aka LH limit
#help-36
proper dagger
proper dagger
no
proper dagger
but f(x) approaches 6 from only one side
ok by one side i don't mean left/right
i mean top/bottom
proper dagger
you only need to consider one of them
because the inner f(x) approaches 2 only from one side as x approaches 6
proper dagger
can you explain why it would be 5?
leaden pond
i input the two 🥀
proper dagger
just that?
proper dagger
no i mean the reasoning is just that?
leaden pond
yeah
proper dagger
incomplete reasoning but the answer is correct
the reason why it's 5 is because the inner f(x) approaches 2 only from above when x approaches 6 from either side.
since the inner f(x) can only approach 2 from above, the outer f(x) becomes a one-sided limit (that is, lim x -> 2+)
the RH limit of f(x) at x = 2+ is 5
using this, can you finish the rest? remember to check what directions the inner function can approach the limit from when doing a limit of a composite function
leaden pond
ok i was not taught this top or bottom stuff 😭
leaden pond
using this, can you finish the rest? remember to check what directions the inner...
yeah
proper dagger
aight, good luck!
leaden pond
oke thanks for your help
final saddleBOT
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leaden pond
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pine crater
I could really use some guidance, I'm trying to review Surface Integrals and how to evalutate them (my specific application is for classical electrodynamics). could anyone go through a couple examples with me and kind of show me the ropes again? This has been something i've been embarassingly bad at for years and I really need to address it as an inadequacy
final saddleBOT
I could really use some guidance, I'm trying to review Surface Integrals and how...
pine crater
i just found a starting example problem from "Paul's online math notes" since it was one of the first entries to pop up with problems
vital crag
i just found a starting example problem from "Paul's online math notes" since it...
do you have a question about this example
pine crater
i'm sorry this was an inappropriate help post to make
i'm really just looking for some guidance as how to set up the problem in a way that's approachable
i can close it, i know i'm not giving anyone any room to work with here
.close
final saddleBOT
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tall kindle
according to the answer key it's A but i am just unable to get that
final saddleBOT
according to the answer key it's A but i am just unable to get that
static fractal
what are you getting instead/what working do you have so far?
tall kindle
i've been trying to graph the two on desmos but im not sure i can get any of the answers
other than C and D when c = 0
static fractal
c is given to positive, so that excludes c = 0
tall kindle
oh thats right
static fractal
the two lines intersect at (x,y) when the two equations hold
so just plug in the coordinates into the two equations
and check which ones agree (i.e. both equal c)
for example, for B, we get 20x+19y = 200 + 7410 = 7610, and 19x-20y = 190 - 7800 = -7610
and these are not equal, so B cannot be an intersection point
static fractal
the equations still need to agree
tall kindle
20c = c so D is incorrect
static fractal
yeah
since the only solution is c=0
and for C, it would be 20x+19y = 20c + 19c/39 = 799c/39 and 19x-20y = 19c - 20c/39 = 721c/39
so 799c = 721c, which again gives c=0
tall kindle
yes got that too
and plugging in A gives ~195
so they intersect
that helps a lot thank you
.close
dull grotto
ABC is a triangle with AB > AC. Let I be the incenter of ABC. The incircle touches AC and BC at E and D respectively. AI and DE intersect at P
Prove that the midpoint of AB, BC and P are collinear
final saddleBOT
ABC is a triangle with AB > AC. Let I be the incenter of ABC. The incircle touch...
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delicate fiber
final saddleBOT
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
2
in the question they are talking about f6 so in the formula n = 6?
and then u plug it in, f7(k)/7! * x-c ^7
and the f7(k)/7! is a7?
that means you do need a7?
rocky tusk
in the question they are talking about f6 so in the formula n = 6?
they said p_5 though
rocky tusk
yes
rocky tusk
indeed
delicate fiber
but doesnt that mean it would never be needed since n+1 is 6 and not 7
rocky tusk
yes
delicate fiber
thats wrong though
delicate fiber
rocky tusk
then it’s a typo and they meant p_6
delicate fiber
that means the answer is is always neccesary?
delicate fiber
what does that mean
rocky tusk
like it’s only valid on a specific interval
rocky tusk
f^(n + 1)(k) is the max on some interval
so it’s only valid within that interval
the error bound
rocky tusk
why is it the max
in the theorem you have that the f^{n + 1} is bounded
|f^{n + 1}(x)| <= M for some M > 0
rocky tusk
hmm well
they gave a graph though
clearly f^(6) is an increasing linear function so you know the min occurs at -1.1 and max occurs at -0.9
delicate fiber
rocky tusk
clearly f^(6) is an increasing linear function so you know the min occurs at -1....
this is probably what they want
rocky tusk
so i’d say B
delicate fiber
how did you get B from that
rocky tusk
the only reason we would ever need a_7 is if we didn’t know where the max of f^6 occurred and what it was
but we know a_6 and can find f^6 at that point since it’s the center of our taylor expansion
clearly it’s the max from the graph
delicate fiber
but when u use the lagrange formula theres still the f^7 term no? which is a7
rocky tusk
but we know a_6 and can find f^6 at that point since it’s the center of our tayl...
recall that a_6 = f^6/6!
rocky tusk
but when u use the lagrange formula theres still the f^7 term no? which is a7
but we’re estimating p_5 so it’s not relevant here unless we didn’t know the max of f^6
rocky tusk
knowing a_7 gives us the slope of the line that defines f^6 and seeing as we have a point on that line since we’re given a_6 and can thus find f^6 at -0.9, we can then find any point on that line
rocky tusk
because f^6(c) is the max here
on the interval [-1.1, -0.9]
f^6 has a max at -0.9 which is the center
and you can find f^6(c) from a_6 * 6!
rocky tusk
recall that a_6 = f^6/6!
as i said here
were you confused because i didn’t say f^6(c)?
i didn’t mean max there
if that’s what you were thinking
it just happens to be the max in this problem
delicate fiber
and you can find f^6(c) from a_6 * 6!
Ok i get that part
delicate fiber
f^6 has a max at -0.9 which is the center
but why u specifically care about -0.9 arent we approximating for -1.1
rocky tusk
x is -1.1, c is -0.9
delicate fiber
does c mean center
rocky tusk
yes
delicate fiber
rocky tusk
yea they’re using a different notation here
x_0 is c in the image you sent before
and c is k
delicate fiber
oh did u choose -0.9 because thats where the maximum value of f^6 is, so u use that because thats the biggest gap you can have between the real and approximated value?
rocky tusk
oh did u choose -0.9 because thats where the maximum value of f^6 is, so u use t...
yes
delicate fiber
okkkk
that means when u use lagrange approx theorem u always choose the biggest fc possible
rocky tusk
yes make sure you say f^{n + 1} though
not just f
also dont say c, ai has poor notation here
c is commonly used for the center
delicate fiber
ok i should use k for the random point and c for center
rocky tusk
z is also common
for the max point
delicate fiber
Ok
yeah the correct answers are B and D
for this part i just sub the values in and calculate
delicate fiber
no its a first year
rocky tusk
for this part i just sub the values in and calculate
yea
delicate fiber
soft zealotBOT
rocky tusk
you’re welcome
cosmic sleet
help
dense coral
what's your question?
cosmic sleet
dense coral
do you understand how function notation works?
cosmic sleet
no
tired walrus
dense coral
maybe a video would help more than I can 
tired walrus
dunno who you are calling "bro" and hope it ain't me but
you're looking at a function that you're supposed to evaluate at three different pts
cosmic sleet
ok let me see
tired walrus
[also i recommend not using the word "equation" to mean "problem"/"question"]
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warm python
$\textbf{Let $V$ be the set of real numbers. Regard $V$ as a vector space over the field of rationals, with the usual operations. Prove this vector space is not finite dimensional.}$
Suppose there existed a finite basis of $V$. Let a basis be $e_1,e_2,\dots, e_m; m \in \N$. We then consider the element $\frac{\sum_{i=1}^{m} e_i}{\sqrt{2}}$. This element however doesn't lie in the span of this basis over $\Q$ ( As $\frac{1}{\sqrt{2}}$ is irrational, as the basis was arbitrary, this can be shown for any basis. However, this element must lie in the span as $V$ is a vector space. Therefore the basis of $V$ is not finite dimensional
final saddleBOT
$\textbf{Let $V$ be the set of real numbers. Regard $V$ as a vector space over t...
desert mantle
that doesnt work
warm python
that doesnt work
the proof?
desert mantle
yes
soft zealotBOT
wai
desert mantle
eg e_1=1, e_2=sqrt2. then (1+sqrt2)/sqrt2 = 1/sqrt2+1 = 1/2*sqrt(2)+1
warm python
right, but (1+√2)/√2 doesn't lie in the span of e_1,e_2 over the rationals
desert mantle
I literally demonstrated that it does
warm python
oh, right. I see.
tired walrus
$\textbf{Let $V$ be the set of real numbers. Regard $V$ as a vector space over t...
if you're going for a \textbf then why did you enclose it in dollars 
warm python
if you're going for a `\textbf` then why did you enclose it in dollars <:utah:10...
I didn't work without it when I did that for some reason
tired walrus
\textbf{Let $V$ be the set of real numbers. Regard $V$ as a vector space over the field of rationals, with the usual operations. Prove this vector space is not finite dimensional.}
soft zealotBOT
Ann
tired walrus
dunno what you're talking about.
warm python
I must have made a mistake
tired walrus
do you want a nudge in the right direction
warm python
ooh, would dividing by √π instead work
tired walrus
e_1 = 1, e_2 = sqrt(pi) - 1
i think you're kinda SOL if you want such an explicit construction that works for any finite LI set
warm python
lemme think in a different direction then
warm python
i think you're kinda SOL if you want such an explicit construction that works fo...
so like if I consider a basis e_1,e_2,\dots, e_m, saying that √2e_1 doesn't lie in the span wouldn't work?
I think that should work?
desert mantle
e_1 = 1, e_2 = sqrt2
tired walrus
so like if I consider a basis e_1,e_2,\dots, e_m, saying that √2e_1 doesn't lie ...
duplicated "wouldn't"
do you want a nudge in the right direction
warm python
yes please
tired walrus
what is one fundamental difference between R and Q that you can point to even if you know like 0 analysis
tired walrus
yeah you're thinking way too complicated
R is bigger than Q in a certain sense. what am i talking about?
tired walrus
R is uncountable , Q is countable
that's the one
now think: to what is an m-dimensional Q-vector space isomorphic?
warm python
now think: to what is an m-dimensional Q-vector space isomorphic?
$\Q^m$
soft zealotBOT
wai
tired walrus
do you want to think some more about this or do you want another hint
warm python
I'll think
final saddleBOT
@warm python Has your question been resolved?
warm python
do you want to think some more about this or do you want another hint
it seems to be isomorphic to R^m \times Q
tired walrus
what's "it"
warm python
what's "it"
m-dimensional Q-vector space
tired walrus
then that is extremely wrong
sorry, i think i should have clarified earlier
my fault for not being affirmative enough
an m-dim Q vector space is iso to Q^m.
thats gonna be key for your original question.
final saddleBOT
warm python
.reopen
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stone wagon
stone wagon
xd
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exotic shale
can someone tell me why its suffecient according to this source to prove it with k=1?
final saddleBOT
can someone tell me why its suffecient according to this source to prove it with...
stone wagon
can someone tell me why its suffecient according to this source to prove it with...
if it converges at k = 10000, then it converges at k = 1 as well
since you just add finitely many things
desert mantle
other perspective, you could just shift the index so that k=1
exotic shale
other perspective, you could just shift the index so that k=1
yes but why would n+1-k
also converge
if a_n converges
desert mantle
well you are adding the same numbers
exotic shale
is it not more
desert mantle
whether you call 1/10^2+1/11^2+1/12^2+... as a_10+a_11+a_12+... or a_1+a_2+a_3+... doesnt matter
exotic shale
how can all integers from 1 to infinity and from 10 to infinity be the same amount
desert mantle
its the same sequence
just shifted index
why should the name of the index matter
I mean feel free to pull out epsilon
exotic shale
I mean feel free to pull out epsilon
how is epsilon related
exotic shale
why should the name of the index matter
alr
desert mantle
how is epsilon related to convergence?
exotic shale
oh we are talking about case it converges only
yea ok
ight thank you for your time
.close
zenith cedar
final saddleBOT
zenith cedar
What is the general method for solving such questions
I tried to subtract different numbers of 78 and luckily 77 worked but I was looking for something better
opal plinth
I'm not sure but I feel like finding a sum that isn't divisible by a small number (which would be k) might be enough
So for example 77 works here because it's divisible by 7 but not by anything less (well except 1 of course)
So if you have 11 consecutive integers in the subset, you can't partition that into 7 subsets to get the sub-sum 11, and partitions into fewer subsets won't have equal sums
For a "general method", you'd need to mention what kinds of constraints are always present in such questions
Like is the main set always consecutive integers? Always starting with 1?
Otherwise I feel like it could be related to the hard knapsack problem
zenith cedar
I have another doubt but it is not related to this question
So should I post it here
zenith cedar
Like is the main set always consecutive integers? Always starting with 1?
This was an objective question so I guess starting from the end wroked
opal plinth
Not sure what you mean, if this is solved and you have a different question, it's best to close this and open a new channel
zenith cedar
.close
solar crest
how should i start this question bruh
final saddleBOT
how should i start this question bruh
solar crest
expand?
night raft
Yeah expand it, you end up with some parts of it that are just standard integrals
actually
I'd maybe do a u substitution of u = 3x first so it makes the definite integral a bit easier to solve when you expand it
boreal smelt
ig there's no need to substitute 3x=u. expanding will do the job
it's easy after expanding
severe canyon
ig there's no need to substitute 3x=u. expanding will do the job
I would feel more comfortable by doing the substitution first
Especially at the beginning
boreal smelt
I would feel more comfortable by doing the substitution first
yea that would be the same thing cuz you have to divide by 3 anyways.
severe canyon
Well, but then you only have u as argument, which make it somehow easier
Also, you divide by 3 just once, and put it outside of the integral and you can kinda forget it till the end
night raft
Also, you divide by 3 just once, and put it outside of the integral and you can ...
yeah that's essentially what I was thinking
boreal smelt
yea like the chances of error are less when you substitute it right.
then ig substitution is more comfortable.
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@solar crest Has your question been resolved?
night raft
<@&268886789983436800>
solar crest
I'd maybe do a u substitution of u = 3x first so it makes the definite integral ...
can yui
explain
wht i needa
do to dx
im a bit confused how substitution changes the entire expression T-T
night raft
im a bit confused how substitution changes the entire expression T-T
alright so, our u sub is u = 3x right?
what would du/dx be
severe canyon
Yes...
solar crest
okay good...
i though it was gonna be a trick question]
wait
why do i even care abt du/dx.
severe canyon
Have you... just ever seen integration by substitution?
solar crest
Have you... just ever seen integration by substitution?
yea i just watched my prof's video
but like
it doesnt rlly make sense
like the way we're sup[osed to adjust the dx
night raft
like the way we're sup[osed to adjust the dx
you adjust it from du/dx
you can treat it as a fraction
solar crest
ok so i rewrite it with function * du/dx instead of just dx?
night raft
then you just rearrange it to get du/3 = dx
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untold kiln
These complex numbers have a full real-only representation: i=sqrt(-1), i^2=-1, i^4=1. Does i^3 have a full real-only representation?
final saddleBOT
These complex numbers have a full real-only representation: `i=sqrt(-1)`, `i^2=-...
umbral hamlet
to add to that, every single complex number has a real-only representation in the sense you're defining
and all reasonable functions of complex numbers also have real-only representations
powers and exponentiation for sure
this is kind of an ill-posed question though
like if i dont tell you how to compute sin(i), but i just write it as sin(sqrt(-1)), would you count that as a real-only representation
in that sense just replacing i with sqrt(-1) in anything gives you a real-only representation
soft zealotBOT
Allen
scenic walrus
is that right?
umbral hamlet
are you asking if you can write particular complex numbers z in the form (a + bi), for real a and b?
e.g. i^3 = (0 + (-1) i), or just -i
untold kiln
.close
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coral wedge
Guys when u can’t use the theorem to find the composite functions limit, what do you do?
final saddleBOT
Guys when u can’t use the theorem to find the composite functions limit, what do...
coral wedge
excuse the dirty laptop
what I understand is lim g(x) x—> 0=2, and then on f(x)’s graph you look at the y value at x=2
which is 0
and then on g(x) u look at x=0? Since that’s what you got at f(x)?
I think I’m just a bit confused of how u check it
khan academy didn’t do a good job explaining it
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@coral wedge Has your question been resolved?
proper dagger
this problem here might be similar to yours?
jagged flare
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jagged flare
whats the motivation of transforming it into a probability? is it some fact that its a common thing to do with the average value of something? is there some telltale sign? or is it like something to try for any combinatoric problem
wet warren
whats the motivation of transforming it into a probability? is it some fact that...
The first thing that came to my mind is transforming it into a probability and trying to get the expected value
But on what I thought to get there I don't know
Maybe because average of some random events happen to be related to the expected value of that random event
jagged flare
ok fair
i need experience on expected value stuff i basically never get them lol
alr ty!
.solved
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robust mulch
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gleaming geode
Hi, I was wondering if it is possible to have a k cell that is a subset of a n cell such that k > n for a cw complex. In words, can you have a lower ordered cell that is containing a higher ordered cell
final saddleBOT
Hi, I was wondering if it is possible to have a k cell that is a subset of a n c...
vital surge
no the open cells are disjoint
static fractal
each k-cell has to be homeomorphic to an open ball in R^k; if it's contained in a lower n-dimensional cell, you can't possibly have such a homeomorphism
final saddleBOT
@gleaming geode Has your question been resolved?
analog owl
please help me to solve the iCauchy nequalty
final saddleBOT
please help me to solve the iCauchy nequalty
@analog owl Has your question been resolved?
vital surge
please help me to solve the iCauchy nequalty
starting with titu works
glad talon
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tired walrus
do you know the symmetry identities for sine and cosine?
sin(pi-x) = sin(x), that sorta stuff
glad talon
yep
tired walrus
and also sin(-x) = -sin(x)
yeah use those for #5
do you also know how the unit circle works (i.e. cos = x-coord, sin = y-coord)? cause that's what you'll need for #6
glad talon
i thought to find the reference angles you add or minus 360 degrees so u add or minus 14pi/7?
that was my initial approach
glad talon
do you also know how the unit circle works (i.e. cos = x-coord, sin = y-coord)? ...
yeah, i know that but idk how to use that knowledge /what the problem is asking for
tired walrus
the problem's asking you to locate the angles (given in radians) on that conveniently 0.1 radian-graduated circle and read off the x and y coords of their points
e.g. this is 0.9 rad
tired walrus
i thought to find the reference angles you add or minus 360 degrees so u add or ...
not sure it's worthwhile to think in terms of "reference angles" or attach yourself to those here
glad talon
wait im still confused 😭
tired walrus
read off the x- and y-coordinates of this black point that i highlighted
glad talon
(0.6, 0.8)
tired walrus
yeah good enough
now do the same for the other thetas that they give you in the question
glad talon
yeah good enough
wait thats so easy
glad talon
now do the same for the other thetas that they give you in the question
oh ok i gotchu now
glad talon
what abt for this one
tired walrus
make a unit circle and mark off intervals of pi/7 on it (there should be 14 in all), then locate 12pi/7 as well as its reflections in both axes and the origin
glad talon
using ur sin(pi-x)=sin(x)
sin(12pi/7) = sin(7pi-12pi/7)=sin(-5pi/7)
omg yas thats one of the answers
how did they conjure up 8pi/7 and -2pi/7
for the reference angle what i did is i figured out where 12pi/7 would be (in what quadrant) so its in quadrant four
so then i did (28pi-12pi)/7 to find the refernce angle, which is 16pi/7
but the asnwer key shows 2pi/7
glad talon
omg yasss
glad talon
16pi/7 is out of range
ok yas i get it thank you, what about for the alternative sin value, why did they minuse 4pi/7?
glad talon
wait why 9pi/7?
glad talon
wait 8pi/7 looks wrong
oh ic
tired walrus
sin(3pi - 12pi/7)
tired walrus
pi-12pi/7 but then add 2pi to get back in range
pliant shore
Ann finished her masters' degree recently
glad talon
Ann finished her masters' degree recently
oh wowzers
gracias ann
and congrats on masters completion boi
glad talon
... i am not a "boi".
sorry thats just how i talk
tired walrus
a habit that needs dropping!
glad talon
a habit that needs dropping!
mkay ig
tired walrus
in case it was not clear, i am in fact a girl.
glad talon
yeah but in case i wasnt clear when I say boi, I don’t mean it literally like boy or bro. I’m using it as a playful slang, kind of like a casual nickname or vibe word. It’s not about gender, it’s more about the tone. I'm sorry nonetheless
soft heart
what does fetch mean here
final saddleBOT
what does fetch mean here
soft heart
hindi words are also..little difficult
digital wave
It just means "you would get 600 more"
soft heart
It just means "you would get 600 more"
what? Simple interest or Principle
soft heart
yes
digital wave
Yeah then your principal is always constant
proper dagger
the time is always constant too
digital wave
But you'll get 500rs more interest because of the 5% higher rate
proper dagger
600*
digital wave
So basically in scenario one we had x% interest rate and you got M amount of interest, but in scenario 2 you have x+5% interest rate and get M+600 interest
proper dagger
or you can just use 5% interest = 600rs in whatever time given
pliant shore
yes and also note that (x + 5%) - x% = 5% is the difference of 1 years' interest
so what must you do to get the difference for 3 years?
soft heart
3× 5%
pliant shore
bingo
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drifting mauve
can someone help me with this question pls i only understand it up to a point
drifting mauve
for context this is what the chapter includes
thin cloud
for context this is what the chapter includes
Treat 3x as x like in this one
pliant shore
well if you set 3x = u
now you just need to write cos(u) - sin(u) in the form r sin(u - alpha)
drifting mauve
im having trouble finding alpha
severe canyon
Have you found cos and sin of alpha?
thin cloud
You could use calculator or trig table
shut granite
alpha is -pi/4 i think
craggy plank
alpha is -pi/4 i think
Nah, let them do it themselves
shut granite
k
drifting mauve
if using cos-1 wouldnt alpha be -pi/4 but if using sin wouldnt it be +pi/4
why is the sign flipped
severe canyon
if using cos-1 wouldnt alpha be -pi/4 but if using sin wouldnt it be +pi/4
Because you have to look at both sin and cos if you wanna identify the angle uniquely
shut granite
replace -alpha with beta and then use sin(beta) = a/r
u will find beta from there and then replace it with -alpha
drifting mauve
so beta = pi/4
because in the answer says this and i dont understand where 5pi/4 came from
pliant shore
because in the answer says this and i dont understand where 5pi/4 came from
so then you have (sin alpha)/(cos alpha) = tan alpha = a/b from this right
in other words you're solving tan(alpha) = -1
yeah the problem is that there are two equally valid solutions every 360 degrees
severe canyon
can someone help me with this question pls i only understand it up to a point
sin(β) = a/r = 1/√2
cos(β) = b/r = -1/√2
What's the angle (btw 0 and 2π) that has these values of sin and cos?
pliant shore
sin(β) = a/r = 1/√2
cos(β) = b/r = -1/√2
What's the angle (btw 0 and 2π) that h...
so you need the two values of sin, cos together, yep
thanks Alberto
then you can subtract 2pi from the angle you got and there you go
severe canyon
sin(β) = a/r = 1/√2
cos(β) = b/r = -1/√2
What's the angle (btw 0 and 2π) that h...
Or you can use tangent, but then choose the angle carefully, eventually by adding π. That can be done by checking the signs of the coefficients in front of sinx and cosx
pliant shore
sin(β) = a/r = 1/√2
cos(β) = b/r = -1/√2
What's the angle (btw 0 and 2π) that h...
yeah this way is much neater
I hate the tangent method cause it never worked for me lol
drifting mauve
hmmm
im still having trouble trying to understand
i think a lot of this is new to me
pliant shore
i think a lot of this is new to me
you need some revision on trig values in all 4 quadrants then
pliant shore
sin(β) = a/r = 1/√2
cos(β) = b/r = -1/√2
What's the angle (btw 0 and 2π) that h...
if you can't tell us which quadrant beta must be from this
severe canyon
im still having trouble trying to understand
I would stick to what the teacher has explained to you in class, as well (assuming they did explain it lol)
drifting mauve
i know if the cos is - and the sin + shouldnt it be in the 2nd quad
pliant shore
i know if the cos is - and the sin + shouldnt it be in the 2nd quad
yep!
so now do you know what the corresponding 1st quadrant angle would be?
drifting mauve
pi/4?
pliant shore
but isnt 5pi/4 3rd quad?
one step at a time
so the 2nd quadrant angle is 3pi/4 indeed
+3pi/4 and -5pi/4
do you see what's going on now?
drifting mauve
isnt the sqrt2/2 negative though
drifting mauve
wait so how come the alpha in the question was 5pi/4
pliant shore
wait so how come the alpha in the question was 5pi/4
not 5pi/4, but -5pi/4
well I mean
drifting mauve
oh yeah my b
pliant shore
yes alpha is 5pi/4, but they are writing it as sin(3x - 5pi/4)
open shore
There isn’t a single correct value of r or alpha you should take
pliant shore
but then your notes have it as sin(3x + 3pi/4), with the +
drifting mauve
im just having difficulty understanding why the alpha was chosen to be -5pi/4
pliant shore
im just having difficulty understanding why the alpha was chosen to be -5pi/4
oh, cause the question asked you to write it in that form
so you can't use sin(3x + 3pi/4), even if it's the same thing
you have to do sin(3x + 3pi/4 - 2pi) = ....
pliant shore
ohhh because its askingfor a - instead of a +
yeah!
drifting mauve
hmm righy
pliant shore
that's literally the only reason why
pliant shore
no worries!!
drifting mauve
.close
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drifting mica
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drifting mica
why is this not continuous?
barren hound
gotta work from the definition of continuity here
barren hound
from a graph perspective it's continuous i guess sure
opal plinth
If you try to get a value for 0, you can end up with anything between -1 and 1
barren hound
that doesn't mean much
plain rain
,tex
Suppose we have a line with an equation; $ Ax+By+C=0 $. How can we show that for any point with coordinates $ (x_0,y_0) $ , the distance from the line is equal to $\frac{\lvert A x_0 + B y_0 + C \rvert }{\sqrt \left( A^2 + B^2 \right) } $ ?
final saddleBOT
,tex
Suppose we have a line with an equation; $ Ax+By+C=0 $. How can we show th...
soft zealotBOT
fijokazż
Suppose we have a line with an equation; $ Ax+By+C=0 $. How can we show that for any point with coordinates $ (x_0,y_0) $ , the distance from the line is equal to $\frac{\lvert A x_0 + B y_0 + C \rvert }{\sqrt \left( A^2 + B^2 \right) } $ ?
```Compilation error:```! Missing { inserted.
<to be read again>
\left
l.49 ... \rvert }{\sqrt \left( A^2 + B^2 \right) }
$ ?
A left brace was mandatory here, so I've put one in.
You might want to delete and/or insert some corrections
so that I will find a matching right brace soon.
(If you're confused by all this, try typing `I}' now.)```
final saddleBOT
@plain rain Has your question been resolved?
bleak granite
a lot of algebra... 😅
final saddleBOT
open shore
You can try using vectors
final saddleBOT
Available help channel!
plain rain
You can try using vectors
how would that help us?
final saddleBOT
how would that help us?
plain rain
sorry to reopen like this btw
I tried to make an equation up, but I can't avoid using the coordinates of the point in our line
proud jolt
Take any analitycal geometry textbook and you will find that
proper dagger
i think for any given point we can consider the vector from the point to the line normal to the line itself
plain rain
Take any analitycal geometry textbook and you will find that
I read it from there, but it didnt specify any proof lol
plain rain
i think for any given point we can consider the vector from the point to the lin...
and the norm is our distance?
That's basically what I tried to do, but I kept having to use unnecessary coordinates
proper dagger
what have you tried thus far
plain rain
uhm
Like
brute forcing
using the distance formula
between any two points
I also tried using right triangles
this is the shape I tried to use
proper dagger
i was thinking of projections and stuff
proper dagger
i don't think i can articulate my solution well enough for me to be helpful here
sorry
plain rain
it's alright, this problem seems really weird lol
thanks for trying
If theres anyone who knows the method please reply
open shore
Projection is definitely a easy way
Or
Consider this
Consider a point (a,b)
And construct a line l: r=a+lamba b where a and b are vectors
Now actually you should use the variables given by the line equation
Yeah I can’t find the original q but it was something like ax+by+c=0 right?
@plain rain
plain rain
And construct a line l: r=a+lamba b where a and b are vectors
So, by this equation you mean we take any vector (a,λb) and we extend it to infinity to be a line?
I'm here
plain rain
Well, we just take a line l_1: Ax+By+C = 0 and we wanna find the distance to it from any point M(x_0,y_0)
open shore
ok sure
There are actually 2 ways I can think of that we can do this
One would be, find a general expression for the distance between the line and the point, and minimise this expression
plain rain
I found one, but it involves coordinates of a point in the line too
open shore
Or, it would be to construct a vector connecting a point on the line and the point, then project that vector onto a vector perpendicular to the line. The magnitude of this vector will be precisely the formula
Let me try to help you out a bit
plain rain
the vector thing sounds more cool
open shore
Yeah sort of. Are you well versed?
plain rain
What's that mean?
open shore
Are you familiar with projection
plain rain
I know the definition yea
Im bad at explaining
of B in ABC
here, arent AD and DC projections?
plain rain
of B in ABC
this one comes second lol
Sorry, I've always hated geometry and the formal definitions kinda snuck out my brain one night
plain rain
Well, I suppose it is a side created by a line that goes perpendicular to another line
I'm so fried
You can start giving me clues and I will tell ya if I dont get it
I am dying to know the proof to this
the whole book is filled with answers and no proofs and this is the only one I couldnt prove myself
up to now
plain rain
lemme latex it
,tex
$ d= \frac{ \lvert A x_0 + B y_0 + C \rvert }{\sqrt \left( A^2 + B^2 \right) } $
soft zealotBOT
fijokazż
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
plain rain
where x_0, y_0 are the coords of the point we are counting the distance from
plain rain
Interestingly my expression is a bit off
Maybe you used different coordinates
I had like 4 different coords and it was so ugly
plain rain
spill the tea
open shore
As in my working looks fine but it’s not matching the answer
Perhaps another helper can look
plain rain
ok so it turns out I never used projections as in vectors
But I think I see what youre doing
open shore
You could ask helpers perhaps
final saddleBOT
@plain rain Has your question been resolved?
hard stratus
the line with equation Ax+By+C=0 is the line with equation (A,B) • r + C = 0 ie. all the points that are -C units away from the origin in the direction of (A,B).
To find out how far our point (x0,y0) is from ts line you can measure how far it is from the origin in the direction of (A,B) and find the difference between that distance and -C
To find the distance between the point and the origin in the direction of (A,B) you can take the dot product with the unit vector in the direction (A,B) ie. with the vector (A/√(A²+B²), B/√(A²+B²))
Wh
ive messed up somewhere bcs the C is in the wrong place ☠️
But thats the general idea
plain rain
I'm gonna check this out a bit later
hard stratus
what i did wrong was that Ax+By+C=0 are not -C distance away from the origin but -C/√(A²+B²) distance
And (this wasnt a mistake, just a clarification) when i say "distance in the direction of" i mean ignoring any distance perpendicular to it, ie. the distance projected onto that direction
final saddleBOT
plain rain
thank you for your time
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tardy pendant
I finished „solve for local extremum” problem and when I checked I was told (by chatgpt) that I am wrong and that it was a local minimum
final saddleBOT
I finished „solve for local extremum” problem and when I checked I was told (by ...
tardy pendant
!status
final saddleBOT
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
3
coral thunder
(2,-2) is one solution for a stationary point.
What was your conclusion about whether (-2,2) is an extremum?
tardy pendant
I got a local maximum
coral thunder
Ah okay. $f_{xx} < 0$ does make it look like a local maximum in the $x$ direction. What about the $y$ direction? From the working above it doesn't look like you checked that one.
soft zealotBOT
Hardy
tardy pendant
No I did not check y direction
Should I also check that? I was told to only check the fxx and the D
coral thunder
Ah I see what happened.
coral thunder
$(6x)(6y) = 36xy$ but you appear to have calculated $36(-2)(2) = 144$.
soft zealotBOT
Hardy
coral thunder
(instead of -144)
tardy pendant
So I did miss a - sign
tardy pendant
One of the most common and most annoying problems 🙂
But then it has no solutions? (Yeah that I know better than anyone)
If we have -144 - 36 it will be a negative number
coral thunder
Yes, there are no extrema. (0,0) and (-2,2) are both stationary but not extrema.
tardy pendant
That feels wrong not gonna lie but I feel like there is no other mistake
Hope my tutor won’t say otherwise here
Thank you for your time and have a great one✨
!done
final saddleBOT
If you are done with this channel, please mark your problem as solved by typing .close
tardy pendant
.close
final saddleBOT
coral thunder
Actually there was one more sign error. The second stationary point is (2,-2), not (-2,2)
It might pay to check whether that's an extremum.
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Available help channel!
tranquil pine
Could somebody help me with this problem
I started with the usual congruence
3^n= 3 (mod 1000)
3^n-1=1(mod 1000)
n-1=4k
But where do I go from here?
Like I am utterly clueless
pliant shore
Could somebody help me with this problem
I started with the usual congruence
3...
you mean 3^n = 3 (mod 1000)
and n = 4k
tranquil pine
you mean 3^n = 3 (mod 1000)
Yes
Sorry I mistyped
pliant shore
anyways, try using Hensel's lifting lemma
tranquil pine
anyways, try using Hensel's lifting lemma
I am sorry what?
pliant shore
I am sorry what?
tranquil pine
https://math.stackexchange.com/questions/3310998/what-is-the-smallest-positive-i...
Alrighty I'll check it out
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soft heart
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soft heart
is there any shorter version of solution for this question
proper dagger
if you mean something shorter than the solution you gave, i don't think so
proper dagger
didn't a helper do that already?
soft heart
proper dagger
what do you understand from what that helper said?
soft heart
didn't a helper do that already?
i was not active back then
vital surge
i do not rly like this solution
vital surge
the division by 100 makes things a bit messy
proper dagger
how do you find that message so fast?
from:you has:image
vital surge
let r be the annual growth rate
after two years customers double
so (1 + r) * (1 + r) = 2
then solving r is solving a quadratic, and your answer is 100r%
but there is nothing to solve rly
since all u do is take the square root of both sides (1 + r)^2 = 2 becomes 1 + r = sqrt(2)
i don't know why this solution pulls out the quadratic formula and divides by 100
everything is more confusing than necessary
proper dagger
didn't think of that, point taken
soft heart
can i ask doubts?
proper dagger
just ask
proper dagger
if r is the growth in decimal form, 1+r gives the customer count of the next year as a percentage (but in decimal form) of the customer count of the current year
and yes, 1 (or 1.00) is 100%
tired walrus
1 means 100%?
yes and you should make yourself comfortable with converting between % changes and multiplicative factors like that
+50% increase <=> ×1.50 for example
+36% increase <=> ×1.36
-20% decrease <=> ×0.80
and so on
soft heart
okay..
let me move forward from there..need a sec
soft heart
let r be the annual growth rate
after two years customers double
so (1 + r) * (1...
why multiply..
not add
strange coral
@soft heart
proper dagger
because we are increasing by a certain percent per year, which means the number of customers on year 2 is r% more than the number of customers on year 1
proper dagger
and the number of customers in year 3 is again r% more than those in year 2
soft heart
because we are increasing by a certain percent per year, which means the number ...
oh ok..
soft heart
i don't know why this solution pulls out the quadratic formula and divides by 10...
so using quadratic eq. is not necessary
?
tired walrus
well you could go straight from (shit)^2 = 2 to shit=sqrt(2) so yes
using the quadratic formula was excessive
soft heart
how 100( root 2 -1)..
will be formed.
sorry
i am not with my notebook
right now
i was free and mobile was also free
.so i thought of just asking this query
soft heart
how 100( root 2 -1)..
will be formed.
so...
tired walrus
since all u do is take the square root of both sides (1 + r)^2 = 2 becomes 1 + r...
do you get this line
fluid agate
guys can yall help me out with this
tired walrus
guys can yall help me out with this
!occupied
final saddleBOT
guys can yall help me out with this
Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).
soft heart
do you get this line
yes
tired walrus
ok so you get 1+r=sqrt(2) yes?
soft heart
ok yes
tired walrus
so r=sqrt(2)-1
but to get back to a percentage as the question wants, you multiply by 100 and that's how 100(sqrt(2)-1) happens
soft heart
100 is going to be multiplied both sides..no?
tired walrus
sure.
tired walrus
let r be the annual growth rate
after two years customers double
so (1 + r) * (1...
here mqnic defined r as the annual growth rate
which means the growth percentage will be 100r
eg r=0.01 would mean 1% growth
so ultimately the question wants the value of 100r
soft heart
can i ask what percentage the question want us to find out?
my questions might not be in order...but that's what i am getting from this.
soft heart
which means the growth percentage will be 100r
also i still don't get it
soft heart
annual growth rate is (1 + r)..
then how 100r
soft heart
oh
now i got it
r is in decimals..
so to change it in percentage we multiplied with 100
am i right.
?
@tired walrus ?
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astral prawn
Why is it the case that, when multiplying or dividing measures with a different range of accuracy and, thus, certainty, we cut the result basing ourselves off the amount of significant figures the measure with the least amount of sig figs has; in contrast with adding or subtracting measures, which only considers the amount of decimal places the measure with the least amount of decimal places has and, off that, cuts the amount of decimal places in the resulting measure?
Why are they different?
final saddleBOT
Why is it the case that, when multiplying or dividing measures with a different ...
final saddleBOT
@astral prawn Has your question been resolved?
astral prawn
Any help would be appreciated!
drowsy epoch
Any help would be appreciated!
You may ping helpers
astral prawn
<@&286206848099549185>
astral prawn
You may ping helpers
Thanks, mate. Forgot 'bout that.
final saddleBOT
@astral prawn Has your question been resolved?
astral prawn
Snowflake, if you're writing an answer I 'preciate it a lot, but I'm off for the day. Gotta go work and sleep.
Perhaps I can ask it again tomorrow?
umbral hamlet
the main point of truncating decimals is that we never want to add information that we don't have, we'd always prefer removing excess information to keep all things equal.
with adding and subtracting it's straightforward: if i have 3 decimal places on one number, and 6 decimal places of information on another, i have no way of determining what those "missing 3 decimal places" are on the first number. so, i just drop the decimals on the second to align the numbers and give me a meaningful sum/difference
with multiplying and dividing, we have a little more flexibility with our information. Our digits aren't really stuck in any position, we can shift the decimals however we want, perform the multiplication, and just keep track of our decimal shifts. you can't really do the same with addition and subtraction
because of this, all we really care about when multiplying is the number of significant digits, since "digits after the decimal point" is a relative thing in this context
oh yeah sure but thats basically the full answer
astral prawn
Thank you very much then, mate!
Is it fine if I contact you later if I have any doubts on your answer=
umbral hamlet
yeah sure
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strong flax
help
tired walrus
what's your concern about it and why do you think you should change it
strong flax
i mean
look at thursday
i have to be at school from 8:30 am to 8:20 pm
isnt that kind of insane?
tuesday 8:30 am to 6:20 pm
like this is just insane
ill literally
not leave the campus
final slate
Good luck
tired walrus
you have a 3h gap on Tue and you can't/won't leave campus during that time?
proper dagger
there's also a 4h gap on thursday + 2h more in the evening
strong flax
no
quiz is like
my school is weird
quiz is where you attend a smaller class size
like 1 : 30 teacher to student ratio
odd seal
oh so a tutorial session?
strong flax
and we get hw help
tired walrus
strong flax
sometime we might get a quiz
tired walrus
this physics class sticks out like a sore thumb
strong flax
so
is this doable
like 8:30 am to 8:20 pm is kinda crazy
and 8:30 am to 6:20 pm
like wtf?
tired walrus
dunno, you haven't said why you have to be on campus throughout every single one of those gaps
strong flax
ugh
this is already too overwhelming
i want to drop out and return to the NEET life
proper dagger
...are you even interested in listening to us or
vital surge
how is this schedule?
here is a more realistic schedule
strong flax
i guess help
tired walrus
ok in that case my next question
strong flax
ok
tired walrus
you have several multi-hour breaks in your schedule as it is currently
strong flax
yes
tired walrus
do you have to be on campus for the entirety of each and every break, and if so why
strong flax
i dont have to
but think about it
if i ever do leave campus
that means ill have to drive to campus again
and waste so much gas money
tired walrus
how far away is it between your campus and home
in terms of time
it's not 3 hours, is it
proper dagger
15+15 = 30 and you have 3.5 hours to chill
tired walrus
so it's a 15 minute drive
tired walrus
,calc 9/26 * 2
soft zealotBOT
Result:
0.69230769230769
,calc 4.50 * 0.6923
soft zealotBOT
Result:
3.11535
so the gas money you'd spend extra on these return trips is like 3 bucks a trip.
4 if you're being pessimistic.
tired walrus
that's per return trip, mind you.
tired walrus
...and that's an issue how?
strong flax
do you know how leasing cars work
tired walrus
do you have like a limit on total distance driven according to your lease
tired walrus
how much is it