cursive kraken
tq
#help-23
lean otter
safe radishBOT
lean otter
ok so
for part b
it's
(x+5)^2 + (y-5)^2 = (x-6)^2 + (y-3)^2
right?
because
z = x+iy
obsidian oracle
yes
lean otter
and u do the magintude thingy for the real number and the imagery number
obsidian oracle
no it's y-3
obsidian oracle
chatgpt unreliable
burnt notch
Prove $$\lim\limits_{x \to 0^+} \frac{3\log^2 x + \log x}{1 + \log^2 x} = 3$$
using the epsilon-delta definition of a limit
flat frigateBOT
Alberto Z.
burnt notch
**Alberto Z.**
Can someone guide me through this?
I've already got to this step: $$\lvert\log x - 3 \rvert < \varepsilon\left(1 + \log^2 x\right)$$
flat frigateBOT
Alberto Z.
burnt notch
And I let $t = \log x$
flat frigateBOT
Alberto Z.
safe radishBOT
@burnt notch Has your question been resolved?
burnt notch
<@&286206848099549185>
supple python
yeah
burnt notch
<@&286206848099549185>
supple python
wat m8
burnt notch
I'm lost with this proof, I don't know how to proceed
fathom jewel
I've already got to this step: $$\lvert\log x - 3 \rvert < \varepsilon\left(1 + ...
Can you explain how you arrived there?
burnt notch
Can you explain how you arrived there?
By doing |f(x) - L| < eps
And then I multiplied both sides by the denominator
fathom jewel
And then I multiplied both sides by the denominator
So essentially you have
[ \abs{\log x -3} < \varepsilon (1+\log^2 x) \Leftrightarrow \frac{\abs{\log x - 3}}{1+\log^2 x} < \varepsilon ]
Since $1 + \log^2x \geq 1$ you could do
[\frac{\abs{\log x - 3}}{1+\log^2 x} < \abs{\log x -3 } < \varepsilon ]
flat frigateBOT
bacc (unhelpful)
safe radishBOT
@burnt notch Has your question been resolved?
still kettle
I need help with the Chinese remainder theorem! Could someone explain it to me?
still kettle
yay!
indigo bay
yay!
so what's up
I think you should start off with the proof
if you have never used it before
still kettle
ok
I have book right now
that has a part about it
and im sure I could learn the proof
but
still kettle
what will it actually tell me?
Like the book just gives me a formula with no explanation
still kettle
😄
still kettle
valid
Comparing this to the one in my book: the thing you sent me is the general form of mine
btw im not special ive just started learning number theory
.close
safe radishBOT
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devout glacier
How my process should look like in checking if this PDE is linear?
PDE is linear if it has two propeties:
L(u+v) = L(u) + L(v)
L(cu) = cL(u)
devout glacier
Would it be then:
- y L(ux) + L(uy)
- rL(ux+uy)
fathom yacht
No, you would need to take two functions u,v and check the conditions
L(u+v) = y(u+v)x + (u+v)y
devout glacier
Okay, where the v came from in y(u+v)x and (u+v)y
Or is it like property that i can use when appliying rule?
fathom yacht
Simply from the definition of L
devout glacier
Okay, and then i just open the parthesises and i should get:
yux + yvx + uy + vy
fathom yacht
Yes
devout glacier
Then i combine same terms and i should get:
(yux + uy) + (yvx+vy)
fathom yacht
Yes
devout glacier
Now im not sure, how do i determine based on this equation is it linear or not
fathom yacht
Now you need to check the other condition: set c a constant and u a function, compute L(cu) = ycux + cuy
devout glacier
Should be:
c(yux + uy)
fathom yacht
Yes
So you showed the two properties of linearity, which means the equation is linear
Good job
fathom yacht
You're welcome
devout glacier
.close
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muted laurel
i understand how to do it if b was larger than a (a=234) and (b=165) but I do not understand how it works the other way, or if i am reading the textbook wrong
safe radishBOT
i understand how to do it if b was larger than a (a=234) and (b=165) but I do no...
safe radishBOT
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dense hatch
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dense hatch
Is there something wrong with my calculator? Or am I typing it wrong?
fresh birch
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crude bear
Is there any other source that proves this result?
safe radishBOT
Is there any other source that proves this result?
crude bear
this is known as exponential law
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real turtle
where do i even begin
safe radishBOT
where do i even begin
alpine cedar
do you not just calculate the mean mass for each stone since theres 20 in total.but 9 fits on the scale. 10 on each pile. so maybe just measure the mean mass and compare on both sides?
long oracle
A naive idea i have is we can atleast order the stones in "weight", if we do so and we've labeled each stone in respective pile (e.g. stone 7 from pile 2) then we can just balance the 9 lightest stones from respective pile to eachother all at ones.
Suppose (the 9 stones in) pile 1 weighs more than pile 2 in this instance, and we know that of the last two stones that remains the stone from pile 1 weighs more, then we've deduced that pile 1 weights more, however if the last stone from pile 2 is heavier then we cant directly say anything and this is where my naive idea stops
real turtle
:(
long oracle
i guess its possible to build upon this idea of ordering but in groups, like we can classify that certain groups of stones weigh more than other groups or singletones etc
real turtle
it's gonna be hard to make an argument out of that
long oracle
hm yeah
i think reducing the amount of stones might be helpful
see if you can spot a pattern or if you can conjure up examples where theres bound to be problems possibly
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molten hawk
I know you can prove this by doing (A+A^T)^T = (A+A^T) but if we know A^T = A, then can't we just say A+A^T = 2A and A+A^T = 2(A^T) and imply symmetry?
safe radishBOT
I know you can prove this by doing (A+A^T)^T = (A+A^T) but if we know A^T = A, t...
last wren
you aren't given that A is symmetric, just that it's n x n
molten hawk
ah
last wren
if it were symmetric, sure that would work
gusty ether
Number 8 pretty please
safe radishBOT
Number 8 pretty please
gusty ether
see if you can spot a pattern or if you can conjure up examples where theres bou...
OMG ASLAN????? LIKE THE LION FROM NARNIA
raw tusk
Number 8 pretty please
is there a way you could get that image to be a bit clearer
flat frigateBOT
gusty ether
Hello 🧍🏻
<@&286206848099549185>
I’m stuck on where to go once I get to the factoring
plucky elk
where even is the question
plucky elk
yea i can't follow your work at all
gusty ether
Add the 1 to the other side to get rid of it
Square both sides
Foil the x + 1
Then it’s 7x -3 = x^2 + 2x + 1
That combines into 5x -4 = x^2
And that’s where I’m stuck
plucky elk
when you have a quadratic and solving for x, you should move all the terms on the same side to get x^2 + (something) x + (something else) = 0
then either factor or use quadratic formula
gusty ether
Which side should I move it to
gusty ether
Either way I’ll have a negative
spice silo
Either way I’ll have a negative
having a negative is fine :)
spice silo
It depends what the original question was
plucky elk
use a * b = 0 implies a = 0 or b = 0
plucky elk
i don't know how you got (-1) * (-4) = 0
gusty ether
The factors of 4 that add to -5
plucky elk
(-1) * (-4) = 4
plucky elk
i don't know what you're doing anymore
plucky elk
Ok so my factors are x -1 and x -4
,w expand (x-1)(x-4)
flat frigateBOT
gusty ether
I don’t either ma
plucky elk
use a * b = 0 implies a = 0 or b = 0
apply this to your factors
gusty ether
-1 = 0
-4 = 0
plucky elk
apply this to your factors
do you know what your factors are
gusty ether
-1 and -4!!!
plucky elk
nope
gusty ether
How
plucky elk
Ok so my factors are x -1 and x -4
.
gusty ether
Therefore they are -1 and -4
plucky elk
i see where you're confused
gusty ether
???????
gusty ether
The only factors of 4 that add to -5 are -1 and -4
plucky elk
but you thought your factors were just the numbers -1 and -4
the factors are (x-1) and (x-4)
gusty ether
You’re confusing me
plucky elk
Ok so my factors are x -1 and x -4
i thought you understood that your factors are (x-1) and (x-4)
plucky elk
did you arrive at (x - 1) * (x - 4) = 0
gusty ether
What is the next step
plucky elk
I don’t really care how I label them
you should because that's why you keep getting confused
gusty ether
What is the next step
plucky elk
use a * b = 0 implies a = 0 or b = 0
.
gusty ether
So you want me to foil them and set that equal to zero
gusty ether
Which can be rewritten as simply (x-1)(x-4) which is foiling
Unless you’re telling me to just combine in which case it’s 2x and -5
cedar owl
tin foil
gusty ether
Funny
cedar owl
this is actually a metaphor
gusty ether
Huh
plucky elk
use a * b = 0 implies a = 0 or b = 0
in words, this means if you have a product of two terms, one of them must be zero. which two terms do you have
gusty ether
….-1 and -4
plucky elk
no
gusty ether
Is failing this class an option 🧍🏻
plucky elk
did you arrive at (x - 1) * (x - 4) = 0
what two things being multiplied in this equation
gusty ether
X-1 and X-4
gusty ether
So..replace x with zero-
plucky elk
absolutely not
gusty ether
I’m entirely lost
plucky elk
if they multiply to zero, one or both of them must be zero
do you know what are "they" in this sentence
plucky elk
X-1 and X-4
i was responding to your answer here
gusty ether
Yeah
plucky elk
You’re probably gonna have to dumb this way down for me
just doing what you asked
spice silo
The equation you got to was x^2 - 5x + 4 = 0. You factored x^2 - 5x + 4 into (x - 1)(x - 4), so now we have (x - 1)(x - 4) = 0, he's telling you that if you have two numbers a and b, and a*b = 0, then one of the numbers must equal 0
gusty ether
You can dumb it down without being a dick about it 😭😭😭
Ok one of the numbers has to be zero
How do I find which one
spice silo
(x-1) and (x-4) don't look like numbers, but you can put them into what he showed you the same way, if both of those multiplied together equal 0, then one of them has to be 0
plucky elk
Ok one of the numbers has to be zero
it says "one or both"
plucky elk
You can dumb it down without being a dick about it 😭😭😭
don't think this was fair
spice silo
How do I find which one
we still haven't exactly solved it, it turns into two more equations because we know that either x-1 = 0 or x-4 = 0, but x could be either one of them and either could be the answer, or both
gusty ether
Then at least one of the numbers has to be zero
Ok that was my original thought- split them into 2 and add to each side
spice silo
Ok that was my original thought- split them into 2 and add to each side
what do you mean by that?
or oooh i see
gusty ether
So my two possible answers would be 1 and 4-
Then plug it into a calculator to figure out which?
spice silo
you can put them back into the original problem to check
gusty ether
That’s what I meant
spice silo
but sometimes both will work and you'll just end up with two answers
gusty ether
My teacher told us putting it in a calculator was fine
spice silo
gusty ether
See that was easy to explain when you aren’t talking in RIDDLES
Since I don’t have a calculator or enough room on my paper to do that rn I’ll do it when I get to class tomorrow
And now I need help with number nine (Beatles reference 🤯🤯🤯🤯)
plucky elk
See that was easy to explain when you aren’t talking in RIDDLES
nothing i said was a riddle. very straightforward for your benefit
gusty ether
Not rly 😭😭
plucky elk
i thought you understood that your factors are (x-1) and (x-4)
if anything, you just kept confusing yourself
gusty ether
I needed specifics on what to do
cedar owl
see what i meant by tinfoil now
gusty ether
No
plucky elk
Ok so my factors are x -1 and x -4
if you only just remembered your own words, you wouldn't think i'd be talking in riddles
gusty ether
I still need specifics I wouldn’t have understood
I’m autistic 😎
I didn’t get math autism (clearly) or trains autism I got the Beatles autism
@spice silo can you come back idk how to even start number 9
spice silo
You can move them to different sides and then square both sides :)
gusty ether
Would that make the inside pieces negative-?
Nvm
That was a dumb question
But what do I do with the extra 2
spice silo
You have to square all the terms on both sides, something like $(\sqrt{1-m})^2 = (\sqrt{2m+2} - 2)^2$
flat frigateBOT
Zapeta
spice silo
and then if you still have square roots you can square both sides again
after simplifying
gusty ether
So how do I square the side with the 2? My teacher said it was smth different but I don’t remember how
gusty ether
Ok
spice silo
$(\sqrt{2m+2} - 2)(\sqrt{2m+2} - 2)$
flat frigateBOT
Zapeta
spice silo
I'm gonna go to bed I know you can do it from here 😴
silver pelican
do you know the difference of squares
gusty ether
I believe so
silver pelican
so apply that
gusty ether
**Zapeta**
Well from here I got 2m + 2, -2-/2m + 2 (two of these), and 4
silver pelican
yeah
(a+b)(a-b)=a^2-b^2
so doing this gets
(sqrt2m+2)^2-(2^2)
simplyfying
we get 2m+2-4
which is 2m-2
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gusty ether
which is 2m-2
Do I set this equal to zero?
silver pelican
Do I set this equal to zero?
wait what's the question
gusty ether
^
silver pelican
oh
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loud sinew
The answers say BAT is 20 and OAB is 70
safe radishBOT
The answers say BAT is 20 and OAB is 70
loud sinew
Can someone explain the reasoning on how to get there
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hollow coyote
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hollow coyote
how do i show this using e-d proof
molten hawk
^
vapid glen
simple way to see, approach (0,0) along the line x=0 and you have the limit of y. Approach along the line y=0 and you have the limit of 1/x^2
first one goes to 0 and second one goes to infinity
hollow coyote
simple way to see, approach (0,0) along the line x=0 and you have the limit of y...
i did that
i also got 0 and infinity
but i need to show it with an e-d proof as well
how can you use an e-d proof if it dne
can we do contradiction?
vapid glen
yeah I guess so
hollow coyote
yeah I guess so
how would that work?
molten hawk
assume the limit exists
and you do the paths along x and y = 0
just add in the epsilon delta definition
and write it in epsilon delta form
hollow coyote
and you do the paths along x and y = 0
how is that e-d?
molten hawk
because the end result is put into epsilon delta
hollow coyote
can you help me write it out
we have:
sqrt(x^2 + y^2) < delta
let that function be f(x, y)
|f(x, y)| < epsilon
assume the limit is b
what now
molten hawk
I'd use the original function instead
and put the end results of the routes into epsilon
hollow coyote
and put the end results of the routes into epsilon
can you elaborate
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glossy cypress
So i was feeling goofy and wanted to mess around with the floor function because... why not (💀). I saw that floor(a+b) is not always equal to floor(a) + floor(b), which seems pretty intuitive. so i wanted to see what happens when you put in complex numbers and WHY THE FUCK DOES IT DO THAT
safe radishBOT
So i was feeling goofy and wanted to mess around with the floor function because...
plucky elk
does it become based on distance to the closest gaussian integer?
That's just a desmos implementation choice. Read the documentation to see why
glossy cypress
That's just a desmos implementation choice. Read the documentation to see why
crisis averted let me go find that rq
median vigil
there isn't necessarily a "natural"/"obvious" definition of the floor function for complex numbers, so it's up to the developers. in this case they seem to have defined it componentwise
glossy cypress
there isn't necessarily a "natural"/"obvious" definition of the floor function f...
my problem is it isnt component wise, and it's like.. something different. see how in the first one it's 5+7i, but setting b to 6.6 makes it 5+6i
hollow coyote
there isn't necessarily a "natural"/"obvious" definition of the floor function f...
hi cloud
can you help me after this guy 😭
glossy cypress
though based on my like 2 minutes of testing it seems to be based on distance to closest gaussian integer or something like that
update im horribly wrong 😭
im gonna try to find the documentation and close this
.close
inner olive
inner olive
can someone double check (f)
plucky elk
Close one of your channels
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inner olive
.close
wraith swift
what property did they use in the second line again for the det power
safe radishBOT
what property did they use in the second line again for the det power
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kind dove
Hello. I was assigned this homework and I have never learned about differential equations. I need to solve it in Wolfram.
So far I've managed to extract those equations to variables and tried to run NDsolve on them. However NDSolve was unsuccessful, I got many errors and those errors are not very helpful. Also ChatGPT does not help.
sp = 100;(* setpoint *)
kp = 0.1; (* proportional gain *)
kd = 0.01; (* derivative gain *)
km = 0.05; (* motor constant *)
(* Define equations as variables *)
errorEq = e[t] == sp - tau[t];
voltageEq = V[t] == kp * e[t] + kd * D[e[t], t];
motorEqCirc = D[tau[t], t] == -kp *tau[t] - kd * D[tau[t],t] + km* V[t];
motorEq = D[tau[t],t] == (-kp * tau[t] + km * V[t])/(1+ kd);
(* initial condtions *)
tau0 = tau[0] == 0;
e0 = e[0] == 100;
v0 = V[0] == 10;
Does anybody have any ideas how to approach this problem?
safe radishBOT
Hello. I was assigned this homework and I have never learned about differential ...
@kind dove Has your question been resolved?
safe radishBOT
@kind dove Has your question been resolved?
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@kind dove Has your question been resolved?
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@kind dove Has your question been resolved?
robust river
Hello. I was assigned this homework and I have never learned about differential ...
It seems like your system is overdefined
kind dove
It seems like your system is overdefined
Hmm, what do you mean by that?
robust river
Hmm, what do you mean by that?
First of all, tau directly depends on e:
tau[t] = sp - e[t], so essentially you have only 2 unknown functions: e[t] and V[t]
However, you have 3 equations for those 2 functions which is too much
kind dove
Hmm
And how to deal with it in Wolfram?
I thought that Wolfram can figure out things like this automatically
but tau[t] can be also expressed from that third equation
robust river
I thought that Wolfram can figure out things like this automatically
Well, no. It's like asking "solve x' = 1, x' = 2"
It's an incorrect question
First you should figure out a well-defined system of ODE
kind dove
uff
robust river
In this case it should be a system of 2 equations
kind dove
Let me think about it
robust river
Hello. I was assigned this homework and I have never learned about differential ...
I suspect that here some equations are dependent on others. If not, then it's an incorrect system
kind dove
Yeah, that's the problem I've been struggling with but I'm not sure If I can get rid of the dependence
So system of linear ODE can't have dependence?
robust river
It can, but after all reductions it should look like (x', y') = A(x, y) + b, where A is a square matrix
kind dove
ufff, this is very difficult for me since I have never solved single ODE 🙁
Also V(t) can be expressed in two ways. How to deal with it?
robust river
Well, the point I'm making is that an system like e.g. x' = 1, y' = 2 is of course equivalent to x' = 1, y' = 2, x' + y' = 3, but it's reasonable to work with the reduced system (so the first one)
robust river
Also V(t) can be expressed in two ways. How to deal with it?
Choose the most convenient expression for you
kind dove
To sum it up, should I write it on paper and somehow rearrange that to system of 2 ODE and then I can put it into Wolfram?
@robust river
robust river
To sum it up, should I write it on paper and somehow rearrange that to system of...
Yes
Or try using e'[t] instead of D[e[t], t], etc.
Maybe NDsolve will notice all simplifications then
However, if your system is incorrect, then it won't help either
Because imagine a system like x' = 1, y' = 2, x' + y' = 4
kind dove
Yes
Hmm, ok. Thanks for help and I will try to figure it out somehow. Please, can I ping you then?
robust river
Hmm, ok. Thanks for help and I will try to figure it out somehow. Please, can I ...
Yes, of course
safe radishBOT
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scenic valley
.close
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fathom plume
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fathom plume
pls help need roots
derived above from that if it looks wrong lmk
i assume its not optimal to expand the (1-12/13a^2)^-1?
<@&286206848099549185>
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fathom plume
<@&286206848099549185>
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fathom plume
fathom plume
can you resend the question
^
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inner tapir
\[
a^{ \log_a \left( \frac{ 6656 }{ \frac{ 52 - 48 a^2 }{ 69 a } } + 1 \right) }
= \left( \frac{ 13 }{ 16 a \frac{ 52 - 48 a^2 }{ 69 a } } \right)
a^{ \frac{ 13 }{ 16 a \frac{ 52 - 48 a^2 }{ 69 a } } }
\]
citrusmunch
inner tapir
just making sure i have the setup correct?
inner tapir
ok i think i misread your handwriting, so i'm using 64 instead of 69. i get something close to what you have. i'll have to recheck my work...
flat frigateBOT
citrusmunch
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hard crest
what happened to the iargz? or the /z?
well there is that pesky /z that you've ignored thus far...
yeah probably, i don't remember all of complex analysis but you do seem to be right so far with that
safe radishBOT
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Available help channel!
arctic mountain
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arctic mountain
I have no idea after this
like I have no idea to continue sloving
and whether the direction is correct
safe radishBOT
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safe radishBOT
arctic mountain
.reopen
safe radishBOT
✅
hasty wagon
is there anything in part (a) that is useful for this question?
sullen mirage
is there anything in part (a) that is useful for this question?
I don't think it would be useful...
hasty wagon
I don't think it would be useful...
oh...
sullen mirage
Why are you assuming x is even
sullen mirage
Your first assumption is fine
If x is rational then we can write x=a/b where a and b integers
then replace this is the equation and you will find a contradiction
(probably)
arctic mountain
is there anything in part (a) that is useful for this question?
nope
that is another question
hasty wagon
that is another question
i see
sullen mirage
then replace this is the equation and you will find a contradiction
you should end with a wierd looking equation but you can rprove that no integer solution exists with arithmetics
hasty wagon
you should end with a wierd looking equation but you can rprove that no integer ...
that's so true
remember the condition x=a/b where gcd(a,b)=1
sullen mirage
remember the condition x=a/b where gcd(a,b)=1
yh
arctic mountain
If x is rational then we can write x=a/b where a and b integers
thanks let me get a try
sullen mirage
thanks let me get a try
alr gl
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@arctic mountain Has your question been resolved?
pine oar
Yea
sullen mirage
like this?
umm why did b^5 disappear in the last line
sullen mirage
Beacuse it wanted
im talking of the b^5 in a^2b^5
arctic mountain
arctic mountain
and factorise the maximum possible
like this? And say it is contradict? it seems wierd : (
sullen mirage
like this? And say it is contradict? it seems wierd : (
try to put all of the a in one same place
sullen mirage
like this? And say it is contradict? it seems wierd : (
wait actually yes it works
wait a min
sullen mirage
Anxiety: 📈 📉 📈 📉 📈 📉
no wait
u can do it another way
b divides a^2 or a^5+b^5 (which means b divides a^5)
so by gauss's theorem b always divides a
which is a contradiction from our original assumption (gcd(a,b)=1)
@arctic mountain
arctic mountain
which is a contradiction from our original assumption (gcd(a,b)=1)
this theorem is out of my syllbus
sry it can't be used
sullen mirage
this theorem is out of my syllbus
well then use what you said
it still works
just ugly as hell
twin solar
hey can someone please help me understand, how can i find rank of such matrix?
twin solar
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gentle socket
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gentle socket
I tried using natural logarithm in both questions, but I cannot solve them.
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dire pond
why do we know that? I understand the first one but the other two don't make sense to me
safe radishBOT
why do we know that? I understand the first one but the other two don't make sen...
fathom jewel
why do we know that? I understand the first one but the other two don't make sen...
The 2nd is chain rule
faint seal
the second equation is just differentiating the first equation along with the use of chain rule
and the third equation is just dividing both sides by f'(x)
dire pond
can someone pls explain why the second one is chain rule? sorry, my brain is completely fried and we werent taught this stuff in class
but thanks for the help
fathom jewel
it's literally defined like that, just look it up
shy lynx
You should know the chain rule
dire pond
i know the chain rule, but how does that relate to inverse functions?
shy lynx
Taking the derivative of g(f(x)) and getting g'(f(x)) * f'(x) is exactly the chain rule, unrelated to inverse functions
The identity g'(f(x)) = 1/f'(x) where g = f^{-1} (i.e. the identity proven here) is one you should also know.
shy lynx
ah no, that's the product rule
shy lynx
easy to get them confused at first
dire pond
i just had my vocab mixed up
dire pond
easy to get them confused at first
the thing is, this isn't at first 😭
fathom jewel
im a dumbass
no
dire pond
my brain has just been so messed the past couple days
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median canopy
for a) tried comparing coeff of the real components when expanding (cis theatre) ^6 but got lost. How else would u do it?
safe radishBOT
for a) tried comparing coeff of the real components when expanding (cis theatre)...
indigo birch
expand (cos(x) + i sin(x))^6
you may have gone wrong when determining powers of i
and replace even powers of sin(x) with cos(x)
using sin^2x = 1 - cos^2 x
median canopy
ohh kk thxx
broken wyvern
It is known that $f(x + 2) + f(x + 1) = x + 2$ for every real number $x$. If $f(9) = 12$, then the value of $f(62)$ is . . .
safe radishBOT
It is known that $f(x + 2) + f(x + 1) = x + 2$ for every real number $x$. If $f(...
flat frigateBOT
m13
broken wyvern
Is there another way of getting f(62) without bashing?
vagrant ice
Is there another way of getting f(62) without bashing?
f(x + 1) + f(x) = x + 1
hence f(x + 2) - f(x) = x + 2 - x - 1 = 1 for all reals x
oh yeah we're nearly there, so you thus know what f(61) is and hence you know f(62)
broken wyvern
f(x + 1) + f(x) = x + 1
hence f(x + 2) - f(x) = x + 2 - x - 1 = 1 for all reals...
To get f(x + 1) + f(x) = x + 1, is subsituting x = -1 right?
How can i get f(x + 2) - f(x) = x + 2 - x - 1 = 1 then
vagrant ice
To get f(x + 1) + f(x) = x + 1, is subsituting x = -1 right?
How can i get f(x ...
just subtract the equations then
$f(x + 2) + f(x + 1) - (f(x + 1) + f(x)) = f(x + 2) - f(x)$
flat frigateBOT
south's secret twin brother
vagrant ice
same for the right hand side
broken wyvern
I'll try wait
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@broken wyvern Has your question been resolved?
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@broken wyvern Has your question been resolved?
snow stag
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hazy spruce
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exotic charm
i had a question about lagrange's theorem
safe radishBOT
i had a question about lagrange's theorem
exotic charm
if you have a finite group G then if H is a subgroup of G, doesnt that mean there are as many right cosets of H as there are elements of G
and since |Hx| = |H| for any x in G and the right cosets partition G, then doesnt that mean |H| + |H| ... + |H| = |G| = |G| |H| = |G| which implies |H| = 1?
but that's obviously not true cz |H| doesnt have to always = 1
nvm i just realised something
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native robin
can someone help me understand this
safe radishBOT
can someone help me understand this
plucky elk
What
native robin
i dont understand the first graph
plucky elk
What is the math you're plotting
native robin
What is the math you're plotting
a trajectory of a gradient vector field
plucky elk
Show the gradient
native robin
Show the gradient
now what
plucky elk
What exactly don't you understand
native robin
What exactly don't you understand
this part
what does xtrajectory and ytrajectory mean
plucky elk
Trajectory is the solution in terms of a function given an initial condition which in this case is random
native robin
Trajectory is the solution in terms of a function given an initial condition whi...
ye
native robin
Are you still confused about something
so is xtrajectory just the x-coordinate of the gradient vector
the x partial derivative of f
plucky elk
Yes
native robin
Yes
so would the explanation for this "Use your knowledge of the fact that when you plot gradf[x, y] with tail at {x, y}, the direction of gradf[x, y] is the direction of greatest initial increase in f[x, y] to explain why this curve had no choice but to go up." just be that xtrajectory specifies the x-direction needed to get the greatest increase of x and ytrajectory specifies the y-direction needed to get the greatest increase so the plot of f[xtrajectory,ytrajectory] goes in the exact direction of the gradient vector, meaning f will always increase
ok ty
i think i understand it now
.close
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fervent hatch
15
fervent hatch
Is this not saying that dT/dt is constant and that T(t) is linear??
It isn’t but I don’t know how else to interpret being given a slope of 2/1
modest hazel
isn't it just f(t)=20+2t?
median vigil
it's just giving you the derivative of temperature in that moment. given the context i don't think you should assume the derivative is constant
fervent hatch
Okay so it’s just horrible wording then 💀
median vigil
it might not even be the derivative actually, that might just be saying its temperature is 22 degrees at t = 1 second
fervent hatch
Bro what is this question wording im so confused
Lemme assume that it’s the rate at that moment
I’ll solve it and see if the answer is right
Nope
The answer I got is 8.31 seconds
The book says 82 seconds
@median vigil
median vigil
maybe try the other interpretation
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fervent hatch
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lean otter
i have no idea how to do this
safe radishBOT
i have no idea how to do this
lean otter
<@&286206848099549185>
hollow tulip
<@&286206848099549185>
Hello
red delta
i have no idea how to do this
Do you know what range means?
lean otter
no...
hollow tulip
It's the difference between the lowest and highest function
red delta
Range is the set of all defined outputs of a function.
hollow tulip
Oh
lean otter
okay
red delta
It's the difference between the lowest and highest function
In statistics
lean otter
Well do you have any idea now?
im still stuck
red delta
im still stuck
Okay so do you know what domain means?
lime lark
im still stuck
have u figured out $g^-1$ ?
flat frigateBOT
localbossman
red delta
$g^{-1}$
flat frigateBOT
David
lime lark
oops
lean otter
Okay so do you know what domain means?
like all the possible numbers for the function
lean otter
have u figured out $g^-1$ ?
nope
red delta
like all the possible numbers for the function
Yes all numbers which give a defined output
lean otter
Yes all numbers which give a defined output
yes
red delta
Now that the definition of domain and range has been established
red delta
do that first, it’s the opposite of g(x)
Do this
lean otter
do that first, it’s the opposite of g(x)
so i input the -3 into the equation?
lime lark
nooo
austere cypress
do that first, it’s the opposite of g(x)
i don't think that's how it would be solved, it's a cubic so it will have a pretty complicated imverse function
lean otter
what
red delta
so i input the -3 into the equation?
No, replace x with $g^{-1}$(x) and replace g(x) with x
Then make $g^{-1}(x)$ stand alone.
lime lark
so i input the -3 into the equation?
So basically you can sub in y for g(x) and solve for x. Whatever you have equal to x is ur inverse function
flat frigateBOT
David
lean otter
So basically you can sub in y for g(x) and solve for x. Whatever you have equal ...
so y is greater than or equal to -3?
austere cypress
$x² + 6x = y$ solve for x to get the inverse function
flat frigateBOT
kaue
lime lark
yes what kaue said
austere cypress
is what he means
red delta
We could just replace the variables though.
red delta
so do i just make x the subject
Yes
red delta
$\sqrt{y}$
flat frigateBOT
David
austere cypress
so that means $g^{-1}(x)= sqrt{y/7}$ i think
thats not the inverse function tho
lime lark
what is it?
red delta
x=$(g^{-1}(x))^2$ +6$\cdot$$g^{-1}(x)$
flat frigateBOT
David
lean otter
imso confused
red delta
):
red delta
imso confused
I think we should start from the beginning.
lime lark
fr
austere cypress
what is it?
$x^2 + 6x = y$ then $x^2 + 6x - y = 0$ then $x = \frac{-6 \pm \sqrt{6^2 - 4(1)(-y)}}{2}$ which can be simplified further
lean otter
I think we should start from the beginning.
okay
red delta
Firstly, replace x with $g^{-1}(x)$
flat frigateBOT
David
lime lark
what’s going on kaue😭
red delta
Then replace g(x) with x
lean otter
yes
flat frigateBOT
kaue
austere cypress
what’s going on kaue😭
latex is hard
flat frigateBOT
David
red delta
<@456226577798135808>
Okay in order to avoid confusing the OP we have to agree on a method to use.
red delta
is this right
No
lime lark
is this right
you aren’t supposed to plug in numbers yet…
lean otter
oh
red delta
c=x
austere cypress
we really should start from the beginning
red delta
where did c come from
The quadratic formula
lean otter
The quadratic formula
ohhh yhh
lean otter
ur right
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Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).
lime lark
oh uh
lean otter
Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they ...
oh
i’m cooked them
then
lean otter
I don’t condone the use of chatgpt I swear
i do
lime lark
ngl
lean otter
ts is long fam
lime lark
if only the function was $g(x)=x^2 +6$ instead
red delta
Can we get back to the question?
lean otter
Can we get back to the question?
yes pls
red delta
Okay I think we stopped here
Replacing variables
Make sure you use c=x for the quadratic formula.
lean otter
Make sure you use c=x for the quadratic formula.
$$so it’s x^2 + 6x + x$$
flat frigateBOT
mj
red delta
No
x=$(g^{-1}(x))^2$ +6$\cdot$$g^{-1}(x)$
flat frigateBOT
David
red delta
Replace x with g^-1(x) and replace g(x) with x to get this
lean otter
Replace x with g^-1(x) and replace g(x) with x to get this
okay got it
austere cypress
**David**
in this case c would be -x, not x, wouldn't it? because you need to move x to the other side making it equal to 0
lean otter
Now you can use the quadratic formula
C=-x remember
but how do i put that in the calculator
austere cypress
in this case c would be -x, not x, wouldn't it? because you need to move x to th...
i think the method i suggestet would be easier
red delta
but how do i put that in the calculator
You don't
Just simplify as much as possible
red delta
i think the method i suggestet would be easier
Okay then
austere cypress
let's start over
lean otter
okay
austere cypress
so you have the function $x^2 + 6x$
flat frigateBOT
kaue
lean otter
yes
austere cypress
to find its inverse you set it equal to y and solve for the input (x)
like this
$x^2 + 6x = y$
flat frigateBOT
kaue
lean otter
yes
austere cypress
you can move y to the other side to make the other side equal to zero so we can apply the quadradic formula
$x^2 + 6x - y = 0$
flat frigateBOT
kaue
austere cypress
now you need to figure out who are the coefficients (a, b and c)
austere cypress
actually, c is -y
lean otter
oh ok
austere cypress
because its the constant part
lean otter
okay
austere cypress
so applying the quadratic formula:
$x = \frac{-(6) \pm \sqrt{(6)^2 - 4(1)(-y)}}{2(1)}$
flat frigateBOT
kaue
austere cypress
do you understand this part?
lean otter
yes
austere cypress
now we can simplify it
first let's look at just the part inside the square root
$(6)^2 - (4)(1)(-y) = 36 + 4y = 4(9 + y)$
flat frigateBOT
kaue
austere cypress
yes
lean otter
2(3+sqrt y)
austere cypress
Actually, for the part inside the bracket, you can't distribure a square root to a suk, only to products
austere cypress
$\sqrt{4(9 + x)} = \sqrt{4} \cdot \sqrt{9 + x} = 2\sqrt{9 + x}$
flat frigateBOT
kaue
lean otter
oh
austere cypress
6+2y
No because you can't take the square root of each term, $\sqrt{p + q} \neq \sqrt{p} + \sqrt{q}$
flat frigateBOT
kaue
lean otter
ohhh okkk
austere cypress
So, replacing that root part we have $\frac{-6 \pm 2\sqrt{9 + y}}{2}$
lean otter
yh
flat frigateBOT
kaue
austere cypress
simpliying by 2 we have $-3 \pm \sqrt{9 + y}$
lean otter
okay
flat frigateBOT
kaue
austere cypress
so thats the inverse function
lean otter
thats the g^-1
austere cypress
$g^{-1}(y) = -3 \pm \sqrt{9 + y}$
flat frigateBOT
kaue
lean otter
yh
austere cypress
what was the question asking for again
austere cypress
So it can output values as small as you can by choosing a big value of y on $-3 - \sqrt{9 + y}$ and it can be as big as you can by choosing big values of y in $-3 + \sqrt{9 + y}$, so the range is $(-\infty, \infty)$
flat frigateBOT
kaue
austere cypress
and i am very dumb because the range of the inverse function is just the domain of the function so there was no need to find the inverse function
goddamn
lean otter
and i am very dumb because the range of the inverse function is just the domain ...
so i js replace the x w y
?
austere cypress
**kaue**
and thats not even the answer smh
can you forget everything that was said before
lean otter
okay
austere cypress
my brain isnt even braining anymore <@&286206848099549185> sum one help
lime lark
U guys are STILL on this question?😭
lime lark
not in a rude way
lean otter
ikik
lime lark
it looks simple on paper but like
lean otter
brobro icli might js skip dis
lime lark
we can’t even get $g^{-1}$ let alone the range
flat frigateBOT
localbossman
austere cypress
the thing is, the range of the inverse function is just the domain of the function itaelf, but the question is already saying then domain is [-3, infinity) so i dont even know anymore
lime lark
can’t we just put $g^{-1}(x) = sqrt{y}/7$
flat frigateBOT
localbossman
lime lark
even if it’s wrong
austere cypress
even if it’s wrong
why would we do that 😭
austere cypress
the thing is, the range of the inverse function is just the domain of the functi...
wull, thats what i would answer
i guess the objective of the question was just realizing that the range of the inverse function is the domain of the function
lean otter
sooooooooooo whats the answer
austere cypress
$[-3, \infty)$ because the range of $g^{-1}$ is the domain of $g$ which is any value $\ge -3$
flat frigateBOT
kaue
austere cypress
damn how much time did this take
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burnt night
is this sufficient proof to show this:
safe radishBOT
is this sufficient proof to show this:
burnt night
my proof goes
B = A * A = [A*a1 A*a2 ... A*ai] where a1...ai denote the columns of A
each column of A, by definition, denotes what vertex j flows into in a walk of size 1
the entire matrix A denotes where an inputted set of vertices will go. for example, if we multiply A by a vector that has an entry in its first component, it will tell you where vertex 1 will go in a walk of length 1
therefore the columns of [Aa1 Aa2 ... A*ai] denote where the vector a_i (representative of where an arbitary vertex j flows into) will flow into
defining a walk of length 2
oh shit man im stupid
i can just multiply thr matrices themselves to get every b_ij
LMFAO
short kelp
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indigo bay
I’m an idiot
no you are not
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austere cypress
is this the only info given?
short kelp
yes
austere cypress
i don't see how one would calculate the radius with just time information
it would need the velocity or something
frozen jolt
hi im wondering how to find a formula for area of the triangle using the radius of the circles inside it
safe radishBOT
hi im wondering how to find a formula for area of the triangle using the radius ...
toxic path
hi im wondering how to find a formula for area of the triangle using the radius ...
I have no idea if it’s true but assuming that’s an equilateral triangle, could you find the tangent of the 3 inner circles to the triangle, make that a seperate triangle, find the area of that, and then multiply by 4?
frozen jolt
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granite cape
Hey, does anyone know how to calculate angle theta and x?
safe radishBOT
Hey, does anyone know how to calculate angle theta and x?
twilit moat
a few things
first the total angle of a straight line is 180 and a right angle is 90 degrees
granite cape
Yes
twilit moat
hmm actually im struggling a bit with your drawing
granite cape
Oh, which part?
twilit moat
I was thinking that the angle x would be equivalent to 70
considering it looks like a reflection question
but nvm
what's the force distribution
granite cape
I’ll send the original question hold on, might be easier that way.
The correction part is just saying that the left support is a roller
I'm just mainly confused with how to resolve the 100kN force into it's components
twilit moat
could you draw a triangle here?
smth like this
and then the 100kN is the downward force
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