what are all the possible pairs of x and y
#help-41
proud nebula
(0,0), (0,1) , (1, 0) , (1,1)
pseudo crescent
(0,0), (0,1) , (1, 0) , (1,1)
now it's a time to check which of these work, because you mightve got some extraneous solutions
so e.g. we can see that the first one works
because 0^2 = 0
but the second one doesnt
because 0^2 isn't 1
(so x^2 = y isn't true)
proud nebula
so ur plugging these into g_x?
pseudo crescent
what about the third one and fourth one?
pseudo crescent
and by possible I mean that both
x^2 = y and y^2 = x
im plugging them into this
into x^2 = y and y^2 = x
to see if all these pairs really solve the system
and as we can see, some of them dont
proud nebula
can u plug into either?
pseudo crescent
you should check that both of them work
pseudo crescent
(0,0), (0,1) , (1, 0) , (1,1)
we already found out that these are the only possible solutions to the system x^2 = y, y^2 = x
now we need to check if all of them actually are solutions
proud nebula
(0,0) AND 1,1 work
proud nebula
okay
pseudo crescent
now just do the second derivative test
pseudo crescent
im not sure if its called discriminant
but you basically need to compute this quantity
proud nebula
pseudo crescent
no
to compute that, first do partial derivative of f wrt y and then do it wrt x
$\frac{d}{dx}\frac{d}{dy}f\left(x,\ y\right)$
grizzled pagodaBOT
MæthIsAlwaysRight
pseudo crescent
i forgot how to write that partial derivative symbol in latex
proud nebula
of the orginal function?
pseudo crescent
so just imagine that instead of d
proud nebula
f(x,y)
pseudo crescent
of the orginal function?
Yes
we already know the extrema
(0, 0) and (1, 1)
now you just need to know if they are minima / maxima / saddles
proud nebula
Dont we already have those 2
pseudo crescent
Dont we already have those 2
ye, we do
pseudo crescent
i already know the deriv of f(x,y0 wrt x
you can use that to your advantage
do one more derivative wrt x and you will have the 2nd derivative
proud nebula
so are they just being multipled together?
proud nebula
fx and fy?
proud nebula
pseudo crescent
if you differentiate fy again wrt y, you get fyy
and if you differentiate fy again wrt x you get fxy
pseudo crescent
i i am looking for the 2nd deriv of fxy right?
by fxy i meant $f_{xy}$
grizzled pagodaBOT
MæthIsAlwaysRight
pseudo crescent
so we have that gx = 3x^2 - 3y
if we differentiate that again wrt x, what do we get?
this is not correct
proud nebula
lme try
pseudo crescent
yep
perfect
and for gxy, you do it like this:
take gx and differentiate it with respect to y
or take gy and differentiate it with respect to x
the result should be the same
pseudo crescent
no, just once
by each differentiate you are basically adding a letter
we already have gx and we only need to add y to make it gxy
so we differentiate it once with respect to y
pseudo crescent
sounds right
now compute that discriminant for (0, 0)
and then compute it for (1, 1)
pseudo crescent
looks fine
proud nebula
how do i determine which is the max and which is the min tho
pseudo crescent
what is weird that it looks like both are saddles according to that calculation
gimme a moment to verify it
Oh I see
proud nebula
D is greater than 0 here tho
pseudo crescent
D is greater than 0 here tho
oh, yeah
but (0, 0) should be a saddle
because it should be - (-3)^2
which is -9
and D(1, 1) = 6 * 6 - (-3)^2 = 27
proud nebula
if fxx > 0 and d> 0 local min
if fxx less than 0 and d greater tan 0 local max
if d neg saddle
proud nebula
okay got it, thanks
@pseudo crescent could you teach me this one as well?
I know the limit def of the deriv
pseudo crescent
sorry, ill have to go soon
maybe try opening a new channel
someone else might answer
amber waspBOT
@proud nebula Has your question been resolved?
split sail
Can someone tell me what I'm doing wrong?
amber waspBOT
Can someone tell me what I'm doing wrong?
amber waspBOT
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woeful relic
can someone help me pls:
for which values of m does the equation mx^2-2mx+3=0 have two solutions for x
woeful relic
i have gotten up to m(m-3)>0
sage chasm
you study the sign of m²-3m
and its positive when m is in (-oo, 0) U (3, +oo)
amber waspBOT
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keen pawn
[
f(x,y) =
\begin{cases}
\frac{x^2 y^3}{2x^2 + y^2} & \text{if } (x,y) \neq (0,0) \
1 & \text{if } (x,y) = (0,0)
\end{cases}
]
Find the points of discontinuity
grizzled pagodaBOT
ƒ(Why am. I here)=I don't Know
keen pawn
How do I do this
I first approached it along x=y
that gives me a llimit of 0 at 0
how do I check if there are any other points
I guess it's continuous everywhere else as it;s the quotient of two continuous functions?
hmm
thanks
All the continuity theorms from single variable calculus carry over to multi var
right
Like the product of two continuous functions is continuous
[
G(x,y) = \tan^{-1} \left( (x+y)^{-2} \right)
]
grizzled pagodaBOT
ƒ(Why am. I here)=I don't Know
amber waspBOT
@keen pawn Has your question been resolved?
sterile nymph
how do I check if there are any other points
if you approach along the line y = mx, then the function becomes m^3 x^5 /(m^2 + 2)x^2 = m^3 x^3 / (m^2 + 2) => 0
So this function is discontinuous at (0, 0) because its value is 1 at this point.
However, I would be careful of this justification in general, you can see that if the numerator was instead something like xy then the limit would be dependent on m, i.e. the direction you were approaching from
sterile nymph
And your second question I believe you are correct
It should be continuous everywhere except at the points where it is not defined due to division by 0
amber waspBOT
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timid wraith
If each of the integers 1,2,3...,325 is assigned red or blue colour, prove that there exists 3 integers which are assigned the same color and form an AP
timid wraith
Now this is how I did it but I'm not sure if it's correct
So I first wrote the numbers in the form of a circle with 163 in the middle and 162 numbers on the either sides
We have to prove that if any 163 numbers are chosen from these 325 numbers,we always get an AP of 3 numbers
Now if 82 numbers are chosen from any side, we will always get a 3 numbers which will be consecutive
So the only case left to prove is 81 chosen from each side and the number '163' is also chosen
rich latch
so this is a pegionhole argument yes? at least half of them must be of the same color, so at least 163 of them are the same color.
timid wraith
so this is a pegionhole argument yes? at least half of them must be of the same ...
Yes
This question is based on php and graph theory ig
rich latch
and your second argument says you have to choose 163 numbers of the same color, so if more thanhalf of 163 is on one side, then we would have 3 consecutive numbers?
timid wraith
and your second argument says you have to choose 163 numbers of the same color, ...
Yes
But it is wrong ig
Coz if 3 numbers are chosen from 1,2,3,4
They necessarily don't have to be consecutive
rich latch
yeah. There is a simpler idea here. the biggest number is 325, and the smallest number is 1. so the absolute difference between any two numbers is at most 324. imagine constructing a complete graph, with 325 vertices labelled 1 to 325, and coloring the edges based on the absolute difference. now there are at least 163 vertices with the same color, consider the complete subgraph induced by them.
amber waspBOT
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fierce edge
coin w is tossed once and coin X is tossed twice. Assume both coins are fair the what is the probability that coin W returns heads and coin x returns tails twice
fierce edge
I have a reasoning for why we can't treat this is a binomial experiment but i'm curious what the more rigorous reasoning would look like
my main question is why don't we multiply by the number of arrangements * 1/8
amber waspBOT
@fierce edge Has your question been resolved?
supple locust
How many possible results are there? How many satisfy what you want?
supple locust
my main question is why don't we multiply by the number of arrangements * 1/8
You may have to account for arrangements in some capacity if you wanted e.g. one of X to be heads and one tails, because there are two ways to do that, but there is only one way to get tails twice
amber waspBOT
@fierce edge Has your question been resolved?
oak sky
The chance of coin w landing on heads is 1/2.
As the problem states, it will only flip coin w once coin x has been flipped twice, and with 2 possible outputs, and 2 flips, that gives us a 1/4th chance of getting 2 tails in a row. This times the chance of getting heads on coin w would give us 1/8.
So the answer would be 1/8
amber waspBOT
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keen pawn
,, f(x, y) = \ln(x + \sqrt{x^2 + y^2}); \quad f_x(3, 4)
amber waspBOT
,, f(x, y) = \ln(x + \sqrt{x^2 + y^2}); \quad f_x(3, 4)
grizzled pagodaBOT
ƒ(Why am. I here)=I don't Know
keen pawn
so this should be $\frac{1}{\sqrt{x^2+y^2}}$
grizzled pagodaBOT
ƒ(Why am. I here)=I don't Know
keen pawn
so 1/5
stuck vapor
,w d/dx ln(x+sqrt(x^2+y^2))
grizzled pagodaBOT
keen pawn
Do I usually have to show my steps for such probelms
stuck vapor
in an exam you'd write the computation to find the answer anyway
cause you only get this form after some simplifications
keen pawn
I mean the reason I got this is , I was thought this as a standard integral
okay
thanks
stuck vapor
I mean the reason I got this is , I was thought this as a standard integral
indian power
but turkish
primal holly
you could probably just write this as an inverse sinh and skip most of the tedious algebra
amber waspBOT
stuck vapor
I mean the reason I got this is , I was thought this as a standard integral
later on in an exam, that may be a sufficient argument if it's indeed well known (I'd still deem it a bit of a flex)
keen pawn
.reopen
amber waspBOT
✅
stuck vapor
reminds me of a question in my latest assignment, that no one could do
We asked the prof and she said "yeah this exercise comes from a very cited paper where they said this computation was obvious. I haven't managed to prove it but I also don't have a counter example"
The exercise became a bonus
don't take the habit of stating too many things as obvious, please
grizzled pagodaBOT
ƒ(Why am. I here)=I don't Know
grizzled pagodaBOT
ƒ(Why am. I here)=I don't Know
keen pawn
Is this right
proven vapor
reminds me of a question in my latest assignment, that no one could do
We asked ...
That's hilarious, do profs regularly add questions like this to their papers
That seems like torture for students
proven vapor
**ƒ(Why am. I here)=I don't Know**
Why no dy/dx?
keen pawn
Why no dy/dx?
partial derivatives
stuck vapor
That's hilarious, do profs regularly add questions like this to their papers
the paper is a paper
it gets published
Then profs take papers and turn them into assignments/exams
another assignment recently was from a paper, which said "obviously" and "through simple algebra"
it used 2 tricks and took 1.5 page
keen pawn
**ƒ(Why am. I here)=I don't Know**
so $\pdv{z}{x} = \frac{2x-3xy}{3yx-2z}$
grizzled pagodaBOT
ƒ(Why am. I here)=I don't Know
proven vapor
partial derivatives
Did the book imply before that z is a function of x and y, where x and y are independent variables?
proven vapor
another assignment recently was from a paper, which said "obviously" and "throug...
Lol
keen pawn
Did the book imply before that z is a function of x and y, where x and y are ind...
yes
.close
amber waspBOT
keen pawn
Thanks
amber waspBOT
Available help channel!
limpid coyote
I got (3x-2)^x=(1/3)^(3x-2) and I am pretty sure it's not how it should be done
graceful badger
I got (3x-2)^x=(1/3)^(3x-2) and I am pretty sure it's not how it should be done
How did you get that?
sweet imp
I got (3x-2)^x=(1/3)^(3x-2) and I am pretty sure it's not how it should be done
there's a common factor!
limpid coyote
How did you get that?
Got rid of first log then turned x into power of 3x-2 then put 3x-2 on the right as the power of 1/3 and got rid of all logs
That may sound confusing but I don't know how to explan math on english because I'm learning it in different language
graceful badger
How could you get rid of the first log?
sweet imp
Got rid of first log then turned x into power of 3x-2 then put 3x-2 on the right...
ohh it's not a composite function
limpid coyote
How could you get rid of the first log?
Because they're both log 1/3 I think
sweet imp
that means log_1/3(x) * log_1/3(3x - 2) = log_1/3(3x - 2)
it doesn't mean log_1/3(x*log_1/3(3x-2)) = log_1/3(3x - 2)
graceful badger
ohh it's not a composite function
Wait is this supposed to be $\log_{\frac 13} \Bigg(x \log_{\frac13 } (3x - 2)\Bigg)$ or $\Big(\log_{\frac 13} x\Big) \cdot \Big( \log_{\frac13 } (3x - 2)\Big)$
grizzled pagodaBOT
Kepe
graceful badger
Ah
sweet imp
ohh
graceful badger
ohh it's not a composite function
That's what taurine thought aswell
Since there are () around 3x -2, so there should be () around the whole thing if it's meant to be inside log_1/3
limpid coyote
Are you sure? The notation would strongly suggest 2 to me
Nah, in other questions there is • in between
graceful badger
Nah, in other questions there is • in between
If it's really 1, then this is the numerical solution and WA can't give an exact one, either because there doesn't exist an exact form or because it involves some Lambert-W machinery
If it had a * between, it has a 'nice' solution
I think it should be *
limpid coyote
If it's really 1, then this is the numerical solution and WA can't give an exact...
Well I asked chatgpt and he just calculated and at one point he just went "let's assume a = 0"
And at last he got 1
graceful badger
!nogpt
amber waspBOT
Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).
graceful badger
And at last he got 1
You can input 1 into the first expression and see that it yields a false expression (left side != right side)
graceful badger
there's a common factor!
If you got convinced that your book means the other one (with *), you can use this and that will give you a 'nice' solution
graceful badger
If it's really 1, then this is the numerical solution and WA can't give an exact...
This is the only correct solution if it's really the first expression
And there is no exact form (or atleast it couldn't be determined by a computer algebra system)
If it's the second expression, there isn't only a numerical solution you can give, and the whole exercise becomes quite simple
sweet imp
Nah, in other questions there is • in between
it could be just a typo
because even if you cancel out the log_1/3 if you treat the left side as one composite fraction (raise both sides as a power of base 1/3)
you're left with x*log_1/3(3x-2) = 3x-2
which is going to need lambert w
in my opinion i think treating the two logs in the left side as factors is much more intuitive!
amber waspBOT
graceful badger
Okay then what will be the answer if you'll treat it differently?
Did you figure it out?
amber waspBOT
Available help channel!
glad pulsar
amber waspBOT
glad pulsar
How to do iii and vi
limpid coyote
Did you figure it out?
Yeah, the answer still was 1
gloomy tide
for vi maybe start by breaking it down by the singleton element?
glad pulsar
for vi maybe start by breaking it down by the singleton element?
how
gloomy tide
well... how many such pairs exist if B n C = {1}?
glad pulsar
well... how many such pairs exist if B n C = {1}?
so u choose 1 first and rest can be treated in 3^n-1 ways?
3 ways being 1) one element goes to B 2) one element goes to A 3) goes to neither
is this correct
gloomy tide
yeah that seems good to me
glad pulsar
alr
wow the vi th one was ez
do yk how to to the iii
im gonna open a new channel
.close
amber waspBOT
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ruby dome
[
M = \begin{pmatrix}
-2 & 1 & 0 \
0 & 1 & 2 \
0 & 0 & -1
\end{pmatrix}
]
grizzled pagodaBOT
Task Bot
ruby dome
How can I see if it is diagonalizable?
rapid peak
Check the eigenvalues. Since it’s upper triangular, the eigenvalues are just the diagonal terms.
tough mica
if all the eigenvalues are distinct then we can conclude is diagonalizable
ruby dome
Check the eigenvalues. Since it’s upper triangular, the eigenvalues are just the...
Are there any other cases to see the eigenvalues right away, based on the structure of the matrix?
rapid peak
Iirc it’s mostly just triangular (and therefore also diagonal) matrices.
There may be others im unaware of though.
ruby dome
I found $\lambda=-2,\lambda=1,\lambda=-1$
grizzled pagodaBOT
Task Bot
ruby dome
As it should have been
But now how do I find the diagonal matrix?
<@&286206848099549185>
amber waspBOT
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ruby dome
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amber waspBOT
kindred sluice
write
day 1, day 2, day 3, ...
and then count how many vegetables are in each
so for the first part you should have term number: 1
term value: 1
visual oyster
You should also note that the number of vegies she eats on any given day it’s equal to the previous day +4, that should help you get how many she eats on day 10 and 25
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hidden swan
Hi there I am trying to implement a spaced-repetition algorithm in a program of mine
hidden swan
Now I don't need any help code wise, I'll figure that out
but I would love some help understanding it?
The algorithm is called FSRS
https://github.com/open-spaced-repetition/fsrs4anki/wiki/The-Algorithm#fsrs-45
There is also a technical document which outlines the workings of fsrs-4.5, I haven't found something similar for fsrs-5, so 4.5 will do
Now I understand the core principles, essentially each card has three variables:
R - Retrievability
S - Stability
D - Difficulty
**Retrievability **represents the probability the user will be able to retrieve the meaning of a given card at any given moment. This depends on S
**Stability ** represents the time in days it will take for R to decrease from 100% to 90%.
Difficulty represents the difficulty of a particular card
its most of these formulas I get caught up in:
amber waspBOT
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hidden swan
.close
amber waspBOT
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tough mica
What is the greatest whole number that MUST be a factor of the sum of any four consecutive positive odd numbers?
fervent ibex
the sum of 4 consecutive positive odd numbers is just the number in between the 2nd and 3rd one times 4
tough mica
what?
fervent ibex
example: 5 + 7 + 9 + 11 = 4 * 8 (because 8 is between 7 and 9)
fervent ibex
the sum of any set of numbers is just their average times the length
now for consecutive odd numbers you can subtract the first one from the rest and you will always get 0, 2, 4, 6
now the average of these is 3
adding back the first one we get 8
(for the example)
fervent ibex
so lets say that you have any 4 consecutive odd numbers a, b, c, d
subtract a from them
and you will always get a - a = 0, b - a = 2, c - a = 4, and d - a = 6
now you can average these to get 3
so the average of a, b, c, and d is just a + 3
tough mica
why is that
fervent ibex
since they're consecutive odd integers, b = a + 2, c = a + 4, and d = a + 6 right
tough mica
okay
fervent ibex
so the sum is a + a + 2 + a + 4 + a + 6 = 4a + 12
tough mica
tough mica
so the sum is a + a + 2 + a + 4 + a + 6 = 4a + 12
what is the greatest factor then?
fervent ibex
so you have 4a + 12 and a is odd
you can factor into 4(a + 3)
now a + 3 is even so
tough mica
hmm
tough mica
now a + 3 is even so
so what
fervent ibex
so you can factor it out again
8((a + 3)/2)
(a + 3)/2 is garunteed to be odd
so its 8
amber waspBOT
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gentle dove
Hi can someone help me graph this
amber waspBOT
Hi can someone help me graph this
gentle dove
on desmos
hidden basalt
what do each sections of the curve remind you of
hidden basalt
yeah that looks about right
gentle dove
can log funciotns have horizontal asymptotes
vocal wagon
amber waspBOT
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spark grotto
I have this packet as a review/study guide I have to do before my finals and i would love some guidance!
spark grotto
spark grotto
its like 8 more questions after this so if ur ok to stay with me for a bit i would love that
spark grotto
lets start with question a
mk!
halcyon skiff
its like 8 more questions after this so if ur ok to stay with me for a bit i wou...
I gotta study too but i dont think this will take really long
spark grotto
I gotta study too but i dont think this will take really long
ok !
halcyon skiff
so what do we know about a parallel line
spark grotto
they never cross right?
spark grotto
they are pos and neg?
grim forge
they are pos and neg?
Their gradients are equal
halcyon skiff
Their gradients are equal
this
spark grotto
oh mb Xp
spark grotto
so just write an equation that are pos or neg?
halcyon skiff
what do you exactly mean with an equation that is positive or negative?
grim forge
They already gave you the gradient
grim forge
K bro
spark grotto
for the function in question a
imma take a guess and say 3 and 4?
grim forge
You might be cooked man
halcyon skiff
imma take a guess and say 3 and 4?
it is 3, you can find the gradient when a function is in the from y=ax+b. Then the a gives you the gradient (in this case the 3)
cuz for every +1 x you go up 3, you understand that?
spark grotto
mhm
faint sedge
imma take a guess and say 3 and 4?
Look at the parallel line to the right and tell, what do their equations have in common
spark grotto
wait are we trying to make a new equation or the graph?
faint sedge
New equation, which graph will be parallel to the old one
grim forge
I think you're sleep deprived
You'd have more luck getting some sleep
Right now you can't even tell what the gradient is
I think you actually need to like sleep
spark grotto
I think you're sleep deprived
trust my adhd medicine will def helpout for my finals rn i need this study guide!
grim forge
trust my adhd medicine will def helpout for my finals rn i need this study guide...
Are your finals tomorrow
spark grotto
Are your finals tomorrow
yeah and my brother told me if i get a single c on my finals grades i get my stuff taken away :p
grim forge
yeah and my brother told me if i get a single c on my finals grades i get my stu...
Then go to sleep man
spark grotto
and in math i have a 79.89% and im leaning into a c
grizzled plume
and in math i have a 79.89% and im leaning into a c
Damn
spark grotto
With less than 3 hours of sleep
ill sleep during lunch ☺️
or i should do my chem
i think ill do my chem
faint sedge
so have you finished with your life story, can we return to math?
spark grotto
wait am i just stupid is it not just -3x and -4 or something like that?
faint sedge
wait am i just stupid is it not just -3x and -4 or something like that?
right
grim forge
wait am i just stupid is it not just -3x and -4 or something like that?
The gradient is 3
Not -3
spark grotto
ok so i am just sleep deprived its whatever
faint sedge
o, wait, not
spark grotto
The gradient is 3
ohh mk mk
grim forge
ok so i am just sleep deprived its whatever
Ok so first
Let's try to graph 3x+4
You know your line equation ax+b right
What does the b represent
spark grotto
run?
grim forge
Uh the y intercept
spark grotto
rise
grim forge
No the y intercept
spark grotto
oh mb
grim forge
Do you know what the y intercept is
spark grotto
yeah the up and down of the graph!
faint sedge
look, as I said. Just compare the lines at the right. Look what their equations have in common and what is different
then just write the similar for your own equation
grim forge
yeah the up and down of the graph!
Actually nevermind I think I get what you're tryna say
spark grotto
so y = 3x -4?
faint sedge
so y = 3x -4?
yes. Or 3x+2 or 3x-50
spark grotto
50?
faint sedge
any number here works
faint sedge
if you do not know, how to graph the lines, you can use https://www.geogebra.org/calculator to do it. Then play with parameters (3 and -4) and see the effect it causes on the line. Then copy it to the sheet
spark grotto
yeah so what i said mk got it :3!
faint sedge
mk?
faint sedge
so what are your thoughts here?
spark grotto
uhh its looks like its telling me to follow the formula
faint sedge
yes, it is the formula for the slope (gradient)
spark grotto
ok wait so do i combine a and b to get one AB?
spark grotto
mk!
faint sedge
no. For AB you have two sets of coordinates: A(x_1, y_1) and B(x_2, y_2). Plug them into the formula above, and you will get the slope of the segment AB
naive aspen
note that for 33, y2-y1 means you subtract (-5) from 7, with would be 7-(-5) with would become positive (7+5)
spark grotto
oh whoops
naive aspen
and 4-(-2) applies the same rule
spark grotto
so 12 over 6?
faint sedge
so 12 over 6?
yep
faint sedge
yes. Then compare
faint sedge
try again. Write the full caclulations here
faint sedge
0 x1 2 y1
8 x2 -2 y2
this is right. Now substitute into the formula
faint sedge
yes
spark grotto
and 8 is fine or?
faint sedge
8-0=8
faint sedge
carefully compare them with theese two cases
spark grotto
there is not way its parallel so perpendicular?
faint sedge
you tell me. Is the condition met?
What are the slopes again? Give the final numbers
faint sedge
simplify to the final number. What is 12/6?
faint sedge
and the second one?
spark grotto
-0.5?
faint sedge
yep. So is the condition of perpendicularity met?
faint sedge
how do you check?
spark grotto
its asking if its parallel , perpendicular or none
faint sedge
do you see the conditions here?
spark grotto
no?
faint sedge
faint sedge
How much is 2*(-0.5)?
spark grotto
oh you wanted me to do that?
faint sedge
?
spark grotto
its -1
faint sedge
so do the slopes meet the second circled condition here?
spark grotto
yea
faint sedge
sorry, have to go now. Good luck on the test!
spark grotto
ok!
ok i have 5 more question so if anyone can help me i would love it!
Mk i have 2 hour and 20 min before go to school i got to ping someone
sorry!
<@&286206848099549185> I have this packet as a review/study guide I have to do before my finals and i would love some guidance!
tawdry verge
64?
spark grotto
hm?
tawdry verge
hm?
Both the geometry (non coordinate) ones use the same concept
That angles opposite to equal sides are equal to one another
spark grotto
wait what question are you on?
tawdry verge
..
tawdry verge
the last 2?
Yeah
tawdry verge
Yeah
spark grotto
mk
tawdry verge
And the first three
You just have to take the general equation for a line
Then find the general like slope equation
Yeah
And divide that by the slope of given line
spark grotto
oh mk
tawdry verge
Uh
Sorry mb actually
Find their product
If the product = -1 then they are perpendicular
But since we need them to be perpendicular
We equate the product to -1 and
Yeah
tawdry verge
and for the other ones?
And for the one where
It asks parallel
The slopes must be equal
tawdry verge
and for the other ones?
Like we can find slope of given line
Then that'll be equal to slope of the other line
We get some equation which still requires bit of solving
Cuz unspecified coefficient
But we can find that by plugging in the point that it is given to pass through with
And then solve just like linear equation
You get it right
Uhh I suck at explaining lol
tawdry verge
You got the last 2 right
@spark grotto
spark grotto
huh
halcyon skiff
where are you stuck?
spark grotto
all thesse
halcyon skiff
okay i have like a 5-10 min break
so ill try to help as much as possible too
if soul doesnt mind of course
tawdry verge
For the first image
Let the line which we have to find be
y=mx+c since we know it's a line so we can represent it like this
So it's slope is m
But we know for a fact that if two lines are perpendicular then the product of them should be -1
And the slope of the given line is 4/7
So 4/7 * m = -1
So we find m = -7/4
Plug that in
And we get the equation for the line we have to find is y=-(7/4)x +c
Now we know it passes through (16,9)
So we plug those values for x and y and so we can find c
Eh
halcyon skiff
all thesse
you did 30b and c right btw
tawdry verge
if soul doesnt mind of course
No why'd i
halcyon skiff
No why'd i
idk some dont like that ig
tawdry verge
idk some dont like that ig
I mean I'm pretty bad at explaining stuff anyways
halcyon skiff
For the first image
Let the line which we have to find be
y=mx+c since we know ...
ye cant really add anything to this
tawdry verge
idk some dont like that ig
Oh
tawdry verge
For the first image
Let the line which we have to find be
y=mx+c since we know ...
Similarly you can do the second one
And for the third one take a line y=mx+c but since parallel so their slopes are equal so we find m
tawdry verge
And for the third one take a line y=mx+c but since parallel so their slopes are ...
Then plug in the point it passes through so you find c
spark grotto
ok done now what?
halcyon skiff
did you put in the values for x and y to get the intersection point? so the constant?
spark grotto
wait is 9= -7/4 (16) + 2 fine?
halcyon skiff
uhm let me check
no
so we have y=mx+c
We have x= 16 y= 9 and m= -7/4 so you must plug that in to solve for c=
and u will know your c value
halcyon skiff
you want to solve: 9=-7/4*16 + c
spark grotto
but how?
<@&286206848099549185> I have this packet as a review/study guide I have to do before my finals and i would love some guidance! i have an hour to finish this!
hollow sphinx
<@&286206848099549185> I have this packet as a review/study guide I have to do b...
which part of the questions do you not get or have trouble with
hollow sphinx
uh i mean everything looks right, just solve for c using the 9=(-7/4)16 equation (i think you forgot to add c to the right side)
spark grotto
but howdo i solve do i use only (-7/4)16?
hollow sphinx
yeah well the problem is you forgot to include c after substituting in the x and y values
the equation should be 9=(-7/4)16+c
tawdry verge
but howdo i solve do i use only (-7/4)16?
You forgot to add the c
spark grotto
so (-7/4)16+c?
spark grotto
so c is 9?
hollow sphinx
no, (-7/4)16+c equals 9
spark grotto
and thats it?
hollow sphinx
well no, you have to isolate c and simplify the fraction
and then plug the solved value of c in to the linear equation
spark grotto
huh?
hollow sphinx
out of curiosity how did your teacher tell you to solve this type of problem
amber waspBOT
summer bone
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Available help channel!
keen pawn
Prove or disprove that every group containing 6 elements is abelian
small pendant
do you know what the dihedral gruop is
look at the rotations and reflections of a triangle
patent raptor
What about S_3
keen pawn
Let $a,b$ be elements of a group $G$, prove that $ab^na^{-1} = (aba^{-1})^n$ for $n \in \Z$
grizzled pagodaBOT
ƒ( wai ina teacup)= I don't know
limpid sun
$(aba^{-1})^n=aba^{-1}aba^{-1}\ldots aba^{-1}=ab^na^{-1}$
grizzled pagodaBOT
everg
limpid sun
keen pawn
.close
amber waspBOT
limpid sun
what is the problem?
unborn plaza
because sinx is symmetrical about x=pi/2
i didn't do it algebraically, i looked at a graph
i think it can
oh you mean why it isn't arcsin(pi-y)...
i don't know, i don't know how you got arcsin(pi-y)
limpid sun
oo I see axe
unborn plaza
arcsin(pi-y) isn't defined when y is in (0,1)
limpid sun
cryolite don't cry
unborn plaza
look at the picture to find the bounds and do y first instead of x first
limpid sun
perhaps you should contact your professor
modern marsh
He didn't give a shit
limpid sun
He didn't give a shit
report him then!
amber waspBOT
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Available help channel!
hasty yacht
amber waspBOT
hasty yacht
According to the info above, m(ADE)=x is how many degrees
i found <bdc=50 and dont know how to progress from here
amber waspBOT
@hasty yacht Has your question been resolved?
hasty yacht
do my emotes not count
sweet imp
since |ab| = |ac|, therefore ?
sweet imp
i found <bdc=50 and dont know how to progress from here
just one more step and u can get ADE
use the fact that DBC is 50
amber waspBOT
@hasty yacht Has your question been resolved?
sweet imp
bdc was 50 mb
isn't bdc 80, and dbc 50?
hasty yacht
no
sweet imp
ahh omg mb
hasty yacht
m(bac) should therefore be 20
yeah
hasty yacht
so to get ade?
no idea
sweet imp
and also with supplementary angles
hasty yacht
i still dont see how to progress further
sweet imp
mb
sweet imp
now you can use this triangle
to isolate angle aed, because currently you have aeb
then after that, 180 - bac - aed = ade
hasty yacht
thanks for participating anyways
hasty yacht
<@&286206848099549185>
amber waspBOT
@hasty yacht Has your question been resolved?
hasty yacht
<@&286206848099549185>
lunar pine
Gayhiya can k help
amber waspBOT
@hasty yacht Has your question been resolved?
split sail
so basically you want to cube root pi then divide by golden ratio
unkempt echo
For a, b, c that satisfies a/c = (a-b)/(b-c). Proves that 1/a + 1/(a-b) = 1/(b-c) - 1/c
amber waspBOT
For a, b, c that satisfies a/c = (a-b)/(b-c). Proves that 1/a + 1/(a-b) = 1/(b-c...
@unkempt echo Has your question been resolved?
vocal pewter
What did you try
amber waspBOT
@unkempt echo Has your question been resolved?
remote egret
.close
amber waspBOT
amber waspBOT
Available help channel!
keen pawn
Given the groups $R^{\ast} = R \setminus {0}$ and $\Z$. Define a binary operation $\circ$ on $G$, by $(a,m) \circ (b,n) =(ab,m+n)$ Show that $G$ is a group under this operation
grizzled pagodaBOT
ƒ( wai ina teacup)= I don't know
keen pawn
We start by proving the existence an an identity. $(a,m) \circ (g,h) = (ag,m+n) = (a,m)$ . It follows that $(1,0)$. Is this set's identity
unborn plaza
G is R* under multiplication crossed with Z under addition?
keen pawn
G is R* under multiplication crossed with Z under addition?
oops, yes
grizzled pagodaBOT
ƒ( wai ina teacup)= I don't know
keen pawn
We start by proving the existence an an identity. $(a,m) \circ (g,h) = (ag,m+n) = (a,m)$ . It follows that $(1,0)$. Is this set's identity. We now prove the existence of an inverse$(a,b) \circ(g,h)=(1,0) \implies (ag,b+h) = (1,0)$. Thus the inverse is $(\frac{1}{a},-b)$, which belongs to $\R^{\ast} \times \Z$.
grizzled pagodaBOT
ƒ( wai ina teacup)= I don't know
keen pawn
I now suppose I prove associativity and closure
unborn plaza
ye
keen pawn
It's closed as the product of two reals is real and the sum of two integers is an integer
$(a,b) \circ [(c,d) \cric(e,f)] = (a,b) \cric (ce, d+f) = (ace, b+d+f)
\
((a,b) \cric (c,d))\circ (e,f) = (ac,b+d) \circ (e,f) = (ace, b+d+f)$
grizzled pagodaBOT
ƒ( wai ina teacup)= I don't know
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
unborn plaza
looks good
amber waspBOT
amber waspBOT
Available help channel!
keen pawn
Prove if $(ab)^{-1} = a^{-1} b^{-1}$ then the group is abelian .
\
$(ab)^{-1}=. b^{-1}a^{-1}= a^{-1}b^{-1}$
This proves the group is abelain
grizzled pagodaBOT
ƒ( wai ina teacup)= I don't know
unborn plaza
what's the justification for the last step?
b^-1 a^-1 = a^-1 b^-1
oh it's given
the order of stuff just confused me
unborn plaza
it's fine
what you wrote probably works but i want to share mine too lol:
$ab=(a^{-1})^{-1}(b^{-1})^{-1}=(a^{-1}b^{-1})^{-1}=(b^{-1})^{-1}(a^{-1})^{-1}=ba$
grizzled pagodaBOT
Axe
amber waspBOT
amber waspBOT
Available help channel!
keen pawn
Prove that the intersection of two subgroups of $G$ is a subgroup of $G$
\
For this we only have to check 3 properties. The existence of the identity element, the inverse of a given element and closure,
\
The identity belongs in both subgroups as it's an element of any group, and thus an element of any subgroup
\
If $a$ belongs to the intersection of two sub groups, then $a^{-1}$ automatically belongs to their intersection too, as $a^{-1}$ is present in both subgroups.
\
\
We now prove closure , if $a, b$ are in the intersection of the two groups, then so is $a \circ b$,as $a \cric b$ belongs to both subgroups, it belongs to their intersection too
grizzled pagodaBOT
ƒ( wai ina teacup)= I don't know
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)
keen pawn
.close
amber waspBOT
amber waspBOT
Available help channel!
meager gale
amber waspBOT
meager gale
this is for part e
this is what i have so far. I’m stuck on what to do next, or if ive messed up already
pallid canopy
For large k, approximate the log term
meager gale
sub k—> infinity?
pallid canopy
You just replace it with inequalities
pallid canopy
That's true but you need to argue it prove why
meager gale
ok
how would i approximate the log term for large k?
ik itd be like ln(e^k+hreallysmall decimal or smth)
pallid canopy
This is really all you need to do. The other stuff was to get you here
meager gale
That's true but you need to argue it prove why
ye just stuck on how to do this properly
meager gale
will probs use the k=log(e^k), but for der, is it cause the rate of increase is exponentially faster than the other?
pallid canopy
No
pallid canopy
will probs use the k=log(e^k), but for der, is it cause the rate of increase is ...
This was almost right. Your rate of increase is just wrong
The slope of k is 1. The slope of the right side is slightly bigger than 1
amber waspBOT
amber waspBOT
Available help channel!
split sail
yo
amber waspBOT
@split sail Has your question been resolved?
sterile thistle
Literally anyone help this has been my most confusing question
viral osprey
What was ur initial approach
grim forge
Literally anyone help this has been my most confusing question
It doesn't seem like this is the full question
Can we see the the whole thing
amber waspBOT
@sterile thistle Has your question been resolved?
amber waspBOT
coarse hemlock
.reopen
amber waspBOT
Available help channel!
subtle mirage
show that the derivative of this is equaol to k*sec(kx) and it's given that 0<kx<pi/2
spiral zealot
amber waspBOT
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Available help channel!
floral perch
Hello, It is in french but I think I made mistakes and I'm not sure of my answers, help please
amber waspBOT
Hello, It is in french but I think I made mistakes and I'm not sure of my answer...
@floral perch Has your question been resolved?
Available help channel!
split sail
amber waspBOT
split sail
Does this require another substitution?
ionic zodiac
let u = e^x +1
split sail
let u = e^x +1
Hm is this based on experience or theres a strategy to let u be
ionic zodiac
well u = e^x alone wouldnt rlly be helpful here
i mean u could but
if you look at the denominator
u can see that the derivative of the denominator is exactly the numerator
so u = e^x+1 is the best choice
split sail
Cuz in this one, i tried letting u be csc(piA) but it didn't work out lol
split sail
u can see that the derivative of the denominator is exactly the numerator
Indeed ohhh
ionic zodiac
well the integral of cot*csc is a standard integral actually
integral of cot*csc by itself is just -csc
split sail
Hmm this is not only coincidence, ur inference makes sense , ty
ionic zodiac
integral of cot*csc by itself is just -csc
so if you know this fact all you have to do here is let u = A*pi
split sail
integral of cot*csc by itself is just -csc
😭i didnt realize but i guess the piA part requires substitution? Doesnt it
ionic zodiac
yea
ionic zodiac
hm im not sure whats going on with the du/u-1 bit
if you let u = e^x + 1, du=e^x dx
you seem to have that part right
the 1/u du part is right
as long as ur using u = e^x+1
$\int_{ }^{ }\frac{e^{x}}{u}\cdot\frac{du}{e^{x}}=\int_{ }^{ }\frac{du}{u}$
grizzled pagodaBOT
water beam
ionic zodiac
ur bounds are incorrect though
split sail
ionic zodiac
,rcw
grizzled pagodaBOT
ionic zodiac
yes ln(1+e) - ln(2) is right
ionic zodiac
well what do you think?
split sail
1+tanx?
ionic zodiac
hint: usually for u-sub problems the stuff inside a bracket with a large power for example (.......)^9 is usually a good place to start
split sail
I forgot how to do tan
How do i set the bounds
d/dx 1+tanx = sec^2x
Wat is tan 0 and tan pi/4
ionic zodiac
How do i set the bounds
the same way youve done them for previous problems
ionic zodiac
tan(pi/4) is just 1
ionic zodiac
do you know the unit circle
split sail
I recalled dw
ionic zodiac
great
split sail
ionic zodiac
,w integrate (1+tanx)^5*sec^2(x) from 0 to pi/4
grizzled pagodaBOT
ionic zodiac
ye
split sail
oh thats neat
ionic zodiac
nice
split sail
cheating trick learned
ionic zodiac
lol only use it to check
grizzled pagodaBOT
split sail
,w integrate xsqrt(1-x^2) from 0 to 1
grizzled pagodaBOT
ionic zodiac
u better be doing the working out
split sail
u better be doing the working out
I know😇😇
ionic zodiac
you didnt integrate sqrt(u) yet
ionic zodiac
go to sleep
split sail
,w integrate cos(3x) from 0 to pi/2