There's still some x after your sub, so it wasn't completed
#help-33
whole sleet
red nimbus
You forgot a few things, you need to write dx in terms of du
Not thinking, I don't think sub was a good idea
Guess doing ibp straight away then
red nimbus
Hmm okay
whole sleet
That's a pretty wild sub lol
keen girder
whole sleet
But yeah I see how it works
red nimbus
Now I am curious
whole sleet
You've still got the same problem here, where you proceed to ibp before finishing your sub. It doesn't work that way
keen girder
hmm okey wait then
whole sleet
Just, finishing the sub is difficult for u = x^2 + 1. That sub doesn't work very well
elfin berryBOT
red nimbus
Okay they did ibp right away
And then you proceed with long division since you get a rational function
The key is in getting rid of the log
red nimbus
In the first line on the left integral you forgot a dx else it looks good.
In the second line, I would rather factor 1/5 before the integral, but still good so far.
In the third line you forgot the dx again and you need to do long division.
smoky wigeon
For me I will do polynomial division
for the integral x^6/(1+x^2) you can transferred into integral of [(x^4 - x^2 + 1) - 1/(1+x^2)]
and from that you could integral the 1/(1+x^2) using inverse trig, it should be arctan(x) + C
actually no need for long division I believe
elfin berryBOT
red nimbus
marsh citrusBOT
@keen girder Has your question been resolved?
short marsh
for the integral x^6/(1+x^2) you can transferred into integral of [(x^4 - x^2 + ...
Good Based on the division you performed, the integration becomes as follows:
marsh citrusBOT
Available help channel!
winged garnet
i’m confused how to solve this
marsh citrusBOT
i’m confused how to solve this
keen fern
what is the average rate of change of an arbitrary function c(x) over the interval [-3,5]?
in terms of c
remote barn
i’m confused how to solve this
what topics do you have
marsh citrusBOT
@winged garnet Has your question been resolved?
winged garnet
what is the average rate of change of an arbitrary function c(x) over the interv...
8 and 4
keen fern
i didn't mean pick an example c
remote barn
i’m confused how to solve this
the average rate of change between the function is the average between the endpoints of the interval it is at
keen fern
if i had a function c i wasn't allowed to reveal to you, and you had to explain to me how to compute its average rate of change over the interval [-3,5], what would you tell me?
remote barn
if i had a function c i wasn't allowed to reveal to you, and you had to explain ...
average, it'd just be the (f(x2)-f(x1)) / (x2-x1), how much it changes for the interval on average so we just look at the interval here
I phrased it wrong mb
winged garnet
the average rate of change between the function is the average between the endpo...
im confused
Gng
js explain how i can do this pls
remote barn
im confused
sorry read my followup message
winged garnet
-3, 5
keen fern
average, it'd just be the (f(x2)-f(x1)) / (x2-x1), how much it changes for the i...
are you an alt of draco?
remote barn
js explain how i can do this pls
the rate of change of f(x) over the interval x is how much f(x) changes by this much per difference in x
for the interval, how MUCH did it change from x=-3 to x=5
winged garnet
are you an alt of draco?
no
keen fern
then why did nautilus answer my question for you lol
winged garnet
the rate of change of f(x) over the interval x is how much f(x) changes by this ...
8 units
winged garnet
then why did nautilus answer my question for you lol
i’m not sure
winged garnet
how much did f(x) change
4 units
remote barn
then why did nautilus answer my question for you lol
not the question, to you doing the ❌ thing
remote barn
4 units
what's the values you have for f(x) for x=-3 and x=5
oh
oh yeah yeah it's 4 units
woops
I was looking the wrong way
remote barn
4 units
find the function with the closest to 4 units changed from x=-3 to x=5
if I'm phrasing this rihgt
winged garnet
find the function with the closest to 4 units changed from x=-3 to x=5
yh
remote barn
yh
yeah that's it u just check every choice for whichever one has an increase of 4 units for x=-3 and x=5
remote barn
4/8
because 4/8 yeah disregard that since it doesn't matter because you use the same bounds throughout
that is the average rate of change however
marsh citrusBOT
@winged garnet Has your question been resolved?
shell python
draco can i ask you something
winged garnet
draco can i ask you something
what
winged garnet
idk
winged garnet
??
tardy nebula
i’m confused how to solve this
do you still require assistance with this Q
umbral folio
marsh citrusBOT
@winged garnet Has your question been resolved?
marsh citrusBOT
marsh citrusBOT
Available help channel!
steady sequoia
Hey, could someone explain to me what the symbol |=, defined as "\models" in latex, means?
marsh citrusBOT
Hey, could someone explain to me what the symbol |=, defined as "\models" in lat...
steady sequoia
More precisely, something like that.
flat basalt
If i recall right, this might be "is a theorem of", but I might be totally wrong.
Nevermind haha.
steady sequoia
Oh, thanks, where did you find this? It's very useful
flat basalt
Oh, thanks, where did you find this? It's very useful
https://en.wikipedia.org/wiki/List_of_logic_symbols
Full list of logic symbols, pretty useful.
steady sequoia
https://en.wikipedia.org/wiki/List_of_logic_symbols
Full list of logic symbols...
Thank you very much
.close
marsh citrusBOT
marsh citrusBOT
Available help channel!
novel juniper
marsh citrusBOT
novel juniper
Its irreducibility is immediate , as $f(0)=1=f(1)$.
Let $\theta$ be a root so that $\theta^3+\theta+1=0$
I now must examine the structure of $\mathbb{F}_{2}[x]/(p(\theta))$
elfin berryBOT
Wai
novel juniper
Not sure how to proceed from here
marsh citrusBOT
@novel juniper Has your question been resolved?
devout mauve
just compute theta^2, theta^3, theta^4,... using the recursion
novel juniper
just compute theta^2, theta^3, theta^4,... using the recursion
so I just say $\theta = -(\theta^3+1)$ and start from there?
elfin berryBOT
Wai
novel juniper
so $\theta^4= \theta^2+ \theta$ and so on
elfin berryBOT
Wai
devout mauve
yes
novel juniper
coool
thanks
and then once I get to $\theta^5+ \theta^3$, I can factor out the $\theta^3$ to cycle back
elfin berryBOT
Wai
novel juniper
you should never get powers bigger than 3
I mean $\theta^3(1+ \theta^2)$ so $(1+ \theta)(1+ \theta^2)$ and then simpliffy
elfin berryBOT
Wai
marsh citrusBOT
marsh citrusBOT
Available help channel!
buoyant jetty
marsh citrusBOT
elfin berryBOT
Renato
knotty trellis
What have you tried?
marsh citrusBOT
@buoyant jetty Has your question been resolved?
buoyant jetty
I dont know how to start
brave marsh
Write down your base cases and your induction hypothesis.
That's how you start any proof by induction.
brave marsh
and your induction hypothesis.
buoyant jetty
> and your induction hypothesis.
I did
brave marsh
You insisted earlier on using strong induction on a problem which was tailored for weak induction. Now this problem needs strong induction and you're not using strong induction.
So what is your induction hypothesis?
What are you assuming holds?
buoyant jetty
what was strong indu
brave marsh
You've only assumed P(n) holds for your induction hypothesis, so this is weak induction, not strong induction.
buoyant jetty
You've only assumed P(n) holds for your induction hypothesis, so this is weak in...
pk holds for k in [1, n]
buoyant jetty
What of a_(n+1)?
unsure
buoyant jetty
What of a_(n+1)?
@brave marsh
brave marsh
Try things to be less unsure.
buoyant jetty
Try things to be less unsure.
dude please help me
brave marsh
You want to write a_(n+1) in terms of previous terms. How could you possibly do that?
buoyant jetty
what is n?
buoyant jetty
You want to write a_(n+1) in terms of previous terms. How could you possibly do ...
why
brave marsh
Because that is how you argue with induction.
brave marsh
You don't need my input to continue
buoyant jetty
I dont know how to continue tho
buoyant jetty
pk
an < 3^n
2an < 2.3^n
2an + an-1 < 2.3^n + an-1
2an + an-1 + 1 < 2.3^n + an-1 +1
an+1 < 2.3^n + an-1 + 1
@brave marsh
brave marsh
Is n-1 <= n?
buoyant jetty
y
brave marsh
pk holds for k in [1, n]
.
buoyant jetty
an+1 < 2.3^n + an-1 + 1 < 2.3^n + 3^n-1 + 1
brave marsh
Now try to use inequalities to show that this is < 3^(n+1).
buoyant jetty
how
an+1 < 2.3^n + 3^(n-1) + 1
a_(n+1) < 2.3^n + 3^(n-1) + 1 < 2.3^n + 3^n + 1
a_(n+1) < (3^n)(2+1) + 1
a_(n+1) < 3^(n+1) + 1
@brave marsh
brave marsh
hint: ||you can do better than 3^(n-1) + 1 < 3^n +1||
buoyant jetty
hint: ||you can do better than 3^(n-1) + 1 < 3^n +1||
how?
buoyant jetty
hint: ||you can do better than 3^(n-1) + 1 < 3^n +1||
care to elaborate
nothing of this appears to be legal
@brave marsh
buoyant jetty
hint: ||you can do better than 3^(n-1) + 1 < 3^n +1||
for all the things you are suggesting we would need more induction
are you here?
brave marsh
You don't need induction to know that a tighter upper bound exists.
Notice that 1 < 3^(n-1) + 3^(n-1).
young owl
a trick that I like when the exercise didn't necessarily have the best approach, like here, is to prove some intermediate step
even if you don't want to do the trivial induction about a_n being a sequence of integers, you can still decide to show a_n <= 3^n - 1 for all n
brave marsh
That works too
brave marsh
Notice that 1 < 3^(n-1) + 3^(n-1).
If you want to prove this sure you can use induction, but it should be fairly obvious that this is the case since 3^(n-1) >= 1 for any n >= 1
marsh citrusBOT
@buoyant jetty Has your question been resolved?
marsh citrusBOT
@buoyant jetty Has your question been resolved?
buoyant jetty
.close
marsh citrusBOT
marsh citrusBOT
Available help channel!
kind mulch
I’ve got no clue on how to start this question. For two vectors a and b, a cross b = b cross a. Write a relationship between a and b and explain your reasoning
marsh citrusBOT
I’ve got no clue on how to start this question. For two vectors a and b, a cross...
sleek hawk
what have you learnt about the cross product of two vectors so far?
kind mulch
Not that much
sleek hawk
that implies you've learnt a bit. can you explain what you've learnt, however little?
because maybe you already have a key piece of information you need.
sleek hawk
okay. and have you learnt about what happens if you swap the order of the two vectors in a cross product?
sleek hawk
yes. and what does that tell you about the signs of the cross products?
like, suppose we have c, d \in R^3. what is the relationship between c x d and d x c?
kind mulch
A cross b = -(b cross a) ?
sleek hawk
absolutely.
but now you are told that a x b = b x a. yet you know that normally, a x b = -(b x a).
can you figure out where to go from here?
kind mulch
For b x a = -( b x a) the components of b and a are even functions but this is just referring to numbers not functions so I’m not too sure
sleek hawk
okay, that's the right first step, but you need not overthink about this.
suppose you treat b x a as a variable. how would you solve for its value?
kind mulch
Get it by itself
sleek hawk
try working it out here.
kind mulch
So like a x b = -V for instance
sleek hawk
mm, I fear you may be straying off track, hence I suggest you focus on b x a = -(b x a).
kind mulch
b x a for the LHS?
kind mulch
b x a = 0
kind mulch
When their components are the same I think, or like ones positive a and the other is negative a. So we just get [0,0,0] when we take the cross product
sleek hawk
this makes one of the two circumstances. what are these vectors called in general, where you can scalar multiply one to get the other?
kind mulch
Parallel
kind mulch
Perpendicular ?
sleek hawk
definitely not.
kind mulch
Is it still if it’s a scalar multiple
sleek hawk
I would advise you to think simple. in regular algebra, how do you get a product of 0?
kind mulch
If one is 0
sleek hawk
one or both, yes.
so the second condition is that either a or b, or both, are zero vectors.
and that's it.
kind mulch
So either a and b are parallel or a and or b are the zero vector
sleek hawk
yes.
kind mulch
And that’s our relationship then
sleek hawk
mhm.
kind mulch
So it’s just recognising what makes a x b = b x a then. And knowing our relationships like 0 = 0 ( if either a or b is 0) and parallel vectors being a scalar multiple
sleek hawk
correct. it ultimately boils down to the definition of the cross product and its anticommutativity (the fact that swapping the order of the operands of the cross product flips the sign of the product).
sleek hawk
glad to have helped!
kind mulch
.close
marsh citrusBOT
marsh citrusBOT
Available help channel!
alpine basalt
Zetamac sequence
marsh citrusBOT
Zetamac sequence
alpine basalt
What is the pattern
weak surge
Zetamac sequence
Polynomial Lagrange interpolation
(+12, then *4, then +12, then *4 again)
exotic valley
(+12, then *4, then +12, then *4 again)
!nosols, on top of misformatting
marsh citrusBOT
(+12, then *4, then +12, then *4 again)
As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.
weak surge
!nosols, on top of misformatting
😭 its been a while since i was factoided
fair, but i need them to decipher my message
im not "giving them the answer directly"
they still need to think about what i mean
indigo nest
I have no idea what to do cuz like sure technically this is nosols but also this isn't homework, and this also doesn't even belong here technically
weak surge
!nosols, on top of misformatting
also what misformatting
exotic valley
you used *s for multiplication. look at your text.
weak surge
yea but whats the issue
indigo nest
Anyway to answer OP's question, it's actually the roots of the polynomial $(x-5)(x-17)(x-68)(x-80)(x-320)(x-42069)$ in ascending order
elfin berryBOT
Zavier 🌺
weak surge
you used *s for multiplication. look at your text.
isn't multiplication supposed to be denoted as *
devout mauve
yes
marsh citrusBOT
@alpine basalt Has your question been resolved?
marsh citrusBOT
reef fable
It is basically what Carbonite said
marsh citrusBOT
Available help channel!
fast geyser
I genuinely don't know what the question wants
marsh citrusBOT
I genuinely don't know what the question wants
fast geyser
what is the definition of limits? Is it the one where left hand limit equal to right hand limit?
or do we prove using epsilon delta
brave marsh
That depends on the definition of limits you've been working with. If you've recently learned about epsilon-delta proofs, then the use of the word "Prove" suggests you should use that.
fast geyser
that's the thing, I learnt both ways, so picking one should be enough right
nvm I will start with epsilon delta
thanks bro
.close
marsh citrusBOT
marsh citrusBOT
Available help channel!
valid nova
$$ \int \sqrt{a^2 - x^2} , dx $$
marsh citrusBOT
$$ \int \sqrt{a^2 - x^2} \, dx $$
elfin berryBOT
Rahil
valid nova
Is there a way to think
Let x = asin(theta) 😭 ?
reef fable
A is a constant?
valid nova
A is a constant?
Yup
reef fable
Then make it in form of A^1/2 where A is a poly
reef fable
Wait ignore this method it wont be robust
harsh scroll
Do it geometrically <:sotrue:826971355542192148>
-# that's what I'm thinking twins 
valid nova
I have seen a method where they take x=asin(theta) and then solve but...
supple tundra
I think there is a direct formula using logarithms
valid nova
would x = asintheta not work?
It works but how can I think, consistent practice or something?
bleak thorn
It works but how can I think, consistent practice or something?
wdyym
stone pelican
It works but how can I think, consistent practice or something?
its a common trick, trigonometric substitutions
valid nova
its a common trick, trigonometric substitutions
Ic
stone pelican
how one would normally spot a trig sub is phytagorean looking things
valid nova
wdyym
Idk much calc just know some basic diff and int, still learning so I found it hard to guess about this thing
.close
marsh citrusBOT
marsh citrusBOT
Available help channel!
livid skiff
i need to prove that the local maximum exist only when the parameter 4 > a > 5
livid skiff
what i did first is
then
and then
then I made a new function, that is the delta
would it be right to apply IVT here? there aren't any clues really aside from it's a positive maximum value
like i can know that a < 4.14 from the delta, but never if a>4
can i simply apply IVT on m(a), would it be releevant?
astral bough
unfortunately i dont think it will be relevant
since you also need to prove that the solution of f'(x) is the local maximum and not for example, the minimum, or the turning point
but now that i think about this problem
it could be actually wrong
i think a<4.14 and |a|>3 is suffice
there's no interval actually
marsh citrusBOT
@livid skiff Has your question been resolved?
astral bough
i think a<4.14 and |a|>3 is suffice
nevermind, only a<4.14 is suffice
so conclusion
without any further conditions
this is the result
livid skiff
yeah
a<4.14
and then the prof puts 3 choices where a is also < 4.14
there's also all wrong option, but i can be too sure if all are wrong
hewlp
hollow blaze
a cubic can have 0,1 or 2 points where its derivative is zero, since we have a monic cubic here, we can say that for a maxima to exist , f’(x) must have 2 real solutions
and hence your method is right
but it’s important to know why it works
livid skiff
really?
actually if i plug the max a then it'd be a zero, not a negative number, is IVT still doable?
hollow blaze
actually if i plug the max a then it'd be a zero, not a negative number, is IVT ...
D > 0 and not D>=0
livid skiff
maybe '\greater'
plain trellis
\ge for >=, '>' for the normal one
hollow blaze
because D=0 gives 1 root
livid skiff
hmmm.. so i can do use IVT? even than m(a) is just delta.. and delta is the x-axis...
to find the range of a local maximum value
livid skiff
and then the prof puts 3 choices where a is also < 4.14
if it's a < 4.14 does that answers if it's option C, B, or A
i mean it eliminates D and that's about it
oooo wai, you're onto something
hollow blaze
how did you arrive at a < 4.14
livid skiff
i solved delta = 0.. indirectly
so the graph would only have 1 maximum value? i have seen the AI one with 1 maximum and 1 minimum
hollow blaze
isnt delta just a linear function in terms of a after some simplification
so you can just solve for delta > 0 quite easily…
hollow blaze
lets restart cuz , i think youve strayed away
why in the first place did you say that for f(x) to have a maxima, the only necessary and sufficient condition is for f’(x) to have real and distinct roots?
livid skiff
the derivative changes sign around each root
hollow blaze
and…
livid skiff
there is a local max
hollow blaze
yeah but why should f’(x) = 0 have two roots
livid skiff
so it has one maximum and one minimum
hollow blaze
but we dont want a minimum right
livid skiff
not at all
hollow blaze
so there has to be a reason why we are saying that if f’(x) only has one real root
it would be a minimum
livid skiff
since you also need to prove that the solution of f'(x) is the local maximum and...
if x_2 is the same number + the delta > 0 then it should be maximum right?
marsh citrusBOT
@livid skiff Has your question been resolved?
livid skiff
<@&286206848099549185>
visual bolt
<@&286206848099549185>
Yo yo yo
livid skiff
sup
visual bolt
What does my bro need help with?
livid skiff
i need to prove that the local maximum exist only when the parameter 4 > a > 5
i need to find the range of a parameter 'a' so this function can have a maximum on the positive axis
we already figured a<4.14, but 3 of the options we were given satisfy this condition
livid skiff
visual bolt
we already figured a<4.14, but 3 of the options we were given satisfy this condi...
For some reason, I can’t scroll up in this channel. Can you repost the question?
livid skiff
i need to prove that the local maximum exist only when the parameter 4 > a > 5
what i did was that i derivatived the function
made Delta > 0 so i can have 2 distinct roots so we have a maximum to begin with
found out a<4.14
but i still don't know how to eliminate choices
i omitted option E) none
it could be the correct tho
visual bolt
found out a<4.14
So, in order to have a maximum, would you agree that f’(4) > 0 and f’(5) < 0
Once you do what you did, and take the derivative, plug in 4, and use the quadratic formula to find a range of values for a. Plug in 5, and do the same thing. Once you do that, you’ll hopefully see a clear range
visual bolt
Once you do what you did, and take the derivative, plug in 4, and use the quadra...
By finding the intersection of your inequalities
@livid skiff, does that make sense? Or do you want me to go through it step by step?
lone gate
cubic function
visual bolt
cubic function
Ok?? What about that
livid skiff
So, in order to have a maximum, would you agree that f’(4) > 0 and f’(5) < 0
how do we know which is positive and which is negative
the maximum is at a=4.14, not x=4.14
visual bolt
the maximum is at a=4.14, not x=4.14
But whatever a value we get corresponds to an x value (4, 5)
Do you catch my drift?
visual bolt
how do we know which is positive and which is negative
Let’s take it one step at a time
What did you get for the derivative of the function?
livid skiff
visual bolt
I think smth might be wrong with that derivative
visual bolt
i will redo
Again, don’t quote me on it, lol
livid skiff
$$f'(x)=3x^2+6(a-2)x+3(a^2-9)$$
visual bolt
So, the 3 doesn’t just disappear
elfin berryBOT
Amer
visual bolt
<@&268886789983436800>
livid skiff
why mrbeat appearing in my channel particularly
visual bolt
why mrbeat appearing in my channel particularly
Because he wants to do stuff to you. Kim
visual bolt
you have to factor out the 3
But anyways…..
If you have 3x + 6
That doesn’t become x + 2
You still have a factor of 3 that you took out
<@&268886789983436800>
livid skiff
You still have a factor of 3 that you took out
ooh..
visual bolt
You still have a factor of 3 that you took out
So you have to multiply your whole expression by that factor
So, 3x + 6 would become 3(x + 2)
livid skiff
right
going back to og f'(x)
alright plugging 5 next
nice
so -13.8 < a < -12.062
visual bolt
Not so fast…..
I want you do draw a number line
And draw line segments for each interval
visual bolt
And draw line segments for each interval
And see what region of the number line both of the line segments occupy
visual bolt
how can i draw if i don't know 'a'
You know the range of a
So, this might be a stupid question, but you know what a number line is, right?
visual bolt
yeah
I want you to draw a number line. And draw two lines.
One that does from
-12.062 to 4.062
And the same thing for the other one
-13.8 to 3.8
Send a pic once you’ve done that
visual bolt
oops srry, i was drawing with notebook on my keyboard
All good. Lmao
livid skiff
they are not aligned oops
there we go
now they are aligned
so -12.062 < a < 3.8
i am taking it 1 step at a time
this makes it kinda hard
i was wishing for a smaller interval
visual bolt
this makes it kinda hard
In pretty sure you did smth wrong when you did f’(5)
visual bolt
i thought it was fine here
Again, can I see your process for f'(5)
livid skiff
visual bolt
And I agree with you up until the very final line
The value you get for a that will make it = 0, is right
But now you have to test out values on either side
If you actually graph it, you can see where it's greater than 0
visual bolt
how did bro graph a function with a parameter and x
I didn't. This is the graph of the parameter
a^2 + 10a - 54
livid skiff
But now you have to test out values on either side
so like, more values into f'(x)
visual bolt
And so, f'(5) actually becomes
a <−13.8 or a > 3.8
visual bolt
so like, more values into f'(x)
Yep.
visual bolt
And so, f'(5) actually becomes
a <−13.8 or a > 3.8
Does this make sense?
livid skiff
so 3.8 < a <4.062 ?
visual bolt
so 3.8 < a <4.062 ?
YESSS!!!!!
livid skiff
this does not fall under 3 < a < 4
so it's none ? option E
what if at maximum value it is 4 < a < 4.062
need more values into f'(x) right
i made AI rework my work and it said 4.062 < a < 4.142
main idol
i made AI rework my work and it said 4.062 < a < 4.142
!nogpt
marsh citrusBOT
i made AI rework my work and it said 4.062 < a < 4.142
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).
livid skiff
something does align
livid skiff
i think if i plug f'(4)
i shoulda have -12.062 > a > 4.062
which is enough by itself
since i should dismiss the negative interval because i am looking for a positive max, as the question states
this should leave me with 4.062 < a < 4.142
livid skiff
if 'a' is a parameter, i am not sure how it's relevant between 'a' being positive and f(x) having a positive max
like if 'a' is positive and then somehow f'(x) is also positive, that means we are yet to come across the axis of the maximim
as x axis progress and get bigger, i have no idea how 'a' is also supposed to move
is it wise that if we want x > 0, we also take a>0
how are they even related
or are they
since my function has a^2 which means 'a' can be what the f it wants to be
livid skiff
i think i might be wrong
maybe instead i need to make a=4
and then plug into x
and then f(x)
which honestly i have no idea what i will achieve
young owl
,w d/dx x^3 + 3(a-7)x²+3(a²-9)x-1
elfin berryBOT
young owl
you solved for x1 and x2 when they exist ig
as long as there is a positive delta you have two roots and the min one is where there is a local max
as others mentioned, it's not the (4, 5) interval
if you take a = 0
,w plot x^3-21x²-27x -1
elfin berryBOT
young owl
see, you still have a local max for a = 0
more precisely, you have a local max as long as a < 29/7
you should show the original question
marsh citrusBOT
@livid skiff Has your question been resolved?
livid skiff
i'll translate it, though the question kinda states a 'limit' instead of 'maximum value'. probably a mistake by prof
elfin berryBOT
Amer
livid skiff
options:
A. a< -3
B. -1 < a < 1
C. 3 < a < 4
D. 5 < a < 6
E. None of the aforementioned
young owl
,w plot x^3 + 3(-4-7)x²+3(25)x-1 from x = 0 to x = 5
elfin berryBOT
livid skiff
its... everwhere
young owl
the answer is a
livid skiff
all over the place
livid skiff
the answer is a
that'd make 80% of my classmates failures
young owl
I mean I'm just trying to understand your weird translation
from what I understand, the prof asks you for a positive local max
and it does happen when a < -3
livid skiff
and it does happen when x=4 no
young owl
the value of x where the local max is depends on a
livid skiff
like the previous helper pointed me
i started to believe the myth that prof sells homework and collaborate with students
young owl
,w -(2a-14)-sqrt((2a-14)²-4a²-36)) > 0
livid skiff
who sell them
the literal translation would be "Positive maximum limit"
and it does not make any sense in the original language as well, other than a typo
young owl
,w x²+(2a-14)x+a²-9 = 0
elfin berryBOT
young owl
,w -a-sqrt(2)*sqrt(29-7a)+7 > 0
elfin berryBOT
elfin berryBOT
red nimbus
maximum limit 
young owl
I'm not sure what wolfram cooked bc I'm tired
but I think the term local maximum was intended
livid skiff
same
young owl
the solving method is that f has a local maximum only when a < 29/7 iirc
young owl
and then f'(x) = 0 has two sol
the local maximum is when x_0 = minimal one
and then you want f(x_0) > 0
livid skiff
some students said "plug and try every 'a' and none works cuz the maxima is 2.65 or some shi"
i have no idea how they know i think the prof give them private sessions
which doesn't make much sense if u gon eliminate choices instead of finding answers
young owl
f(-a-sqrt(2(29-7a))+7) > 0 should imply one of the answers
but the exercise may be flawed
or maybe the translation isn't good enough idk
wait you did a sign error
aaaaaaaaaaa
you wrote two different things
it's either a²+9 or -9
livid skiff
oh shi
true
it's the former
sprry abt that, made u run a bunch of commands for nothin
young owl
yeah and I think there is a problem with the exercise anyway
like, the max limit stuff, the translation, idk
I can't find a way to really make sense out of it
livid skiff
i'll write an email to the professor
nothing gon change since half the peeps submitted the assignment
red nimbus
like, the max limit stuff, the translation, idk
we are starting to also reach our max limit and capacities 😔
livid skiff
the public guide, made by students, stated the following steps:
- derivative f(x)
- solve for f'(x)
- find 2 values, one max one min
- plug in f(x) to verify the value is max and positive. if that step is true then there are values for a
young owl
I mean that's litterally what I did
and I don't find an answer that is in your sheet
so
livid skiff
if you don't find an answer, doesn't it mean it's E
livid skiff
(none)
young owl
well it's not clear
there are solutions
it's just that it's not an interval written on the sheet
so idk
livid skiff
solutions that include the intervals in the options, but not all of them
young owl
don't even know that this means
young owl
<@&268886789983436800>
livid skiff
out of interest, what is ur method? if i may ask
because when i plug max x into f i end up with bunch of a's and x's
and i draw blank
young owl
f'(x)/3 = x²+(2a-14)x+a²-9 = 0 for two roots
the smallest one being x = -a - sqrt(2) sqrt(29 - 7 a) + 7
so you need to solve f(-a - sqrt(2) sqrt(29 - 7 a) + 7) > 0
f(x) = x^3+3(a-7)x²+3(a²-9)x-1
-a^3 - 21 a^2 + (321 - 28 sqrt(58 - 14 a)) a + 116 sqrt(58 - 14 a) - 876 > 0
livid skiff
the old helper who told me to plug f'(4) and find intervals for a
we narrowed down, but i assume it was for that specific parameter value
😵💫
young owl
can you show the answer sheet again?
the multiple choices
I find 3 sol
that are these ones
but we should exclude 29/7
for me it doesn't match the sheet
but I don't speak your language so there is definitely a translation issue
can't do more than that
livid skiff
a
livid skiff
unless you are arab
alright, but it confuses me as much
even that c is a part of the same question with a and b, i dont think they are connected
young owl
did the prof give the intended answer?
livid skiff
no, students believe its none
young owl
I would say none too, but who knows, it could be that the prof made a very unclear question
livid skiff
so is my method of solving f'(4) and finding "a = -4± √65"
and then ignoring the negative of a
young owl
I don't think this "method" has meaning
livid skiff
I don't think this "method" has meaning
how many lines it took u to find the 3 sol
young owl
I gave the eq to the solver, because it's atrocious
young owl
yeah, I would eliminate d first since a < 29/7
and then I would eliminate the others one by one
by checking I can find a such that f has a negative local maximum
in each interval
livid skiff
A is cool, but it eliminates all positive a's where the condition is satisfied
young owl
well not all, you would need to check between -3.5 and -3 to see the problem
the answer b would be the most painful
because you won't find negative local maximum
depending on what the prof meant, the answer could be b too
livid skiff
maybe he meant an actual limit where it's also the positive maximum
young owl
a limit doesn't make sense in this context
the upper limit of f(x) is +inf, and inferior is -inf
naive plover
Hey, this is grade 11 maths?
young owl
degree 3 polynomial
livid skiff
Hey, this is grade 11 maths?
yes, but flawed
livid skiff
degree 3 polynomial
maybe i could use IVT and narrow down where the sign flips
naive plover
yes, but flawed
Egp Thanawi ?
livid skiff
no
young owl
maybe i could use IVT and narrow down where the sign flips
I mean, I solved it, it's on the bounds of my intervals
and that's the problem
what the prof proposes doesn't make sense to me
livid skiff
i think they meant positive maxima as in on the positive x axis
young owl
doesn't make sense either because it would be the bounds of my intervals
livid skiff
i mean he put the word positive in bold blue
i think that's the key
perhaps second interval
young owl
he didn't talk about positive a tho
and he proposed negative ones
wouldn't make sense either
only the prof can explain what he meant imo
livid skiff
plug these a's into f(x) and find which are positive
which would be a hell of a question for 5 marks out of 100
young owl
the exercise isn't asking about f(a) either
young owl
if you plug a in f(x), it's f(a)
livid skiff
f(x)
livid skiff
yeah this one
young owl
yeah that's why it's not worth it anymore it becomes a guessing game about what a prof meant
you can't read the prof thoughts so
livid skiff
f(-a - sqrt(2) sqrt(29 - 7 a) + 7)
at this state as is, is it possible to know whether its positive or negative
young owl
no, you need to solve this > 0
and it gives what I gave
we're circling around and I can't guess what the prof meant
so I'll be dropping it here
livid skiff
f
marsh citrusBOT
@livid skiff Has your question been resolved?
meager hemlock
I’m trying to understand the connection between vectors and systems of linear equations.
For example:
x * [1, 1] + y * [1, -1] = [5, 1]
I know this corresponds to:
x + y = 5
x - y = 1
My confusion is about how to interpret the vector [1, 1].
In geometry, I think of [1, 1] as moving 1 unit right and 1 unit up.
But in the system-of-equations view, it seems like the entries are just coefficients in each equation, not really “directions”.
So my question is:
Are these two interpretations actually the same thing (just viewed differently), or should I think of vectors in linear systems differently from geometric movement?
Also, what’s the best way to think about what [1, 1] represents in this context?
marsh citrusBOT
I’m trying to understand the connection between *vectors* and *systems of linear...
cunning fiber
I’m trying to understand the connection between *vectors* and *systems of linear...
Yeah these are the same thing. It's the arrow from the origin to $(1,1)$ and is also one of the column vectors being scaled and added to make $\begin{bmatrix} 5 & 1 \end{bmatrix}$. You can think of it as a direction/arrow that's also a building block in the equation rather than something different from a (geometric) vector.
elfin berryBOT
Civil Service Pigeon
static quarry
at the risk of saying something that seems obvious, you can think of (1,1) as what you would get if you plugged in x=1 and y=0 to the left hand side of the system of equations
saying this in a somewhat more abstract way, the linear map represented by the left hand side maps the basis vector (x,y) = (1,0) to the vector (1,1)
meager hemlock
But here's what I am confused about. If “x-components” are supposed to correspond to horizontal movements only, why does the vector [1,1] (for the x coefficients) include a second entry that seems like it corresponds to a vertical direction?
I asked AI it mentioned this:
The core mistake in your thinking
You’re mixing two different uses of “x”:
1. Geometry “x-axis”
horizontal direction in a plane
moving left/right
2. Algebra variable “x”
just a number (a coefficient)
not a direction in space
They are unrelated except for notation.
💡 So why does “vertical movement” appear?
Because:
The second equation is not “vertical space”
It is just the second coordinate in a vector
We are not physically moving up/down.
We are just placing numbers in positions.
🧠 The key unifying idea
A vector like:
[1,1]
does NOT inherently mean:
horizontal + vertical motion
It means:
“first component = 1, second component = 1”
So now I'm confused as to what to understand
How can x components mean that we also move vertically as well, I thought they only move horizontally only? If so then we just say that the x component x[1, 1] is just us placing numbers in the x componetns in each of the two linear equations.
Geometrically: yes, it means 1 right, 1 up
In this system: the two entries correspond to two equations, not x and y directions
When you're in the system-of-equations world, the two slots in the vector don't mean "horizontal/vertical". They mean "equation 1 / equation 2".
So there is no vertical shift happening. The word "vertical" only applies when you're in geometric mode. In equation mode, the second slot just means "the second equation" — it could represent anything.
That's the core of your confusion — you're trying to read geometric meaning into slots that, in this context, are just equation labels.```
fervent rampart
those reflect two different ways of viewing a system of linear equations, what gilbert strang likes to call the "row picture" and the "column picture"
they're not immediately interchangeable, but they complement each other in a way
under the row picture, the vector [1,1] has geometric meaning in that it represents the intersection point of two lines, whereas in the column picture it's just a way of packaging up the coefficients of a linear combination and doesn't have a particular geometric significance
marsh citrusBOT
@meager hemlock Has your question been resolved?
undone agate
marsh citrusBOT
undone agate
I can’t solve this
plain trellis
,rccw
elfin berryBOT
plain trellis
chrome jacinth
Hi! What have you tried so far?
chrome jacinth
I can’t solve this
Hmmm, well, consider trying casework :)
Anjie and Bubay are going to be put in a group no matter what.
Suppose then Anjie goes in the first group - how many ways can you choose the remaining members of that group?
marsh citrusBOT
@undone agate Has your question been resolved?
marsh citrusBOT
marsh citrusBOT
Available help channel!
keen anchor
Saw this and playing around with it gave me the idea to try to show you can't flip the right arrow by flipping the left arrow.
I rephrased the problem as a set of actions on a 2dim vector of elements in {-1, 1}
So
a(v) = v .* [1, 1]b(v) = v .* [-1, 1]c(v) = v .* [1, -1]
And I want to show thatb(v)andc(v)are independent of each other. I have gotten as far as showing there are no real solutions, but I'm a bit stuck on whether there are complex solutions.
My approach thus far:
Repeated operations of b can be expressed as [-1, 1]^n
Suppose that b and c are dependent. This would imply a solution to the equation:
v .* [-1, 1]^n = v .* [1, -1]
writing this out elementwise we get a system of two equations:
v0 * (-1)^n = v0 * 1v1 * 1^n = v1 * -1
Which simplifies to(-1)^n = 11^n = -1
There are obviously no real solutions to 1^n = -1, but this is where I'm not sure where to go next, since there may be complex solutions. And there are real solutions to (-1) ^ n = 1, but I'm not sure if there are complex solutions.
marsh citrusBOT
Saw this and playing around with it gave me the idea to try to show you can't fl...
cedar stone
v .* [x,y] means termwise multiplication, right?, what do you mean a(v) and b(v) are independent
keen anchor
Sorry if my terminology is bad. .* is termwise yes, and I mean you can't get b(v) from a(v) no matter what you do with it, i.e. by repeating it some number of times.
cedar stone
Ok, then why are you worrying about complex solutions? If you only care about repeating a, then isn't n natural?
keen anchor
Once I put the problem in vector form, constraining it to natural numbers seemed... incomplete / un-rigorous
there's no rubric or anything on this I'm doing it for the love of the game. But I'm also bad at the game.
cedar stone
I see, well if you are taking the principal soln of 1^z for complex z, isn't that always 1?
keen anchor
if z is allowed to be non-real I'm not sure tbh
tight furnace
It's pretty hard for me to tell what's going on
cedar stone
if `z` is allowed to be non-real I'm not sure tbh
Ok, because if you allow non-principal solutions, 1^0.5 can be -1.
tight furnace
Is the arrow thing a Cayley diagram?
keen anchor
Yes
tight furnace
Whatever action corresponds to flipping the left arrow squares to the identity
So it can only generate the action of flipping the left arrow or the trivial action
cedar stone
Whatever action corresponds to flipping the left arrow squares to the identity
I think they are considering what happens if you only do half an operation, or a 2+pi*i of an operation, whatever that is supposed to mean
keen anchor
Trying to reframe the problem in a linear-algebra ish way was probably a mistake, it just led me down what seemed like an interesting rabbit hole.
tight furnace
hmm, a horizontal reflection doesn't have a square root as a matrix operation
Maybe over C it does
keen anchor
I'm not sure what half a flip, or a complex flip would look like, but we can do math on arbitrary dimensions even though I can only visualize up to R^3. Sorry if this question is dumb
cedar stone
I'm not sure what half a flip, or a complex flip would look like, but we can do ...
I would imagine it like rotating the first arrow 90degrees
tight furnace
applying it z times is the same as applying the matrix e^zA where A =
[πi 0
0 2πi]
or something
cedar stone
I guess if you are allowing weird enough stuff, you could have one base thing,
like consider Z+root(2)Z under + and quotient it by 2Z+2root(2)Z
If you allow like irrational amount of a operation
Then 1 would be a "generator" (in quotes because not exactly true but ehh)
keen anchor
there's probably already a name for the group of things you can get by multiplying the elements of [1, 1] by -1 and 1, but I haven't gotten that far into groups yet
cedar stone
In particular, applying the left flip operation "root(2) times" would do a right flip, in this case
cedar stone
there's probably already a name for the group of things you can get by multiplyi...
It's called (Z/2Z)^2
I think
tight furnace
Also V_4 or occasionally K_4
keen anchor
If we were to try the system of equations approach, where we get
(-1)^n = 11^n = -1
is this the part where I need to learn how to use euler's identity or some such?
cedar stone
In this part, if all you cared about was group behaviour, you could say n is an integer and get done with it
keen anchor
yeah that's true. If n has to be an integer for this to stay constrained to the original group, then 1*n = -1 is proof enough
dusty eagle
I really need help understanding aerospace engineering.
glass silo
I really need help understanding aerospace engineering.
I'm going to close this one because you already have #help-44｜stanley-🌲-v2-dans (and it's likely this is out of scope for our server).
If you do have questions closer to maths, you can open a channel for that purpose, do make sure that the occupied category is expanded so you can see where your channel is 
.close
marsh citrusBOT
Available help channel!
signal escarp
-cos 4t - 1/2 sin 4t
I am trying to find auxiliary angle for this equation I have got
Sqrt(5)/2 sin(x -alpha)
How do I find alpha?
I did:
Cos alpha = -0.5/ R
signal escarp
But that is not the right ans pls help
marsh citrusBOT
@signal escarp Has your question been resolved?
marsh citrusBOT
@signal escarp Has your question been resolved?
digital field
$-\cos 4t - 1/2 \sin 4t $
I am trying to find auxiliary angle for this equation I have got
$\frac{\sqrt{5}}{2 \sin(x -\alpha)}$
How do I find alpha?
I did:
$\cos \alpha = -0.5/ R$
@signal escarp is this what you mean?
elfin berryBOT
Xetrov
signal escarp
$-\cos 4t -\frac{ 1}{2} \sin 4t $
//
I am trying to find auxiliary angle for this equation Here's what I got
$\frac{\sqrt{5}}{2} \sin(x -\alpha)}$
How do I find alpha?
I did:
$\cos \alpha = -0.5/ R$
$\cos \alpha = \frac{-0.5} {\frac{\sqrt(5)}{2}}$
digital field
oof xD
elfin berryBOT
màe
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
digital field
i can't help, because i don't know what it is. just wanted to help with formatting
although it'll be better if you write it on paper then take a picture
signal escarp
Oh thank you!!
marsh citrusBOT
marsh citrusBOT
Available help channel!
astral olive
marsh citrusBOT
astral olive
I'm not 100% understanding whether these are the same formula but just rearranged a certain way?
Thank you
twin girder
copper lava
marsh citrusBOT
copper lava
Can someone walk me through this?
I need to figure out the vertex by changing it into vertex form by completing the square
so first I did $h=-16(t^2 -2t) + 6$
elfin berryBOT
H-Mobile
elfin berryBOT
H-Mobile
copper lava
which is $h=-16(t^2-2t+1)+5$
elfin berryBOT
H-Mobile
copper lava
im not sure what to do from here though
marsh citrusBOT
@copper lava Has your question been resolved?
night ridge
@copper lava im not sure what you were trying to do but for question a all you need is the derivative of the initial function, find where that is 0 and put that number in the place of t
marsh citrusBOT
<@!745478303590383616> im not sure what you were trying to do but for question a...
@night ridge Has your question been resolved?
night ridge
.close
loud aspen
marsh citrusBOT
loud aspen
my brain aint workin
marsh citrusBOT
@loud aspen Has your question been resolved?
proven sentinel
is there any other way other than graphing to solve this equation?
marsh citrusBOT
@loud aspen Has your question been resolved?
frigid raven
And idk how
cos^-1(x) is the inverse of cosx, thus if you multiply both sides with cos^-1, you get cos^-1(x)=cos^-1(cos(2x)) -> cos^-1(x)=2x
marsh citrusBOT
@loud aspen Has your question been resolved?
coarse gorge
Hii guys, how can I get the angle in 360degrees given the x and y when either or both is negative? lets say (-4,5), shouldnt it be between 90 and 180degrees? but when i use arctan y/x it just makes the angle negative
marsh citrusBOT
Hii guys, how can I get the angle in 360degrees given the x and y when either or...
hardy slate
coarse gorge
do i use the negative x when using the arctan?
arctan(5/-4)?
also what if its the other way around, (5,-4)? or (-5,-4)?
if you guys know of a video tutorial in yt it would be really appreciated, as i can understand more in video form
or tell me what this topic is called
so i can search
hazy lion
they have codomains/ranges
@coarse gorge
that you need to correct for
you may look up like "inverse trig functions quadrants"
but i might be able to explain more
hazy lion
sure
what you should know is like
well just to say because i feel like it lol
you only capture -pi/2 to pi/2 in the output of arctan
so x is only positive
so in the output, a positive answer captures the (+,+) and (-,-) case
and in the output, a negative answer captures both (+,-) and (-,+)
hope that helps
if you have a specific exercise or something let me know
shell vigil
marsh citrusBOT
marsh citrusBOT
@shell vigil Has your question been resolved?
marsh citrusBOT
marsh citrusBOT
Available help channel!
stark garden
for this word problem I don't get it. What do you do after you find out the volume of the room?
marsh citrusBOT
for this word problem I don't get it. What do you do after you find out the volu...
cunning citrus
for this word problem I don't get it. What do you do after you find out the volu...
find the cost per month
stark garden
so is it 60 times 0.50?
cunning citrus
yes
frank elm
I need help.
marsh citrusBOT
I need help.
odd anvil
Not sure how to start, exactly.
so it starts at 4 (4 is the 1st term)
and the nth term is equal to the n-1th term times 2/3
frank elm
Okay.
odd anvil
the sum is
$4([\frac{2}{3}]^0 + [\frac{2}{3}]^1 + [\frac{2}{3}]^2 + ...)$
elfin berryBOT
Adavocowana
odd anvil
**Adavocowana**
don't like the rendering, but yeah
@frank elm multiply the series by 2/3, then subtract
frank elm
Okay, subtract what tho?
odd anvil
(btw this series diverges, so idk what the teacher expected...)
serene heart
can someone please explain why this is incorrect?
marsh citrusBOT
can someone please explain why this is incorrect?
@serene heart Has your question been resolved?
serene heart
solved it dw
marsh citrusBOT
Available help channel!
simple pollen
If Tan = -0.75 and Cos is negative, how would I determine Sin
marsh citrusBOT
If Tan = -0.75 and Cos is negative, how would I determine Sin
simple pollen
I figured out that the angle is in quadrant 2, although I am not sure what steps I would take to solve Sin
marsh citrusBOT
@simple pollen Has your question been resolved?
odd anvil
can you find tan(alpha) such that tan(180 degrees - alpha) = -0.75?
hint: find arctan(-0.75) first
marsh citrusBOT
marsh citrusBOT
Available help channel!
winged violet
can someone help me with this, ive been preparing for my test next week and i stumbled upon this problem and i dont know how to do this. u dont have to give the answer as i want to work on it myself, just teach me the process
heres the problem:
Trizia Mae, a gymnast, dismounts the uneven parallel bars. Her height,, depends h, on the time, t, that she is in the air as h = -16t²+8t+8.
a. How long will it take Trizia Mae to reach the ground?
b. When Trizia Mae be 8 feet above the ground?
marsh citrusBOT
can someone help me with this, ive been preparing for my test next week and i st...
short umbra
h is how high trizia is in
for question A, you need to figure out when her height is 0 (as she is on the ground)
this is basically h=0
for question B, you need to figure out when her height is 8
this is basically h=8
so for a, you need to figure out 0 = -16t²+8t+8
and for b, you need to figure out 8 = -16t²+8t+8
ok thank you very much ill try and work with it
marsh citrusBOT
desert widget
i need help
if i have
y(t) = y0*e^(kt)
and i want to linearize this equation
i get ln(y)=ln(yo)+k*t
marsh citrusBOT
Available help channel!
desert widget
then i do a linear regression on the data
marsh citrusBOT
then i do a linear regression on the data
hard sentinel
bright jay
marsh citrusBOT
@desert widget Has your question been resolved?
marsh citrusBOT
@desert widget Has your question been resolved?
tropic leaf
marsh citrusBOT
tropic leaf
I do not know how to calculate the standard deviation of number of items purchased
If possible a link to a forumla or steps would be great
sleek flame
standard deviation
sleek flame
just sub numbers inside
tropic leaf
how do I know which X to use
sleek flame
the first method is sum of numbers square over total number minus the mean sqaure
2nd is sum of every number minus mean squared over total number
and then square root it
tropic leaf
and they give the same thing
sleek flame
yea
tropic leaf
so I can use whichever method seems easier to me
sleek flame
yep
tropic leaf
wait I dont get how the probabilities factor into here then
is X the probability * the number?
marsh citrusBOT
@tropic leaf Has your question been resolved?
marsh citrusBOT
@tropic leaf Has your question been resolved?
near bay
How many numbers of different digits less than 500 and are the multiples of 3 can be formed from the integers 1,2,3,4,5,6,7?
i got that there are 169 digits less than 500
and for multiple of 3, i just divided by three and got 56
dk whether if that's correct
marsh citrusBOT
@tropic leaf Has your question been resolved?
marsh citrusBOT
marsh citrusBOT
Available help channel!
surreal shore
marsh citrusBOT