spring crystal
@vocal lark
#help-39
cobalt hinge
Suppose that the coefficients of the equation $x^n+a_{n-1}x^{n-1}...a_1x+a_0=0$ are real and satisfy $0<a_0\le a_1\le\ldots\le a_{n-1}\le 1$. Let $z$ be a complex root of the equation with $|z|\ge 1$. Show that $z^{n+1}=1$.
jolly parrotBOT
;(
cobalt hinge
!status
pearl pondBOT
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
4
Also, I’m gonna be going soon, so if my post closes PLEASE reopen it as soon as possible
Now for the solution
We know that $z^n+a_{n-1}z^{n-1}...a_0=0$. Multiplying by $(z-1)$ on both sides, we get $$z^{n+1}+a_{n-1}z^n...a_0z-z^n-a_{n-1}z^{n-1}...-a_0=0\implies z^{n+1}+(a_{n-1}-1)z^n+(a_{n-2}-a_{n-1})z^{n-1}...a_0=0$. Solving for $z^{n+1}$ we get $$z^{n+1}=(1-a_{n-1})z^n+(a_{n-1}-a_{n-2}...(a_1-a_0)z+a_0=0$$ Applying the Triangle Inequality we get $$|z|^{n+1}\le (1-a_{n-1})|z|^n+...(a_1-a_0)|z|+a_0$$ We know that the right hand side is less than $z^n$, since all of the coefficients of the $z^i$ terms are less than 1, so $|z|^{n+1}\le |z|^n\implies |z|=1$. Thus (since when $|z|=1$, the Triangle Inequality becomes an equality), we get $$|z|^{n+1}=1-a_{n-1}+a_{n-1}-a_{n-2}...a_0=1$$ Looking back at the original equation, we thus have $|z|^{n+1}=1\implies z^{n+1}=1$
jolly parrotBOT
;(
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brittle tinsel
i'm not entirely sure how I would do this myself, but I will point out that the modulus of a complex number being 1 does not mean the complex number itself is 1. i also don't see how the RHS of that one inequality is less than |z|^n just because all of the coefficients are less than 1 in magnitude
pearl pondBOT
@cobalt hinge Has your question been resolved?
cobalt hinge
i'm not entirely sure how I would do this myself, but I will point out that the ...
- “Looking at the original equation”
- If $|z|\ge 1$, then $|z|^n\ge |z|^j$ for $i\ge j$
jolly parrotBOT
;(
brittle tinsel
ah i see, well your last step makes sense then. i would mention that |z| >= 1 at some point in your proof then to make it more readable
brittle tinsel
but my point about the RHS of that inequality still stands i think, why is that large sum <= |z|^n? unless i'm missing something simple
cobalt hinge
1. “Looking at the original equation”
2. If $|z|\ge 1$, then $|z|^n\ge |z|^j$ fo...
Oh shit I meant i instead of n
cobalt hinge
but my point about the RHS of that inequality still stands i think, why is that ...
Okay
tiny lintel
Do you know how to rewrite z^n+1 -1
cobalt hinge
So we know $|z|^i\ge |z|^j$ for $i\ge j$
jolly parrotBOT
;(
cobalt hinge
Do you know how to rewrite z^n+1 -1
What?
tiny lintel
cobalt hinge
And from the problem statement we know that (1-a_{stuff}) is less than 1 but greater than 0
tiny lintel
Can you use this ?
cobalt hinge
So it will “decrease” the value
cobalt hinge
That doesn’t apply here
brittle tinsel
What they said is a good point, it would be very useful to use
$\sum_{k=0}^n z^k = \frac{z^{n+1}-1}{z-1}$
jolly parrotBOT
doaby
tiny lintel
That doesn’t apply here
Why ?
cobalt hinge
As an alternate proof probably
cobalt hinge
So it will “decrease” the value
And we also know (by triviality) |z|^n=|z|^n
So the entire sum will be less than or equal to |z|^n
Well here is a better demonstration
We can replace |z|^n with every other power of |z| by the relation stated above and say “\le than”
Since 1-a_{n-1}+a_{n-1}-a_{n-2}...a_1-a_0+a_0=1 it is \le than |z|^n
brittle tinsel
there you go, you just need to say that in your proof
cobalt hinge
Alright
cobalt hinge
there you go, you just need to say that in your proof
It looks good otherwise?
This was an MOP problem so I’m praying I got this right
brittle tinsel
yeah, just after you use the triangle inequality, mention somewhere that |z|^i <= |z|^n for i <= n and that the sum of the coefficients is 1
i would just continue the inequality to show that personally
then mention why the inequality works after for readability and you should be fine
earnest flint
guys can someone guide me pls u dont have to tell me the answer i just need to know how to get the answer 😭🙏
- Using the bell-shaped curve above. Shade the region bounded by the lowest 23% of the data values. What must be the z score to be part of this group? (5pts)
earnest flint
i dontknow where to start 😢
earnest flint
guys can someone guide me pls u dont have to tell me the answer i just need to k...
guys do i look for the closest percentile and then i would get -0.74
is that it?
versed mica
yea so from the picture you provided with all of the necessary context i would say it’s right about at the 23rd percentile
earnest flint
yea so from the picture you provided with all of the necessary context i would s...
thank uuu so its -0.74 right
versed mica
yes
earnest flint
thank you so much!!
dapper knoll
pearl pondBOT
@earnest flint Has your question been resolved?
pearl pondBOT
pearl pondBOT
Available help channel!
alpine current
In the function = {x² for x < 1;
1 for x = 1; 2 - x for x > 1}
at x = 1 the function is discontinuous but what is the type of discontinuity?
wary bane
why is it discontinuous
alpine current
there is a hole in the graph
alpine current
Is this pic wrong?
wary bane
thats not what you wrote
plush bramble
Is this pic wrong?
Next time just upload the screenshot rather than type
alpine current
thats not what you wrote
oh my bad
alpine current
Is this pic wrong?
Why is this considered as removable discontinuity?
plush bramble
Do you know what a removable discontinuity is
alpine current
Not fully because I don't understand why this is considered removable
my understanding of removable discontinuities is when you can remove something by factoring
plush bramble
my understanding of removable discontinuities is when you can remove something b...
Factoring is only sometimes involved in removable discontinuity
If you change one point in the function to make the function continuous, that's a removable discontinuity
But that requires the function to be discontinuous at a single point and the left and right sides limits to be equal
alpine current
I think I understand now, the right and left side limits are equal
plush bramble
Is this pic wrong?
Keep reading the wiki
wary bane
If you change one point in the function to make the function continuous, that's ...
(make the function continuous at that point)
alpine current
What is meant by the second sentence? I don't understand the explanation.
"continuity and discontinuity of a function are concepts defined only for points in the function's domain."
What does this mean?
wary bane
continuity is a property of the function at specific points in the domain
to call a function continuous is a shorthand way to say the function at all points of its domain are continuous
===
as for singularity
this is in reference to a point not in the original functions domain
the domain of x/x doesnt contain 0 but you talk about the removable singularity at 0
alpine current
So I understand those to a certain degree but I still don't understand why it is an abuse of terminology
and perhaps a dumb question but can you also use range to define the continuity of a function?
wary bane
the point isnt in the domain
if its not in the domain, then its neither continuous nor discontinuous
alpine current
So continuity is strictly defined within the domain?
wary bane
yes (at least according to that text)
there might be more general definitions around this
alpine current
there might be more general definitions around this
like?
wary bane
idk. there just might be
alpine current
also why is it an abuse of terminology again?
wary bane
yes (at least according to that text)
generally speaking though yes
wary bane
So continuity is strictly defined within the domain?
if u take this to be true
then talking about the continuity of a point not in the domain is abuse of terminology
wary bane
What is meant by the second sentence? I don't understand the explanation.
>"co...
what exactly is this source anyways
There are ways to phrase the definition so it doesnt matter
alpine current
what exactly is this source anyways
wiki (Ik it's not the best source)
wary bane
wikipedia?
alpine current
yeah
alpine current
then talking about the continuity of a point not in the domain is abuse of termi...
I'm sorry but I still don't understand. Isn't the point in the domain of that piecewise function?
wary bane
Let f : A -> B, and s is a singularity of f.
If there exists y such that g : A u {s} -> B with
g(s) = y
g(x) = f(z) everywhere else
and g is continuous at s
then this singularity is removable
wary bane
I'm sorry but I still don't understand. Isn't the point in the domain of that pi...
f : R - {0} -> R
f(x) = x/x
0 isnt in the domain
alpine current
What is that notation called?
(f : A -> B)
wary bane
its just standard notation for functions
alpine current
I'm not familiar with that
wait Imma look it ip
Okay I don't think I can sustain this conversation, the knowledge discrepancy between us is quite big I think, thank you for engaging though and I need to study functions more.
sharp vigil
$f \colon A \to B$ means that $f$ is a function with domain $A$ and codomain $B$
jolly parrotBOT
cloud
alpine current
idk what is a co domain
sharp vigil
the codomain is the set of numbers which the function could possibly output. we use it because you have to, strictly speaking, include the domain and codomain as part of the definition of a function
and finding the range/image is difficult
alpine current
are range and codomain synonymous?
sharp vigil
not necessarily
the image of a function is the set of all possible values that a function does output, whereas the codomain may include some values that the function doesn't
and the word "range" usually refers to the image, although some sources use it to mean codomain so it's a little ambiguous
sharp vigil
the image of a function is the set of all possible values that a function does o...
this is a useful concept when you start working with functions that can output something other than just real numbers
and it's useful to specify what type of thing it outputs, without getting bogged down in exactly which of them it will or won't output
loud minnow
subject: piecewise function definition
could someone explain what the last sentence is saying? I’m not familiar with some of the notation it’s using
loud minnow
humble lintel
all the X_is are subsets of R and they are pairwise disjoint sets
For example, you could have X1 = [1,2] and X2 = [3,4] then their intersection is empty
loud minnow
is the symbol before the R the same as the “is an element of” symbol?
loud minnow
For example, you could have X1 = [1,2] and X2 = [3,4] then their intersection is...
ohh
humble lintel
no because that would imply that the Xi's are singletons
loud minnow
sharp vigil
$\subseteq$ means "is a subset of"
jolly parrotBOT
cloud
loud minnow
ahh I see
I understand now, thank you for the help!
btw do you guys recommend a good resource to learn these symbols?
plush bramble
btw do you guys recommend a good resource to learn these symbols?
Wikipedia
rough forge
pearl pondBOT
pearl pondBOT
Available help channel!
flint musk
Where did my integral go wrong here? The correct answer should be 27pi and I got 26pi
honest spear
didnt distribute the 2 in the 4th last step
pearl pondBOT
pearl pondBOT
Available help channel!
fickle mantle
I need help again 🧍🏻♀️
pearl pondBOT
I need help again 🧍🏻♀️
fickle mantle
Im helping my little sister on her long division but she needs to it in the BIG 7 method
But I lowkey forgot how to do long division
The problem is 61065 divided by 45
toxic lichen
the "BIG 7 method"?
toxic lichen
The problem is 61065 divided by 45
are you sure you meant to put a dot there? the dot means multiplication
did you maybe mean 61065 / 45 ?
are we talking about something that looks like this
fickle mantle
are we talking about something that looks like this
Yes!!
toxic lichen
ok right
fickle mantle
Im so confused on it 😂
toxic lichen
this is how you would begin, yes?
fickle mantle
I feel bad bc she needs my help and I never learned this 😔
fickle mantle
this is how you would begin, yes?
Yes
toxic lichen
wait, you never learned long division? or did you just forget it entirely
fickle mantle
wait, you never learned long division? or did you just forget it entirely
No I have
toxic lichen
or did you learn it in a different format and find yourself unable to adjust
fickle mantle
Just not the big 7 method
toxic lichen
the thing about these graphical devices is that they're different across countries but ultimately all display the same basic idea of subtracting multiples of the divisor from the dividend
whichever way you put it on the paper
here, the biggest obvious multiple of 45 we can subtract is 45 * 1000
or 45,000
we could not subtract 45 * 2000 (which is 90,000) bc that's too big
and we get this
do you follow so far
toxic lichen
yes?
fickle mantle
Wait nvm i understand it 😭
It took me a minute for my brain to get it
Hers looks a bit different tho
toxic lichen
hers as in?
i am following that layout i grabbed off google images.
if your little sister uses a different layout then please share it.
fickle mantle
Ok I will
Let me just re write it bc her hand writing is messy 🧍🏻♀️
That’s how she solves her problems 🧍🏻♀️
toxic lichen
ok well that aint too different from what ive got
it's just that instead of writing =5*120 she just puts 120
is she gonna be graded on strict adherence to the method?
fickle mantle
is she gonna be graded on strict adherence to the method?
Yes
toxic lichen
ah crap.
fickle mantle
She’s in advance math
toxic lichen
how old is she
fickle mantle
12
toxic lichen
do i take it that a solution which is correct by the numbers, but deviates slightly in how it's formatted, might receive zero credit?
anywya whatever. adjusting for the format, here is how the second step will look. we can't fit any more 45*1000 so we see how many times we can fit 45*100 into what's left. turns out we can do it 3 times.
fickle mantle
Omg yesss that’s exactly how she does it
How r you so smart 😭
I feel like a lost puppy😭
toxic lichen
i mean i'm a literal math teacher
and i am some steps away from having a master's degree in math
fickle mantle
i mean i'm a literal math teacher
That explains it 😌💗
toxic lichen
anyway, do you see how to continue this
fickle mantle
Yes
toxic lichen
ok good
toxic lichen
whats got you stuck
fickle mantle
So would 20 be the next thing I’ll put
toxic lichen
45 * 20 = 900, so you can certainly subtract that much, but you can do more
fickle mantle
2,565 - 2,250 =315
That’s what I have so far 🧍🏻♀️
Btw I really appreciate you helping me
💖
The next one is 7
And the answer is 1357
Thank you so much for helping me 🫶🏼
midnight haven
pearl pondBOT
midnight haven
im lost
i was thinking on
apply cezaro stolz
but i dont think it will work pretty
it might
calm wing
can't you multiply by the conjugate?
midnight haven
thats also an good idea
but
should i try cezaro stolz first
i feel like its begging for it
wait but
its not montonly increasing
its decreasing
calm wing
should i try cezaro stolz first
idk i have very little experience with it
calm wing
yes
midnight haven
what now hmm
calm wing
multiply top and bottom by it
midnight haven
i did lol
midnight haven
calm wing
this isn't right
calm wing
$\sqrt{k+\sqrt{k^2-1}}\sqrt{k-\sqrt{k^2-1}} = \sqrt{(k+\sqrt{k^2-1})(k-\sqrt{k^2-1})} = \sqrt{k^2 - k^2 + 1}$
jolly parrotBOT
artemetra
calm wing
i feel like the denominaotr is wrong
yes
calm wing
$\sqrt{k+\sqrt{k^2-1}}\sqrt{k-\sqrt{k^2-1}} = \sqrt{(k+\sqrt{k^2-1})(k-\sqrt{k^2...
which simplifies to 1
midnight haven
oh wow
calm wing
so your series reformulated is $\sum_{k=1}^{n} \sqrt{k-\sqrt{k^2-1}}$
jolly parrotBOT
artemetra
midnight haven
yes
calm wing
hmm
calm wing
yes
i think you can stolz cesaro it from this point but i can't help you much with that
the final answer should be (spoiler!!!) ||sqrt(2)||
midnight haven
how do you know the final answer
calm wing
desmos lol
calm wing
show
midnight haven
i applied cezaro stolz from the start not from where u said to
but it will be the same thing
oh i see the solution
.close
pearl pondBOT
pearl pondBOT
gleaming moss
definitely not dodgy
sage walrus
ronni and anat both choose a random number between 1 - 10.
What is the probability that the number that ronni chose is larger by 5 than the number anat chose
hasty edge
<@&268886789983436800> Scammer!
cursive wraith
<@&268886789983436800>
sage walrus
whatt?
dapper kraken
<@&268886789983436800> Scammer!
who did you even ping lol
hasty edge
ronni and anat both choose a random number between 1 - 10.
What is the probabili...
Okay. Well, which numbers in this set are 5 apart from each other? i.e., if we were to choose a number a and a number b, which ones would satisfy |a-b|=5?
solid pier
hasty edge
who did you even ping lol
The person who was sending the suspicious link.
dapper kraken
ronni and anat both choose a random number between 1 - 10.
What is the probabili...
if your determined you can make a table, else you can do some trickery as if ronni was 5 higher than annat, then the number annat chose was <=5, so just check the possibilities of annat chose 1,2,3,4,5
sage walrus
Okay. Well, which numbers in this set are 5 apart from each other? i.e., if we w...
1,6
2,7
3,8
4,9
5,10
hasty edge
1,6
2,7
3,8
4,9
5,10
Great! So, what is the probability of choosing these pairs?
sage walrus
I have no idea
hasty edge
What is the probability of choosing 1 number from the set of 10?
hasty edge
Great.
sage walrus
but that still doesn't help
hasty edge
Why?
hasty edge
cuz what am I gonna do? (1/10) ^5 or something?
No. What is the probability of choosing both 1 and 6?
sage walrus
I have no idea
the probability of getting both 1 and 6 is.. 1/10 * 1/10 ?
which is 1/100
right?
hasty edge
So what should you do?
sage walrus
well we've got only 5 options
what if we just multiply that by 5 ?
which is 1/20 ?
no but it's an OR, not an AND
so we gotta add it up?
1/100 + 1/100 + 1/100 + 1/100 + 1/100
??
hasty edge
1/100 + 1/100 + 1/100 + 1/100 + 1/100
??
Yeah!
sage walrus
so that's 5/100
hasty edge
Yep!
sage walrus
wait but..
sage walrus
Yep!
that's not one of the options:
here's a more accurate translation of our problem:
Anat randomly chose an integer between 1 and 10 (inclusive), and Ori randomly chose an integer between 1 and 10 (inclusive).
What is the probability that the number Anat chose is exactly 5 greater than the number Ori chose?
versed wigeon
You just simplify 5/100
sage walrus
You just simplify 5/100
oh, which is 1/20
hasty edge
that's not one of the options:
Simplify 5/100.
pearl pondBOT
hasty edge
You are quite welcome!
pearl pondBOT
Available help channel!
lone wagon
pearl pondBOT
fierce totem
v=y/x?
looks quite troublesome
also you missed the minus sign
so basically you're wrong right from the beginning
this might be easier to manage
pearl pondBOT
@lone wagon Has your question been resolved?
fierce totem
ik
but he still missed the minus sign
vague plover
Could someone help me with this? I can't figure it out
pearl pondBOT
Could someone help me with this? I can't figure it out
wet swallow
Could someone help me with this? I can't figure it out
Have you learned about congruency
vague plover
Have you learned about congruency
Yes
vague plover
Yep
vague plover
Is a parallelogram
How would we know its a parallelogram? I'm not really sure about this part
wet swallow
How would we know its a parallelogram? I'm not really sure about this part
Opposite sides are parallel
Those are what the arrow heads mean
vague plover
Ohh okay
wet swallow
Look at ABC and ADC
vague plover
Look at ABC and ADC
AB = DC
wet swallow
And
wet swallow
You only need to state a pair of congruent triangles
vague plover
You only need to state **a pair** of congruent triangles
Ahh I thought the right angles would be used somewhere
.close
pearl pondBOT
wet swallow
Ahh I thought the right angles would be used somewhere
To find another congruent triangle
But you only need 1
vague plover
.reopen
pearl pondBOT
✅
vague plover
To find another congruent triangle
Could you guide me on the other one also
I just noticed CAB = ACD
wet swallow
Could someone help me with this? I can't figure it out
Rotate to MAKE C coincide with A
Type .reopen
.reopen
vague plover
.reopen
wet swallow
Ah ok nvm
frosty finch
ΣΣΒΟΥ
vague plover
Yeahh I already did it
vague plover
Rotate to MAKE C coincide with A
I think I see it
wet swallow
Rotate to MAKE C coincide with A
Then you will see All the congruent triangles
vague plover
Then you will see All the congruent triangles
How many are there together?
pearl pondBOT
pearl pondBOT
Available help channel!
scarlet trellis
Let y be a positive integer such that y=1.065x. If x=k/w where k and w are positive integers, find the smallest possible value of (k+w).
My solution is to graph x=1.065y and use table of values. I find the smallest sum at x=71 (well, technically y=71) which is 200/3. So my answer is 203. But, that is incorrect, the key states the answer as 201. What could I have done to solve this?
scarlet trellis
I could also use a program like guess and check here but what about mathematically?
light helm
convert 1.065 to a simplified improper fraction
scarlet trellis
y=(213/200)x
toxic lichen
so what's x in terms of y
scarlet trellis
x=(200/213)y
toxic lichen
right
so how might we minimize the sum of num+denom in the simplification of 200y/213
scarlet trellis
find a value of 200y that has a GCD with 213?
which could just be y=213
42600/213=200
so 200/1 = k/w and k+w=201
thanks
light helm
dunno why you'd multiply the 213 to the 200 like that
instead of just cancelling the 213 directly
but yes
pearl pondBOT
pearl pondBOT
Available help channel!
glass hare
pearl pondBOT
glass hare
I need help with this
We have to simplify it to a single positive exponent or whatever
light helm
what have you tried
light helm
can you show your work
hasty edge
$\frac{(2a^3b^2)^3}{(4ab^{-4})^2}=\frac{8a^9b^6}{16a^2b^{-8}}$
light helm
write work on paint, or online sketch tool
jolly parrotBOT
mathisfun
glass hare
$\frac{(2a^3b^2)^3}{(4ab^{-4})^2}=\frac{8a^9b^6}{16a^2b^{-8}}$
then you'd simplify from here correct?
so (1/2)a^7b^-2?
hasty edge
then you'd simplify from here correct?
Yes.
hasty edge
so (1/2)a^7b^-2?
Not necessarily. Note that we have a negative exponent in the denominator for b.
glass hare
I didn't learn what to do when that happens
My past 3 algebra teachers were horrible
😭
hasty edge
I didn't learn what to do when that happens
$\frac{a^m}{a^n}=a^{m-n}$
jolly parrotBOT
mathisfun
pearl pondBOT
jolly parrotBOT
waffle
spare lark
It seems good
proven orchid
Half correct seems no
Incorrect about having no absolute maximum, because the value at x=0 is 1, which is the highest value in the domain.
Unless I am mistaken
inland ivy
that might just be a printing error
spare lark
Well a maximum must be reached
spare lark
"f has an absolute max at x=0"
"... does not have an absolute maximum"
Yeah no sense
They wrong in the answer sheet
Its no abs max
pearl pondBOT
proven orchid
Am I tripping
pearl pondBOT
Available help channel!
proven orchid
But the function is defined at x=0…?
pearl pondBOT
But the function is defined at x=0…?
proven orchid
.close
cunning mauve
Hello ! I'm french I want to say sorry first if my english is bad ... 😅
When an object is thrown at an elevation angle (θ) at a speed v, in meters per second, the horizontal and vertical distances traveled by this object are respectively dh = v(t)cosθ and dv = v(t)sinθ - 4.9². A ball thrown at a speed of 16 m/s reaches a height of 10 m after 2 s.
A) At which angle was this ball thrown?
I try this :
cunning mauve
hearty snow
Is the answer arcsin(0.925) or approx 68 degrees??
cunning mauve
Is the answer arcsin(0.925) or approx 68 degrees??
yes
hearty snow
well all you gotta do is put the vertical distance equal to 10.
s = 10 = VsinA - 0.5gt^2
here VsinA will be your vertical velocity component and t is the time.
V = 16 as per the given information.
cunning mauve
well all you gotta do is put the vertical distance equal to 10.
s = 10 = VsinA ...
I get it ! Thanks, now that I see it, it feel like it was so obvious
.close
pearl pondBOT
cunning mauve
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cunning mauve
well all you gotta do is put the vertical distance equal to 10.
s = 10 = VsinA ...
I have another problem, if you could help me that would be great!
The paddle wheel of a ship has a diameter of 4.4m and rotates at a speed of 10 rpm counterclockwise. Its center is located 1.1m above the surface of the water. One of the blades of the paddle wheel is broken. When the wheel is started, the end of the broken blade is 2.2m above the surface of the water
- Determine the rule of the function representing this situation
the answer is f(x) = -2.2sin(π/3(x-2.5))+1.1
hearty snow
Sorry. Neither can I get this either. Physics really is not my cup of tea.
pearl pondBOT
@cunning mauve Has your question been resolved?
pearl pondBOT
@cunning mauve Has your question been resolved?
pearl pondBOT
@cunning mauve Has your question been resolved?
nocturne night
I have another problem, if you could help me that would be great!
The paddle wh...
What part ya need help with?
(Bit of a late response)
cunning mauve
Determine the rule of the function
plush bramble
<@&268886789983436800>
pearl pondBOT
@cunning mauve Has your question been resolved?
tiny lintel
I have another problem, if you could help me that would be great!
The paddle wh...
Partage le problème en FR stp
cunning mauve
Partage le problème en FR stp
La roue à aubes d'un navire a un diamètre de 4,4 m et tourne à une vitesse de 10 tr/min dans le sens inverse des aiguilles d'une montre. Son centre est situé à 1,1 m au-dessus de la surface de l'eau. Une des pales de la roue à aubes est cassée. Lorsque la roue est mise en marche, l'extrémité de la pale cassée se trouve à 2,2 m au-dessus de la surface de l'eau
- Déterminer la règle de la fonction représentant cette situation
tiny lintel
La roue à aubes d'un navire a un diamètre de 4,4 m et tourne à une vitesse de 10...
On ne connaît pas la position initiale de la pale cassée ? Quel angle fait elle avec la surface de l’eau à t=0 ?
cunning mauve
On ne connaît pas la position initiale de la pale cassée ? Quel angle fait elle ...
non
tu comprend pourquoi j'ai envie de me pêter le crane xD
La réponse nous démonte qu'on doit procéder avec une fonction sinus, mais je comprend pas comment on y arrive
tiny lintel
La réponse nous démonte qu'on doit procéder avec une fonction sinus, mais je com...
Tu peux faire ça, mais il te faut quand même l’angle initial
tiny lintel
La réponse nous démonte qu'on doit procéder avec une fonction sinus, mais je com...
Il y’a un schéma avec ce beau bordel ?
tiny lintel
Ok je vois le truc
L’amplitude de ta fonction est de 4,4. Dans la réponse
Ce qui veut dire que ta pale cassée n’a pas un rayon « diminué »
Donc tu peux considérer que la longueur de ta pale est 2,2
Maintenant, le fait qu’elle soit à 2,2 m au dessus de l’eau à t=0 te donne sa position initiale (considère qu’elle est du côté droit)
cunning mauve
ok
pearl pondBOT
@cunning mauve Has your question been resolved?
errant shoal
pearl pondBOT
errant shoal
is it okay to say the series is geometric series, that have a_n as 2/3^n-1
golden bramble
It is
golden bramble
Yes
errant shoal
then the formula for it would be a(1-r^n) over 1-r
and i would get 2(1-1/3^n-1) over 1-1/3
then simplify
golden bramble
Ye
fringe ice
Is the mark scheme wrong? Speed cannot be negative right?
pearl pondBOT
Is the mark scheme wrong? Speed cannot be negative right?
hushed heath
Is the mark scheme wrong? Speed cannot be negative right?
Seems to me like a printing error
Speed can indeed not be 0
umbral socket
I believe it negative when youre going backwards
pearl pondBOT
@fringe ice Has your question been resolved?
tropic herald
For part c, Is it mathematically correct and rigorous to take the expectation of both sides?
tropic herald
E(y)= y_bar for instance
pearl pondBOT
@tropic herald Has your question been resolved?
pearl pondBOT
@tropic herald Has your question been resolved?
obtuse tangle
For part c, Is it mathematically correct and rigorous to take the expectation of...
Sorry bro never did that yet but will in my 3rd yeat
chrome patio
For part c, Is it mathematically correct and rigorous to take the expectation of...
!showwork
pearl pondBOT
For part c, Is it mathematically correct and rigorous to take the expectation of...
Show your work, and if possible, explain where you are stuck.
tropic herald
Sorry I meant part b
tropic herald
!showwork
For part b
I want to take the expectation of both sides
But I don’t know if it is correct to do so
chrome patio
The "x̅" (resp. "y̅") means the "mean of observed x-values (resp. y-values)"
The expectation "E" can be regarded as a linear operator from the space of random variables to a field.
Given an equation involving random variables on both sides, then of course you can take expectation on both sides
pearl pondBOT
@tropic herald Has your question been resolved?
viral matrix
I've got this problem to find the integral functions of f(x) which have a minimum value of 2 when f(x)=x^2+(2/x^4)
viral matrix
I got the one where F(x)=x^2-(2/3x^3)+(1/3) when x<0
My book says that there's also x^2-(2/3x^3)+(1/3) when x>0 though
Which I don't get
,tex $f\left(x\right)=2x+\frac{2}{x^4}$
jolly parrotBOT
ZormaX
viral matrix
Here's the original function
Since it's not defined when x=0 it's got two integral functions right?
The one that's on the left side of 0 has a minimum at x=-1 but the one on the right doesn't
this is how it looks
The function has an integral function that is
$x^2-\frac{2}{3x^3}+C$
jolly parrotBOT
ZormaX
viral matrix
and I'm suppose to find the integral functions that have a minimum value of 2, but since the right side doesn't have a minimum value it's just said that it's
$x^2-\frac{2}{3x^3}+D$
jolly parrotBOT
ZormaX
viral matrix
So my question is why is it stated that that function has a minimum of 2 when x>0?
pearl pondBOT
@viral matrix Has your question been resolved?
steep pasture
After calculating the limit of F(x) for example, how can we determine the asymptote? When is it a Vertical asymptote, horizontal asymptote, and oblique asymptote?
hasty edge
After calculating the limit of F(x) for example, how can we determine the asympt...
Show the original problem.
steep pasture
After calculating the limit of F(x) for example, how can we determine the asympt...
When we have a function, and after calculating the limit on boundaries
steep pasture
Show the original problem.
Alright, one second
My handwriting is bad so I'll just write it
Function F(x) = (x-1)/(x+1)
Thus, x isn't equal to -1
After calculating the limit at boundaries:
Limit of F(x):
At -infinity its 1
At +infinity it's 1
At 1^- it's +infinity
At 1^+ it's -infinity
For the first 2, the asymptote turned out to be
Y=1 horizontal asymptote
For the last 2 it's x = -1 Vertical asymptote
But how did we determine that?
toxic lichen
Thus, x isn't equal to 1
dont you mean x ≠ -1 instead
it's the denominator that can't be 0.
hasty edge
For the first 2, the asymptote turned out to be
Y=1 horizontal asymptote
For th...
Yeah $\lim_{x\to\infty}\frac{x-1}{x+1}=1$
jolly parrotBOT
mathisfun
hasty edge
$x+1\neq 0$ when $x\neq -1$, so that was your mistake as ann pointed out
jolly parrotBOT
mathisfun
toxic lichen
yeah so vertical asymptotes $x=c$ happen precisely when $\lim_{x \to c^{\pm}} f(x) = \pm\infty$
jolly parrotBOT
ann.in.a.teacup
toxic lichen
where $x \to c^{\pm}$ is shorthand for referring to a limit that may be one-sided from either direction
jolly parrotBOT
ann.in.a.teacup
steep pasture
**ann.in.a.teacup**
Oh
Thanks!
So if the answer of the limit of F(x) when x -> 1 is infinity, it's a vertical asymptote
Where x = 1
hasty edge
Thanks!
So if the answer of the limit of F(x) when x -> 1 is infinity, it's a ve...
Yes
steep pasture
ooh alright
Soo for the horizontal asymptote, it's because the limit of F(x) at x-> infinity is a constant number ?
pearl pondBOT
@steep pasture Has your question been resolved?
proud frost
im kinda confused on what this question is asking for.... specifically we already know that R^d is compelte, so then why do we need to prove a seuqnce in R^d is ocnvergent iff it is cauchy?
proud frost
isnt that like the defintion of complete....
proud frost
but not 100% the definition
how so? isnt the definition if a sequence is cauchy then it convegres?
proud frost
but not 100% the definition
and it can be proved that if a sewqunece converges then it is cauchy
proud frost
and it can be proved that if a sewqunece converges then it is cauchy
for any general metric space
elfin stirrup
how so? isnt the definition if a sequence is cauchy then it convegres?
yes, but if you start with that you only get
For all Eps > 0, there Exists N, such that for all k >N |S_k - S |< eps
proud frost
yes, but if you start with that you only get
For all Eps > 0, there Exists N, s...
like if a sequenec converges?
elfin stirrup
I mean its not expressed as a Sum
proud frost
are you saying this is for proving convergent => cauhcy?
elfin stirrup
I mean completeness of R^d is that if a sequence is Cauchy then its convergent
but yeah the other direction can be shown
proud frost
like can i just shwo for arbiteayr metric space
proud frost
but yeah the other direction can be shown
let take e/2 > 0
elfin stirrup
also its quiet trivial
proud frost
then there exists N in natural such that for all n >= N d(a_n, L) < e/2
L is limit
then just tkae m, n > N
and then d(a_n, a_m) <= d(a_n, L) + d(a_m, L) < e
and we are done
like that would be all right?
elfin stirrup
maybe the 2nd statement is a bit less immediate
elfin stirrup
like that would be all right?
yes
proud frost
maybe the 2nd statement is a bit less immediate
ah okay. is it okay if i use the result that for R^d a suqene converges to R^d iff each coordiantewise sequence converges to R?
and then i think we are good
because then we know each coordiante wise sequecne is cauchy
then we take m = n
so then |a_n| = \sum_{k = n}^n a_k < e
so then each coordiante wise sequence converges to 0
thus the sequnece (a_n) in R^d must convegre to (0, 0 ,0....) = 0
elfin stirrup
yeah we only need to assume that R is complete, it then follows that R^d is also complete.
(We show that Normed Vector spaces are Banach spaces by reducing it to the known cases that R or C are complete)
proud frost
yeah we only need to assume that R is complete, it then follows that R^d is also...
i see. Would what i have be alr then? I think when they say lim a_n = 0, pretty srue they mean (0, 0, 0 ,0)
yeah or like sorry, I mean maximum N_i in N where for all n >= N_i ....
elfin stirrup
yeah define N as the maximum of all N_i
then every coordinate is smaller as epsilon
meaning the vector in R^d is close to the limit in terms of its metric
it will probably be epsilon*d or something like that
elfin stirrup
i see. Would what i have be alr then? I think when they say lim a_n = 0, pretty ...
yes that is what is meant with it
that the sequence converges to the zero vector in R^d
proud frost
yes that is what is meant with it
and the reason R, R^k is complete follows from Bolzano Weiestrass Theorem right
elfin stirrup
I mean the Completeness Axiom is that Every nonemepty set S⊂R that is bounded above has a least upper bound (supremum) in R.
Assuming this we can prove the Bolzano Weierstrass Theorem. (In R)
This gives us that every Cauchy Sequence in R converges.
Now if we consider a Cauchy Sequence in R^d then its a Cauchy sequence in R in all the d components, therefore every component converges (in R).
And therefore the vector in R^d converges.
But it is not immediately obvious, why this should be true by just stating the Completeness Axiom of the Real numbers.
But the completeness in R or R^d is often just seen as every Cauchy sequence convergences
there are many equivalent ways of stating completeness in R
proud frost
I mean the Completeness Axiom is that Every nonemepty set S⊂R that is bounded ab...
ahi see hmm
pearl pondBOT
@proud frost Has your question been resolved?
pearl pondBOT
pearl pondBOT
Available help channel!
quartz yoke
pearl pondBOT
quartz yoke
i got the right answer, but im not sure if i did it the right way
so i know one root is 0
because we can factor
that leaves 4
and i know the other roots will be irrational
so they have to be conjugates
so i just figured that it would be -5 (the opposite) and assumed that the irrationals would cancel out
nevermind im just stupid
.close
pearl pondBOT
quartz yoke
.reopen
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quartz yoke
wait a minute
so then like the sum has to be 0
so why is the other root -5
when the sum of the irrational roots could be completely other than 0?
pearl pondBOT
@quartz yoke Has your question been resolved?
quartz yoke
.close
pearl pondBOT
pearl pondBOT
Available help channel!
hollow grotto
largest local end
smallest local end
increased duration
decreased duration
https://cdn.discordapp.com/attachments/1265983124634996838/1341789322629873694/image.png?ex=67b7462f&is=67b5f4af&hm=c64652347f8133211d00c45b19360e38ac69cf27cee4c2598a7d647538aaa0dd&
quick star
largest local end
smallest local end
increased duration
decreased duration
https...
i'm guessing this is translated
anyway
did u get started with this?
sharp smelt
Trying to find the partial derivatives of this function
pearl pondBOT
Trying to find the partial derivatives of this function
sharp smelt
$\pdv{w}{x} = \frac{1}{x+2y+3z}; \pdv{w}{y} =\frac{2}{ln(x+2y+3z} ; \pdv{w}{z} = \frac{3}{x+2y+3z}$
jolly parrotBOT
What a wonderful world it is !
finite thistle
Problem:
Let f(x) = e^x and g(x) = \sin(x) . Find the value of the following integral:
I = \int_0^\infty \frac{e^x \sin(x)}{x^2 + 1} , dx
jolly parrotBOT
What a wonderful world it is !
sharp smelt
is this right?
summer gorge
Trying to find the partial derivatives of this
do you know what x,y are in terms of t?
jolly parrotBOT
What a wonderful world it is !
summer gorge
$2xcos(x^2)$
why is it 2x\cos{(x^2)} and not 2x \cos{(x^2)} \farc{\partial x}{\partial t}- 2y \cos{(y^2)} \farc{\partial y}{\partial t} ?
jolly parrotBOT
What a wonderful world it is !
summer gorge
ok then what will you do?
summer gorge
oh I get it you are differentiating with respect to x ,I thought it was wrt to some other variable
midnight haven
pearl pondBOT
midnight haven
So a Linear Transformation function isnt always onto ?
toxic lichen
indeed
easy example of a linear map that is nowhere near being onto:
T : R -> R^3 given by T(x) = (x, 0, 0)
pearl pondBOT
toxic lichen
well your assumption was wrong then
midnight haven
yeah lol
pearl pondBOT
Available help channel!
midnight haven
pearl pondBOT
midnight haven
T is a linear transformation
and ker(T) would be
any matrix in M_2(R)
such that a+2b=0 and c+2d=0
so
would i write it was
{A belongs to M_2(R) | a+2b=0 and c+2d=0}
?
toxic lichen
any matrix in M_2(R)
not "any matrix" but "the set of all matrices".
midnight haven
okay
midnight haven
{A belongs to M_2(R) | a+2b=0 and c+2d=0}
this is correct ?
toxic lichen
yes i suppose
neon herald
Hello i need help because I am trying to get out of an escape room but like on google forms i am in algebra and i am learning system of equations and a little matrix thingys
neon herald
you also say the answers like 1234 but instead of the number its the letter answer you got
im also stuck
he also said to show work but i dont know how too
sorry if im retarted
pearl pondBOT
@neon herald Has your question been resolved?
bleak atlas
whats the question
neon herald
its up there
bleak atlas
these?
neon herald
ys
neon herald
these?
yes
im not getting my help am i
<@&286206848099549185>
its been a hour or somethin
neon herald
can you help no
w
ok bro
😭
alright bruh
<@&286206848099549185>
bro
i waited 1 hour
bronze veldt
I have answer
neon herald
but its not 9th grade math
bleak atlas
alright bruh
shit im so osrray
gimme a min
so the rule of thumb i use for finding points is, Along the corridor, Up/down the stairs
so the first coordinatre you find by going along the 'corridor' (x axis}
and then you go up/down the 'stairs' (y axis)
@neon herald
neon herald
so how do i input that into letters
neon herald
but like your supposed to get an answer with 4 words
bleak atlas
youre not supposed to answer with 4 words?
youre supposed to answer with coordinates
neon herald
my tecaher said this: Input your letter answers for questions 1234 as ABCD, changing the letters for the correct ordered pairs. Be sure your responses are capitalized and there are no spaces!
so you would get the answer for answer
but i dont know how to do
how to get the answer
neon herald
america
so your supposed to get like abcd as the answer depending on the answers
Input your letter answers for questions 1234 as ABCD, changing the letters for the correct ordered pairs. Be sure your responses are capitalized and there are no spaces!
wrong thing
neon herald
yes
bleak atlas
using the ABCD etcetc as the answers
neon herald
yes
theres letter on the answers and i keep on getting wrong even though i think its right
neon herald
g
neon herald
bruh
cunning swallow
neon herald
wow so i just got number one wrong
pearl pondBOT
Helpers are just people volunteering their time to help you. Be polite and patient.
pearl pondBOT
Helpers are just people volunteering their time to help you. Be polite and patient.
bleak atlas
what grade u in boss
neon herald
my teachers smarter tho
pearl pondBOT
pearl pondBOT
Available help channel!
solar vault
Start is 0 and its inclusive
Stop is 5 and its exclusive
Step is 2
Is there an math solution to find how many times i can step?
open rivet
yeah, its (end-start)/stepsize
and then round up
if you land on a whole number and its inclusive, add 1
solar vault
sorry if i ask this question but im from programming world and sometimes round up mean trunicate the end, but i hope you mean 1.5 -> 2
open rivet
round up means you go to the lowest integer thats higher than the number
crystal lion
if you land on a whole number and its inclusive, add 1
In that case floor it and + 1 is easier to write
open rivet
or that lol
remove the decimals and do +1
actually, thats better, cuz it already includes the edge case
solar vault
whats floor it?
jolly parrotBOT
Bonk
open rivet
$\ceil{x}$
jolly parrotBOT
Bonk
open rivet
ever seen this?
solar vault
im not from math world and my math skill very limited
open rivet
okay, so you have a number
92374982734.89723459213 or whatever
just gibberish
you remove everything after the decimal
92374982734
is what you get
then add 1
crystal lion
Unless its a negative
open rivet
92374982735
ye okay, but ive ignored negatives for now
cuz i assume that the end is larger than the start
solar vault
i do have a case when step is negative and stop value can be -1
solar vault
Is there like an equation where i could read more in depth and not waste your time? For now i understand everything
open rivet
its simply the length of the itnerval
and you divide that interval into your step size
if your step size is negative
you go the other way around instead
solar vault
Right, thank you.
What i have to do to close this post?
lunar osprey
solar vault
.close
stiff mauve
2b?
stiff mauve
im stuck on this part
the limit has to equal zero but idk how to continue after this
i wrote x^2 + y^2 + (z-1)^2 as the norm off x,y, z-1 to the power of 4
so| | x, y , z-1| |^4
and then sqrt at the bottom i put it as | | x, y , z-1| |
open rivet
why do you divide by sqrt(stuff)
stiff mauve
open rivet
isnt showing $\lim_{(x,y,z)\to(0,0,1)}\frac{x^2y^2}{x^2+y^2+(z-1)^2}+5z=5$ enough?
jolly parrotBOT
Bonk
open rivet
ah i see what youre doing now
then just show $\lim_{(x,y,z)\to(0,0,1)}\frac{x^2y^2}{(x^2+y^2+(z-1)^2)^{\frac32}}=0$
jolly parrotBOT
Bonk
open rivet
you can use $x\leq\sqrt{x^2+y^2+(z-1)^2}$
jolly parrotBOT
Bonk
stiff mauve
was what i did with the norm wrong
stiff mauve
**Bonk**
so with this we basically have (stuff)^2 / stuff^1.5
which is sqrt(stuff)
and if x goes to 0 0 1 sqrt stuff goes to zero
but why can i not use the norm
or did i just mess up somehwere
open rivet
so with this we basically have (stuff)^2 / stuff^1.5
yup
open rivet
but why can i not use the norm
you did use it
right here
||(x,y,z)-(0,0,1)|| is that
open rivet
this looks prime for polar coordinates
stiff mauve
how did we find the bounds 5pi/4 and pi/4
stiff mauve
this looks prime for polar coordinates
it sure does
open rivet
do you know what the area G looks like?
stiff mauve
prolly a circle then no>
