this would be (m+n+1)c3-(m+1)c3-(n+1)c3 i believe
#help-49
rough birch
rough birch
if you expand it out you will get that
rough birch
wait just expand (m+n+1)choose 3 and (m+1)choose 3 and (n+1)choose 3
rough birch
when a is included
they are taking the answer from the previous part, and adding all triangles which contain A
rough birch
all the triangles which contain A will contain point A, one point along AB which isn't A, and one point along AC which isn't A
rough birch
which triangles contain a
i mean where A is one of the 3 verticies of the triangle
barren onyx
i mean where A is one of the 3 verticies of the triangle
how is that not included in the answer ?
rough birch
how is that not included in the answer ?
it is included in the second problem, but not the first problem
read carefully, there are 2 problems in what you sent
barren onyx
read carefully, there are 2 problems in what you sent
they should have the same answer
bcz a is already a vertice in first question
rough birch
bcz a is already a vertice in first question
no it isn't one of the verticies of the first problem. read it carefully
barren onyx
if anything the answer should be less than the first
rough birch
"none of them being A"
rough birch
if anything the answer should be less than the first
no
barren onyx
bruh
barren onyx
"none of them being A"
i didn't read this
rough birch
np
rough birch
how do we get mc1 nc1 additional triangles?
because we choose A, one point along AB, and one point along AC
barren onyx
because we choose A, one point along AB, and one point along AC
how does that make it a triangle?
you're only choosing a point right
rough birch
you're only choosing a point right
we chose 2 points in addition to A, making it 3 points?
one point along AB, one point along AC, and A itself
rough birch
yeah, they are just adding the "new" triangles here
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sage heath
There are 10 employees in a company and they are to be assigned to four distinct tasks,
with each task requiring at least two employees. How many ways can this be done if no task should
be left empty?
midnight plankBOT
There are 10 employees in a company and they are to be assigned to four distinct...
gleaming edge
there has to be at least 2 employees in each task and there’s 4 tasks so that’s 8 employees needed to satisfy this condition then there’s 2 employees left for the other 4 tasks so just think of the question as how many ways can two employees work on 4 tasks where they can either work on the same one or different ones
You can use the counting formula allowing repetition and with ordering needed
polar pebble
is this correct way to do this problem? 8x+5/6-x
midnight plankBOT
is this correct way to do this problem? 8x+5/6-x
@polar pebble Has your question been resolved?
rough birch
is this correct way to do this problem? 8x+5/6-x
no
rough birch
ok so we have our original equation is y=(6x-5)/(x+8) right?
so now, to calculate the inverse, switch around x and y
and we get x=(6y-5)/(y+8)
and now try to simplify this equation into y="something"
polar pebble
ah ok
polar pebble
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pure wraith
I've been asked to compute the area of an ellipse using Green's Theorem
midnight plankBOT
I've been asked to compute the area of an ellipse using Green's Theorem
pure wraith
I'm not sure how exactly I'm supposed to do that
grand pondBOT
jewels!
pure wraith
Do I need to just choose a field such that the right integrand is 1?
grand pondBOT
jewels!
grand pondBOT
jewels!
grand pondBOT
leaden matrix
midnight plankBOT
leaden matrix
can someone explain how to go from the second line to the third
where'd the sines and cosines go?
frozen talon
you evaluate this expression at 2L and 0
things go away because n is just a positive integer, supposedly
it's easier to see if you write it out yourself
leaden matrix
right but I dont see why the sins and cos go away
is it because the inside is 2pi n x?
frozen talon
take for instance the expression L/(npi) * x * sin(xnpi/L)
at x=0, the x outside of the sin(-) kills the expression
at x=2L, the argument in the sin(-) function is 2npi
provided n is an integer, sin(2npi) is just the vertical component of the point (1,0) after rotating counterclockwise around the unit circle n times (clockwise if n is negative)
frozen talon
it's similarly nice for the cosine bit, but it's different because it doesn't evaluate to 0 in either way
leaden matrix
wait nvm
im still confused about the first one
whys it n squared and pi squared?
doesn't the sin(2npi) term just become 1?
leaden matrix
provided n is an integer, sin(2npi) is just the vertical component of the point ...
so wouldn't it be 2L squared / n pi?
frozen talon
no, sin(2npi) is not 1
it's the same as sin(0), which is the vertical component of the point (1,0)
which is 0
leaden matrix
so wouldnt the whole thing be 0 then?
whys there an L squared / pi squared * n squared
frozen talon
because that's being contributed from the rightmost term involving cosine
the third line is the result of computing
$$\frac1L\left[\frac{L}{n\pi}(2L)\sin\left(n\frac{\pi}{L}(2L)\right)+\frac{L^2}{n^2\pi^2}\cos\left(n\frac{\pi}{L}(2L)\right)\right]-\frac1L\left[\frac{L}{n\pi}(0)\sin\left(n\frac{\pi}{L}(0)\right)+\frac{L^2}{n^2\pi^2}\cos\left(n\frac{\pi}{L}(0)\right)\right]$$
grand pondBOT
Flip
frozen talon
we effectively see four terms being added together
the first and third terms involve sine and return the first and third 0's that we see
the second and fourth terms involve cosine, and they are supposedly contributing L^2 / (n^2 * pi^2) and its negation respectively -- why?
frozen talon
whys there an L squared / pi squared * n squared
ping, in case you didn't know I responded
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obtuse totem
midnight plankBOT
obtuse totem
and answers
Im not sure how they got v1(t) and v2(t), because for some reason, I have dv1(t)/dt as 0
because flow in -flow out =0
robust isle
6gal/min * 2gal/min, why are you multiplying them
obtuse totem
2lbs
conc times rate
also, I just realized, for dv/dt, its 6-3 for T1
cause I guess you dont count the outflow when its flowing into another container?
cause what I initially did was (6gal)(1gal from other container)- (2 to other container)(3 outflow out of the container)
robust isle
nah count all the inflows, all the outflows
robust isle
6 (external) + 1 (from T1) - 2 (to T1) - 3 (external)
like the numbers you wrote are correct
robust isle
I just don't understand why you multiply the inflows/outflows together
robust isle
6 (external) + 1 (from T1) - 2 (to T1) - 3 (external)
that is +2gal/min
obtuse totem
oh
wait
omg
for dv/dt
im just accounting for the rate
so I dont have to multiply
oh my bad😅
thank you @robust isle
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quartz swift
just did this but my friend got something else? what i do wrong lol
midnight plankBOT
just did this but my friend got something else? what i do wrong lol
small jasper
,w \frac{3}{2} x-x+4=2\sqrt{x^2-64}
grand pondBOT
small jasper
,w 7x^2-16x-576=0
grand pondBOT
small jasper
just did this but my friend got something else? what i do wrong lol
$2\sqrt{x^2-64}$, not $\sqrt{2(x^2-64)}$
grand pondBOT
Civil Service Pigeon
brazen moth
just did this but my friend got something else? what i do wrong lol
Here is your error
@quartz swift
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gritty hatch
yo
gritty hatch
Been a while.
```How does the factorization process work again?``
How do you know what part of the problem to factor? Arent there typically multiple pieces (more than two) and the goal of splitting it into two binomials?
Why did they split 5x into 25/4?
lethal path
factorisation and completing the square are completely different methods
gritty hatch
can you explain both
lethal path
can you explain both
what do you find unclear in prof Dave's video about completing the square?
cause YouTube or OpenStax (online textbook) can explain both
what complicates things is that there are a million methods to factorise
if I show you how I usually do it, it might make you even more confused
gritty hatch
Uncertain, I've been at this for around half a year now though, so I believe any resource I can gain is needed.
lethal path
I can explain completing the square cause there's basically only one process for that
gritty hatch
Uncertain, I've been at this for around half a year now though, so I believe any...
Tbf, that's because mentally things kind of broke down at some point and I had to develop a very specific level of self-awareness to get around that, but shrug.
gritty hatch
I can explain completing the square cause there's basically only one process for...
alr
lethal path
alr
why you do those steps is because:
$(x + a)(x + a) = x^2 + ax + ax + a^2 = x^2 + 2ax + a^2$
using FOIL / distributive law
grand pondBOT
southlander!
lethal path
so now you want to compare this with x^2 + 5x + any constant
are you with me so far?
gritty hatch
lethal path
okay I'll give you time to think
gritty hatch
**southlander!**
I think I understand, but I don't understand how in the world you would read that explanation format.
Like, the only way I can decipher that isnt by looking at that, its by trying to figure out what combination of concepts led to that.
lethal path
Like, the only way I can decipher that isnt by looking at that, its by trying to...
ahhhh I see
do you want to see how the Babylonians completed the square then
they did so very literally
gritty hatch
man that guy lookstired
gritty hatch
**southlander!**
If I am correct, this is stating that when you break a polynomial into two equivalent binomials itd be equivalent to both of them.... what??? what is that
why are they adding?
lethal path
why are they adding?
to complete the square
literally, if you watch the videos I sent you'll see why
it's a very literal completing of the square
gritty hatch
alr
slim eagle
any hints?
midnight plankBOT
any hints?
slim eagle
i tried using scalar triple product manipulations but didnt make out anything of it useful
midnight plankBOT
@slim eagle Has your question been resolved?
midnight plankBOT
@slim eagle Has your question been resolved?
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snow iron
midnight plankBOT
snow iron
what would the answer to this be?
fallen vigil
!nosols
midnight plankBOT
As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.
snow iron
yes and i am stuck which is why im looking for an answer as i am on a limit
fallen vigil
What did you try and/or where do you get stuck
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humble wraith
where did the 2r/1 go? can it just be ignored if arg() is used?
midnight plankBOT
where did the 2r/1 go? can it just be ignored if arg() is used?
cunning gulch
it results into fibonacci sequence and we still must find the number that is divisible with 13
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last slate
midnight plankBOT
last slate
could did they do it ?
long dagger
What
upper crag
who
last slate
why
sage helm
are you asking why they chose that parametrization
last slate
yeah
last slate
yes
sage helm
This is basically ρ = a
last slate
oh wait
oH
it all starts here
one mroe thiing
for the cylinder
x^2 + (y-3)^2 = 9 and 0=<z =< 5
why is r equal to 6 sin(theta) ?
grand pondBOT
nœn
sage helm
^2 not ^3
last slate
geometrically
sage helm
you can plot the polar curve yourself and see lol
last slate
tying
midnight plankBOT
@last slate Has your question been resolved?
humble wraith
yooo i dont get the answer key and how they got those values
midnight plankBOT
yooo i dont get the answer key and how they got those values
runic hamlet
what do you not understand about it
its the quadratic formula
$X=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
grand pondBOT
Denascite
humble wraith
yooo i dont get the answer key and how they got those values
oh its the coefficents of these
ohhh
i forgot
thanks thanks
.close
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runic nest
can someone help me with this question, for part B, how did they draw the graph like that based on part A's graph? im confused how they drew it
midnight plankBOT
can someone help me with this question, for part B, how did they draw the graph ...
rain wasp
note that $(f+g)(x)=f(x) + g(x)$
grand pondBOT
buoyant linden
can someone help me with this question, for part B, how did they draw the graph ...
they added up both values
runic nest
they added up both values
thanks
midnight plankBOT
@runic nest Has your question been resolved?
chilly cobalt
midnight plankBOT
chilly cobalt
"Suppose a1,a2,...,a2023 are positive reals, k is a positive integer where […]
i need to prove the following
crisp sun
😱
chilly cobalt
crisp sun
Im dumb duh
chilly cobalt
โหด
crisp sun
real
chilly cobalt
i need to prove the following
ive done cauchy schwarz and i got k >= 2 which probably doesnt help at all
gaunt otter
what level is this
spring wave
imo sl you mean?
chilly cobalt
nah tmo
spring wave
or tmo?
gaunt otter
thailand?
chilly cobalt
thai math olym
spring wave
oh
chilly cobalt
isnt hm just like C-S
midnight plankBOT
@chilly cobalt Has your question been resolved?
gaunt otter
chilly cobalt
oh
chilly cobalt
<@&286206848099549185>
midnight plankBOT
@chilly cobalt Has your question been resolved?
dusty tendon
midnight plankBOT
dusty tendon
Used AI to translate it, if anything is unclear I´ll explain need to resolve it, I tried, wasn´t capable of that
full leaf
if the length of the cable that goes by land costs 2400 dollars and the one that goes by water costs 6000 per meter, what was the smallest cost?
is that what the last question says
dusty tendon
Says 1/3 it´s "1/sqrdrootof3", dunno why it changed it
dusty tendon
if the length of the cable that goes by land costs 2400 dollars and the one that...
Yep and it gives this
rare hollow
bye
tepid field
dusty tendon
What
dusty tendon
dusty tendon
if the length of the cable that goes by land costs 2400 dollars and the one that...
Yeah
dusty tendon
"If the length of the cable that goes by land costs 2400 dollars and the one that goes by water costs 6000 per meter, what was the smallest cost?"
Is what I need to answer from the paragraph, the question and this
.close
midnight plankBOT
tepid field
Here reopened with the question more organized.
Not reopened until you type .reopen
dusty tendon
.reopen
midnight plankBOT
✅
dusty tendon
"If the length of the cable that goes by land costs 2400 dollars and the one that goes by water costs 6000 per meter, what was the smallest cost?"
dusty tendon
Yes
last slate
Label the horizontal coast section as x.
Then the horizontal water section is 1 - x.
So, the length of Cable 2 is L^2 = (1 - x)^2 + (1/3)^2
And the length of Cable 1 is just x
Use that to write a function for the total cost of the line
dusty tendon
Label the horizontal coast section as x.
Then the horizontal water section is 1...
What is L?
last slate
L = length of cable 2
dusty tendon
oh that´s pytagoras ok
dusty tendon
Label the horizontal coast section as x.
Then the horizontal water section is 1...
The translation changed the number into 1/3, the river´s width measures 1/rtsqrd3 so that´s elevated by 2 instead of 1/3. You read the AI right? that´s why you put the 1/3?
last slate
If it's actually 1/root(3) then use that instead
I thought it was 1/3
last slate
(1 - x)^2 = x^2 - 2x + 1
Then + 1/3 = x^2 - 2x + 4/3
dusty tendon
oh it´s -2x and +4/3, my bad I had written it over like twice, since when I did it, it was uglier and I worried you wouldn´t understand it, so I wrote it more ordered again and changed the signs by mistake lol.
Now I actually use the quadratic formula since I´m not seeing an easy way to factorize this?
last slate
Your total cost in millions for the lines would be:
You have to find the x value that makes that as low as possible
dusty tendon
Your total cost in millions for the lines would be:
Wait wait, wait, how did you get here?
last slate
The land section costs 2400 per meter
The water section costs 6000 per meter
last slate
Youll have to find the minimum x for that function
I recommend using a graphing calculator
dusty tendon
Youll have to find the minimum x for that function
I recommend using a graphing...
Oh god, right, so much stuff I don´t remember, this is hurting my head. Okay how do I get the minimum from here?
dusty tendon
And without a graph calculator, can I get this?
last slate
Maybe but you would need calculus
dusty tendon
Maybe but you would need calculus
Might as well start to learn... fuck
last slate
But the point is, the y-value when x = 0.748 is approximately 5.6
That's the minimum cost (in millions of dollars) to build this project
dusty tendon
Aight, thanks, I am a bit overwhelmed but that gets me the answer so I´ll probably stop for now.
.close
midnight plankBOT
subtle torrent
I wonder why it's xxxx b-sqrt - xxxx b+sqrt in the soln
Why isn't it the other way around
midnight plankBOT
@subtle torrent Has your question been resolved?
midnight plankBOT
@subtle torrent Has your question been resolved?
last slate
How to find all equations of lines that pass through (x_0, y_0)?
midnight plankBOT
How to find all equations of lines that pass through (x_0, y_0)?
novel lion
How to find all equations of lines that pass through (x_0, y_0)?
there's infinite lines that pass through a point
what you are probably looking for is equation of family of lines
novel lion
y - y_0 = m(x - x_0)
last slate
Oh
Got it, since for every m (x_0, y_0) satisfies
thanks
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tidal turret
midnight plankBOT
tidal turret
why is it relevant that is strictly increasing
exotic pelican
how many times does f cross y=4?
tidal turret
i am not sure
midnight plankBOT
@tidal turret Has your question been resolved?
tidal turret
how many times does f cross y=4?
only once
exotic pelican
yeah
exotic pelican
and because its continuous
grand pondBOT
exotic pelican
what do you mean?
tidal turret
you are clever
like it took me long time before I realize what you meant by that
however, I managed to see what you are saying, if its strictly increasing or strictly decreasing the function is injective and every f(x) = k is unique there is no f(something else ) = k , if the y value is k then it must be x evaluated under f
exotic pelican
oh yeah, that's a fact you just gotta see applied once somewhere and you will be able to apply it in future whenever you come across strict monotone functions
tidal turret
what about the asymptotes
exotic pelican
well what types of asymptotes can there be? What limits would be interesting to take a look at?
tidal turret
well, first of all
from the definition of horizontal asymptote
,, \textbf{horizontal asymptote} \ \lim_{x \to +\infty} f(x) = k \ \textbf{or} \ \lim_{x \to -\infty} f(x) = k \ \textbf{ where k is a constant}
grand pondBOT
Renato Chavez
exotic pelican
and the limits of g are?
tidal turret
well
from the original question g(x) is defined as
,, g(x) = \frac{f(x)}{f(x) - 4}
grand pondBOT
Renato Chavez
tidal turret
now, to see the horizontal asymptotes of this g(x) function we need to check both limits as x approaches -infinity and positive infinity
,, \textbf{basically check this two limits:} \ \lim_{x \to +\infty} g(x) = \lim_{x \to +\infty} \frac{f(x)}{f(x) - 4} \ \textbf{and} \ \lim_{x \to -\infty} g(x) = \lim_{x \to -\infty} \frac{f(x)}{f(x) - 4}
grand pondBOT
Renato Chavez
tidal turret
thats to find the Horizontal asymptotes of g(x)
it says y = 7 is a horizontal asymptote of f(x) as x approaches positive infinity
so for the first horizontal asymptote of g(x)
,, \lim_{x \to +\infty} \frac{f(x)}{f(x) - 4} = \lim_{x \to +\infty} \frac{7}{7-4} = \frac{7}{3}
grand pondBOT
Renato Chavez
tidal turret
we got the first horizontal asymptote of g(x) as x approaches positive infinity to be y = 7/3
exotic pelican
yeah
tidal turret
Now as x approaches negative infinity the horizontal asymptote for g(x) would be
,, \frac{2}{2-4} = \frac{2}{-2} = -1
grand pondBOT
Renato Chavez
tidal turret
y = -1 is the horizontal asymptote for g(x) as x appraches negative infinity
well from the definition of vertical asymptote
x = k is a vertical asymptote if
,, \lim_{x \to k} f(x) = \pm \infty \ \textbf{or } \ \lim_{x \to k^+} f(x) = \pm \infty \ \textbf{or} \ \lim_{x \to k^-} f(x) = \pm \infty
exotic pelican
wheres the difference
tidal turret
well know we either need to find the critical values or check the domain for g(x)
the domain of g(x) is R except for when the denominator is zero
exotic pelican
not quite still
tidal turret
exotic pelican
now we getting there
grand pondBOT
Renato Chavez
tidal turret
to get f(x) as x approaches some k to equal infinite we need to check which values make this g(x) function undefined
from the original question g(x) is defined as
,, g(x) = \frac{f(x)}{f(x) - 4}
grand pondBOT
Renato Chavez
tidal turret
Dom(g(x)) = R - { f(x) - 4 = 0}
Dom(g(x)) = R - { f(x) = 4}
and since we know is continuous and injective and f(5) = 4
there is a vertical asymptote at x = 5
Dom(g(x)) = R - {5}
to get other vertical asymptotes we would need to find the critical values but we are not given info about f'(x) so I would call it as finished
exotic pelican
what do you mean? As long as the function is defined and continuous at a point, it can't have a vertical asymptote there
tidal turret
what do you mean? As long as the function is defined and continuous at a point, ...
thats for f(x), for g(x) we can have vertical asymptotes
exercise is asking us to find the HA and VA of g(x), not of f(x) btw
exotic pelican
yeah but g is continuous on all of its domain so it cannot have any other vertical asymptotes
tidal turret
continuous at all except x = 5 I pressume
exotic pelican
its undefined at 5 yeah
tidal turret
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solid pilot
hello, can someone dm me so i can send my answers to be checked? its for a extra credit assignment and i dont want to flood this chat. (there are 10 that i need to double checl)
midnight plankBOT
hello, can someone dm me so i can send my answers to be checked? its for a extra...
steep hinge
this might be against rule
fallow scarab
just post one at a time here
solid pilot
this might be against rule
oh my bad
i just finished reading them i and didnt think it meant that too 😭
carmine sigil
!show
midnight plankBOT
Show your work, and if possible, explain where you are stuck.
lavish venture
oh my bad
i just finished reading them i and didnt think it meant that too 😭
you’ve been in the server for nearly 2 years
solid pilot
you’ve been in the server for nearly 2 years
i never asked about going into dms
so i read over the rules again
💀
lavish venture
💀
carmine sigil
sqrt(9 + 81) sqrt(4 + 36) = sqrt(90) sqrt(40) = sqrt(10)^2 * 3 * 2 = 60. Seems good to me.
Is this how you arrived at the value?
actually, you can also get it from recentering the corner to (0, 0) and then making a matrix.
then finding the determinant of it
which is probably less calculation, but if you're not taking linear algebra it might not make sense.
solid pilot
Is this how you arrived at the value?
i had the teacher help me get to that so yeah?
carmine sigil
wait, if a teacher told you the answer, then why are you asking us to double check your work?
lavish venture
just making sure you know how to do it
carmine sigil
man, I'm so glad this helpee is so thoughtful to make sure I still remember how to do the pythagorean theorem.
I'm going to take my kitties outside now
solid pilot
wait, if a teacher told you the answer, then why are you asking us to double che...
she didnt give me the answer, i had to solve it myself but i meant she helpped me get thro the equation/steps ykwim
im not confident in math so i ask alot 😭
i fall behind easily in this subject unfortunately
ok i gtg rn..ill be back later if im able to. thank you for that tho!
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alpine gyro
How did f went to f'?
midnight plankBOT
How did f went to f'?
alpine gyro
n=1 in the second picture.
junior flower
just term-wise differentiation
write out the first few terms and differentiate them and you should see
it isn’t obvious you should be able to differentiate like that with infinitely many terms but you can
alpine gyro
My bad, didn't see the rule for derive that you need to add 1 to the lower-bound.
Silly, thanks for the help.
junior flower
well it just depends how you index stuff
alpine gyro
Yeah, I just didn't see that $$f(x)=\sum_{n=0}^{\infty}a_{n} (x-a)^n$$ $$f'(x)=\sum_{n=1}^{\infty}na_{n} (x-a)^{n-1}$$
grand pondBOT
Good
alpine gyro
Thought it were also n=0 for the derivate of f.
This was still wrong...
Should probably take a rest.
junior flower
when n is 0 the first term in the derivative is 0
alpine gyro
How did f went to f'?
Yeah, but shouldn't the exponent of f' be x-1?
Say that's the day. Thanks for help.
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gritty hatch
-
Can someone explain this process?
-
Please explain the factorization process aswell as how completing the square works.
midnight plankBOT
1. Can someone explain this process?
2. Please explain the factorization proces...
gritty hatch
1. Can someone explain this process?
2. Please explain the factorization proces...
original problem:
oak slate
(x + a)² = x² + 2ax + a²
you want to make the LHS in the form x² + 2ax + a² so you can factor it as (x + a)², for some value a
the coefficient of x should be 2a
and you know the coefficient is 5
so what should 'a' be?
dusky grove
hello
dusky grove
someone pls really quickly
tell me if the 18.1 is right
or my answer
because im going to look like a fool
if my one is not right
hard shard
,calc (12/52)(40/51)
grand pondBOT
Result:
0.18099547511312
Result:
0.17751479289941
im unsure how you got your answer
hard shard
the face cards are K Q J
if you are doing 36/51, then you arent considering A as a number
dusky grove
mb for the mixup
dusky grove
if you are doing 36/51, then you arent considering A as a number
yeah exactly
but thats the thing
is ace not like a seperate thing
neither face card nor number
hard shard
was this given by your teacher?
it kinda looks like a standardized test so im not sure
if it is given by your teacher, if you explain your thoughts calmly and preferably after or before class, they should be sympathetic
dusky grove
was this given by your teacher?
honestly i dont think so
dusky grove
it kinda looks like a standardized test so im not sure
this is probably it
so wish me luck
thanks
hard shard
youre welcome
dusky grove
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neat canyon
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neat canyon
Hi guys!
This is not true right?
im supposed to prove this identity but i think its wrong
in the RHS I think it should be 4*cos^2(2x)
then it’d be true
next bramble
Have you got a better quality image, I'm strugling to read that
dull yoke
$\cot^2 (x) - \tan^2 (x) = \frac {4\cos (2x)}{\sin^2 (2x)}$
grand pondBOT
dull yoke
@neat canyon what have you tried
neat canyon
$\cot^2 (x) - \tan^2 (x) = \frac {4\cos (2x)}{\sin^2 (2x)}$
yes
neat canyon
<@255708580810981378> what have you tried
hold on
neat canyon
<@255708580810981378> what have you tried
and this doens't simplify to what it should
grand pondBOT
lethal path
you should know the identity that can simplify cos^2 x - sin^2 x
also $\sin x \cos x = \frac{1}{2} \sin(2x)$
grand pondBOT
southlander!
lethal path
so square both sides of that
dull yoke
wait ur going in the wrong direction south lol
we started on the right side of the equation
trying to get to the left
neat canyon
$\frac {a-b}{c} = \frac ac - \frac bc$
i did i think
lethal path
oh wtf
lethal path
okay I see now
dull yoke
well show me what u get
neat canyon
well show me what u get
dull yoke
ok now what
dull yoke
turn 1/sin^2 into another trig
neat canyon
I don’t think so
dull yoke
before u rule it out
use ur pythagorean trig identities
rewrite csc^2 in terms of ctg^2 and write sec^2 in terms of tg^2
lethal path
okay I can finally jump in
$\frac{c^2}{s^2} - \frac{s^2}{c^2} = \frac{c^2 c^2 - s^2 s^2}{s^2 c^2}$
$= \frac{(c^2 + s^2)(c^2 - s^2)}{s^2 c^2}$
$= \frac{4(c^2 - s^2)}{(2 sc)^2} = \text{RHS}$
grand pondBOT
southlander!
lethal path
@neat canyon if you wanted an alternative solution
lethal path
I just think it's easier to start by converting one side into sines and cosines only
and that it's easier to combine a + b or a - b into one fraction
rather than try to split a fraction into a + b or a - b
hence my thought process
also yeah you better keep an eye out for (a + b)^2 or a^2 - b^2, identities like these
dull yoke
if theres a side with double angle i generally just start there
but this is a great approach
lethal path
wow interesting
also yeah does rely on a good deal of pattern matching, the multiplying by 4 part
on top and bottom
neat canyon
.solved
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last slate
Idk where to start 💀
midnight plankBOT
Idk where to start 💀
rain wasp
hint, it starts with q and ends with uadratic formula
twin rock
Idk where to start 💀
move the 24 to the left side
and then factorise or quadratic formula and find the zeros
i think...
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twilit field
Let $T$ be a linear operator in a finite dim vector space, let $c$ be any scalar.
Determine the relationship between the eigenvalues and eigenvectors of $T$ and those of $T-cI$
midnight plankBOT
Let $T$ be a linear operator in a finite dim vector space, let $c$ be any scalar...
grand pondBOT
A dense set(Ping when reply)
twilit field
The eigenvalues of T are the solutions to $det(T- \lambda I) =0$
\
The eigenvalues of $T-cI$ are the solutions to $det(T-(c I+ \beta I))=0$ , where we solve for $\beta$
grand pondBOT
A dense set(Ping when reply)
twilit field
<@&286206848099549185>
hard umbra
cI + βI = (c + β)I
hard umbra
yes
grand pondBOT
A dense set(Ping when reply)
twilit field
is the relation
Thanks
I'll figure out wth is happening to the eigenvectors after lunch
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subtle torrent
Why does x just vanish
midnight plankBOT
Why does x just vanish
fleet axle
happy t hanksgiving
fleet axle
Why does x just vanish
it'w becuase of
the periodicity of your function
which is pie, right?
subtle torrent
Hmm you mean the period of cos x+y ?
fleet axle
Hmm you mean the period of cos x+y ?
yes 👍 in particular the period in y
where you imagine x is a constant
fleet axle
well the first step i guess
is to accept (but not without question)
that x is just an "arbitrary" constant
subtle torrent
I mean like, ∫cos(x)dx is different from ∫cos(2+x)dx
fleet axle
as far as the integration over y is concerned
fleet axle
I mean like, ∫cos(x)dx is different from ∫cos(2+x)dx
okok this is good
so
the limits are over a period
cos(x) has a period of two pies (where your actual function, |cos(x + y)|, has a period of pie)
but
yeah, consider if the integral was over a whole period
subtle torrent
I mean like, ∫cos(x)dx is different from ∫cos(2+x)dx
So you mean like, it's just a shift by a small units, since it's still an integral in a whole period so this shift doesn't matter? Like, whatever it's 2 or 3 etc, I mean like, ∫cos(x)dx = ∫cos(2+x)dx in the period of cosx?
fleet axle
So you mean like, it's just a shift by a small units, since it's still an integr...
yeah that sounds basically perfect!
idk exatly what you mean by "small" units
but yes it's just a shift
yes exactly
subtle torrent
Thx!!
grand pondBOT
Ginger
fleet axle
np! 🎉
subtle torrent
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ebon kiln
How do I prove that this number is natural? <@&286206848099549185>
midnight plankBOT
How do I prove that this number is natural? <@&286206848099549185>
spring wave
which number?
spring wave
actually
(a-root(3) )^2 = ?
figure out a for both sqrts
and show the difference of a is natural
formal orchid
$7 - 4\sqrt{3} = (\sqrt{4}- \sqrt{3})^2$
spring wave
yes
grand pondBOT
Roman_Garland
formal orchid
I think you already figure it out in the spam of time I was fixing this
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lone hound
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lone hound
worthy wing
Is there any difference between the first and second image?
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@lone hound Has your question been resolved?
lone hound
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sturdy kraken
u need help?
lone hound
yes
lone hound
which grade r u in
10th
lone hound
what is igcse
lone hound
oh
midnight plankBOT
@lone hound Has your question been resolved?
empty tide
known function f(x) = 5 - 2x determine the:
a. function range, if domain = { x| -1 ≤ x ≤ 4, x e integer }
b. shade of 5
c. f(a) = 9, determine the value of a
midnight plankBOT
known function f(x) = 5 - 2x determine the:
a. function range, if domain = { x| ...
empty tide
i dont understand the definition of b, "shadow of 5" by the way
the raw translation is shade
i especially dont know how to do a
empty tide
known function f(x) = 5 - 2x determine the:
a. function range, if domain = { x| ...
if any indonesians would like the raw
diketahui fungsi f(x) = 5 - 2x tentukan
a. range fungsi, jika domain = { x| -1 <(with line under) x <(with line under) 4, x e bilangan bulat }
b. bayangan dari 5
c. f(a) = 9, tentukan nilai a
i have a remedial tomorrow for functions i might flunk it again since i dont know any of these 😭
deft sage
ok, for part a do you know what range is
empty tide
no......
one of my friends told me that
it was like
an arragement from big to small
@deft sage ^
hoary sparrow
range is the biggest number - the smalles number
empty tide
okayy, what about the symbols?
i dont know any of them
by the way i didnt know how to find the 'e' symbol
sorry
deft sage
so the range is the outputs of the function
hoary sparrow
are you indonesian
empty tide
yes]
deft sage
what outputs can you get from f(x) = 5 - 2x
if the smallest x is -1 and the biggest x is 4
hoary sparrow
bayangan itu hasil jika dimasukin nilai x nya
jadi kayak f(x) = 2x+3, kalo x =4 jadi f(4) = 11, maka bayangannya 11
empty tide
what outputs can you get from f(x) = 5 - 2x
if the smallest x is -1 and the bigg...
uh 4, 3, 2, 1, 0, -1?
empty tide
jadi kayak f(x) = 2x+3, kalo x =4 jadi f(4) = 11, maka bayangannya 11
oh okok
empty tide
jadi kayak f(x) = 2x+3, kalo x =4 jadi f(4) = 11, maka bayangannya 11
so bayangan is supposed to be the result?
deft sage
uh 4, 3, 2, 1, 0, -1?
yes those are the x you can plug in, so to get the range you just plug them in and you get your outputs
your set of outputs is the range
empty tide
yes those are the x you can plug in, so to get the range you just plug them in a...
what do the symbols mean?
likethis thing <
and the e
empty tide
kelas brpp
hoary sparrow
8
deft sage
< means less than
≤ means less than or equal to
deft sage
when it says x e Integer
it means x is an integer
empty tide
when it says x e Integer
it means x is an integer
integer is like negative right?
deft sage
yes
empty tide
anak olim ni bos
omg congrats bro
deft sage
integer are all numbers like ..-3,-2,-1,0,1,2,3...
hoary sparrow
yang bagian c. its 5 - 2x = 9, so x = -2
empty tide
known function f(x) = 5 - 2x determine the:
a. function range, if domain = { x| ...
i still dont really understand what its asking me though for a
hoary sparrow
becanda
empty tide
yang bagian c. its 5 - 2x = 9, so x = -2
gw bagian c udh bener tp lupa aja cause like the test was 2++ weeks ago
deft sage
ok so your domain or inputs are -1, 0, 1, 2, 3, 4
your outputs are f(-1), f(0), f(1), f(2), f(3), f(4)
your range would be all of the outputs
hoary sparrow
yeah like what @Epic said
empty tide
ok so your domain or inputs are -1, 0, 1, 2, 3, 4
your outputs are f(-1), f(0),...
oh then am i supposed to write the outputs from f(4)-f(-1)?
deft sage
yeah thats the range for part a
empty tide
am i allowed to ask more questions for function here?
deft sage
sure
empty tide
for question 2 its asking me: a function has a formula of h(x) = ax + b and h( -5) = 26 and h(4) = -1, determine:
a. value of a and b
b. map ofrom 6
c. h(-2m+ + h(3m)
what im confused about is what they meant by map
cause i dont think i've learnt anything about it
and im pretty sure im not supposed to draw a literal map
empty tide
for question 2 its asking me: a function has a formula of h(x) = ax + b and h( -...
to find the values i just insert h(x) with h(-5) and h(4) while using SPLDV right? (spldv is Systems of Linear Equations in Two Variables) i used google translate i dont know if its accurate for SPLDV
hoary sparrow
itu buat grafiknya gk?
hoary sparrow
to find the values i just insert h(x) with h(-5) and h(4) while using SPLDV righ...
iya pake SPLDV
empty tide
itu buat grafiknya gk?
gkk
hoary sparrow
masukin aja itu
deft sage
how would you use SPLDV
empty tide
how would you use SPLDV
like h(-5) = a(-5) + b
then it'd be
-5a + b
and theres another one so
h(4) = a(4) + b
= 4a + b
-5a + b = 26
4a + b = -1 -
deft sage
yeah
deft sage
yes thats how you do part a
empty tide
i got b. and c. in my test wrong tho
i still dont know what map means after aksing
😭
empty tide
the raw is
peta dari 6
@hoary sparrow hello
do you know how to say peta dari 6 in english
oh wait i think i know how to do c. as well
the only thing i dont know is the definition of map
deft sage
ok i see
so it means image of 6
when x = 6
so it's just asking for h(6)
after you've found a and b
empty tide
ohh ok so i just switch x with 6 right and do the formula normally?
like f(x) = fx + b? but switch x with 6?
deft sage
image in math means the output
empty tide
OHHHHHH
ohh okk i've never heard of output and input before
actually
whenever i used youtube to learn functions i wouldnt get it cause i didnt know what output n input was
deft sage
what words do you use in english instead of input and output?
empty tide
what words do you use in english instead of input and output?
image = map = output right?
its like peta/bayangan
which translates to shadow idk
input i dont know
oh input is just domain
deft sage
input = preimage if youve seen that word
empty tide
no i havent 😭
empty tide
input is what you put inside the function
i just call it x
input is just x
i dont think thats right but thats how i was taught
empty tide
wait is output f()? and then input completes it?
empty tide
yea..
deft sage
i guess
empty tide
i thought input was x from this
deft sage
yes x = 6 is one input
and the output is h(6)
empty tide
when theyre asking for the value of f(-4) in a formula of f(3x + 1) = 6x - 3 they just want me to switch the f(x) right?
deft sage
you put the 6 inside the function
empty tide
yes x = 6 is one input
and the output is h(6)
ooh okok
what does it mean if theyre asking me for formula function f(x)?
deft sage
what does it mean if theyre asking me for formula function f(x)?
like in what example
empty tide
like in what example
we know that f(3x + 1) = 6x - 3, determine
a. formula function f(x)
b. value f(-4)
deft sage
its asking you to find f(x) given f(3x + 1) = 6x - 3, i think you should watch a video it's kinda long to explain
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last slate
The exterior diameter of an iron pipe is 25cm and it is one cm thick. Find the whole surface are
of the pipe it is 21cm long.
midnight plankBOT
The exterior diameter of an iron pipe is 25cm and it is one cm thick. Find the w...
lilac sage
The exterior diameter of an iron pipe is 25cm and it is one cm thick. Find the w...
split it into 2 endless cylinders and 2 rings
midnight plankBOT
@last slate Has your question been resolved?
last slate
Can we simplify this?
1/(x+1)
midnight plankBOT
Can we simplify this?
1/(x+1)
last slate
For example, we can simplify (5+x)/(x) = 5/x + 1
By simplifying I mean to bring it as a combination of fractions whose denominator is a monomial.
rose trout
Not really.
Splitting a numerator is easy, as your example showcases. Splitting a denominator isn't so much
fervent ember
$\frac{1}{x+1}=\frac{\frac{x}{x+1}}{x}$ is a fraction with a monomial denominator.
grand pondBOT
ThM
last slate
Condition 2: The numerator should not contain fractions.
fervent ember
changing the requirements afterwards isnt really fine.
last slate
Wdym?
midnight plankBOT
@last slate Has your question been resolved?
dreamy lichen
By simplifying I mean to bring it as a combination of fractions whose denominato...
If that was possible, the result would be defined at x = -1
while our original function 1/(x+1) is not
so it cant be possible
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rotund hull
hi i need to solve this laplcae transform, i've gotten to the part where you then need to do an inverse laplace transform but im not sure how to do the partial fraction for this
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hi i need to solve this laplcae transform, i've gotten to the part where you the...
@rotund hull Has your question been resolved?
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dusty tendon
If a coin is thrown 5 times what´s the probability of getting 3 tails?
midnight plankBOT
If a coin is thrown 5 times what´s the probability of getting 3 tails?
next rover
yes the last one
junior flower
yes
dusty tendon
Thanks, thanks oh god, I just went through an hour of videos, + practice on 4 topics after getting help in the morning, because I didn´t know jack shit about how to do this
That´s it.
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tidal turret
let f(x) = 4x - x^4, find Range of f
midnight plankBOT
let f(x) = 4x - x^4, find Range of f
tidal turret
Dom(f) = R
grim vector
So range f ?
junior flower
image was fine before the edit
tidal turret
yeah is whatever, how do I find Range/Image guys/gals
tired vapor
the range is the set of values that the function can take, so because this is defined for real numbers, try finding the maximum point/minimum
tired vapor
yep
junior flower
yes
tired vapor
yeah
grim vector
Wunderbar
tidal turret
we have a maximum at f(1) = 3
why
tired vapor
essentially at that point, the function goes from increasing to decreasing, so it's maximum. (you learn this later with calculus, unless you already know).
tidal turret
yeah, but is the only critical value
junior flower
this is irrelevant
tidal turret
f''(x) = -12x^2
i mean if we plug in x = 1 to the second derivative we can see is negative output
?
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tidal turret
The number of real solutions to the equation
[
4x + \cos^2(x) = 0
]
is:
[
\boxed{\text{infinite}} \quad \boxed{1} \quad \boxed{2} \quad \boxed{3}
]
midnight plankBOT
The number of real solutions to the equation
\[
4x + \cos^2(x) = 0
\]
is:
\[
...
grand pondBOT
Renato Chavez
tidal turret
bolzano + monotonically decreasing
crystal dagger
Guys I need help with solving linear equations using Matrix method , is it tough ??? Exam is after 4 hours . And I need to learn 10 questions