river aurora
Yeah I didn't pick up on that
#help-39
drowsy tendon
Hey Guys
Can I get some steps to how to solve this? ( just want to understand how to get to the answer)
inland lantern
in some order:
- combine fractions
- multiple the x^2 into the parenthesis
- simplify
pearl pondBOT
@drowsy tendon Has your question been resolved?
pearl pondBOT
drowsy tendon
.reopen
pearl pondBOT
✅
drowsy tendon
in some order:
- combine fractions
- multiple the x^2 into the parenthesis
- sim...
you said multiply the x^2 into the parenthesis and got 2x^2 - 2x^2 which if we simplify more will equal to 0, right?
inland lantern
you said multiply the x^2 into the parenthesis and got 2x^2 - 2x^2 which if we s...
ya
narrow solstice
Don't know what to do with this tangent in this differential equation
narrow solstice
devout sluice
Don't know what to do with this tangent in this differential equation
tan^-1 ?
narrow solstice
I just need to bring the tan over without bringing the y
devout sluice
np
narrow solstice
.close
rancid mirage
Need help
rancid mirage
I got 13 as the answer but it's wrong
pearl pondBOT
@rancid mirage Has your question been resolved?
pearl pondBOT
@rancid mirage Has your question been resolved?
frosty badge
Need help with this system
pearl pondBOT
Need help with this system
shrewd basin
You could rearrange for y in the first equation and plug that into the second equation.
pearl pondBOT
@frosty badge Has your question been resolved?
craggy zephyr
pearl pondBOT
warm current
Try u=ln x
pearl pondBOT
@craggy zephyr Has your question been resolved?
craggy zephyr
idk where the du goes... but this dont look like the right answer
what do i do after the dx=wtv is found... do i plug it into the dx??
warm current
This is incorrect
derp no it's not
Where you wrote du here, you needed to write du*x
warm current
secret powers
craggy zephyr
SHEESH
jolly parrotBOT
warm current
good calculator, but you'll still have to show your work 
craggy zephyr
yeah i didnt know that existed...
pearl pondBOT
@craggy zephyr Has your question been resolved?
upbeat glade
Hey, one thing you should always do whenever you find an antiderivative is to simply differentiate it to see if we obtain the function that we were integrating. If we do then we know know its right.
upbeat glade
On the right handside what I did was first differentiate the antiderivative obtained wrt to ln(x) and then simply differentiated the composition, so that we have it wrt x and boom.
pearl pondBOT
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upbeat glade
.close
pearl pondBOT
pearl pondBOT
Available help channel!
burnt minnow
pearl pondBOT
@burnt minnow Has your question been resolved?
burnt minnow
<@&286206848099549185>
pearl pondBOT
@burnt minnow Has your question been resolved?
pearl pondBOT
pearl pondBOT
Available help channel!
hybrid field
How can i show that this time series is invertible? X_t - X_{t-1} = 2Z_{t-4) + Z_{t}
strong atlas
you need to check that the autoregressive (AR) polynomial has all its roots outside the unit circle
strong atlas
the AR polynomial should not have any roots with magnitude less than 1
strong atlas
ye
hybrid field
ok, so b lays inside on the unit circle
hence its not invertible
Ok @strong atlas dont think i provided enough context
initially, this was a SARIMA model however with some conditions of coefficients, i was left with the above time series
So in that case, i would check the MA terms right?
pearl pondBOT
@hybrid field Has your question been resolved?
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@hybrid field Has your question been resolved?
upper shoal
Hello
upper shoal
Part B,
Im unsure how to find it, as you cant just dy/dx it find for r and throw it in
Also I have No Idea what the green means
This is the answer
versed mica
well the perimeter of the sector would be 2 times the length of the radius plus the length of the sector
and it tells u the area of the sector is 100
u know the area of a circle is pir^2
since a sector is a portion of a circle
r>sqrt(100/pi)
because if the total area was 100
then pi * r^2=100 and r=sqrt(100/pi)
the area of sector is 1/2 * r^2 theta
=100
the length of a sector is r theta
where did u get stuck
the length of the sector will be 200/r because
1/2 *r^2 theta is 100
therefore r^2 theta is 200
therefore theta is 200/r^2
since u know the length of the sector is r theta
then length of the sector is
r(200/r^2)=200/r
so total perimeter is 2r+200/r
to find the minimum value
differentiate with respect to r
=2-200/r^2
=0
so 2r^2-200= 0
r^2-100=0
r= plus or minus 10 but u can’t have negative radius
so r=10
then plug that value of the radius back into the formula for perimeter
does that make sense
@upper shoal
versed mica
👍🏻
upper shoal
Ah, I did what you did first, and differentiated it. I just lost after
2-200/r^2 becuase it was in a weird form.
This makes very much sense how you got r^2-100=0 lol stpid me
upper shoal
well the perimeter of the sector would be 2 times the length of the radius plus ...
And for your explination on this sry its a bit like a foreign language didnt understand :<
But I will go through it in school with a tchr , so thank you for all the explination
@versed mica
versed mica
yo
versed mica
Ah, I did what you did first, and differentiated it. I just lost after
2-200/r...
yes just get a common denominator
what didn’t u understand
upper shoal
yeah, very straight forward, I just left it as -200r^-2
versed mica
upper shoal
and I forgot that it is also -200/ r^2
and then you do a bit of switcheroo
Got that
upper shoal
The whole worded text basically
upper shoal
No I understand this
upper shoal
Just how area of circle > area of sector
Therfore the conclusion r > 100/Pie
Is made
upper shoal
Yes
upper shoal
yep
versed mica
the fraction can be found with the angle it makes
upper shoal
rTheta?
versed mica
that would be the length
think about it like this
area= pi r^2 * theta/360
where theta is in degrees
does that make sense
because there’s 360 degrees in a circle
upper shoal
Yes
versed mica
ok then if u just converted throat into radians
u would get
1/2 r^2theta
because 360 is 2pi radians
pi cancels
upper shoal
ah ok
versed mica
and u get 1/2 r^2 theta
and u we don’t know theta
but we can express it as a function of the radius
versed mica
which is hinted at because the perimeter formula they give u is in terms of r
all of these optimization problems
r all about expressing one variable in terms of another
upper shoal
OH I understand now
versed mica
then differentiating with respect to that variable to find extrema
max/min
by setting it equal to zero
the derivative
upper shoal
The only part Im not sure is,
How do you know, know which formula to use
Of sector
upper shoal
ok
versed mica
but then
u would have to make sure
the length of the sector was in degrees too
because s= r theta is for radians
the angle has to both be in terms of radians or both in terms of degrees
radians is simpler imo
and used more as u get more advanced
upper shoal
and used more as u get more advanced
Ok good to know, bc we dont have any of that degree question stuff anymore
versed mica
s= r theta is simpler than s= 2 pi r * theta/360
and similarly u can easily convert this into radians by rewriting 360 as 2 op
pi
then 2pi cancels
and u get r theta
versed mica
day
upper shoal
haha day then
versed mica
only 130
upper shoal
7:30 for moi
versed mica
have a good night then
waxen falcon
help
waxen falcon
help pls !?
tough mica
Notice that the ratio of cups of lemon juice to cups of water is 3:5, and this ratio won't change no matter how large the batch is
Let the cups of lemon juice used be x
Do you see how to solve for x?
pearl pondBOT
@waxen falcon Has your question been resolved?
torn willow
can someone help me with this question pls
pearl pondBOT
can someone help me with this question pls
nimble lily
can you make some equations linking various bits of the question
like how many women are there, w, compared to the men, m
how many of the women were unmarried
and so on
and then let n = number of people there
n = ????? (some formula in a bunch of variables)
pearl pondBOT
@torn willow Has your question been resolved?
torn willow
k
so the number of unmarried women = 40% of four-fifth of men?
unmaried women = (40/100)(4x/5), where x is the no. of men
the question says there were thirty women unmarried
so 30 = (40/100)(4x/5)
<@&286206848099549185>
yes!!!!
idk if that equation is right?
did u read the actual question?
pearl pondBOT
@torn willow Has your question been resolved?
torn willow
<@&286206848099549185>
hasty patio
Just to make sure I don't misread, 40% of the women or men are unmarried?
@torn willow
torn willow
i think its 40% of four-fifth of men are unmarried
cuz the question says 40% of "whom" were unmarried
and "whom" refers to four-fifths of the men
i guess
does it make sense @hasty patio
hasty patio
Kay
Wait ya
@torn willow
Why not, take it slow, and do some modelling, rather than trying to come up with an algebraic equation that might go south with a careless mistake
pearl pondBOT
@torn willow Has your question been resolved?
mighty canopy
The area of a sector of a circle is 64sq.cm . Find its radius if the perimeter is a maximum
pearl pondBOT
The area of a sector of a circle is 64sq.cm . Find its radius if the perimeter i...
last summit
Don't close and reopen a new channel for the same question
mighty canopy
yeah sorry
last summit
!original
pearl pondBOT
Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
mighty canopy
I'm just in a rush
last summit
Doesn't change the rules unforunately
mighty canopy
mighty canopy
Doesn't change the rules unforunately
yeah im really sorry about that
mighty canopy
I guess its saying that the perimeter is already at its maximum value?
<@&286206848099549185>
last summit
What is the formula for the area of a sector
mighty canopy
A = 1/2 r^2 θ
last summit
do you know how to maximize the perimeter of that sector using calculus
fallow cosmos
This is a strange question. The perimeter of the section is at least 2r, and no matter how large an r we choose there will be a theta that gives the right area. So there's no such thing as a maximum perimeter here.
last summit
Possibly there was a part 1 of this question?
mighty canopy
no this is all there is
last summit
I would ask your teacher about it then, or whoever assigned it to you
mighty canopy
alright
cursive wraith
Well I'm guessing if you let $A = {(r,\theta), \frac{1}{2} r^2 \theta = 64}$, you're trying to find $\arg \max_{(r,\theta)\in A}(r\theta + 2r)$
jolly parrotBOT
rafilou2003
light helm
maybe they actually wanted min
fallow cosmos
That would make sense.
zealous oriole
pearl pondBOT
merry carbon
,rccw
jolly parrotBOT
cinder flower
slay
merry carbon
first question, is part a correct?
How did you do part a? it's not very clear 
zealous oriole
so basically I just used the volume of a cone formula
(pi)(5^2)(-50/3)
the -50/3 came from -3/10 cm/hr but when it reaches 5 cm
so i set up a proportion for -3/10 x = 5
and then i got -50/3
merry carbon
part (a) doesn't need your -3/10 cm/h, it's just volume
merry carbon
so basically I just used the volume of a cone formula
What's the volume of a cone formula? 
merry carbon
Yep, r isn't 5 in that case, but the height h is 5
If the height is 5, you can use that you're in the bigger cone to find r from there 
zealous oriole
if the height is 5,
then
r = h/5?
wait ik i need to find the relationship bween h and r first, but how do i do it
merry carbon
Note that you're in a right circular cone...
zealous oriole
ohh h/2
merry carbon
Yea the larger cone has radius 5, so you know that 5/10 = r/h (similar triangles), so r = h/2
zealous oriole
righttt i just realized 😅
so then itd become pi*(5/2)^2 * 5/3
which is 125pi/12
merry carbon
looks good to me 
zealous oriole
now hb part b?
dV/dt = 2pi(r)(dr/dt) * 1/3(dh/dt) ?
would i or would i not have a dh/dt after the 1/3?
merry carbon
Well, bear in mind that you can find the volume of water V in terms of h
zealous oriole
?
merry carbon
You can then of course use chain rule nicely
Basically you have that $V = \frac{\pi}{12}h^3$ - and that you were given $\dv{h}{t} = -\frac3{10}$
jolly parrotBOT
@merry carbon
merry carbon
<@788085606483361802>
You can find dV/dt from there ⛓️
zealous oriole
Im getting a very wrong answer by doing that...
i mean my dV/dt was 3(pi/12)^2 * (pi/12 * dh/dt)
is that even right in the first place?
merry carbon
i mean my dV/dt was 3(pi/12)^2 * (pi/12 * dh/dt)
how did you get that? 
zealous oriole
idk thats what im saying
merry carbon
Well note we want dV/dt, and we have dh/dt, but we have V in terms of h...
pearl pondBOT
Available help channel!
unique flame
Hey, I'm new to logarithms (doing Pre-Calc 12), and can't seem to solve 5^x=2^(x-3). Logging both sides with a base of 2 leads me to log5/log2=(x-3)/x, and base 5 gets the reciprocal of both. Can I have some steps through the problem please? I got some help from someone else, but I need confirmation.
unique flame
$\frac{log5}{log2} = \frac{x-3}{x}$
jolly parrotBOT
Just_Iced
pearl pondBOT
@unique flame Has your question been resolved?
calm root
pearl pondBOT
calm root
solved this onne
where m = 1 and x = -1
then i solved the second case (x^2 + 2x + m = 0) but was just unsure why its (x-1)(x-m) and not (x+1) (x+m) if the common root is -1 for both quadratics
answers^
i got both but i dont understand how the x^2 + 2x + m quadratic can't be expressed as (x+1)(x+m)
but its (x-1)(x-m) if the root is -1
calm root
yeah i know
but from the first part
it gives us a root
so we use that
its like (rootm x +1)^2 and then the first quadratic where m = 1 and x = -1
pallid leaf
The root being -1 will give you a value for m
Since the original expression cant be factored like that
It means
That the reason why the root is -1
Is m
sudden dove
pearl pondBOT
sudden dove
im getting x+3y=0
i replaced x^2 y^2 with xx1 and yy1 and i replaced x with (x+x1)/2 and y witth (y+y1)/2
what did i do wrong?
robust sentinel
do you have to correct answer
sudden dove
another approach i tried was finding dy/dx for the eq i got
dy/dx = - [(x+2)/(y+1)]
now for (-1,2) i get dy/dx = -1/3
And line equation at (-1, 2)
y-2 = -1/3 (x+1)
y = -1/3 x -1/3 + 2
y = -1/3x +2/3
which is different but still wrong
sudden dove
do you have to correct answer
the answer booklet says answer is C but im not sure if the booklet is right or no
pearl pondBOT
@sudden dove Has your question been resolved?
pearl pondBOT
@sudden dove Has your question been resolved?
pearl pondBOT
@sudden dove Has your question been resolved?
latent tartan
wouldn't this be derivative ?
and then sub in your (x,y) point to get the slope of the tangent line
and then add a constant accordingly to get it to the point
azure pilot
pearl pondBOT
sly dagger
well lets see
you can only move upwards or to the right
so first determine how many ways can we get from A to B?
azure pilot
3 ways from B to C
sly dagger
and then finally for D?
sly dagger
not necessarily
azure pilot
multiply
sly dagger
yes
sly dagger
yep
azure pilot
pearl pondBOT
strange quiver
!status
pearl pondBOT
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
Ok please show your work
azure pilot
so 0, 1, 2, 3, 4 can be placed in 3 places, [ ] [ ] [ ] ,the first one can be 4 numbers, the seoncd one can also be 4 numbers, the last one can be 3 numbers
then 4*4 = 16 then times 3 is equal to 48
strange quiver
So 48
wraith plinth
when we talk about coloring a graph, are we referring to coloring the verticies of the graph or the faces of the graph
wraith plinth
cuz like I have this problem
if we are talking about coloring the verticies then it is clear to see that the max degree of a vertex is 2 so it has at most 12 edges
but if we talk about faces it is much harder to prove
merry python
It's about coloring the vertices, but I don't think you claim about the max degree of a vertex being 2 is correct
wraith plinth
Actually yea you right that logic wouldn’t work
I’m not really sure how to prove it then
merry python
I don't have full answer either rn, but more than 12 edges would imply a vertex having at least degree 5
I think that might give a contradiction
wraith plinth
How would it give a vertex having degree 5
wraith plinth
Oh yea the sun of the degrees would be >= 26
merry python
more than 12 would actually imply that vertex having exactly degree 5 and needing a unique color
wraith plinth
I think 2 would need degree 5 cuz we need degrees to sum to 26, like we can’t have 5, 4, 4, 4, 4, 4 cuz that is 25
And then if any 2 of the other 4 vertices are connected then we have essentially K4 which needs 4 colours so we have contradiction
^ that make sense?
merry python
yeah, you're right. That would mean those 2 vertices have unique colors though
wraith plinth
Alr makes sense
I was trying to approach it through theorems from class which was much harder than it needed to be ig
Ty for the help
merry python
I don't know a lot of theorems so I had to use this approach
wraith plinth
Yea using contradiction works a lot in graph theory so it’s a pre good approach
.close
pearl pondBOT
pearl pondBOT
Available help channel!
serene hamlet
Any ideas?
pearl pondBOT
Any ideas?
serene hamlet
I do know what a boolean function of n degree means
But for example how am I supposed to know how many different boolean functions of n degree are there? I can make up as many as I want, no?
also im unsure regarding the surjective and injective part
Like for example f(a, b, c, d) = a + b + c + d is ... I dont know what it is lol
pearl pondBOT
@serene hamlet Has your question been resolved?
serene hamlet
<@&286206848099549185>
opal moss
I do know what a boolean function of n degree means
a quick google will tell you that the degree of a boolean function is the number of its inputs
serene hamlet
a quick google will tell you that the degree of a boolean function is the number...
yeah I already said that I do know what it means
serene hamlet
I do know what a boolean function of n degree means
.
opal moss
But for example how am I supposed to know how many different boolean functions o...
if you say define two degree 3 functions, f(a,b,c) = 1 regardless of a,b,c. and g(a,b,c) = 1 regardless of a,b,c, then they're really the same function
serene hamlet
dont we define the function?
like f(a,b,c)= a + b * c
I feel like im about to get a whole lecture about this topic cause I misunderstood the whole idea of these functions
opal moss
I feel like im about to get a whole lecture about this topic cause I misundersto...
ok i typed something that's about 5 times too long so lets try something less insane
when you define a function, you only look at the following:
its domain
its codomain
the mapping from the domain to the codomain
and if 2 functions share the same everything, they're considered the same function
so for example, when you answer the question "the number of functions whose input is a real number, and whose output is always 0"
the answer is 1. you can define it as f: R -> {0}, f(x) = 0
it's not infinite because of some claim that you can change the letter f to be any other label
serene hamlet
codomain is the image right?
opal moss
uh as far as we're concerned here that's a fine assumption
in general the image is a subset of codomain (and codomain is used when the function is not well understood which is not the case here)
serene hamlet
hmm
so back to the question
looking at A
the image of the function can be 1, 0
opal moss
so for your f: {0,1}^3 -> {0, 1}, f(a,b,c) = a+b*c, it has its truth table represented as:
a,b,c | f
---------
0,0,0 | 0
0,0,1 | 0
0,1,0 | 0
0,1,1 | 1
1,0,0 | 1
1,0,1 | 1
1,1,0 | 1
1,1,1 | 0
and any other function that you define for 3 inputs with the same truth table is identical to f
for a lazy example, say g(a,b,c) = a XOR (b AND c), it has the same everything, so g is identical to f
and when you count functions, f and g are not distinct, so they're only going to add 1 to your final count
serene hamlet
so for your f: {0,1}^3 -> {0, 1}, f(a,b,c) = a+b*c, it has its truth table repre...
so this is basically all the possible values that a, b, c can get
opal moss
correct, so the number of distinct truth tables dictate the number of distinct boolean functions you can have
serene hamlet
so its 2 to the power of the number of elements we have
does it work that way for any number of elements or this was just a coincidence
opal moss
this will hold in general for boolean functions of any degree by the same combinatorical argument
serene hamlet
and whats a combinatorical argument?
or do I need to be worried about it at all as a freshman?
opal moss
eh, combinatorical argument is just the mathy jargon for "how to count something"
serene hamlet
oh its that thing where you define how operations work
opal moss
here counting the number of boolean functions for some degree
serene hamlet
so ig I dont need to worry about it for now xd
okay thanks a lot, it really helped!
.close
livid tiger
$\frac{1}{{x^2+4}}$
pearl pondBOT
$\frac{1}{{x^2+4}}$
livid tiger
how would i use u sub in that
stoic path
\frac*
jolly parrotBOT
xyz
wooden lion
Using u sub on that wouldn’t really work, so do you know how to use trig sub?
final bone
u = x^2 + 4?
livid tiger
no du wouldnt resemble the other part i think
would rewiriting it as the derivative of the arctan help?
subtle ginkgo
x = 4u dx = 4 du
livid tiger
or are there better ways?
livid tiger
Using u sub on that wouldn’t really work, so do you know how to use trig sub?
ah you mean convert to an arctan
right
wooden lion
The goal is to change x itself into a trig function using theta. When dealing with addition in the denominator you use tangent(theta). Here’s an example:
jolly parrotBOT
xyz
livid tiger
and then do u =x/4
gloomy hare
an easier way would be doing the sub given by @wooden lion
livid tiger
.close
ebon eagle
Int 0 to infinity x^6/(1+x^2)^4
pearl pondBOT
Int 0 to infinity x^6/(1+x^2)^4
stoic path
$\frac{x^6}{(1+x^2)^4}$
jolly parrotBOT
Lorentz
stoic path
This eh?
pastel flax
by partial fraction
subtle ginkgo
Int 0 to infinity x^6/(1+x^2)^4
you should apply the complex analysis, to be precise: residue theorem, please read about it first
ebon eagle
you should apply the complex analysis, to be precise: residue theorem, please r...
We are supposed to solve it using reduction formulas
I cannot find the way to simplify it to that point first
wooden lion
That looks like a trig sub. Have you tried setting x=tan(theta)?
pearl pondBOT
@ebon eagle Has your question been resolved?
pearl pondBOT
@ebon eagle Has your question been resolved?
pearl pondBOT
@ebon eagle Has your question been resolved?
midnight haven
pearl pondBOT
pearl pondBOT
Available help channel!
thin ginkgo
Find x and y
xy + x = -3
2x + 5y = 8
pearl pondBOT
Find x and y
xy + x = -3
2x + 5y = 8
thin ginkgo
Steps
-
Make y subject of the formula in the 2nd equation
-
Substitute the answer to (1) into the 1st equation
-
Solve
limpid lily
,rccw
jolly parrotBOT
limpid lily
I don't think there are any rational roots.
shrewd basin
Try 15 and 1
limpid lily
No, I was wrong.
thin ginkgo
But -15 + 1 don’t add up to -13
shrewd basin
It’ll be -15+2 because of the 2x
jolly parrotBOT
shrewd basin
thin ginkgo
.close
bleak wyvern
hi, how to prove that x^2+x-2 is a square in Fp* please ?
pearl pondBOT
hi, how to prove that x^2+x-2 is a square in Fp* please ?
jolly parrotBOT
90umut ☪
pearl pondBOT
Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
bleak wyvern
we are in the corps Fp
and Fp*
show that if x+x+1=0 has a solution, then -3 is a square in Fp*
p is prime
tropic saddle
its called field in english
bleak wyvern
ok sorry
tropic saddle
ok, so basically a completely different question than what you originally asked
bleak wyvern
why ? if x^2+x+1 has solution, then -3 is square, so x^2+x+1 - 3 = -3
then x^2+x-2 is square
tropic saddle
you completely neglected the if part
without it this question is false
what happens if you complete the square for x^2+x+1
tropic saddle
completing the square?
never heard of it?
its the way the quadratic formula is derived
pearl pondBOT
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gilded minnow
Can someone help me prove it?Can a number 44...41, whose decimal representation consists of an odd number of digits 4 followed by a digit 1, be a square number?
pearl pondBOT
Can someone help me prove it?Can a number 44...41, whose decimal representation ...
@gilded minnow Has your question been resolved?
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fickle canyon
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fickle canyon
If anyone could help that'd be great I fear I'm not sure what to do after dividing the outcome by the original amount
manic wadi
fickle canyon
i do have a calculator
can u use logs for this
i had the idea to but it doesnt seem like u could randomly slap a log on it
fickle canyon
truthfully this doesnt feel right lol
it might be i got 3.5
wow it is nice
thnx for the help
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torn pendant
pearl pondBOT
torn pendant
how do i do this i rly have no clue
pearl pondBOT
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fair frost
im trying to help my sibling with his homework, is it 22?
pearl pondBOT
im trying to help my sibling with his homework, is it 22?
autumn topaz
yes
vivid pilot
how do i find the graph of an antiderivative?
pearl pondBOT
how do i find the graph of an antiderivative?
@vivid pilot Has your question been resolved?
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vivid pilot
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vivid pilot
upbeat glade
hmmmm, it seems we have to infer it based on the graph of the derivative
d/dx [F(x)] = f(x), we need F(x)
What I would first start doing is labelling the sections
where the derivative is 0 (critical point)
subtle ginkgo
graph f is a graph of the derivative of F, i.e. it shows monotonicity intervals and local extremes
upbeat glade
mhmmm
vivid pilot
is F going to be curved, or have lines?
upbeat glade
yep we can infer whether its concave downwards based on this test
If f' goes from - --> + f'' > 0 (concave upwards)
If f' goes from + --> - f'' < 0(concave downwards)
also do note the inflection points
vivid pilot
so should i graph part b first?
@upbeat glade im still pretty confused on what to do
upbeat glade
f'(x) would be the second derivative of F(x)
pearl pondBOT
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final briar
A circle is definied as the set of all points equidistant from a fixed point. A circle intersects the horizontal line y = 6 at two points. The center of the circle is (-2, 3). Find the length of the radius if the two points of intersection form a line segment inside the circle that has the length of 10 units.
final briar
Im really confused on this question
ik that the answer is root 34 from the answer key
but idk what to do for this question
pearl pondBOT
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final briar
<@&286206848099549185> (sorry)
near echo
did you draw it out?
final briar
idk how to exactly draw it out
near echo
you could try something and then send it here
final briar
I drew the center of the circle and y =6, but idk where to stay from here
Ik it involves Pythagorean theorm
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summer hearth
Is this correct?
pearl pondBOT
Is this correct?
summer hearth
Divide using synthetic division
autumn topaz
I don't know synthetic division, but multiplying x^2 - x- 3 by x -2 doesn't give you the numerator
summer hearth
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waxen sequoia
how would i do this?
pearl pondBOT
how would i do this?
regal herald
if A^-1 x=b then AA^-1 x=Ab Ix=Ab, x=Ab
waxen sequoia
if A^-1 x=b then AA^-1 x=Ab Ix=Ab, x=Ab
so just multiply A^-1 and x?
waxen sequoia
[17,3]
regal herald
you meant A and x, but yeah
regal herald
is there a reason you doubt it?
waxen sequoia
if A^-1 x=b then AA^-1 x=Ab Ix=Ab, x=Ab
im not sure with this
a bit hard to understand
regal herald
if i have an invertible matrix A with Ax=b
if i multiply on the left by the inverse of A, A^-1
A^-1 (Ax)=A^-1 b like so
(A^-1 A)x=A^-1 b, A^-1 A=I which is the identity matrix, since thats how inverse matrices work
so youre left with Ix=A^-1 b hence x=A^-1 b
regal herald
indeed
waxen sequoia
its just a bit confusing that the question gives you x and then asks for x
waxen sequoia
whats x = [2 3] then
waxen sequoia
would this be the same?
regal herald
same process yeah
waxen sequoia
and when multiplying, the inverse matrix comes first right?
regal herald
you have to multiply on the left
waxen sequoia
answer is a?
regal herald
,w ((1,5),(0,1))*((0,2),(1,3))
jolly parrotBOT
regal herald
seems so
regal herald
if you multiplied on the right, youd just get AxA^-1=bA^-1, which doesnt really help
you want to multiply on whichever side will put the inverse and its matrix next to eachother
regal herald
would this be the same?
if you want to verify, replace A with your result
and see if you get the right hand side
but itll require you to actually work out that inverse matrix
normal wyvern
So you got it how to compute this
waxen sequoia
yes A^-1b
normal wyvern
Matrix is already in inverse so you multiply the Matrix by 2,3 like this you have done?
waxen sequoia
yea
and you get [17 3]
rn im looking for a way to verify ive got the correct answer
normal wyvern
Well it's correct no need to verify
autumn topaz
if you really want to verify, invert the matrix, then multiply by [17 3]
waxen sequoia
wdym by invert?
normal wyvern
wdym by invert?
Find inverse matix he means
waxen sequoia
oh like A^-1 times A^-1 will give me A?
autumn topaz
no
you might want to review matrix inverse: https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:matrices/x9e81a4f98389efdf:practice-finding-inverses-of-2x2-matrices/v/inverse-of-a-2x2-matrix for example
waxen sequoia
i know how to do that
you're asking me to find inverse of the inverse?
ok yea i just did that im getting [2 3]
ty
autumn topaz
no, I'm saying if you want to verify your answer, find the inverse of $\begin{bmatrix} 1 & 5 \ 0 & 1 \end{bmatrix}$, then multiply that by $\begin{bmatrix} 17 \ 3 \end{bmatrix}$. you should get $\begin{bmatrix} 2 \ 3 \end{bmatrix}$
jolly parrotBOT
cwatson
normal wyvern
you're asking me to find inverse of the inverse?
No he means to verify it by find inverse of 1,0 5,1
waxen sequoia
No he means to verify it by find inverse of 1,0 5,1
but that still means finding the inverse of a inverse matrix or am i missing something
autumn topaz
no?
you are given some matrix A already. they are saying to solve the equation $A^{-1} x = b$ for $x$. so you left-multiply both sides by $A$
jolly parrotBOT
cwatson
autumn topaz
so you just compute A* b, both of which are given
waxen sequoia
oh wait this is not already in inverse form?
that threw me off
how would i compute this
what i did was to find the first col of A i did:
[ 1 3 1
1 2 -1]
and found x1 = -5, x2 = 2 but thats wrong so not sure
autumn topaz
you could invert the matrix on the left, and right-multiply both sides of the equation by that inverse
waxen sequoia
Im not sure what you mean by that
autumn topaz
you have an equation $AB = C$. to solve for A, you would do $A = CB^{-1}$.
jolly parrotBOT
cwatson
waxen sequoia
oh
this time the inverse is on the right side. For the last 2 questions, the inverse on the left side. This is why i wanted to verify my answers
solution should be b
waxen sequoia
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timid cedar
which is greater, a or b if number a = (23^2022+1)/(23^2023+1) and b = (23^2023+1)/(23^2024+1)
pearl pondBOT
@timid cedar Has your question been resolved?
timid cedar
also do you mean (23^2022)+1 or 23^(2022+1)
(23^2022)+1
timid cedar
which is greater, a or b if number a = (23^2022+1)/(23^2023+1) and b = (23^2023+...
so what's more? It seems to me that they are equal
normal wyvern
Wait I am about to solve this question
And the answer is a is greater than b
Am I right
last wing
plugging into wolfram alpha gives the same result
normal wyvern
Do I send the solution
last wing
yeah that would probably be good
normal wyvern
Ok try to understand my handwriting pls
I multiplied both numerator and denominator by 23 in a
Where d is the denominator part
You can do same for b I have only written the process for a
normal wyvern
which is greater, a or b if number a = (23^2022+1)/(23^2023+1) and b = (23^2023+...
Got it
timid cedar
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prisma pier
one thing: are you sure the sum starts at n = 0?
that looks like it would cause issues
right
then what can we say about each term?
$a_n\le:?$
jolly parrotBOT
lifefuel
prisma pier
given the even-odd condition, it'll be one of two formulas, right
both converge but one is even smaller
we could show the sum converges if we show it's less than or equal to an already convergent sum
if $a_n$ is equal to one of two things, but one is always less than or equal to another...
prisma pier
since the square of the reciprocal is equal and the cube of the reciprocal is going to be smaller (or equal for n=1, right)
so the entire sum is less than or equal to a sum of only squared reciprocals
yup
i'm not sure how rigorous this would be, but wouldn't it suffice to notice that
jolly parrotBOT
lifefuel
prisma pier
and
jolly parrotBOT
lifefuel
prisma pier
then notice that
jolly parrotBOT
lifefuel
prisma pier
redundant parentheses mb
prisma pier
**lifefuel**
holds for n ≥ 1 obviously, which we're considering
midnight haven
Let $n \in \N$, $a \in \R$, $b \in \R$, with $a > b > 0$. Where $r$, $\theta$, and $\varphi$ are spherical polar coordinates for $\R^3$ (with colatitude $\theta$ and azimuth $\varphi$), find the volume enclosed by the surface
[
r = a + b\cos(\theta)\cos(n\varphi).
]
jolly parrotBOT
midnight haven
I need some guidance with this one. Any hints on how I should do this because I'm clueless
unborn abyss
i would start by bullying a computer into drawing that surface for me
midnight haven
i would start by bullying a computer into drawing that surface for me
I probably need to python or something right
unborn abyss
desmos
unborn abyss
pretty sure desmos can do spherical coordinates
midnight haven
what information am I meant to retrieve from its graph
vital estuary
if you cant use a computing program (e.g. on a test)
try making one constant and vary the other one
unborn abyss
what information am I meant to retrieve from its graph
vibes
vital estuary
then when you have a whole buncha values, interpolate
vital estuary
yeah idk how you would figure this out
this is a=b=n=1
it gets wackier for different values
midnight haven
looks like a liver
vital estuary
kinda cool tho
midnight haven
the liver function
unborn abyss
this surface is cursed but i love it
anyway the most important thing here is that it seems like the origin is inside the surface always
so you can just do a fairly standard integral over the whole sphere
vital estuary
my dumbass was messing around with values and was like "oh wow, b = 0 makes a sphere"
unborn abyss
yes very good garlic
midnight haven
midnight haven
so you can just do a fairly standard integral over the whole sphere
sphere?
unborn abyss
like do a double integral of rho, where 0 < theta < 2π and 0 < phi < π
with the jacobian
unborn abyss
the surface isn't in multiple parts, or floating off in space somewhere etc
and I think rho is always nonnegative? haven't checked that
midnight haven
okay thanks I got an idea
I'll big brain this
and in case I don't big brain this
I'll reopen
👍
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pearl pondBOT
vital estuary
:D
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rancid prawn
Does anyone have an example where it is possible to still find a finite solution for a problem with an open simplex?
pearl pondBOT
Does anyone have an example where it is possible to still find a finite solution...
@rancid prawn Has your question been resolved?
twin ravine
@Reinout Has your question been resolved?
rancid prawn
no
pearl pondBOT
@rancid prawn Has your question been resolved?
pearl pondBOT
@rancid prawn Has your question been resolved?
unique ibex
Does anyone have an example where it is possible to still find a finite solution...
Can you provide mroe context?
rancid prawn
Can you provide mroe context?
I've read in a textbook about optimization that even though a simplex might be open, there can still be a way to find a feasible finite solution.
unique ibex
I've read in a textbook about optimization that even though a simplex might be o...
Is this linear programming?
rancid prawn
Is this linear programming?
yes
pearl pondBOT
@rancid prawn Has your question been resolved?
pearl pondBOT
@rancid prawn Has your question been resolved?
pearl pondBOT
@rancid prawn Has your question been resolved?
wanton depot
You need to find the minimum n for which such a condition is fulfilled. If we take any n integers, we can choose 18 numbers so that their sum is divisible by 18.
pearl pondBOT
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feral sedge
Minimize x+y subject to x≥0, y≥0
fossil drum
why is x discontinuous at -2
pearl pondBOT
why is x discontinuous at -2
fossil drum
isn’t as x approaches -2 from the left -2^2 is 4 and as it approaches from the right isn’t it also 4
fossil drum
yeah but the answer says it’s discontinuous
timid spindle
Your function doesn't exist at -2
pallid leaf
It is not defined for x=-2
fossil drum
i thought it not being defined doesn’t affect the continuity as long as they approach 4 from both sides like this
timid spindle
Depends on your definition of continuity, really it makes no sense to even talk about continuity at a point not in the domain of a function
stoic path
Your function doesn't exist at -2
Yeah mb, the function needs to be defined at that point as well
fossil drum
oh i see
fossil drum
Depends on your definition of continuity, really it makes no sense to even talk ...
ok thanks
.close
pearl pondBOT
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stark crag
I'm a bit confused, isn't the y intercept just the last number since this is in y=mx+b form already
midnight haven
[
y=\underbrace{m}_{\mathclap{\t{the slope}}}x + \overbrace{b}^{\mathclap{y-\t{intercept}}}
]
jolly parrotBOT
stark crag
so then would the y intercept just be 13/5? also the subtraction sign is kinda throwing me off
does that subtraction sign matter or no
stark crag
oh ok
and it's just that simple? The question was just seeing if I knew the difference from the slope and the y intercept?
nimble lily
and it's just that simple? The question was just seeing if I knew the difference...
well the y-intercept is "where does this line cross the y-axis"
which, obviously always happens at x=0
so it's always the point (0, a)
and it should be obvious that when x=0, the equation just becomes...y=b
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alpine elbow
What is the maximum number of sides that can be longer than the largest diagonal?
Have a convex polygon?
pearl pondBOT
What is the maximum number of sides that can be longer than the largest diagonal...
@alpine elbow Has your question been resolved?
alpine elbow
<@&286206848099549185>
midnight haven
In a convex polygon, no side is longer than largest diagonal
alpine elbow
Why
midnight haven
If you have a convex polygon with the vertices V1,V2,…,Vn,
where n>3
, then the diagonal V1V3¯¯¯¯¯¯¯¯¯¯
is certainly longer than any of the edges V1V2¯¯¯¯¯¯¯¯¯¯
and V2V3¯¯¯¯¯¯¯¯¯¯
. A similar assertion holds also for any other three consecutive vertices. This means that in any convex polygon there exists at least one diagonal that is longer than any of its edges. However, the longest diagonal must certainly have this property. This means that in a convex polygon there can never exist any edge that would be longer (or have the same size) than the longest diagonal of this polygon.
alpine elbow
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gilded hollow
Does someone have a good explanation on how conversion between bases work?
gilded hollow
Or perhaps a good reference where i could learn it
pearl pondBOT
@gilded hollow Has your question been resolved?
crystal thicket
Give me an example of a number and a base you want to convert it to from base 10
vestal tapir
the trick is to use base 10, even if it's not supposed to be involved, like if you need to convert base-11 to base-7, you avoid the confusion if you convert to base 10 and then to base 7
gilded hollow
Interesting, i read something similar about that
If i want to convert to one base to another i always make it base 10 and then make it the sought after base
is that correct?
vestal tapir
right
gilded hollow
Okay let me bring an example so i can double check that
With my current knowledge i know that
1 0001
2 0010
3 0011
4 0100
5 0101
6 0110
7 0111
8 1000
9 1001
A 1010
B 1011
C 1100
D 1101
E 1110
F 1111
I remember this by heart, what does it actually mean?
0001 <-- is this in base 2 ?
vestal tapir
yes it is
gilded hollow
If i have 432 base 4 and want to convert it to base 7
$432 base 4 = 4^2 3^1 2^0 \
\implies4 * 4^2 +34^1+24^0 \
= 78$
jolly parrotBOT
Merineth
vestal tapir
you can do this in your head if you read left-to-right
4 * 4 + 3 = 19
19*4 + 2 = 78
gilded hollow
**Merineth**
I ideally prefer this way since it's a method i can apply to other examples
vestal tapir
there's like × 4 + between the digits
gilded hollow
but i am thinking correct?
vestal tapir
yes