#help-36
gaunt dome
If i have an area like the shades region if red. I know i can rotate it around the x axis for example and find the volume of that. But can i rotate it around none straight lines like sin(5x)-3. If not can i atleast do it with linear functions
Wait this channel closed or something while i was typing this?
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gaunt dome
If i have an area like the shades region if red. I know i can rotate it around the x axis for example and find the volume of that. But can i rotate it around none straight lines like sin(5x)-3. If not can i atleast do it with linear functions
final saddleBOT
If i have an area like the shades region if red. I know i can rotate it around t...
onyx peak
If i have an area like the shades region if red. I know i can rotate it around t...
what would it even mean to be rotated around a curve? I cant really imagine that
gaunt dome
Idk thats why im asking
To make things simple heres my attempt im gonna rotate it around sin(5y)-3 to make it simpler for me
We know that each width will be dx and the height will be the integral of ln(5x)+5 - int of x^3
The circumference will be changing based on sin(5y)-3
I don't know how i would find the area of the thing when the line is wiggiling
Cuz each shell would jud in and out
But yeah i dont realy know where to go from here i feel like it must be possible
final saddleBOT
@gaunt dome Has your question been resolved?
proud jolt
If i have an area like the shades region if red. I know i can rotate it around t...
https://www.geogebra.org/calculator/ypmg7evn
I think this could be a way
No, it isn't right
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@gaunt dome Has your question been resolved?
gritty flume
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gritty flume
Question b1)
For the lhs, I’m having trouble w knowing if I should put the right hand side answer in a bracket
opal pelican
For the lhs, I’m having trouble w knowing if I should put the right hand side an...
What do you mean?
gritty flume
What do you mean?
My last line
opal pelican
My last line
Yes, what about it
gritty flume
Yes, what about it
Do I put the right side in a bracket
opal pelican
Do I put the right side in a bracket
I don't think you need to?
From reading the problem you gave, I assume all you're meant to do is algebraically manipulate both sides so they're equal
Don't see why you'd need brackets on any of the manipulations
Also, you can still do more with that expression
gritty flume
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runic heart
Determine the perimeter of the area, if 2–√=1.41
final saddleBOT
Determine the perimeter of the area, if 2–√=1.41
grave trout
!status
final saddleBOT
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
yes
runic heart
i cant 😭
rugged merlin
Don't have to
Its one way
nimble basin
Just put the value
grave trout
Its one way
You need to find the perimeter
nimble basin
And calculate
grave trout
@runic heart what have done so far?
nimble basin
Cm
runic heart
so do i do all the sqrt then i add it all up
runic heart
<@1402507076403466310> what have done so far?
ive tried to do all the sqrt thrn i add it al up
nimble basin
ive tried to do all the sqrt thrn i add it al up
Just multiply and add
grave trout
ive tried to do all the sqrt thrn i add it al up
First try adding it all tgether
runic heart
Just multiply and add
multiply what?
nimble basin
Yea
runic heart
uhm ok holdon
nimble basin
Or u can take common
runic heart
common?
runic heart
15√2 + √2 + 7√2+7√2+10√2
Take √2 common
√2(15+1+7+7+10)
√2(40)
wait where did u get 7√2 and 10√2
nimble basin
wait where did u get 7√2 and 10√2
√98=√2x√49
7√2 and √200=√100x√2
10√2
Whats the answer??
If u have ?
@runic heart
nimble basin
i found 76.14
Is there answer given along with the q?
runic heart
Is there answer given along with the q?
no 😔
nimble basin
Length of 3 sides r not given
nimble basin
the answer is 76.14
How
runic heart
How
so
left: 15√2
up: √98 = √49x2 = 7√2
inner vertical cutout : √2
inner horizontal cutout:√98 = √49x2 = 7√2
bottom side: √200 = √100x2 = 10√2
then we find the missingright vertical side
right vertical side: 15√2 -√2 = 14√2
then calculate perimeter
7√2 + √2 + 7√2 + 14√2 + 10√2 +15√2
(7+1+7+14+10+15)√2
54√2
then we the provided value:
√2 = 1.41:
perimeter: 54 x 1.41 = 76.14cm
i literally typed that for 10 minutes 😭
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@runic heart Has your question been resolved?
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modest sequoia
<@&268886789983436800>
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drowsy epoch
Can you possibly show your work
junior elm
oh wait nvm i just forgot that x is constant with respect to y so its 0 lmao my bad
yeah i understand it now its all good
bold turtle
damn you guys are FAST
see cos this character's main colour is pink as well lol
<@&268886789983436800> oh baby a triple
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lime crest
Let $ABC$ be an isosceles triangle with $AB=BC$. Let $D$ and $E$ be points on $AC$ and $BC$ respectively such that $CD=DE$. Let $H,J,K$ be the midpoints of $DE,AE,BD$ respectively. Let the circumcircle of $DHK$ intersect $AD$ at $F$. Let the circumcircle of $HEJ$ intersect $BE$ at $G$. The line passing through $K$ parallel to $AC$ intersects $AB$ at $I$. Let $IH,GF$ intersect at $M$. Show that $J,M,K$ are colinear.
final saddleBOT
Let $ABC$ be an isosceles triangle with $AB=BC$. Let $D$ and $E$ be points on $A...
soft zealotBOT
Copter
lime crest
as <BAD= <DCE=<DEC we know A.B.D.E cyclic. Notice that BAD and KFD have a shared angle and the same side ratio so BAD and KFD are similar, <BAD = <KFD
also clearly I is the midpoint of AB
Since K,F,H,D cyclic we have <DEC=<BAC=<KFD=<KHE, so KH and BC are parallel
other than that i dont really know how to continue ;-;
also i havent really used the fact that H is the midpoint of DE yet, any ideas?
ionic venture
There are a lot of midpoints in this diagram.
Why don't you try redefining the construction with them?
Like
Let I, H, K, J, ||G, F, and M|| be the midpoints of AB, DE, BD, AE, ||BE, AD, and IH||, respectively
final saddleBOT
@lime crest Has your question been resolved?
ionic venture
Here's a problem that will help you with your solution:
Let X and Y be the midpoints be midpoints of arbitrary segments AB and CD, respectively. Given that F and E are the midpoints of segments BC and AD, respectively, prove that M := FE ∩ XY is the midpoint of XY.
final saddleBOT
@lime crest Has your question been resolved?
lime crest
hmm
ionic venture
You got it?
lime crest
XF//EY, hence the triangles are congruent and we are done?
ionic venture
They are parallel, but that doesn't mean they'll be congruent.
You're just missing one fact possibly under your nose... or the triangles
lime crest
They are parallel, but that doesn't mean they'll be congruent.
im mean, youd get XFM is similar to YEM but YM = YE so theyre congruent
ionic venture
YM = YE?
They are congruent
That's right
But I'd just like to know your reasoning for it
Hint: ||Look at BD, XE, and YF||
lime crest
MF=ME, mb
ionic venture
ionic venture
oh theyre similar to BD so youd getXM = MY?
There are some gaps in the intuition
lime crest
BD=2XE =2FY so XE = FY now XE//BD and FY//BD so XE//FY then we can angle chase
XEFY is a parallelogram?
ionic venture
That means their intersection, M, is the midpoint of both FE and XY.
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quartz vigil
Explain 26
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Explain 26
modest sequoia
what have you tried so far?
quartz vigil
I am confused should I integrate
I tried integrating it
with u sub
but it’s a fail
modest sequoia
try setting f(t) = sqrt(t^3 + 1) and then use FTC
quartz vigil
FTC?
modest sequoia
fundamental theorem of calculus
pliant elk
Explain 26
partial differentiation of these function wrt what ?
pliant elk
fundamental theorem of calculus
how does that work for a definite integral with variable limits
leibniz integral rule should be applied here according to me
final saddleBOT
@quartz vigil Has your question been resolved?
junior siren
Minimum value of √x^2+y^2
Where x^2+y^2+xy=1
final saddleBOT
Minimum value of √x^2+y^2
Where x^2+y^2+xy=1
junior siren
Should I apply Am GM but how?
wary juniper
Minimum value of √x^2+y^2
Where x^2+y^2+xy=1
Is that square root of the sum
junior siren
Yeah@wary juniper
grave trout
Is x and y real?
wary juniper
x^2+y^2 ≥ 2xy (AM-GM)
3(x^2+y^2)≥2 (add 2x^2 +2y^2 to both sides and use the equality given)
Also ye I'm assuming x and y are real ofc
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earnest gull
i think solve the quadratic in terms of y and substitute the y value in the first equation and then use maxima minima
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agile dock
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warm python
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warm python
So a few questions here,
TO start the score function is $-\ln(\sigma^2)-\ln(\sqrt{2 \pi}) -\sum_{i=1}^{n} \frac{(x_i-\mu)^2}{2 \sigma^2}$
soft zealotBOT
Wai
warm python
differentiating twice, partially wrt $\sigma^2$
soft zealotBOT
Wai
warm python
$\frac{-1}{2\sigma^3} +\frac{1}{2} \sum \frac{(x_i-\mu)^2}{2 \sigma^4}$ is the first pdv
soft zealotBOT
Wai
warm python
or am I tripping
vital crag
maybe set theta = sigma^2 and differentiate wrt theta instead
first derivative of log(sigma^2) wrt sigma^2 should be 1/sigma^2
warm python
first derivative of log(sigma^2) wrt sigma^2 should be 1/sigma^2
oops, right
I forgot to apply the chain rule
😔
warm python
maybe set theta = sigma^2 and differentiate wrt theta instead
is the 2nd functions' derivative fine?
vital crag
what do you mean by fine? did you send it here somewhere?
warm python
what do you mean by fine? did you send it here somewhere?
I mean$\frac{1}{2} \sum \frac{(x_i-\mu)^2}{2 \sigma^4}$
soft zealotBOT
Wai
vital crag
oh the second term
warm python
well, looks like a stray 1/2?
vital crag
1/sigma^2= sigma^(-2) and use power rule from there to fix your mistake
warm python
1/sigma^2= sigma^(-2) and use power rule from there to fix your mistake
and once I find the pdv, I integrate over the parameter space, right
vital crag
are you following the formula you learned
warm python
yes, this is what I'm using
chrome thunder
Can you help me with this Differentiation question ❓
final saddleBOT
Can you help me with this Differentiation question ❓
@chrome thunder Has your question been resolved?
deep condor
<@&268886789983436800>
deep condor
Can you help me with this Differentiation question ❓
what rule do you think we need to use?
chrome thunder
Assumed X method
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candid pulsar
does $\text{det}(A) = 0$ necessarily imply $\text{det}(A-A^T) = 0$ for a matrix of even order?
final saddleBOT
does $\text{det}(A) = 0$ necessarily imply $\text{det}(A-A^T) = 0$ for a matrix ...
soft zealotBOT
candid pulsar
or at least a 2x2 matrix
winter dagger
Don't think so
rare girder
$A=\begin{pmatrix}2&1\2&1\end{pmatrix}$
soft zealotBOT
Flip
rare girder
does $\text{det}(A) = 0$ necessarily imply $\text{det}(A-A^T) = 0$ for a matrix ...
what do you mean by order btw?
formal trail
0 1; 0 0
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shy socket
how do i do number 9
final saddleBOT
how do i do number 9
radiant igloo
how do i do number 9
what do we do when we want to interchange the limits of an integral?
radiant igloo
Combine them>
int with limits a to b= -(int with limits b to a)
warm python
Combine them>
$\int_{a}^{b} f(x)dx =k \int_{b}^{a} f(x) dx$
soft zealotBOT
Wai
warm python
what is k here
warm python
yes
shy socket
well -1 is a constant
radiant igloo
what happens when we combine int with limit a to b and the same int but with limits b to c
shy socket
subtract them?
radiant igloo
exactly
shy socket
But what about (p(x)+1)/
radiant igloo
simply break
shy socket
p(x) + 1?
radiant igloo
yea
shy socket
and then i gotta integrate it?
radiant igloo
yea
radiant igloo
Wait so it becomes the intragal of like 3 to 8?
this
vernal lynx
well yeah
shy socket
But like this specifically
radiant igloo
@warm pythontake over pls i gtg
shy socket
Like when u combine the stuff
vernal lynx
cmbine what
warm python
<@796939110815629322>take over pls i gtg
even I have work 🙏 ,
shy socket
Like question 9
topaz kite
it's still part of integration. possibly sum rule? I don't think a subtopic is dedicated to this rule in any calculus text.
oak nimbus
Like when u combine the stuff
It's property of integration
radiant igloo
even I have work 🙏 ,
🙏
vernal lynx
its still definite intergation
there are diff types of questions properties
and rules
in this chapter
topaz kite
<@&268886789983436800>
marsh mountain
<@&268886789983436800>
oak nimbus
How do u even rewrute this
Ummm u fully understand that property right ?
marsh mountain
So what do i do after integral 3 to 8
have you been able to see what $\displaystyle\int_3^8 p(x) \dd x$ would be?
soft zealotBOT
blanketism
shy socket
Ummm u fully understand that property right ?
I need some help on this
shy socket
**blanketism**
like 4
marsh mountain
don't think so
shy socket
Oh
oak nimbus
I need some help on this
Bruh
Thing is you have to find p(x) from 8 to 3 so in integration you can break this limits like 8-5 + 5-3
marsh mountain
Oh
do you understand what they're talking about when you combine integrals with similar limits of integration?
if not, we can discuss that first before moving onto your problem :)
shy socket
I need some help on this
marsh mountain
okay
shy socket
do you understand what they're talking about when you combine integrals with sim...
This part
oak nimbus
Thing is you have to find p(x) from 8 to 3 so in integration you can break this ...
Don't understand this?
shy socket
No i only know how to like integrate stuff idk what this is talking about
oak nimbus
Oh ic
marsh mountain
can you conceptually understand
that if i take the integral from 1 to 2 of a function
and then 2 to 3 of the same function
thats the same as the integral from 1 to 3, yes?
shy socket
Yes
marsh mountain
okay, sounds good
more formally, that looks like this
$$\int_a^b f(x) \dd x + \int_b^c f(x) \dd x = \int_a^c f(x) \dd x$$
soft zealotBOT
blanketism
marsh mountain
do you see how the "b" part disappears and we're just going from the entirety of a to c?
shy socket
Yes
marsh mountain
okay great
shy socket
So for the integral we write the smallest value to the biggest
marsh mountain
typically, yes, its always nice to go from smaller to larger
marsh mountain
but do you see how one of them is going from 8 to 5?
marsh mountain
correct
shy socket
So for that one it's negative?
marsh mountain
they have not
shy socket
3-1
marsh mountain
perfect
marsh mountain
so $$\int_3^8 p(x) \dd x = 2$$
soft zealotBOT
blanketism
marsh mountain
but they're asking for $\displaystyle\int_3^8(p(x) + 1) \dd x$
shy socket
Ok but i am confused because the question is asking for like integral of 8 to 3 (p(x)+1))
soft zealotBOT
blanketism
shy socket
Yes
marsh mountain
correct
do you know that $\displaystyle\int (f(x) + g(x)) \dd x = \displaystyle\int f(x) \dd x + \displaystyle\int g(x) \dd x$?
soft zealotBOT
blanketism
marsh mountain
(integration is linear in addition)
shy socket
Yes
marsh mountain
okay cool
so why dont you try splitting up the p(x) and the 1
into two separate integrals
i think you can integrate 1, right?
and you already have the integral for p(x) (that was the 2 we figured out earlier)
oak nimbus
Ig she already got answer ☠️
shy socket
I got 2+5
marsh mountain
np, glad you understand it now!
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shy socket
is this correct
final saddleBOT
is this correct
plucky rover
Assuming that your calculator is correct yeah
final saddleBOT
warm python
.reopen?
shy socket
warm python
.reopen
final saddleBOT
✅ Original question: #help-36 message
warm python
I need help
shy socket
I need help
This is like the enclosed area formula integral b a top function-bottom function dx
The bottom function is literally y=0
Idk how i can incorportate the rate of change of k into the equation
Idk how to write this
shy socket
Idek if this is possible because k is a constant
And u can only do the FTC if it's like unbounded or u dont know the top bound
💀
Plz if anyone sees this please help me
shy socket
dk/dt = pi/4
late gazelle
what have you tried so far? Or is the image above your current progress
shy socket
That's my progress
late gazelle
mhm ok
shy socket
I just know this is one of those
b a
integral
top - bottom function
I know it's related rates too
So like f'(x) = dA/dt
f(x) is the area
of A
late gazelle
shy socket
Ok so the function = F(x)
late gazelle
so thats one part of FTC
shy socket
and then the derivative is f(x)
can't i just write = f(x)
and d/dx = f'(x)
isn't that the same
late gazelle
well in general this replaces your input t with x
but this isnt exactly what we have
late gazelle
but, if we take d/dx of both sides then we get sin(K)/K = f'(K)
late gazelle
yes
late gazelle
so we have: $\frac{\sin(K)}{K} = f'(K)$
soft zealotBOT
KB
shy socket
Ok i. have a thoery
U know if u evaluate this u gotta get two x or y equations
since we have the x=k
do we solve for x in the sin(x)/x function
shy socket
k= pi/6
dK/dt = pi/4
am i onto something
should we actually solve for x
for y=sin(x)/x
proper lion
I need help
You can apply the newton leibnitz rule (simply FTC), you know k is a function of time
shy socket
Ok
Umm
are the bounds from left to right?
and not up to down
if it's h(x) and g(x) for bounds?
late gazelle
im typing up a response rq
shy socket
Ok
late gazelle
We have: $\int_{0}^{K} \frac{\sin(x)}{x} dx = f(x)$ Differentiating both sides gives with respect to time gives us $\frac{dA}{dt} = \frac{\sin(K)}{K} \frac{dK}{dt}$ We know that $K = \frac{\pi}{6}$ from the original question, so: $\frac{dA}{dt} = \frac{\sin(\frac{\pi}{6})}{\frac{\pi}{6}} \frac{\pi}{4}$ The reason why we get $\frac{\pi}{4}$ is because we are given K is changing at a rate of $\frac{\pi}{4}$ Do you think you can go from here?
cursive crag
leibiniz rule?
soft zealotBOT
KB
late gazelle
thats better
final saddleBOT
@shy socket Has your question been resolved?
late gazelle
np
pliant elk
final saddleBOT
pliant elk
,rccw
soft zealotBOT
pliant elk
whats the logic to this theorem
pliant elk
Buncha cross products innit
cant see it clearly , did they divide the polygon into triangles or something
cosmic bolt
I think if you know that the length of the cross products of two vectors is equal to the area of the parallelogram they span, then it becomes clear pretty quickly
pliant elk
I think if you know that the length of the cross products of two vectors is equa...
i know this
cosmic bolt
Its indeed a bunch of triangles, spanned by the vertices and the origin (if you imagine the origin to be placed inside the polygon somewhere)
cosmic bolt
I'm proud that I saw it so quickly ngl hehehe
pliant elk
the criss cross thingy cuz of the determinant
cosmic bolt
Yeah that's another way to see it
final saddleBOT
cosmic bolt
yw ❤️
pliant elk
I'm proud that I saw it so quickly ngl hehehe
"postgraduate math"
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Available help channel!
timber plume
this is right but how do I know there might not be more optiosn to select from these 2 boxes
final saddleBOT
this is right but how do I know there might not be more optiosn to select from t...
timber plume
I dont know how I got this right tbh
keen wolf
this is right but how do I know there might not be more optiosn to select from t...
Its simple, If you're missing some you're wrong
timber plume
Its simple, If you're missing some you're wrong
wdym?
keen wolf
Your asking how do you know that any other one from those options are right?
timber plume
yes
keen wolf
yes
Yeah if you're missing some then your wrong and get an x
Although some you get half a point or so
keen wolf
can u tell me the cases
Generally if you have many options like more than five you usually get half a point per options
timber plume
i want full marks tho
keen wolf
this is right but how do I know there might not be more optiosn to select from t...
This case it'll be 100 or zero so the fact that its right that means there are no other options
drowsy epoch
For the cases you need to consider, when x³-12x+16, for that I recommend working with its derivative
\prpl [ f(x) coloneqq |p(x)| ] and then consider $p'$ for the monotony and possible extreme values, in order to deduce when $p$ chagnes signs.
soft zealotBOT
timber plume
For the cases you need to consider, when x³-12x+16, for that I recommend working...
do u man when that equals 0?
also im confused why i have it on yellow
i thought it would be green
supple jolt
also im confused why i have it on yellow
yes its yellow because its wrong
2 ticks are correct and 1 is wrong
supple jolt
just check them all again
supple jolt
yeah
supple jolt
so A is there is one unique solution
why
supple jolt
how many variables
drowsy epoch
also im confused why i have it on yellow
Can you stop switching always the topic
You want help on one thing, and before I get to it, you already switch the original question, that's not how help channels work
It's exhausting
supple jolt
inserts rolling cat emoji
timber plume
True My bad
timber plume
and how many equations
3
timber plume
yeah so uh, I am not sure, what are the cases?
supple jolt
we talked about this yesterday
drowsy epoch
also im confused why i have it on yellow
timber plume
if he number of equations dont match the nnumber of variables
supple jolt
if he number of equations dont match the nnumber of variables
ok lets make it a bit more absurd
lets say we have 67 variables
and 2 equations
can we solve the system of quations?
can we find a unique solution?
drowsy epoch
Geometrically, you have two planes (hyperplanes) and if they are not parallel then they intersect always. Their intersection is a set of points along a lower-dimensional hyperplane in your IR^67 space
supple jolt
Geometrically, you have two planes (hyperplanes) and if they are not parallel th...
star what is this
oh
yeah its correct
compact laurel
Geometrically, you have two planes (hyperplanes) and if they are not parallel th...
SIX SEVEN!!
final saddleBOT
@timber plume Has your question been resolved?
odd quest
What’s 5x5
final saddleBOT
What’s 5x5
soft zealotBOT
Result:
25
tulip coyote
<@&268886789983436800>
I saw what you typed 
tropic crest
I saw what you typed <:catbit:1458673839897710788>
Jk
tulip coyote
What’s 5x5
As for you, it's 30 
tropic crest
.close
tulip coyote
Minutes of being timed out 
don't troll in the help channels
final saddleBOT
tulip coyote
final saddleBOT
Available help channel!
timber plume
can we find a unique solution?
No we can’t (sorry my wifi ✈️ off)
final saddleBOT
No we can’t (sorry my wifi ✈️ off)
gleaming anchor
also im confused why i have it on yellow
supple jolt
No we can’t (sorry my wifi ✈️ off)
right
so if we have 4 variables
and 3 equations
can we find a unique solution?
supple jolt
we just said
67 variables, 3 equations -> no solutions
so
4 variables, 3 equations -> ...
timber plume
How does number of equations and and variables determine a solution tho?
I only know in the case of no. Of variables
That determines the number of solutions
supple jolt
67 variables, 3 equations -> no solutions
in this example
which is bigger
the number of equations or the number of variables
timber plume
variables
supple jolt
4 variables, 3 equations -> ...
what about here
timber plume
One unique solution?
supple jolt
the number of equations or the number of variables
the question was this
gritty chasm
can we find a unique solution?
usually no.
gritty chasm
usually, if there are more variables and less equations, we can't find unique solutions
supple jolt
One unique solution?
ok so in the first example, variables > equations, so no solution. in the second example, variables > equation so 1 unique solution 
if you still dont understand you can look at pivots
at the fact that there is no pivot on the 4th colomn
timber plume
ok so in the first example, variables > equations, so no solution. in the second...
Isn’t that contradicting
timber plume
Oh
gritty chasm
however in some systems, like the one below:\\
Let $x,y,z \in \mathbb{R}$\
Prove that if $xy+yz+zx = 0$ and $x+y+z=0$ then $x=y=z=0$\\
Here we get a unique solution due to the constraints on the unknowns. Observe that this system is 3 variables, 2 unknowns. So the general rule is that:\
If there are more variables and less equations, usually one of the variables depends on the others and thus no unique solution can be found\\\
However, if there are more equations and less unknowns:\
If there are more equations and less unknowns, usually the system is inconsistent, or the solutions must satisfy some conditions on them(usually provided in the question) to make the system consistent\\
Here, I'm talking of linear equations.(yes ik bad example above, but i gave it for understanding), and the above two hold only if no two pair of equations are essentially the same(eg. if one is $2x+3y = 10$ and another is $2x+3y+5 = 15$, we consider them as one equation in my explanation above; theyre reductible to each other)
soft zealotBOT
Annie Maqionde
gritty chasm
sorry for text wall
final saddleBOT
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clear lodge
hi
night raft
if you have no questions, you can close the help channel then 
clear lodge
bye
sage stirrup
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sage stirrup
I have a rough idea that we need to use inequality here
Like I used AM-GM and got that's it's greater than equal to 2
honest carbon
Any restriction on n?
sage stirrup
But how to get past the none of tbese
honest carbon
If n can take any real number, you might be as well as replace $n^3-4$ with k, because $n^3-4$ is a monotone function
sage stirrup
Momotone?
soft zealotBOT
Xwtek
sage stirrup
Monotone
I mean
?
Idk what this monotone function is but tbh it wouldn't you be able to replace it with k regardless
honest carbon
Wait, I realized that monotone is not enough. I mean autobijection.
sage stirrup
Fr atp r u just saying big words T_T
wary juniper
Like I used AM-GM and got that's it's greater than equal to 2
👍
honest carbon
Autobijection is a bijection, with the domain and the codomain being the same set.
If you can find a suitable k for some m, you can easily find the n.
sage stirrup
Ok o will leave that work to u
honest carbon
Yeah, without restriction on n, m basically can take any positive values (not zero)
sage stirrup
Lol
honest carbon
Maybe, it's supposed to be restricted to integer n?
leaden moon
hold on hold on
sage stirrup
So among the options u think it can take 2
leaden moon
Like I used AM-GM and got that's it's greater than equal to 2
since you used AM-GM and got your result as $\geq 2$
soft zealotBOT
1 divided by 0 equals Infinity
honest carbon
Wait, I graphed it wrong
leaden moon
so option 2 and option 3 aren't good here
leaden moon
Yes sir
the only way that $m = 2$ is when $(7 + 4\sqrt{3})^{n^3 - 4} = (7 - 4\sqrt{3})^{n^3 - 4}$
soft zealotBOT
1 divided by 0 equals Infinity
leaden moon
do you agree @sage stirrup?
leaden moon
**1 divided by 0 equals Infinity**
but look at this
these 2 can only equal to each other if the powers are 0
sage stirrup
Ok.....
leaden moon
in other words $n^3 - 4 = 0$ which you can solve for $n$
soft zealotBOT
1 divided by 0 equals Infinity
sage stirrup
Yea n will be smth ig
leaden moon
yea
leaden moon
Hnmm interesting thought process
good thinking on that AM-GM idea
i wouldn't think of that at first lol
sage stirrup
good thinking on that AM-GM idea
Lol o see reciprocal I think amgm
leaden moon
uhm that's conjugate
sage stirrup
Well reciprocal of each other
leaden moon
no way these guys used some monotones shits when you said there's AM-GM lol 😭
sage stirrup
7+4root 4 =1/7-4root3
leaden moon
hold on
sage stirrup
no way these guys used some monotones shits when you said there's AM-GM lol 😭
Lol helps I am high scooter
Schooler
leaden moon
but can $n$ be smth like $\sqrt{2}$ or $\sqrt[3]{4}$?
soft zealotBOT
1 divided by 0 equals Infinity
sage stirrup
I only know this and cuachy quartz
leaden moon
im not even used to cauchy schwartz lol
onyx peak
wait is n supposed to be integer btw? Or any real n?
final hedge
you can like substitute x = 1 ryt
sage stirrup
but can $n$ be smth like $\sqrt{2}$ or $\sqrt[3]{4}$?
Lol I have used it thrice
sage stirrup
wait is n supposed to be integer btw? Or any real n?
Your guess is as good as mine
leaden moon
wait is n supposed to be integer btw? Or any real n?
yea that's the thing
leaden moon
if $n \in \mathbb Z$ then $m \neq 2$
soft zealotBOT
1 divided by 0 equals Infinity
leaden moon
n = cuberoot of 4 and put in x = 1
i don't see an $x$?
soft zealotBOT
1 divided by 0 equals Infinity
onyx peak
if n is supposed to be integer, i do have an extremely overengineered solution using recursive sequences
leaden moon
if n is supposed to be integer, i do have an extremely overengineered solution u...
there's no need for that
leaden moon
there's no need for that
-# ||didn't realize you said overenginnered 😭||
leaden moon
yall know what AM-GM is lol
leaden moon
yep
final hedge
OP left?
leaden moon
the only way $m = 2$ can happen is at $n = \sqrt[3]{4}$
soft zealotBOT
1 divided by 0 equals Infinity
final hedge
**1 divided by 0 equals Infinity**
yeah and if you substitute that
you get 2
option (a) lol
leaden moon
poor blud's a bitro nooster but not a green helpful
do we summon @onyx peak to close the channel or not
final saddleBOT
onyx peak
like how many messages
nobody knows
leaden moon
how do you get helpful btw
im not even allowed to have one
onyx peak
only god and mods
final hedge
wait its exclusively given by someone?
final hedge
oh
onyx peak
iirc it was said that its semi automatic
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tough shard
i wanna write this in terms of squares only, what do i do?
final saddleBOT
i wanna write this in terms of squares only, what do i do?
stone wagon
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gentle zephyr
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Renato
dense coral
your final conclusion doesn't suffice to prove the induction step, I'm afraid :c
you need to get to a_{n + 1} < 3^{n + 1} for that 
obtuse hedge
That's what I get from this as well ^
dense coral
there's a fairly quick way to get there, if you stare hard at the initial expression of $a_{n + 1} = 2a_n + a_{n - 1} + 1$
soft zealotBOT
higher!
gentle zephyr
you need to get to a_{n + 1} < 3^{n + 1} for that <:sadcatthumbsup:9323628796241...
yes I got stuck
dense coral
as you've already noticed, you can use the induction hypothesis of $a_n < 3^n$ and $a_{n - 1} < 3^{n - 1}$ to turn this into $a_{n + 1} < 2 \cdot 3^n + 3^{n - 1} + 1$
soft zealotBOT
higher!
dense coral
the key trick here is to focus on the term 3^{n - 1} + 1
if you can bound it by an appropriate number, then you will be done 
what number is that? 
obtuse hedge
Oh that's not what I was imagining but I think this is better 
dense coral
this one should be a quick induction 
or actually, you don't need to induct
you can mess around with that equality and arrive at something that's always true
a good first step might be to turn 3^n into a form that's compatible with 3^n-1
gentle zephyr
as you've already noticed, you can use the induction hypothesis of $a_n < 3^n$ a...
a_(n+1) < 2 x 3^n + 3^(n-1) + 1 < 2 x 3^n + 3^n + 1
but I went overboard by 1
dense coral
mhm, but that's not what we wish to prove now, is it?
dense coral
3^n > 3^(n-1) + 1?
this is what we'd like to show
gentle zephyr
yeah so induction
dense coral
a good first step might be to turn 3^n into a form that's compatible with 3^n-1
and this is my suggestion for showing it 
nono, I think induction would be much messier
try turning 3^n > 3^{n - 1} + 1 into a different equality, one that's always true for any n
as I said, it would not hurt to turn 3^n into a form compatible with 3^{n - 1} to do that :p
gentle zephyr
3^(n-1) + 1 < 3^n
3^(n-1) + 1 < 3 x 3^(n-1)
oh then I
3^(n-1) + 1 < 3^n
3^(n-1) + 1 < 3 x 3^(n-1)
3x3^(n-1) - 3^(n-1) - 1 > 0
3^(n-1) x (3 - 1) - 1 > 0
dense coral
keep going!! 
gentle zephyr
since n >= 1
2 x 3^(n-1) >= 2 > 1
dense coral
there you go 
so with that, you have shown that 3^{n - 1} + 1 < 3^n for any n >= 1 
now you can return to your original inequality and prove the induction hypothesis 
gentle zephyr
yes, thankiu
I would have done a weak induction tbh, but most likely I would have to face this eitherway
dense coral
whatever keeps the boat afloat is enough 
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dense coral
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leaden moon
<@&268886789983436800>
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tropic crest
sniped
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cold lark
hello i wanna know how to solve this properly using smichdt thing
final saddleBOT
hello i wanna know how to solve this properly using smichdt thing
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
@cold lark Has your question been resolved?
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grand orbit
find maximas of the function:
y = 2x+sqrt 1-x
final saddleBOT
find maximas of the function:
y = 2x+sqrt 1-x
drowsy epoch
find maximas of the function:
y = 2x+sqrt 1-x
What have you tried
grand orbit
wdym
drowsy epoch
chain rule
grand orbit
doesnt the thing inside the sqrt stays the same?
drowsy epoch
not always
soft zealotBOT
drowsy epoch
If f(x)=x+c then f'(x)=1 so in that case it stays the same
grand orbit
so basically
f'(x)/2sqrtf(x)
drowsy epoch
yea
soft zealotBOT
drowsy epoch
Now set that to 0, bring sqrt term to the other side and square both sides
<@&268886789983436800> mr least
urban wasp
holy
tulip coyote
<@&268886789983436800> mr least
Least likely to remain in the server 
urban wasp
that was so fast
drowsy epoch
so basically
f'(x)/2sqrtf(x)
u here
final saddleBOT
@grand orbit Has your question been resolved?
junior siren
How do I find range of 1/(x(x-1)(x-2)))
final saddleBOT
How do I find range of 1/(x(x-1)(x-2)))
fast kettle
Well domain is always (-inf,inf)
For real functions
Which I’m assuming you’re using
junior siren
it is but we need to exclude some points no?
fast kettle
Maybe for range, but not domain
junior siren
okay so?
fast kettle
Now look at where this cannot have an answer
Where would this function divide by 0
Well, isn’t it when x(x-1)(x-2)=0?
junior siren
Bruh??? You are confusing me
fast kettle
x is real so it is not defined at x=0,1,2
Yes
Sorry, I missed that
So do you know how to get the range from there
junior siren
I don't know that's why i am here
And asking and waiting for some hints
@fast kettle
ruby crypt
And asking and waiting for some hints
hi, I can help
do u know what's the range of x? I want to know if u can write the notations
keen wolf
$\frac{1}{x(x-1)(x-2)}$
soft zealotBOT
Frosty
keen wolf
what values can make bottom zero
final saddleBOT
@junior siren Has your question been resolved?
copper roost
hello so
copper roost
2.1 says f“(2) ≠ 0
show that 2 is an extremepoint
do i just plug it into the first derivative?
dense coral
what are graphs 1 and 2? 
copper roost
what are graphs 1 and 2? <:pikathink:879066872513970227>
thats for 2.2
oh
wait yeah ill ask about that too
but is it correct what i did
dense coral
do i just plug it into the first derivative?
that’s a good idea
copper roost
that’s a good idea
the 2.2 asks which of the graphs is the anti derivative of f
so wait
let me think
id just do the integrating part and then see if the slope is < 0 or >0
and then we could say which one is the correct one right?
supple jolt
so you need help with 2.1
supple jolt
do i just plug it into the first derivative?
yes this should be enough for 2.1
well for 2.2 try calculating the integral of the function
copper roost
well for 2.2 try calculating the integral of the function
yeah
and then check the slope right
not slope but
the number before the x^4 should tell us if its wide or like
the opposite of wide right
sand patrol
<@&268886789983436800>
rugged merlin
<@&268886789983436800>
rugged merlin
oml
supple jolt
the opposite of wide right
so
f is the slope of F right
so for example
if you were to calculate f(2) for example
that would be the slope of F at x=2
so just calculate f(2)
and see which graph has that slope
at x=2
makes sense?
copper roost
wait r u that guy?
supple jolt
what guy
copper roost
who always helps me
copper roost
wait r u german
supple jolt
who always helps me
i dont think so
supple jolt
wait r u german
yes