now draw a line connecting these two vectors
#help-13
pastel vault
yes, if you draw it out, you can find the angle between the two planes
abstract locust
and then look at your cartesian plane
crimson sedge
okay i drew it out
what would come after?
calculation wise to find the distance
i assume i have to use the distance formula at some point
pastel vault
what would come after?
so then, you need to work out the angles
pastel vault
i assume i have to use the distance formula at some point
no the approach we're using is the cosine rule
distance formula also works, I just think it's a bit messy for this question
pastel vault
so then, you need to work out the angles
as in:
- how many degrees is N10W from north?
- how many degrees is S71E from east?
crimson sedge
Well it would be 10 degrees away from north and 19 degrees away from east if im not mistaken
crimson sedge
distance formula also works, I just think it's a bit messy for this question
hmm i see
pastel vault
Well it would be 10 degrees away from north and 19 degrees away from east if im ...
correct!
great from your drawing, the angle between the two planes is 10 + 19 + 90 = 119
now you just need to use the cosine rule
the opposite side to this angle will be the distance between the two planes
crimson sedge
oh oh okay so using the information we just found
I would just plug it all into law of cosines formula and whatever I get would be the answer?
and after that it continues on by asking
how would I finish up this part 2
abstract locust
It’s the angle between vector 1 and 2
abstract locust
Yes
pastel vault
if you wanted to express that, it would be E29S
oh wait it's not 119 cause the 1st plane is the S71E sorry
so it's 360 - 119 = 241 instead
cause the clockwise angle is 360 - 119
bearings are measured clockwise
pastel vault
so the bearing is 241 degrees not 29 degrees?
yes
no 241 degrees would be in the southwest quadrant
pastel vault
s241w
no
south is already 180 degrees
btw this video might help you with the notation
the problems should be just revision
pastel vault
hmmmmm so how would I write it out then if its not that?
so would be S61W
crimson sedge
https://www.youtube.com/watch?v=cSKAqfGXOPI
thanks alot! ill make sure to watch this
pastel vault
go to south, then move 61 degrees towards west
pastel vault
thanks alot! ill make sure to watch this
yeah well the video wouldn't be too useful for you
the thumbnail basically tells you the concept already
but it's good revision if you want it
cedar kilnBOT
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livid hound
This correct?
cedar kilnBOT
This correct?
solemn quail
If you replace 2 with b, it's much better
spare kettle
you should use the definition of the derivative of the inverse of a function
if you know
what it is
livid hound
you should use the definition of the derivative of the inverse of a function
ohh okok that's the 1/f'(f^-1(x)) ??
In that case, would I first need to rearrange (y-2)^2 = x so that y is isolated to one side of the equation beforeee using the derivative of inverse functions?
spare kettle
you can just do it by putting values in definition
livid hound
I'm multitasking between two questions right now and just realized I was referring to a completely different one mb 😭
livid hound
you can just do it by putting values in definition
ok yeaa i get that tyyy
livid hound
.close
barren panther
cedar kilnBOT
barren panther
Unsure why that sent as an image but there we go...
Hi. I’ve been asked to show whether or not the elements 3, 9, 21 and 27 form a group under multiplication mod 60. I thought I had the answer but when I checked it on ChatGPT it said that there is no identity element for the set so it is not a group. Why is 21 not the identity element, as it’s one of the original elements in the set and: 3 x 21 = 3 mod 60, 9 x 21 = 9 mod 60, 21 x 21 = 21 mod 60, and 27 x 21 = 27 mod 60?
rare vault
chatgpt doesn't know math, it's not a checking software
assuming your math is correct then it is the identity element
cedar kilnBOT
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barren panther
.reopen
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barren panther
I just want to make sure
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raven shard
I just want to make sure
it is a group
cedar kilnBOT
barren panther
Ta
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wispy veldt
Not much of a math question , but a latex one although its still related to math in some way.When working with matrixes in Latex:
I wanted to make this look nicer, the matrixes are unaligned and the operations between them make the whole thing look quite bad, is there something i can do to make it look prettier? I've tried using chatgpt but i wasn't satisfied with the results.
idle tusk
Not much of a math question , but a latex one although its still related to math...
try asking in #latex-help
cedar kilnBOT
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quick tree
just had an math exam. couldn`t solve this one:
needed to find high or low points. teacher said there actually is one.
quick tree
$e^x+e^-x$
wraith daggerBOT
ir0nfly
idle tusk
$e^x+e^{-x}$
wraith daggerBOT
artemetra
idle tusk
just had an math exam. couldn`t solve this one:
needed to find high or low point...
have you tried differentiating it?
wraith daggerBOT
ir0nfly
idle tusk
yep
quick tree
right?
idle tusk
when does $e^x - e^{-x} = 0$?
wraith daggerBOT
artemetra
quick tree
e is never zero
idle tusk
that's true
idle tusk
**artemetra**
but this is solvable
quick tree
so how do i find where the derivation is 0
idle tusk
$\iff e^x = e^{-x}$
wraith daggerBOT
artemetra
idle tusk
is there an x that satisfies this?
quick tree
i got this far in the exam. but what is x?
wraith daggerBOT
artemetra
quick tree
x= -x ?
idle tusk
yep
quick tree
teacher said solution is a number
idle tusk
what number is unaffected when you put a minus sign in front?
quick tree
0?
idle tusk
no problem, if you are done type ".close"
quick tree
.close
cedar kilnBOT
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arctic gull
Dhey deposits 5000 pesos at the end of each month into a retirement account that earns 4.5% interest conpounded monthly. How much will she have in the account after 35 years
rocky fern
Take the amount of months in 35 years and use the compound interest equation
pastel vault
and also it should be 4.5% interest annually
so the monthly interest rate is 4.5/12 %
rocky fern
Could be monthly
pastel vault
so the monthly interest rate is 4.5/12 %
you need this one
pastel vault
Could be monthly
pretty unrealistic
pastel vault
yeah well blame the question for not specifying I guess
rocky fern
Take the amount of months in 35 years and use the compound interest equation
also this is wrong because you’re depositing continuously
use a compound interest equation with monthly contributions
pastel vault
fv= p x (1+r)^n-1/r
one of the terms is $p \cdot (1 + r)^n$ for the right $r, n$
interest rate is monthly
n is number of months
oh wait
yeah you have to geometric series this
pastel vault
one of the terms is $p \cdot (1 + r)^n$ for the right $r, n$
it's the sum of this from n = 0 to n = 35 * 12
pastel vault
fv= p x (1+r)^n-1/r
I think the denominator should hence be 1 - (1 + r) = -r
wraith daggerBOT
south, just south
pastel vault
it's the sum of this from n = 0 to n = 35 * 12
yeah instead of making one big formula
just find the first term of this and the common ratio, then sub into the geometric sum formula
rocky fern
$5000(1 + \frac{.045}{12})^{12\cdot 35}\cdot \frac{(1 + \frac{.045}{12})^ {12\cdot 35}-1}{\frac{.045}{12}}$
oh
$5000(1 + \frac{.045}{12})^{12\cdot 35}\cdot 5000\frac{(1 + \frac{.045}{12})^ {12\cdot 35}-1}{\frac{.045}{12}}$
wraith daggerBOT
5ky13rr
rocky fern
latex on the phone not too easy
Subtract 5k pesos from the total if you are starting from zero, if starting from 5k then it’s fine
cedar kilnBOT
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paper edge
So, I thought about tackling a proof of the existence of a basis for any finitely generated vector space(so far, I have no interest on the infinitely generated case)
Have 2 questions, first, can I assume, without proof, that there exists a finite generating set for the space?
And second, my idea was to prove the fact that any generating set contains a basis, and eventually reduce the aforementioned set to a basis of the space, but not sure whether this falls or not on circular reasoning
paper edge
<@&286206848099549185>
(if I Can't get any help this time either I'll probs try to write it in another channel)
And thanks in advance
cedar kilnBOT
@paper edge Has your question been resolved?
lofty topaz
Could you give us an example, probably a visualization if possible.
paper edge
Could you give us an example, probably a visualization if possible.
It's a proof, not sure how I could give an example for you to visualize it
The proposition I'm trying to prove is :“for all finitely generated vector spaces there exists a set of vectors B such that B is a basis of the space“
The rest of my message was me asking whether my reasoning was valid
cedar kilnBOT
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snow quartz
ik its a little basic, but how do we do this one?\
cedar kilnBOT
ik its a little basic, but how do we do this one?\
snow quartz
anyone?
cedar kilnBOT
@snow quartz Has your question been resolved?
snow quartz
guys, anyone?
queen stirrup
have you found the height of the peak through a formula other than this?
snow quartz
no
queen stirrup
draw a diagram and try to find as many relations between the variables as possible
snow quartz
alr
queen stirrup
are you done?
cedar kilnBOT
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paper stone
im not sure what im doing wrong
cedar kilnBOT
im not sure what im doing wrong
paper stone
this is where i am at rn
we dont have any examples to help with this practice problem so im struggling
my approach is basically
if we have an orthonormal basis it is then an inner product space isomorphism
so i start with the standard basis convert it
but when i try prove it they just dont equal each other
some stuff from my course notes im using to help
im a bit stumped and with no example questions dont know if my entire approach is wrong or just some part of it
<@&286206848099549185>
cedar kilnBOT
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paper stone
naur😭
royal finch
Your problem is that if you use example 2, if you apply that L to your two basis elements in P and then take the inner product in L, you see that you don't have an orthonormal set
You need to define L so that it takes an orthonormal set to another orthonormal set
cedar kilnBOT
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night vale
Hello
night vale
Need help with this function
How am i meant to find what it represents
I thought it’s y=tanx but it doesn’t seem right
dull oxide
red almost looks like arctan, but slightly shifted. You should play with graphs on desmos or geogebra to get some ideas
night vale
It’s arctan but like
The lower part is shifted up
And the upper part is down
Here’s the other points if it helps
This what i have rn but there’s like a lil pause in between the left part
And the last arctan
<@&286206848099549185>
cedar kilnBOT
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oak meadow
that looks more like arccos to me
cedar kilnBOT
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obtuse sparrow
could someone explain why the sign on the resident theorem changes if you take a contour with the opposite direction?
obtuse sparrow
warning: incredibly new to complex analysis
learning this in passing for an E&M course
dire geode
warm ravine
I'm pretty sure this thread is already occupied lol
shut magnet
of oops
cedar kilnBOT
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obtuse sparrow
I don't understand the way these like-terms are combined. Can someone explain?
obtuse sparrow
nevermind
plucky owl
I don't understand the way these like-terms are combined. Can someone explain?
Commutative property
ab = ba
obtuse sparrow
sinxcos2x were flipped
plucky owl
So cos^2xsinx = sinxcos^2x
obtuse sparrow
yeah i just saw that but thank you'
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remote glade
need help with this
cedar kilnBOT
need help with this
remote glade
need help with this
<@&286206848099549185> shouldn't be too complicated, I'd like some help please
.close
cedar kilnBOT
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topaz sundial
A hemisphere of fixed radius r has its great circle = boundary resting flat on the ground. A
second, smaller hemisphere is placed above, with its great circle resting horizontally on the first hemisphere. What is the maximum height about the ground for the top of the second hemisphere?
topaz sundial
I'm not good at geometry and I don't get how to do it at all, this is completely different from what we have went over in class.
remote blade
I'm not good at geometry and I don't get how to do it at all, this is completely...
"Great circle" just means a circle that goes across a sphere's surface that maximises the length around the surface. It's basically your 3 dimensional equivalent to the diameter
so like, if your sphere has radius r, the great circle would be a circle on the sphere surface that has a length going around of 2*pi*r
Here the great circle is just where you chop your ball in half to get the hemisphere (half sphere, the sphere equivalent of a semicircle)
a diagram is always a good idea for geometry. what it's asking for should be a lot clearer after that
topaz sundial
This is what I have so far, am i on the right track?
remote blade
yeah pretty much bang on
topaz sundial
how do i find the max height just from this diagram? is it essentially the radius of the big hemisphere + radius of the smaller hemisphere?
remote blade
I would think so. The question is worded confusingly. My interpretation would be you're sliding the smaller one about on the surface of the bigger one and wanting to find how high the top of the smaller hemisphere can get
which just happens when it's directly on top of the other one
cedar kilnBOT
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whole wharf
How is this wrong???
cedar kilnBOT
How is this wrong???
whole wharf
.close
cedar kilnBOT
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undone onyx
why are the two waves in phase again when the first wave goes through exactly one more cycle?
cedar kilnBOT
why are the two waves in phase again when the first wave goes through exactly on...
@undone onyx Has your question been resolved?
earnest roost
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@undone onyx Has your question been resolved?
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@undone onyx Has your question been resolved?
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hollow trail
x_1,...x_k are not required to be integers
sharp veldt
hollow trail
there have to be an integer number of them, but you can have something like 1/2, 1, 3/2, etc
hollow trail
a discrete distribution has a finite number of inputs. in the first case those inputs can potentially be any number, whereas for the second case those inputs are only positive integers
feral jasper
i need help with this
cedar kilnBOT
i need help with this
livid hound
what have you tried?
cedar kilnBOT
@feral jasper Has your question been resolved?
feral jasper
what have you tried?
nothing rlly i tried finding <utv butcouldnt i have no clue
livid hound
first goal would be to find <VUT
(there's a theorem for that)
cedar kilnBOT
feral jasper
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feral jasper
first goal would be to find <VUT
(there's a theorem for that)
i dont think ive been taught it
livid hound
look up alternate segment theorem
feral jasper
so VUT= 72 deg?
livid hound
yes
cedar kilnBOT
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dark escarp
hey guys. can anyone help me? i want to learn how to solve each one so i can prepare for examss T_T
drifting marlin
dark escarp
Ive only written #1 but its only the given and requireddd
drifting marlin
and what do you know about simple interest?
dark escarp
i = Prt, since time it would be t = i/pr
but then i searched the answer online and they used future value tooo
drifting marlin
you aren't given i in this problem, so that won't help you
dark escarp
i will have to get I first?
drifting marlin
sure
drifting marlin
well what, conceptually, does the future value represent?
dark escarp
50,000?
drifting marlin
i didn't ask for a number
the future value is related to the principal in some way
dark escarp
Its the sum of principal and interest
drifting marlin
indeed it is
drifting marlin
Its the sum of principal and interest
is that answer consistent with this?
drifting marlin
yes
dark escarp
okayy thank youu
drifting marlin
why don't we add one more tool to your kit to make future problems easier
drifting marlin
Its the sum of principal and interest
please write out the equation that corresponds with this information
dark escarp
Tysmm omg
dark escarp
please write out the equation that corresponds with this information
F = P + I
dark escarp
I is interesttt
drifting marlin
indeed, and what is the equation used to compute it?
dark escarp
I = PRT ^^
drifting marlin
so F = P + Prt
do a little algebra to the right side of that equation and simplify it a little, would you?
dark escarp
do a little algebra to the right side of that equation and simplify it a little,...
I dont know what u mean sorryy 😭
drifting marlin
factor
dark escarp
f = p(1+rt)? this was in one of my notezz
drifting marlin
well then why did you go through I first?
you're given F, P, r and asked to find t
cedar kilnBOT
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dense merlin
Help yo
dense merlin
.
cedar kilnBOT
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gritty viper
!status
cedar kilnBOT
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
tight stag
is it correct answer?
cedar kilnBOT
is it correct answer?
snow quartz
let me check, wait
snow quartz
the n cancels out during simplification no?
snow quartz
take n squared as common
3/n and 3/n^2 will vanish as n-> infinity(if u take the limit as that)
snow quartz
no, +3/n and +3/n squared will both vanish, leaving -5
steel fractal
the answer is right….
steel fractal
no, +3/n and +3/n squared will both vanish, leaving -5
there is the n that you took out as common
snow quartz
sure
lime tartan
yes it is
I don’t even think you need to solve. The second term will grow faster than the first so always trend - no?
steel fractal
I don’t even think you need to solve. The second term will grow faster than the ...
yea.. i didnt solve too haha
lime tartan
-inf was my guess on sniff test
steel fractal
its 1 power inf models
lime tartan
Think about this one, if n is growing to infinity and you have a negative exponent what would happen to the whole thing
tight stag
Think about this one, if n is growing to infinity and you have a negative expone...
equal to 0 in inf?
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ornate rune
Can anyone give me hint on how do I get the derivative of sin using limit definition of derivative
summer otter
Can anyone give me hint on how do I get the derivative of sin using limit defini...
$\lim_{h \to 0}\frac{\sin {(x+h)}-\sin{(x)}}{h}$
wraith daggerBOT
convergence
crimson sedge
Got sniped by convergence
summer otter
apply the sin(a+b) formula
crimson sedge
And at some point you will have to use
$$\lim_{x\to 0} \frac{\sin x}{x} = 1$$
wraith daggerBOT
Edmund Cloudsley
crimson sedge
This can be determined by the good old 🏥 rule
crimson delta
thats circular
for lhopital you need the derivative of sin but thats the thing you want to compute in the first place
steel fractal
Wait..
crimson delta
you want to use the squeeze theorem
summer otter
yes
crimson sedge
for lhopital you need the derivative of sin but thats the thing you want to comp...
Ohhhh right
Correct
My apologies. Thanks for the correction.
twilit escarp
Maybe the cos(a+b) formula be useful for this ?
earnest thistle
idk why but i thought of writing sin as a taylor series
summer otter
idk why but i thought of writing sin as a taylor series
again that will be circular but i think you know why
earnest thistle
again that will be circular but i think you know why
that'd be hyperbolical lol
blazing zephyr
**Edmund Cloudsley**
the proof if you want it
obtuse mango
again that will be circular but i think you know why
tbf it will depend on how u've defined sin & cos
like at my uni, we just defined it as (exp(ix) + exp(-ix)) / 2 etc.
and you define the exponential as a power series anyway so there's no issue
cold timber
obtuse mango
but yeah you probably don't wanna write sine as a taylor series, there are easier ways
cold timber
The first one
earnest thistle
but yeah you probably don't wanna write sine as a taylor series, there are easie...
yeah
summer otter
like at my uni, we just defined it as (exp(ix) + exp(-ix)) / 2 etc.
i mean if you that definition sure it will be just very long to prove it
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grim umbra
How to solve this by only using quadratic mean, arithmetic mean, geometric mean and harmonic mean:
rad(15)+rad(24)+rad(35)<rad(226)
cedar kilnBOT
How to solve this by only using quadratic mean, arithmetic mean, geometric mean ...
@grim umbra Has your question been resolved?
grim umbra
How to solve this by only using quadratic mean, arithmetic mean, geometric mean ...
<@&286206848099549185>
cedar kilnBOT
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potent fractal
How to solve this by only using quadratic mean, arithmetic mean, geometric mean ...
$\frac{x_1 + \dots + x_n}{n} \le \sqrt{\frac{x_1^2+\dots+x_n^2}{n}}$
wraith daggerBOT
EQUENOS
potent fractal
Consequently, $x_1+\dots+x_n \le \sqrt{n}\sqrt{x_1^2+\dots+x_n^2}$
wraith daggerBOT
EQUENOS
potent fractal
In your case $n=3$ and $x_1=\sqrt{15}$, $x_2=\sqrt{24}$, $x_3=\sqrt{35}$
wraith daggerBOT
EQUENOS
potent fractal
Should be easy to continue from here
spare citrus
Graph theory Question, what kinda matching am I looking for here
cedar kilnBOT
Graph theory Question, what kinda matching am I looking for here
@spare citrus Has your question been resolved?
spare citrus
<@&286206848099549185>
spare citrus
so you got it?
spare citrus
| Uiel Ail >= Σ di
sry i wasn't paying attention to notifications
swift badger
Yeah
swift badger
Tell what?
swift badger
Alright
So what I did
I constructed a bipartite graph
where a perfect matching
Would correspond
spare citrus
is that not just the hint
spare citrus
Are you just trolling?
spare citrus
Alright anyone else?
cedar kilnBOT
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@spare citrus Has your question been resolved?
severe elm
when finding a recursive formula for sequences where the differences increases by a set amount each time (i.e -3, -1, 2, 6, 11, 17...) what would be the general setup for the formula
severe elm
the diff between -3 and -1 is +2, between -1 and 2 is +3, 2 and 6 is +4, etc
<@&286206848099549185>
merry solstice
when finding a recursive formula for sequences where the differences increases b...
hi
when you have an equation where there is a set amount of increase
we can make the start value d
and the equation increases
so we can make the increase n
severe elm
the increase is changing each time tho
merry solstice
yeah
severe elm
should the increase be (n+_)
merry solstice
so A(n)=A(n-1)+d
basically
it's saying
for the nexxt term, take the previous term and add d
severe elm
for the nexxt term, take the previous term and add d
no?
merry solstice
the difference is increasing by one
ohh
Then it would be
A(n)=A(n-1)+n-d
D depends on the starting value
severe elm
like this
merry solstice
well d is the constant
and the equation for your sequence is N^2+N-d
if you don't want to write it as recursive
severe elm
if you don't want to write it as recursive
i want it as recursive
merry solstice
i want it as recursive
ok
then put those two equations down
and see if they're equal
if they are, then this is equal
if they aren't lmk
severe elm
well d is the constant
so for my sequence it would be a_n = a-(n-1) + n + 1
merry solstice
so for my sequence it would be a_n = a-(n-1) + n + 1
a*(n-1)
severe elm
a_{n}=a_{\left(n-1\right)}+n-1
merry solstice
but yes
merry solstice
do they match?
merry solstice
huh
merry solstice
well i need to write in recursive so it would be a sub n-1
The sequence is quadratic
so
it's the previous term +n
cedar kilnBOT
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severe elm
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feral basalt
Could someone please help me with the attached problem? I'll send my work for it in a minute.
ancient lodge
Your vector is in the null space, but the basis has more than one vector
ngl you're better off just solving it parametrically and copying off the basis vectors from there
feral basalt
I do not understand this conceptually. Specifically why there is a t* [1 0 1] in the solution attached. Could you please explain why?
ancient lodge
I do not understand this conceptually. Specifically why there is a t* [1 0 1] in...
$\vec{x}=\begin{bmatrix} -s \ s \ 0 \end{bmatrix}+\begin{bmatrix} t \ 0 \ t \end{bmatrix}=s \begin{bmatrix} -1 \ 1 \ 0 \end{bmatrix}+t \begin{bmatrix} 1 \ 0 \ 1 \end{bmatrix}$
wraith daggerBOT
Civil Service Pigeon
feral basalt
Could you please remind me about how to get parametric form from an RREF?
ancient lodge
Could you please remind me about how to get parametric form from an RREF?
The algebra is explained from "The variables x2 and x3 are free..."
feral basalt
When is a column a pivot column? Is it when the leading entry is 1 in the row and has 0's above and below it?
ancient lodge
Column in which a pivot appears (pivots are the first nonzero entry in a row)
cedar kilnBOT
@feral basalt Has your question been resolved?
ancient lodge
Google the definition of RREF - it doesn’t necessarily have to be a “strict staircase”
feral basalt
Yeah you're right
According to an AI overview
It meets 2 conitions off the bat
All zero rows are below the non zero row
And the leading entry of the first row is 1
cedar kilnBOT
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clever kayak
The question is: During 2020, the membership count increased by a certain amount, by how many?
So, what I know is that the first created members account in 2020 was #347397 at the start of the year, and at the end, the last created was #354682.
The difference between those two numbers is 7285, but do I also include, in the count, both the first created user and the last?
I'm thinking the first, so it becomes 7286.
merry solstice
The question is: During 2020, the membership count increased by a certain amount...
yes
clever kayak
So it's 7286 more members during 2020?
clever kayak
Yeah, I'm thinking that from if I'm 1 in a queue and the queue is 1000 long. 1000-1 = 999, so it has to be one more
merry solstice
Yeah, I'm thinking that from if I'm 1 in a queue and the queue is 1000 long. 100...
oh
clever kayak
to make it 1000, the correct amount
merry solstice
well we can use less number
like if we started with 2 memebers
and ended with 5
how many members did we gain
clever kayak
The question would more similiar if we say
That the first member of that time is #3
and the last one is #5
merry solstice
then the increase is 3
merry solstice
remember we gained the first
clever kayak
ty
merry solstice
yeah
clever kayak
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cedar kilnBOT
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compact knoll
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slate lintel
what are your eigenvectors?
slate lintel
those are either the rows of P or the columns of P\inv, i don't remember which haha
compact knoll
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cedar kilnBOT
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ripe maple
cedar kilnBOT
ripe maple
how do I find the endpoints of this sum?
cedar kilnBOT
@ripe maple Has your question been resolved?
eager sapphire
what do you mean endpoints? do you mean the lower and upper bounds of the resulting integral?
ripe maple
what do you mean endpoints? do you mean the lower and upper bounds of the result...
yeah
im really confused on how to get them
eager sapphire
im really confused on how to get them
so id first look at the form of the riemann integral,
$\int_a^b f(x) ; \text{d}x = \lim_{n \to \infty} \sum_{i=1}^n \Delta x f\left(a + i \cdot \Delta x\right)$
wraith daggerBOT
AlphaNull
eager sapphire
where $\Delta x = \frac{b-a}{n}$
wraith daggerBOT
AlphaNull
eager sapphire
so we can see that $b-a = 2$
wraith daggerBOT
AlphaNull
eager sapphire
are you with me so far?
ripe maple
yeah
eager sapphire
then we can look at $f\left(a + i \cdot \frac{2}{n}\right) = \left(\frac{3i}{n}\right)^3 + \frac{5i}{n} + 2$
wraith daggerBOT
AlphaNull
eager sapphire
its a bit arbitrary where to start a at, but in this case i'd just choose 2 because thats the constant term of the function
ripe maple
So would a = 2?
eager sapphire
so just $x = a + i \cdot \frac{2}{n}$, solve for $i$ in terms of $x$ and plug that in
wraith daggerBOT
AlphaNull
ripe maple
**AlphaNull**
And i would sub a=2 into this right
eager sapphire
yes
eager sapphire
solve for i in terms of x, i = something involving x
eager sapphire
**AlphaNull**
now plug in both sides of this
ripe maple
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cedar kilnBOT
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molten flame
Im a little confused on how to answer this question
cedar kilnBOT
Im a little confused on how to answer this question
quiet plover
Im a little confused on how to answer this question
integral of a function is basically the "algebraic sum" of the area under the grapgh
meaning you add or subtract the area depending on whether it is above or below the graph
molten flame
ok so i get that part which is that you would add anything above the x axis and subtract anything below the x axis but im confused on how they got the values "2 + 2 - 1/2"
omg
i get it
its just the squares
quiet plover
area from x = 0 to x = 4
molten flame
so quick question
lets say i have a different graph for this problem
for this type of question, would i always do the same thing which is add anything above x axis and subtract what ever is below it
cedar kilnBOT
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cedar kilnBOT
@silver coral Has your question been resolved?
cedar kilnBOT
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silver coral
<@&286206848099549185>
cedar kilnBOT
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@silver coral Has your question been resolved?
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gusty spire
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gusty spire
u mean to find the common
gusty spire
idk whats that
alpine knoll
multiply and divide by the congugate
alpine knoll
you don't know this ? and you're doing sequence and series ? lol
gusty spire
am in a greek university sorry
alpine knoll
fine ok so
multiply and divide by (nsqrt(n+1)-sqrt(n)(n+1)
numerator and denominator
then use (a+b)(a-b) formula in the denominator
gusty spire
bruh thats what i was talking about in the chat and they gave me some other ideas
gusty spire
ok let me try it again
alpine knoll
ye
gusty spire
1 sec
gusty spire
do i multiply both 1 and the bottom?
alpine knoll
yeah
alpine knoll
now use (A+b)(a-b)
in denominator
i solved it for you
the final what you get is the simplified version
gusty spire
ok 1 sec
alpine knoll
now that is a telescopic series which means after the first term everything cancels out
gusty spire
so i multiply together the bottom part
gusty spire
so i multiply together the bottom part
.
alpine knoll
wdym
use the difference of squares identity
in the denominator
tha's what i did look at the screenshot
gusty spire
(nsrt(n+1)+(n+1))(srt(n))(n*srt(n+1)*srt(n))
am just trying to do it step by step
@alpine knoll
alpine knoll
i solved it for you
.
gusty spire
gusty spire
ok
alpine knoll
using identity
cedar kilnBOT
@gusty spire Has your question been resolved?
alpine knoll
using identity
.
cedar kilnBOT
@gusty spire Has your question been resolved?
cedar kilnBOT
@gusty spire Has your question been resolved?
cedar kilnBOT
@gusty spire Has your question been resolved?
gusty spire
and how am gonna do partial fractions with a^2-b^2
last jewel
cedar kilnBOT
@gusty spire Has your question been resolved?
cedar kilnBOT
@gusty spire Has your question been resolved?
stiff vigil
hello, let n * sqrt(n+1) = a, and, (n+1) * sqrt (n) = b. Then the fraction would look like 1/(a+b). Notice how a and b both have square roots in the denominator, we do not like that and want to rationalise the denominator. Rationalising is the same as multiplying the numerator and denominator by (a-b) here. So we have 1/(a+b) = (a-b)/(a+b)(a-b). Now use the fact that (a-b)(a+b) = a^2 - b^2. Then we have 1/(a+b) = (a-b)/(a^2-b^2). a^2-b^2 here simplifies to -n(n+1). Then you can break the fraction (a-b)/(a^2-b^2) as a/(a^2-b^2) - b/(a^2-b^2). Then after simplifying, you have 1/sqrt(n) - 1/sqrt(n+1). this becomes a telescoping sum, then you can easily evaluate it as most terms cancel
stiff vigil
in your third line here, in the denominator, the middle terms cancel each other because of the - and +. then you are left as a different of two squares, and you are not left with the extra +2(...) in the denominator of your fourth line
gusty spire
OK thank you I just can't understand how we conclude that the denominator becomes - n(n+1)
stiff vigil
OK thank you I just can't understand how we conclude that the denominator become...
oh by difference of squares. we do (a+b) * (a-b) = a^2-b^2. It simplifies a lot, and you should try it
gusty spire
And the a^2-b^2 becomes -n(n+1)
stiff vigil
And the a^2-b^2 becomes -n(n+1)
yes
gusty spire
Ye but how that's my question
stiff vigil
Ye but how that's my question
like i showed before, do you know what a and b are?
stiff vigil
like i showed before, do you know what a and b are?
here, n * sqrt(n+1) = a, and, (n+1) * sqrt (n) = b
stiff vigil
here, n * sqrt(n+1) = a, and, (n+1) * sqrt (n) = b
from here first find a^2, then find b^2 and finally find what is a^2-b^2
gusty spire
yes ik but we jumb from (n*srt(n+1))^2-((n+1)*srt(n))^2 to -n(n+1)
stiff vigil
yes ik but we jumb from (n*srt(n+1))^2-((n+1)*srt(n))^2 to -n(n+1)
yeah so first find a^2 which is (nsqrt(n+1))^2
gusty spire
Ye I did
gusty spire
(nsqrt(n+1))^2
stiff vigil
(nsqrt(n+1))^2
and you can simplify even more
gusty spire
by commomn?
stiff vigil
by commomn?
basically (nsqrt(n+1))^2 = (n * sqrt(n+1))^2 = n^2 * (sqrt(n+1))^2
stiff vigil
(a*b)^2=a^2*b^2?
yes we use this idea
gusty spire
doesnt make sense'
stiff vigil
doesnt make sense'
so you dont get how (ab)^2 is the same as a^2 * b^2 ?
gusty spire
sense the powers sum each other in multiplication
yes'
btw we talk about the a^2 part only
stiff vigil
btw we talk about the a^2 part only
wheres the n?
gusty spire
n*srt(n+1)
stiff vigil
n*srt(n+1)
yeah this is a
stiff vigil
[n*srt(n+1)]^2=(n^2)*(srt(n+1)^2)?
yes
stiff vigil
disapears with the power of 2
yes
gusty spire
so we get n^2*(n+1) for a
stiff vigil
so we get n^2*(n+1) for a
yes!
stiff vigil
so we get n^2*(n+1) for a
this is a^2 btw
gusty spire
and for b with same steps we get (n+1)^2*n
stiff vigil
and for b with same steps we get (n+1)^2*n
yes, this is b^2
gusty spire
ok so on the denominator now we have n^2(n+1)-(n+1)^2*n
stiff vigil
ok so on the denominator now we have n^2(n+1)-(n+1)^2*n
yes, now try simplify the denominator
stiff vigil
ok so on the denominator now we have n^2(n+1)-(n+1)^2*n
try expanding the brackets
gusty spire
ok so we get then n^3+n^2-(n^2+2n+1)n= n^3+n^2-(n^3+2n^2+N)
stiff vigil
ok so we get then n^3+n^2-(n^2+2n+1)n= n^3+n^2-(n^3+2n^2+N)
yes, now simplify further
gusty spire
ok so now the common comes to play probably
stiff vigil
ok so now the common comes to play probably
yeah
stiff vigil
ok so we get then n^3+n^2-(n^2+2n+1)n= n^3+n^2-(n^3+2n^2+N)
simplify this further. remember we had n^3 + n^2 - n^3 - 2n^2 - n
gusty spire
yea
stiff vigil
simplify this further. remember we had n^3 + n^2 - n^3 - 2n^2 - n
then the n^3 's cancel
gusty spire
oh right
stiff vigil
simplify this further. remember we had n^3 + n^2 - n^3 - 2n^2 - n
n^2 - 2n^2 becomes -n^2
and then we only have the -n as well
gusty spire
-n^2-n
stiff vigil
with the same numerator?
no, you just factorise the -n^2-n. notice -n^2 and the -n both have -n as a factor
gusty spire
(-n)(n-1)
stiff vigil
do i split it?
yes you split it
stiff vigil
ok but what about the top side
the numerator is just a-b, then you split it into a and then -b
gusty spire
ok so we have [nsrt(n+1)/(-n(n-1)] - [(n+1)srt(n)/(-n(n-1)]
stiff vigil
ok so we have [n*srt(n+1)/(-n*(n-1)] - [(n+1)*srt(n)/(-n*(n-1)]
okay, now you can simplify the [nsrt(n+1)/(-n(n-1)] first
stiff vigil
(-n)(n-1)
oh wait, this is incorrect, its (-n)(n+1), not (-n)(n-1)
change your denominator first
gusty spire
ok so we have nsrt(n+1)=A/(-n)+B/(n-1)
stiff vigil
ok so we have nsrt(n+1)=A/(-n)+B/(n-1)
wait, dont do this
the denominator is (-n)(n+1) not (-n)(n-1)
stiff vigil
then the same way
then you split
gusty spire
but with +
stiff vigil
but with +
yes
gusty spire
ok give me a sec am almost home i will do it all at once
stiff vigil
ok give me a sec am almost home i will do it all at once
okay
stiff vigil
sorry took to long
no problem
stiff vigil
you can simplify infinite sum
stiff vigil
on the second series right?
on both series in the last line
stiff vigil
on first the n removed on second the n+1
yes
gusty spire
ok now they remind me of something
stiff vigil
on first the n removed on second the n+1
now in both of them, push the minus from the denominator out
stiff vigil
ok now they remind me of something
yes
stiff vigil
the second
how about the first?
gusty spire
it remains negative
stiff vigil
it remains negative
yes
stiff vigil
it remains negative
now write out the terms of the infinite sum for each one
gusty spire
u mean the couple first and the last
stiff vigil
u mean the couple first and the last
yeah
stiff vigil
the cross out
most terms cancel
gusty spire
yes but
i have 2 1/2
and 1 1
so all cross out
no except srt(2)/2
and srt(n)/n
so now i have only srt(2)/2+srt(n)/n
stiff vigil
so now i have only srt(2)/2+srt(n)/n
instead of sqrt(2)/2 it should be sqrt(1/1), aka the first term
stiff vigil
so now i have only srt(2)/2+srt(n)/n
and also its meant to be -sqrt(n)/n
gusty spire
so 1 remains
stiff vigil
its srt2/2 so it gets canceled later
yes
stiff vigil
so 1 remains
yep, thats the answer
gusty spire
so lim(1+srt(n)/n)=1
stiff vigil
because every other term gets cancelled eventually
stiff vigil
thx a lot man
no problem :)))
stiff vigil
great job! this is correct
stiff vigil
That's not human
lmao
gusty spire
now the real question is if am gonna think of that in the exams
stiff vigil
now the real question is if am gonna think of that in the exams
the main idea is that you always have to simplify the fraction as best as can, and make it into a telescoping sum so the terms cancel
gusty spire
what about the geometric series
stiff vigil
what about the geometric series
yeah, then you can simplify geometric series with a formula
given the ratio and starting term ofc
gusty spire
i noticed in the telescopic they are always 1 or e
and in geometric they might be too
0
stiff vigil
i noticed in the telescopic they are always 1 or e
yeah thats very common
stiff vigil
and in geometric they might be too
yeah
gusty spire
ok man thx a lot
stiff vigil
ok man thx a lot
no problem at all, all the best!
lyric narwhal
cedar kilnBOT
lyric narwhal
Is there a way to prove this inequality without using the Holder Inequality on the counting measure?
a special case of this is the Cauchy-Schwarz Inequality with p=q=2, and that can be proven by considering the discriminant of a polynomial
but I don't think something like that works here
cedar kilnBOT
@lyric narwhal Has your question been resolved?
lyric narwhal
<@&286206848099549185>
raven shard
wait i was looking at this page just yesterday, that’s funny
cedar kilnBOT
@lyric narwhal Has your question been resolved?
lyric narwhal
wait i was looking at this page just yesterday, that’s funny
Wow
Did you find anything useful?
raven shard
Is there a way to prove this inequality without using the Holder Inequality on t...
lagrange multipliers 
mighty drift
Is there a way to prove this inequality without using the Holder Inequality on t...
would the proof not carry over to just rewriting it with sums?
lyric narwhal
lagrange multipliers <:sotrue:826971355542192148>
What are those lol
I'll look them up in a little bit
lyric narwhal
would the proof not carry over to just rewriting it with sums?
What?
How so?
Rewriting what with sums
mighty drift
expectation wrt counting measure is just a sum
raven shard
What are those lol
wait that’s actually wild
i fully expect for you to have seen this before just maybe not called that, don’t know
lyric narwhal
expectation wrt counting measure is just a sum
Well yeah
That's the inequality I'm trying to prove
lyric narwhal
i fully expect for you to have seen this before just maybe not called that, don’...
Probably yeah
Okay apparently I have not
raven shard
that assumption comes from how this concept is usually seen in like, an introductory multivariable calculus course
i don’t actually know how much overlap this has with JEE, i guess
lyric narwhal
Yeah there's absolutely no multivar calc in jee lol
Is it true that $$\left(\sum_{i=1}^n x_i^2\right)^{1/2}\left(\sum_{i=1}^n y_i^2\right)^{1/2}\le\left(\sum_{i=1}^n x_i^p\right)^{1/p}\left(\sum_{i=1}^n x_i^q\right)^{1/q}$$ for $1/p+1/q=1$
wraith daggerBOT
kheerii
lyric narwhal
I feel like we could take the left side as a function of p and prove that it has a minimum at p=2
potent fractal
Is it true that $$\left(\sum_{i=1}^n x_i^2\right)^{1/2}\left(\sum_{i=1}^n y_i^2\...
no
lyric narwhal
that tracks
potent fractal
$x_1=x_2=1$, $y_1=1$, $y_2=0$, $p=3$
wraith daggerBOT
EQUENOS
lyric narwhal
ah right
potent fractal
LHS = sqrt(2), RHS = 2^(1/3)
lyric narwhal
**kheerii**
also that should be y_i^q on the right
potent fractal
oh I didn't even notice that
potent fractal
So the goal is to prove this, right?
