in there
rustic nest
isnt the first hump a local max?
tardy hemlock
isnt the first hump a local max?
What first hump?
rustic nest
in my drawing not urs
lofty walrus
isnt the first hump a local max?
yes it is
rustic nest
ok thx
tardy hemlock
in my drawing not urs
Ah
rustic nest
question since it is asking in between [-1, 2] does that mean i cant have the open circle at x = 2?
lofty walrus
however, at the point of discontinuity (i assume on your picture x = 2) the function still needs to take on some value, since [-1, 2] is closed
rustic nest
so like this instead
lofty walrus
you still need some value for your function at that "hole"
lofty walrus
yes, so just draw a black point beneath that hole somewhere
rustic nest
oh ur saying it has to be continuous
lofty walrus
no
lofty walrus
because your function is defined to be a map from [-1,2] to R and therefore needs to take on some value there
tepid hamlet
it has to be defined for every x in the interval [-1, 2]
lofty walrus
see this earlier example, beneath the "hole" there is a black dot to indicate the value of f at that point
tepid hamlet
so like this instead
for this example, what is f(1.5) (let's just assume that hole is at 1.5)
tepid hamlet
good question, no then you could actually have continuous functions with no absolute max \ min
tepid hamlet
btw this has to do with one of the most important theorems in calculus
rustic nest
can that black dot be the local max??
tepid hamlet
rustic nest
thathsa what im learning now
tepid hamlet
so with extreme value theorem, the premise that the interval is closed is important
rustic nest
brb
tepid hamlet
if it isn't, then it doesn't apply, a simple counter example is what you came up with, with the hole at x = 2 where the function is defined and continuous on [-1, 2)
rustic nest
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rustic nest
This is the graph of 1/x
does it have a local min or max?
tardy hemlock
does it have a local min or max?
Not yet
But you can quickly make one
By drawing in a "hump"
rustic nest
my book has a quesation to draw graph of f to find absolute and local max and min values of f. and the quesation is f(x) = 1/x, 1 < x < 3
it has a similar one f(x) = 1/x, x>= 1 ... and says f(1) = 1 is Abs max
im kind of confused how they want me to draw that by hand
should i make a new help for that question?
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rapid horizon
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rapid horizon
My answer is C
uncut aspen
could you translate to english?
vagrant marsh
also there is no option C
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cosmic meadow
Worst format I’ve ever seen tbh
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rapid horizon
Asking about how many symmetric lines?@cosmic meadow @vagrant marsh
stoic iron
define symmetric line
rapid horizon
Y axis and x @stoic iron axis?
stoic iron
what is a symmetric line
trail ingot
maybe the question is asking for (something equivalent to) the order of the permutation group of that shape
rapid horizon
what is a symmetric line
What did I say?
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odd beacon
The equation of the straight line through the point of intersection of
16x - 10y - 33 = 0 and 12x + 14y + 29 = 0 and
the intersection of x - y + 4 = 0, x - y + 2 = 0 is:
odd beacon
guys tell me are these equation correct
like last 2
their point of intersection ??
<@&286206848099549185>
vagrant prism
guys tell me are these equation correct
what exactly is your question?
odd beacon
I have to find equation of straight line
I find given eq intersect points
but in 3rd and 4th equation i do not got any point so i am confuse
vagrant prism
you have four lines in total, first two intersect at some point and the other two intersect at a different one
vagrant prism
but in 3rd and 4th equation i do not got any point so i am confuse
show your work
actually hold up
yeah those equations don't look correct
odd beacon
you have four lines in total, first two intersect at some point and the other tw...
they do not say
The statment which is above is written
vagrant prism
they do not say
what do they not say
what i wrote wasn't a question, i just rephrased your task
odd beacon
That 2 intersect at one point and an other on an other point
vagrant prism
The equation of the straight line through the point of intersection of
16x - 10...
but the last two equations don't intersect
vagrant prism
The equation of the straight line through the point of intersection of
16x - 10...
you sure you rewrote this correctly?
odd beacon
yup
wait
do you have time
for another question
The equation of hypotenuse of AB of length √2 in isosceles △OAB is:
A) x + y = 1
B) x + √2y = 1
C) √2 x + y = 1
D) x + y =√2
<@&286206848099549185>
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cold pine
ayuda
native raptor
did you sketch graph
cold pine
no
native raptor
do that 😭
cold pine
do i have too
native raptor
no
if u wanna visualize it in ur head you can
but seeing that ur asking how to do it i'd advise you to draw the lines if you wanna solve the problem
unique minnow
They solved for the intersection
native raptor
yea i thought so
cold pine
wait but what are the bounds then
native raptor
i beg u just draw it out
cold pine
lol
native raptor
u dont need calculus
u need like algebra 1
and the formula for a triangle's area
native raptor
bet thanks
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wraith hinge
<@&286206848099549185>
junior saffron
Use formula $(x+y)^2=x^2+2xy+y^2$, identify $x$ and $y$ here.
solid kilnBOT
Crystopher
wraith hinge
?
junior saffron
? Something wrong?
wraith hinge
I don’t get it sorry
civic iron
ok hold on
try to think of it as this
everything
in the parentheses gets squared
there are exponent rules such as:
wraith hinge
Wait parenthesis everything?
pulsar dust
() these
wraith hinge
Yeah
civic iron
hold on let me solve this first and ill get back to you
wraith hinge
Aight
pulsar dust
Now think like this $9a^4= (3a^2)^2$
solid kilnBOT
convergence
pulsar dust
Similarly do it for 4b⁴
languid walrus
try solving (3a^2)^2
civic iron
Use formula $(x+y)^2=x^2+2xy+y^2$, identify $x$ and $y$ here.
use crystophers technique
too
pulsar dust
$(3a^2)^2=3a^23a^2=(33)a^{2+2}$
civic iron
yeah what he said
wraith hinge
I’m not getting it at all. Thanks tho.
solid kilnBOT
convergence
civic iron
youre just trying to get the equation back to its
pulsar dust
This is equal to $9a^4$
solid kilnBOT
convergence
civic iron
original first form
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tropic heart
can someone make a diagram for this question
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can someone make a diagram for this question
strong zinc
can someone make a diagram for this question
It’s weird that they have 360-theta
strong zinc
I can't get the directions right
This is my interpretation
For an example theta=45
But the difference between S and R should always be 90
tropic heart
Wdym
strong zinc
But also (360-t)-(90-t)=(0-t)-(90-t)=-90
strong zinc
like wdym the difference between S and R should always be 90
Yeah regardless of the choice of theta the angle between s and r will be 90
tropic heart
ohh
strong zinc
If we pick theta=45 we get 90 and theta=30 we still get 90
tropic heart
well yea
so
theta will need to be same
no matter what angle, if i subtract it from 360 and 90 as long as they are same, the angle between 90 and 360 will always be 90
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alpine viper
This shape is cut from the right based on the area given. How to calculate the length of the shape, which is represented by the green line, on given area, which is represented by the part coloured in red?
quick hound
do you know integrals?
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alpine viper
do you know integrals?
No.
alpine viper
How can I make a full question? I said everything I want.
alpine viper
The red part, I already answered that.
alpine viper
No, it is this red and gray part. I explained.
And I drew it, so clearly, and expectedly, illiterate still want not to look at anything.
formal sage
Let's stay respectful
alpine viper
I drew the red line, it cuts the shape, and thus I care only about the length of the red part, that is the part that is left from cutting.
alpine viper
I do know just area, and I do want to get the length of the green line, why do not you read the question?!
formal sage
Because you are contradicting yourself
formal sage
I drew the red line, it cuts the shape, and thus I care only about the length of...
Right here you said you only care about length of red part
formal sage
This shape is cut from the right based on the area given. How to calculate the l...
Here you say length of shape (green line)
quick hound
yeah i didnt get him either
he gives 0 values of what we know already
like based on what he gives us use a ruler i guess
alpine viper
Right here you said you only care about length of red part
Which is the same as the length of the green line, I drew it, it is for representation of the length used in drawings. The grey line is a projection for easier look.
formal sage
So let's use different colors to avoid further confusion
Anyways
You know red area
You want to find green line
Do you know the size of red line?
alpine viper
Yes, I do, sometimes I want the area to be 50%, sometimes 20% or other number. It is a percentage of the whole area, and the whole area is red and grey part together, for that is the shape given. I know not the size of red line, just the area.
formal sage
Also how about the gray area
alpine viper
Grey area is nothing, it is to show where the rest of the shape would be, if the area is 100%.
formal sage
I don't think your question is solvable if the only you know is the area and not anything else
formal sage
I don't think your question is solvable if the only you know is the area and not...
Yes there seems to be some relations between the fact that it is a hemisphere inside a square, so for example if it was 50% green line would have been calculated as:
x^2 - 1/4 * pix^2 = area
So you solve for x.
But even in this niche case we have more info than the area. We know the maximum possible length
Also no integrals makes it at some aspects less approachable
Because now we only have to think geometrically and we have not much info
alpine viper
I can imagine the square being area 1, the half square here is 0,5, the full area of my shape is 0,5-(pi*1^2)/2 which I understand not how is it possible since the area of half the circle I just calculated seems to be greater than the half of the square it is in.
toxic flicker
I'm just gonna throw in my two cents, I've studied everything up to calculus including some pretty rigorous geometry and I haven't seen anything similar to this at all. I think you might need to know about integrals dude
formal sage
I can imagine the square being area 1, the half square here is 0,5, the full are...
Problem is you are multiplying pi with square of diameter instead of radius
alpine viper
Yes. Therefore the area of my full shape is: 0,5-(pi*0,5^2)/2, which is close to 0,1073. My problem looks like this.
I will be back.
I just want to calculate the height of the cut circle based on the area.
I will make formula by myself.
Thank you.
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rapid horizon
Can anyone tell me the m,n,r numbers/values?
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wraith hinge
n=2m
r=4m
depends on the value of m
this result was achieved assuming that the starting term of the A.P is 0(to make lives easier)
@rapid horizon
oh its wrong :p
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marsh forum
\item prove that $n!>n^2 \forall n \geq 4 $ and $n!>n^3 \forall n \geq 6$
base case:- $4! \geq 4^2$
\
nth case :- $n!\geq n^2$
(n+1)^{th} case $(n)!(n+1) \geq (n+1)^2$
solid kilnBOT
ƒ(Why am. I here)=I don't Know
\item prove that $n!>n^2 \forall n \geq 4 $ and $n!>n^3 \forall n \geq 6$
base case:- $4! \geq 4^2$
\\
nth case :- $n!\geq n^2$
(n+1)^{th} case $(n)!(n+1) \geq (n+1)^2$
```Compilation error:```! LaTeX Error: Lonely \item--perhaps a missing list environment.
See the LaTeX manual or LaTeX Companion for explanation.
Type H <return> for immediate help.
...
l.1421 \item p
rove that $n!>n^2 \forall n \geq 4 $ and $n!>n^3 \forall n \...
Try typing <return> to proceed.
If that doesn't work, type X <return> to quit.```
marsh forum
I use induction I guess?
or is there a better method
y/n
so here's what I've done so far
$n \cdot n! +n! \geq n^2+1+2n$
solid kilnBOT
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marsh forum
$(n+1)!=(n+1)n!$
solid kilnBOT
ƒ(Why am. I here)=I don't Know
marsh forum
let $\wedge$ stand for any comparison sign
solid kilnBOT
ƒ(Why am. I here)=I don't Know
marsh forum
$(n+1)n! \wedge (n+1) (n+1)$
solid kilnBOT
ƒ(Why am. I here)=I don't Know
solid kilnBOT
ƒ(Why am. I here)=I don't Know
fair bison
I use induction I guess?
yes, use induction
solid kilnBOT
ƒ(Why am. I here)=I don't Know
fair bison
??? what signs
marsh forum
$\geq , \leq =$
solid kilnBOT
ƒ(Why am. I here)=I don't Know
fair bison
why would you need to include ≤ and =
marsh forum
I was thinking of starting from the fact that $n!>n$
solid kilnBOT
ƒ(Why am. I here)=I don't Know
fair bison
so your base case is that 4! > 4^2, that's good
and now you want to assume n! > n^2
and show that (n+1)! > (n+1)^2
solid kilnBOT
ƒ(Why am. I here)=I don't Know
marsh forum
is what I need to prove
fair bison
yes
solid kilnBOT
ƒ(Why am. I here)=I don't Know
fair bison
why are you writing "something"
marsh forum
we don't know if $(n+1)! > (n+1)^2$
fair bison
, not ≥
solid kilnBOT
ƒ(Why am. I here)=I don't Know
marsh forum
assuming $n!>n+1$
solid kilnBOT
ƒ(Why am. I here)=I don't Know
marsh forum
we have $\frac{n!}{n+1} >1 \forall n>4$
solid kilnBOT
ƒ(Why am. I here)=I don't Know
lofty walrus
all you'd have to prove for the induction to work is $n^2 > n+1$, no?
solid kilnBOT
belabutter
solid kilnBOT
ƒ(Why am. I here)=I don't Know
lofty walrus
my thinking is : $n!(n+1) > n^2(n+1)$
solid kilnBOT
belabutter
lofty walrus
by the induction hypothesis
solid kilnBOT
ƒ(Why am. I here)=I don't Know
lofty walrus
and then $n^2(n+1)>(n+1)^2$ because $n^2>n+1$
solid kilnBOT
belabutter
lofty walrus
therefore $(n+1)!>(n+1)^2$
solid kilnBOT
belabutter
lofty walrus
$\Box$
marsh forum
mhm
solid kilnBOT
belabutter
lofty walrus
np
marsh forum
$n!-n = n((n-1)(n-2)(n-3)(n-4)....(1)-1)$
solid kilnBOT
ƒ(Why am. I here)=I don't Know
marsh forum
now if n>4 , it's obvious that this will evuvate to a number greater than 1
is that sufficent too?
marsh forum
my thinking is : $n!(n+1) > n^2(n+1)$
wait, isn't the induction hypothesis if f(x)>g(x) \implies f(x+1) > g(x+1)?
<@&286206848099549185>
lofty walrus
the induction hypothesis is:
neat swan
Hi
lofty walrus
assume $A_n$ is true, then prove $A_n \implies A_{n+1}$
solid kilnBOT
belabutter
solid kilnBOT
ƒ(Why am. I here)=I don't Know
marsh forum
we have to prove $(n+1)!>(n+1)^2$
solid kilnBOT
ƒ(Why am. I here)=I don't Know
lofty walrus
$\implies (n+1)! > (n+1)n^2$
solid kilnBOT
belabutter
marsh forum
wait
lofty walrus
that's what i did and it follows immediately from $n! > n^2$, no?
solid kilnBOT
belabutter
marsh forum
by induction again I guess?
fair bison
You could
solid kilnBOT
ƒ(Why am. I here)=I don't Know
marsh forum
I'd probably show that $x^2-x-1>0$ \forall x> 1
solid kilnBOT
ƒ(Why am. I here)=I don't Know
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marsh forum
using calculus
fair bison
btw it's probably easier to follow if you don't use the latex bot for everything lol
n^2 > n + 1 is already readable
marsh forum
got it
fair bison
Yeah that's another way to show it ^
marsh forum
**ƒ(Why am. I here)=I don't Know**
would this be wrong?
fair bison
A way I was thinking of too was n^2 = n * n ≥ 2 * n = n + n > n + 1
fair bison
would this be wrong?
Well first off the product should end in 1, not (n-n)
fair bison
I thought you were trying to prove that n! > n^2 though, not n! > n?
solid kilnBOT
ƒ(Why am. I here)=I don't Know
marsh forum
that's equivalent
as we multiply both sides by (n+1) to get back to our OG statement
fair bison
I see
Well you're going to want to use the induction hypothesis
n! > n^2
And then all that remains is to prove n^2 > n + 1, which we just did
marsh forum
but why is my method wrong?
fair bison
You want to avoid words like "it's obvious" in a proof
You need to justify every step
Why is that expression greater than 1?
marsh forum
if $n<4$ it turns out to be negative
solid kilnBOT
ƒ(Why am. I here)=I don't Know
marsh forum
$n!-n = n((n-1)(n-2)(n-3)(n-4)....(1)-1)$ (edited)
solid kilnBOT
ƒ(Why am. I here)=I don't Know
fair bison
if $n<4$ it turns out to be negative
I don't think that's true; 3! - 3 for example is 3
fair bison
You should practice writing up your proofs
Here, write your full proof that n! > n^2 for all n ≥ 4 in one message
marsh forum
I'm using overleaf, to do the same
fair bison
I'll read it and give you feedback
marsh forum
ok
\item prove that $n!>n^2 \forall n \geq 4 $ and $n!>n^3 \forall n \geq 6$
base case:- $4! \geq 4^2$
\
nth case :- $n!\geq n^2$
\
$(n+1)! >(n+1)^2$
\implies $n!>n+1$
$n!-n>1$
$n((n-1)!-1) >1!$
solid kilnBOT
ƒ(Why am. I here)=I don't Know
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marsh forum
is what I'm gettinf
fair bison
Okay let me rewrite this in easier to understand LaTeX
First off, a proof should be mostly words
Here's an example
Also, make sure not to confuse > and ≥
You seem to do that a couple times
marsh forum
👍
fair bison
Prove that $n!>n^2$ for all $n \geq 4$ and $n!>n^3$ for all $n \geq 6$.
We'll prove this statement by induction on $n$.
\begin{itemize}
\item Base case: When $n=4$, $4! = 24$ and $4^2 = 16$, so $4! > 4^2$.
\item Induction case: Next, we assume that $n! > n^2$, and we need to prove that $(n+1)! > (n+1)^2$.
[Complete this proof.]
\end{itemize}
solid kilnBOT
kalman_filtERIC
fair bison
It's important not to overuse symbols in proofs and to make it easily readable
marsh forum
thanks!
fair bison
You can copy-paste this LaTeX to see what I did by right-clicking my message and pressing Copy text
Then paste it into your Overleaf document and complete the proof
And then send it back
solid kilnBOT
ƒ(Why am. I here)=I don't Know
marsh forum
**kalman\_filtERIC**
this is the OG problem statement
fair bison
why isn't this equivalent to proving $n! >(n+1)$?
You can also do that
I'm just saying your method of proving it wasn't clear
marsh forum
ah
fair bison
Yeah I mean I don't think you have to prove this statement using induction
But it seems like you need practice with it anyways
solid kilnBOT
ƒ(Why am. I here)=I don't Know
marsh forum
$n((n-1)(n-2).....-1)>1.$
solid kilnBOT
ƒ(Why am. I here)=I don't Know
marsh forum
I'd probably solve it using logs now
fair bison
Don't use logarithms
marsh forum
ok
fair bison
You can do it simpler
marsh forum
hmm
marsh forum
**ƒ(Why am. I here)=I don't Know**
from here?
or do I have to think in a different way
fair bison
**ƒ(Why am. I here)=I don't Know**
Well, can you give me a reason why this would be true
marsh forum
I'm assuming it's true and solving the inequality
fair bison
You said it was obviously true, what's the reason behind that
marsh forum
beyond n=3 , the factorial function increases faster than x
rather n!-n
is increasing
and it becomes positive beyond n=3
I'm only analysing it at natural points
fair bison
I'm going to rewrite it as $n((n-1)! - 1) > 1$
solid kilnBOT
kalman_filtERIC
fair bison
It would suffice to show both factors of the product are greater than 1, yeah?
marsh forum
yeah
fair bison
OK so we want
n > 1
(n-1)! - 1 > 1
In other words
n > 1
(n-1)! > 2
For what values of n is that true
marsh forum
$n\geq 4$
solid kilnBOT
ƒ(Why am. I here)=I don't Know
fair bison
Yeah
In fact we can do a little better
If n > 1, then we don't need (n-1)! - 1 to be > 1, only ≥ 1
So
n > 1
(n-1)! ≥ 2
Does that make sense?
fair bison
sure, what's your point though
fair bison
So
n > 1
(n-1)! ≥ 2
So what values of n satisfy this
marsh forum
$n \geq 4$
solid kilnBOT
ƒ(Why am. I here)=I don't Know
fair bison
There's more
marsh forum
$n \geq 3$?
solid kilnBOT
ƒ(Why am. I here)=I don't Know
fair bison
Yeah
So now our claim is: "If n≥3, then n>1 and (n-1)!-1≥1, so n! - n = n((n-1)!-1) > 1. That means that n! > n+1, so (n+1)! = (n+1)n! > (n+1)(n+1) = (n+1)^2."
We can rewrite this a little bit to shift everything over by 1.
If n≥4, then n-1 > 1 and (n-2)! - 1 ≥ 1, so (n-1)! - (n-1) = (n-1)((n-2)! - 1) > 1. That means that (n-1)! > n, so n! = n(n-1)! > n(n) = n^2.
That gives a proof of the original fact that doesn't use any induction, except we still haven't actually proven that if n≥4, (n-2)! - 1 ≥ 1
How could we prove that part?
marsh forum
uh
I'm not too sure, sorry
I think I'll try this later today when I'm fresh
it's nearly 12:40 here
sorry
can i close this for now?
marsh forum
How could we prove that part?
is that fine?
marsh forum
That gives a proof of the original fact that doesn't use any induction, *except*...
I don't get why solving in inequality isn't sufficent
fair bison
can i close this for now?
Sure
fair bison
I don't get why solving in inequality isn't sufficent
Well we haven't proven yet that (n-2)! - 1 ≥ 1. This fact is pretty easy to prove by induction though.
But, just goes to show you can't entirely escape the induction, even if most of your argument does not involve induction.
trim joltBOT
fair bison
np
grim sparrow
prime wagon
Well we haven't proven yet that (n-2)! - 1 ≥ 1. This fact is pretty easy to prov...
is induction really needed in that case though? Wouldn't showing the fact that (n-2)! achieves its minimum value at n = 4, 2, which is greater than or equal to 2 be enough?
fair bison
How would you go about showing (n-2)! ≥ 2 for all n≥4 without an inductive argument
You can reduce it to showing that (n-2)! ≥ (4-2)! for all n≥4, but even such a simple statement as that seems like it requires induction
grim sparrow
How would you go about showing (n-2)! ≥ 2 for all n≥4 without an inductive argum...
by saying that it's obvious, obviously
frozen plover
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sweet pagoda
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sweet pagoda
so the original equation of the red line was y=-Ix-6I
then there was another question that changed the line to vertically shift down 0.50
and now
theres this question
im not sure how to write the equation that would reflect this change
im also pretty sure the new function equation for this question would be y=-Ix-6.5I+4
if anyone could help that'll be awesome
for this question i think the new function graphed is shifted horizontally
ocean latch
original was y=-|x-6|
shift down by 0.5
so y=-|x-6|-0.5
profit calc:
If the selling price is $2.50 and the desired profit is $4, the cost price must be:
sp-cp=2.50-4=-1.50
this is senseless cuz cp cant be negative
price willigness adjustment: if the maximum price people are willing to pay has increased to $6.50, the function's maximum value should be adjusted accordingly.
the correct approach would be to shift the function vertically to account for the new price. Since people are now willing to pay $6.50, but the desired profit is $4 on top of this new cost, you should reflect this change in our function.
new max val:
y=|x-6|-0.5
to reflect the new maximum price people are willing to pay, the equation should be:
y=-|x-6.5|+4
transformation includes
horizontal shift right by .5 units(from x-6 to x-6.5)
vertical shift upwards by 4( to account for 4$ profit)
**
so new function is
y=-|x-6.5|+4**
ocean latch
yh
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sudden seal
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sudden seal
I keep getting stuck after I combine the fractions in the parentheses
lofty walrus
are you solving for x?
trim hawk
I keep getting stuck after I combine the fractions in the parentheses
Can you send what you have so far
sudden seal
Can you send what you have so far
Solving the equation basically.
I came to
x/2 - ((x+3)/6) = x - 2 ((x-4)/4)
trim hawk
the (x+3)/6 is wrong, and also you should be able to just distribute the - and -2 on both sides instead of combining
sudden seal
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maiden hare
(-1.03x - x)*1.02 - x?
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cold pine
how do i get the answer right but not everything else
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how do i get the answer right but not everything else
limber minnow
Hmm, I believe it's because you did the evaluation in terms of x
So you got the right final answer, but the intermediate steps wanted you to do it in terms of y
cold pine
so what do i do then
limber minnow
Well, lets start with b, the upper bound being x = 4 is right, but we need to put it in terms of y
So, when x=4, what does y equal?
cold pine
2/9(4)^2
limber minnow
Yeah, which you can simplify to 32/9
cold pine
so thats the upperbound
limber minnow
Yup!
cold pine
but then what goes to the integral
limber minnow
Well that will have two parts to it, first we'll need to get the formula for x given y
Essentially you want to manipulate the formula so x is alone on one side
edit: I meant x should be alone
i.e if y = x + 2 then x = y - 2
limber minnow
Yup!
Now there's one problem still
Doing an integral with that equation will give you the blue region, but the region we want to integrate over is the red region
Does that make sense? If so, do you think you can modify the equation to give us the red region?
Note that we're working with y, so the equation changes as we move up rather than move right
cold pine
makes sense but so then we how do i change the formula?
limber minnow
Well you want the distance between the curve and the straight line at x=4
The current equation gives the distance between the curve and the line at x = 0
heady hatch
remember the definition of integral
cold pine
i still dont know
limber minnow
Well let's look at a few points
At y = 0, the equation gives us sqrt(0*9/2) = 0 for the width of the blue region, so the width of the red region would be 4
At y = 32/9, we know that the width of the blue region will be 4, making the width of the red region 0
Can you figure out the width of the blue & red regions at y = 2?
cold pine
where does 2 come from?
limber minnow
Okay you know what, I think I'm actually approaching this from the wrong direction, I'm sorry about that
Give me a moment, I need to do a couple quick calculations to make sure I actually know what I'm talking about
limber minnow
Yeah the thing I was trying to show you was wrong, sorry lol (btw the answer I was pushing you to was 4 minus the equation)
limber minnow
for x = sqrt(y * 9/2)
So lets back up to this point
Integrating over that will still give the incorrect region, but what it will give is actually very close to what we want
The region it would give would look like this
What we want is the area on the other side of the blue curve
But if you think about it, the region that equation will give us and the region we want can be combined to create a solid cylinder
Does that make sense at all?
cold pine
yes slightly
limber minnow
So the volume of the region we've got + the volume of the region we want = volume of a cyclinder
And we can easily figure out the volume of the cyclinder from our bounds (radius of 4, height of 32/9)
Do you see how we can use that to get to the volume of the region we want?
cold pine
yeah
limber minnow
Now this is sort of a guess, but assuming that's what the question wants you to do, the "expression for the integral" should be derived from our current equation
limber minnow
for x = sqrt(y * 9/2)
This equation
cold pine
$2\pi\left(4\right)\left(\sqrt{y\cdot\frac{9}{2}}\right)$
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Big Yannis
limber minnow
Hmm, I think you might be using the surface area equation
Remember, the circumference of a circle is 2pi*r, and the area is pi*r^2
solid kilnBOT
NotABot
limber minnow
Here dx is the height of the very small cylinders that the integral uses, and f(x) is the radius
cold pine
see my modules r so confusing idk which solids of revolution im solving for
idk if its disk and washer or cylindrical shells
limber minnow
Yeah...
cold pine
the questions dont specify
limber minnow
I'd say when we're talking about volume we usually care about disk, if we want the surface area then cylindrical shells, and washer is probable when we want the volume of a donut-like shape
I do agree that a lot of how calculus is taught is pretty confusing
cold pine
like all the questions are formatted like this
i dont even know what shape
its all textbook
limber minnow
Yeah that's kinda weird
cold pine
i hate this section so much
limber minnow
Well, do you know what x^2 + y^2 = 16 looks like?
spiral estuary
i dont even know what shape
It's like a ball with a hole
limber minnow
A really big hole
cold pine
the craziest partis legit solved it
with the completely incorrect steps
like bruh
how does that even work
$\pi\left(\sqrt{y\cdot\frac{9}{2}}\right)^2$
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Big Yannis
limber minnow
Yeah, though you can simplify a bit
limber minnow
Oh you know what, you mentioned washers? That might be what it wants
Blah, I don't miss working with math homework software
I mentioned that to get to our final answer we'd wanna remove the region that equation gives from the solid cylinder right? That sounds like a washer now that I think about it
Do you have any equations for solving for washers?
limber minnow
Yeye
limber minnow
Referencing this graph again
The outer radius is going to be 4, while the inner radius will be found from our equation
Does that make sense?
cold pine
$\pi\left[\left(\sqrt{y\cdot\frac{9}{2}}\right)^2-\left(4\right)^2\right]$
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Big Yannis
limber minnow
Yeah I was about to say
limber minnow
Awesome!
Indeed, sorry about that 20 minute detour earlier 😛
I should honestly review some of my calc 2 if I wanna help people out with it
But anyway, I'm glad you got there in the end!
Need any more help, or do you think you're good for now?
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edgy mural
You repeatedly draw marbles from a bag containing 50 red and 50 blue marbles until there are no more marbles left, recording the order of red and blue marbles drawn. You then count the number of "runs,'' where a run is defined as any number of consecutive marbles of the same color. For example, 𝑅 𝐵 𝐵 𝑅 𝑅 𝑅 𝐵 𝑅 𝑅 contains 5 runs. What is the expected number of runs that you observe?
edgy mural
I tried linearity of expectations but it goes down a rabbit whole that I can't seem to figure out
starting from the 1st marble we have E[X] => E[ 1 + E[X_99 + E[....]] or something like that
shrewd ridge
there are 99 opportunities to get +1 run, and you get it half the time
i would guess 1+44.5
edgy mural
hm, so the answer key was 51
edgy mural
So I guess with after the first marble, there are either 49R / 50 B or vice versa
So the probability of getting another plus 1 would be 50/99 after that
and then I guess this sequence continues down probability of gettin a 1 or a 0 taking out a marble from 99 -> 0, but I guess framing this as a linearity of expectations is the difficulty...
edgy mural
I did find this explanation right here which to 51, https://math.stackexchange.com/questions/4747341/marble-runs-expected-value-question-linearity-of-expectation.
However, i didn't quiet et the explanation maybe it makes sense to you?
shrewd ridge
right, i can't point to answers about coinflips
edgy mural
I guess they're adjacent, but since the sample size reduces with each turn I guess the expectation breaks down?
shrewd ridge
oh it's my idea but i missed that the chance is 50/99
1 of 50 Rs followed by 1 of 50 Bs or vice versa
50*50*2/ 100*99 = 50/99
is the probability that two adjacent marbles are different
edgy mural
is it because order doesn't matter?
shrewd ridge
imagine 100 experiments, so 10,000 marbles
no like 99
9900 marbles
marbles 5 and 6 will be different in 50 of experiments
marbles 92 and 93 will be different in 50 experiments
shrewd ridge
me too
edgy mural
So i'm tryin to think of it as the # of color changes now instead of the number of orderings of each balls
given that the first ball always starts a sequence
so 1 + E[Z] where E[Z] is the expectation of runs with the remaining 99 balls
I guess with 99 balls, whats the probability of a color change happenin i guess
edgy mural
So the probability of the 2nd ball being a color change is i guess 50/99 out of 99 balls
shrewd ridge
yes
edgy mural
but then im wondering now the probability of the 3rd ball being a color change, 4th, 5th... etc
shrewd ridge
right, it doesn't matter that it's not independent
my way of thinking is you imagine infinte experiments, or just 99 in this case
50/99 is the chance that any two adjacent marbles are different, so in 50 out of 99 experiments marbles 34 and 35 will be different
so there will be 50 color changes
per experiment
edgy mural
aha i see
so considering this in the n case
we have 1 + # of ball changes in n - 1 cases
which is the probability that any two adjacent marbles will be different in (n-1) cases, considering there are n/2 marbles of each color
hm explanation says linearity of expecations, but how does that play into this?
I guess thats the confusing part
shrewd ridge
that's waht i'm explaining
all we want is the amount of color changes in 99 experiments, total
we can confidently say that there's 99+(that amount) runs in 99 expriments
that amount is 50×99
because there are 99 experiments, with 99 opportunities for color change, and each time we have 50/99 chance that it changes
50/99*99*99 = 50×99
(99+(50×99))/ 99 = 51
edgy mural
I see, that makes a lot more sense but this part "with 99 opportunities for color change" why is there 99 opportunites for color change?
shrewd ridge
99 adjacent pairs
any adjacent pair is the same
any adjacent pair has 50/99 chance of being two different marbles
from 50*50*2/ 100*99 = 50/99
edgy mural
ah i see
Hm ok, so by linearity of expectation we consider any 99 pairs of marbles that each have a 50/99 chance of being two different marbles
shrewd ridge
totally
edgy mural
So then by that the expected number of changes amonst all those 99 pairs is 50/99 * 99
then given the 1 by the beginning marble, we can see that the total number of runs expected is 1 + 99 * 50 / 99 = 51
aha i see
do you agree?
shrewd ridge
yeah
edgy mural
nice nice, thanks for the conversation
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azure wagon
did i do this right?
trim joltBOT
did i do this right?
rancid flax
did i do this right?
I think your graph of $-\log_2(x)$ is wrong
solid kilnBOT
nw
rancid flax
it should be a reflection of the other one across the x axis
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rapid horizon
Set A has m elements and set B has n elements and m>n
So how many into function can be?
unkempt smelt
into function is injective?
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chilly rivet
did i do this right??
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did i do this right??
unkempt smelt
!show
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solemn thorn
is this a quiz
proper terrace
hey i am meolve
solemn thorn
Yo
chilly rivet
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solemn thorn
?
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valid halo
im lost it takes me 1h and still with no correct reasoning
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im lost it takes me 1h and still with no correct reasoning
@valid halo Has your question been resolved?
valid halo
no
unique minnow
Pour l'implication =>, essaie de tâtonner un peu avec l'expression
c = at + b(1-t).
Si tu peux isoler t là dedans, tu peux partir de cette valeur de t et montrer, par exemple, que t est dans [0,1], puis le reste suivra.
Pour l'implication <=, tu dois essentiellement montrer que si ce t existe, alors a <= c et c <= b.
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wraith hinge
how did they come up with phi(sqrt(2)) in the end with those values of m and sigma??
nova spire
how did they come up with phi(sqrt(2)) in the end with those values of m and sig...
mu = 1000, sigma^2 = 50
1010 - mu = 10
(1010 - mu)/sigma = 10/sqrt(50)
sqrt(100)/sqrt(50)
sqrt(2)
wraith hinge
but they wrote 50²=sigma² no?
wraith hinge
bruh
wraith hinge
bruh okay
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radiant portal
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radiant portal
Im not really sure what the first step here is
I know what a jump discontinuity is.... it means the left limit isnt equal to the right limit
light abyss
I know what a jump discontinuity is.... it means the left limit isnt equal to th...
basically you just have to evaluate f(-1) from both sides using a limit and then solve the resulting equation for a
radiant portal
Ah ok, but why evaluate f(-1) sorry? Its been a long time since I did these haha
light abyss
Ah ok, but why evaluate f(-1) sorry? Its been a long time since I did these haha
because the limit at x=-1 doesnt exist
radiant portal
Ok makes sense, so what are we actually doing by subbing in x=-1
light abyss
basically the parameter a moves the "left side" of the function up and down
so you have to find the value at which this part of the function perfectly aligns with the other branch
and to find that value you have to evaluate both of them at -1 and set them equal to each other
this is a little filthy but it works for this example
if you want to do it orderly you have to plug in -1 into the first branch
and use the limit as x approaches -1 from the left side for the other branch
radiant portal
basically the parameter a moves the "left side" of the function up and down
whats the parameter a sorry 🙃
wraith hinge
i⁹⁷ = i⁹⁶ ×i
= (i⁴)²⁴×i
= (1)²⁴ × i
= 1×i
= i
wraith hinge
i⁹⁷ = i⁹⁶ ×i
= (i⁴)²⁴×i
= (1)²⁴ × i
= 1×i
= i
How it's become i
Like what's maths is it called
radiant portal
Bro this channel is taken
solemn thorn
!occupied
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radiant portal
Ok makes sense
light abyss
you have to find the value for which they align
so treat it like a basic intersection problem
radiant portal
you have to find the value for which they align
Wait but why sorry
light abyss
Wait but why sorry
because the question asks for which values of a the function has a jump discontinuity
in this case there is no jump
radiant portal
But it only has jump discontinuity when they arent aligned, no?
light abyss
But it only has jump discontinuity when they arent aligned, no?
yes exactly
radiant portal
in this case there is no jump
Like here there is no jump
So arent we looking for the opposite?
light abyss
Like here there is no jump
and you have to find the value of a for when this occurs
light abyss
So arent we looking for the opposite?
there is a jump for any number a except for 1 single value yes
radiant portal
Okkkk I think I see
So by finding when there is no jump discont, we just set it =/= to that to get all areas where there is jump discont, right?
light abyss
basically you have to solve this equation $f_{right\ branch}\left(-1\right)=f_{left\ branch}\left(-1\right)$
solid kilnBOT
Jill ♡
light abyss
there is a jump for any number a except for 1 single value yes
you can determine this since a only moves the branch up or down and doesnt distort it in any way
radiant portal
so a =/= -5 right?
radiant portal
YES! I got it 😄
Thank you soooo much
You are a really good teacher thank you for being patient with me haha
❤️
light abyss
yw lol
radiant portal
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gaunt temple
by the monotone convergence theorem a sequence converges if it's bounded and monotone
gaunt temple
does this mean we have to find inf and sup or are upperbounds and lowerbounds sufficient
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opal verge
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opal verge
skglskgl;rek
unkempt smelt
!show
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unkempt smelt
i see where you went wrong
you took the 6x to the other side and then divided by 7 didnt you?
unkempt smelt
almost
there is a -7 in front of the y
so you have to divide by -7
not 7
so you should get $-\frac{14-6x}{7}$
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Flappie
opal verge
ohhhhhhhhhhhh
unkempt smelt
which is equal to $\frac{6x-14}{7}$
solid kilnBOT
Flappie
opal verge
thank you c: fucking - signs are always confusing me
unkempt smelt
you can also do it without having to fuck with the - signs
you take the 7y to the other side
and 14 to the other side
so you get 7y=6x-14
then divide that by 7
rotund grove
if a-b =c
b = a-c
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fiery cedar
how can i multiply two values without suffering from diminishing returns the bigger number they are
fiery cedar
ie a * b, and if i add 10 to b, get the same % increase as if b were 4000 or 40000000
brazen vector
what do you mean by that?
fiery cedar
50 x 300 | 50 x 310 vs 50 x 300000 | 50 x 300010 (adding 10 on each as example)
want to find a way to make both have the same % increase
split chasm
add a percentage of the value instead of a flat amount
fiery cedar
but i want to add a flat amount
split chasm
then you can't get what you want
fiery cedar
damn, no formula can help here? something to get the value proportional to b?
based on a flat number?
brazen vector
how would that work.
if you are increasing the first one to a specifif percent value of 300
then for the second one to have the same percent increase, you need to have the same percent value.
fiery cedar
i just want to add 10 to valueB, and regardless of the amount it has, always have a linear increase
and not decrease overtime
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fiery cedar
<@&286206848099549185>
zenith cipher
?