im wondering, how hard it is and what are the steps to finding the solutions of a cubic function that has an irrational real root and an irrational complex conjugate, such as 2x^3-7x^2+14x-11
#help-33
faint coral
marsh citrusBOT
im wondering, how hard it is and what are the steps to finding the solutions of ...
@faint coral Has your question been resolved?
faint coral
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neat granite
im wondering, how hard it is and what are the steps to finding the solutions of ...
Is this ur question?
faint coral
yes
vivid wren
I can help. you can easily reach me nbox
cosmic veldt
does anyone know the reaoning or can explain why the cross product gives the area of a parreleogram (or half it a traingle in this case)
is the cross product not the normal to a plane made by two vectors
marsh citrusBOT
does anyone know the reaoning or can explain why the cross product gives the are...
cosmic veldt
so how is the magnitude of a cross product the area?
but if its too complciated im fine ig
nvm sigh
.close
untold wyvern
I need to show that for n>6, the product of the first n-1 primes is greater than the cube of the nth prime. Could anyone hint me on what to do?
marsh citrusBOT
I need to show that for n>6, the product of the first n-1 primes is greater than...
untold wyvern
I have tried using bertrand's postulate but it doesn't seem to give me what I want
marsh citrusBOT
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fleet fossil
I have tried using bertrand's postulate but it doesn't seem to give me what I wa...
I think that should give you the proof for n>7
Hint
marsh citrusBOT
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manic brook
marsh citrusBOT
manic brook
Help
lucid zenith
since you have a polynomial here, think of it as a function $$f(x)=(4x^2+x-2)^4(2x^2-x+2)^3=a_0+a_1x+a_2x^2+...+a_nx^n$$
elfin berryBOT
kheerii
lucid zenith
then what's the sum you need?
manic brook
Shld I like take x= 1 in the first part and um and add it with 2 of the second part
lucid zenith
what?
manic brook
Sum of all coefficient with x raised to an even power
But like without expanding it
(condition)
lucid zenith
what is that in terms of the notation I introduced?
marsh citrusBOT
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oblique karma
how do i add 2 vectors in a triangle if they are not like AB+BC? like if they go like this? in my book i only see how to solve if they are like AB+BC or BC+CA or CA+AB
marsh citrusBOT
how do i add 2 vectors in a triangle if they are not like AB+BC? like if they go...
junior copper
how do i add 2 vectors in a triangle if they are not like AB+BC? like if they go...
How else would you be adding them in a triangle??
oblique karma
How else would you be adding them in a triangle??
like AB+BC = AC
but if they are like this idk how to add them
usually the vectors are like this or in reverse
unreal oxide
how do i add 2 vectors in a triangle if they are not like AB+BC? like if they go...
so u have an arrow CA
and another one CB
u move the tail of the arrow CB to the tip of CA
and add them like u normally wud
marsh citrusBOT
junior copper
just the tip 🎶
oblique karma
.reopen
marsh citrusBOT
✅
oblique karma
wait, what if both vectors are going in the same point? like AC+ BC?
gloomy plume
marsh citrusBOT
gloomy plume
can someone explain this to me?
I have solution here but cant make any sense of it
marsh citrusBOT
@gloomy plume Has your question been resolved?
gloomy plume
anyone? 🥺
marsh citrusBOT
@gloomy plume Has your question been resolved?
quiet anvil
@gloomy plume green area is possible, red is impossible
Gray area from original image satisfies the condition
gloomy plume
the inequality confuses me a little
why is it -1/2 < y - x < 1/2
like shouldnt every point have both x & y coordinates?
quiet anvil
"Given their distance is less than 1/2"
x is a number between 0 and 1.
y is also a number between 0 and 1
So the graph above shows pairs of numbers (x, y) between (0, 0) and (1, 1)
The green area is where the distance condition is satisfied (this is B)
The gray area is where the endpoint condition is satisfied (this is A)
gloomy plume
the endpoints are (0,1) and (1,0) ??
quiet anvil
The endpoints are just (0, 0) and (1, 1). However, only one of these is the endpoint specified. The solution assumes the endpoint is (0, 0) without loss of generality.
Or more specifically, the endpoint is just 0 or just 1
And both x and y need to be closer to 0 than 1 (if 0 was specified)
gloomy plume
so the x & y in the equality are meant to be a point's coordinates right?
gloomy plume
ohhh
quiet anvil
The graph is about pairs of (x, y) which we put onto a plane
gloomy plume
tbh I still dont understand
so the two points have one coordinate bc they're both chosen FROM the unit interval?
andd we're supposed to compare the distance if we put one point on the horizontal and the other on the vertical axis??
quiet anvil
So, the two points are independently chosen on the unit interval
We then combine the data from those points to get a single point on the 2d plane
This is only a visualization trick
We compare distances purely 1 dimensionally
And separately
@gloomy plume
gloomy plume
And separately
and what does A and B stand for here?
gloomy plume
A = their distance is less than 1/2
quiet anvil
A is the condition "both points are closer to the endpoint selected"
B is the condition "the distance between the points is less than 1/2"
Note here distance between the points refers to the two points on the number line
When we translate this into 2d each pair of points becomes a single point, so the "distance" between them isn't something that makes sense immediately.
But you can come up with an interpretation pretty quick
If we define the line y = x and measure the distance of our pair to this line then we get the distance between the points
In particular the two lines drawn on the solution graph are exactly the points such that the distance between them is exactly 1/2
Between these two lines is the region where the pairs are closer to the line y=x and so their distances between the two points on the number line are less than 1/2
gloomy plume
how was the area calculated to get the probability for P(B)
silk cave
Trying to prove theorem e6.
My attempt (im not sure if it is good enough)
Suppose o(x) and f(x) are not relatively prime. Then there exits some polynomial g(x) that divides both o(x) and f(x). Since o(x) is irreducible, g(x) can either be o(x) or a constant polynomial. If g(x) = o(x), then certainly g(x) does not divide f(x), since o(x) does not f(x) as given; if g(x) = c for some c in F (the field), a polynomial that is not of positive degree. Two polynomials are relatively prime if and only if no polynomial of positive degree divides each of them; so either case leads to a contradiction.
marsh citrusBOT
Trying to prove theorem e6.
My attempt (im not sure if it is good enough)
Suppo...
royal sable
Trying to prove theorem e6.
My attempt (im not sure if it is good enough)
Suppo...
That's a good attempt
Because you proved it
marsh citrusBOT
@silk cave Has your question been resolved?
silk cave
That's a good attempt
Thanks a lot
silk cave
I’m on theorem e7
My attempt:
let f(x) and g(x) be such polynomials. Supper they are not relatively prime, that is, there exists a polynomial h(x) of positive degree such that h(x) | f(x) and h(x) | g(x). h(x) | f(x) implies h(x) = f(x), since f(x) is irreducible (and h(x) cannot be a constant polynomial since it has positive degree). Similarly, h(x) | g(x) implies h(x) = g(x). This shows f(x) = g(x) = h(x). But g(x) and h(x) are distinct, so this is a contradiction.
Comment:
I think I’m missing something here; I don’t know where to use the monic property of the two polynomials
royal sable
I’m on theorem e7
My attempt:
let f(x) and g(x) be such polynomials. Supper th...
correct
Although the implication h(x) = f(x) is correct only if you choose h(x) to be monic
In general we only have h(x) = c f(x)
But this is just a small nitpick and it doesn't change much
silk cave
Although the implication h(x) = f(x) is correct only if you choose h(x) to be mo...
Ohhh that makes a lot of sense
Tysm again
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marsh citrusBOT
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pure quail
can someone help me with this question
marsh citrusBOT
can someone help me with this question
quaint solar
I think you show that the catalan recurrence is obyed as far as i know
pure quail
obyed?
quaint solar
sorry obeyed
idk if you write it like this i am not an english native
but easiest is the bijection to dyck-paths i think
how have you defined the catalan numbers?
pure quail
the definition is not given in the task, i just googled it
quaint solar
ok cuz you could define the catalan numbers like that
but probably they want you to relate it to the recurrence
pure quail
yes, its from a combinatorics course
but still i dont really have an idea of how to solve it
quaint solar
i would take this definition:
$$C_0=0, \qquad C_{n}=\sum_{i=1}^n C_{i-1}C_{n-i}\quad n\geqslant 1$$
proud arch
The Wikipedia page regarding the catalan number should be able to help you a lot better
elfin berryBOT
cosmic
proud arch
Mostly such questions are done by induction
But choice is yours
You could do this analogically
To the grid problem
quaint solar
yeah i figured that they want you to relate it to the recurrence, which is basically induction
pure quail
i am not really sure of how i can apply this to the paranthesis part though?
proud arch
Basically
Assume start with only assuming a
Count cases
Then move onto a,b
Count again
And so on
After certain time you will begin to see the pattern
As cosmic mentioned
proud arch
**cosmic**
This thing
pure quail
ohh okay got it
proud arch
Which is the catalan number
pure quail
then i prove the recurrence by induction
proud arch
Yet again I urge you to view the Wikipedia page which concerns the same
It is incredible
The insight is amazing
Worth a read in my opinion
They also have a proof exactly through this method
quaint solar
then i prove the recurrence by induction
exactly, you have to split up your counting in the right way
pure quail
great yeah i will check it out in more detail
quaint solar
ping if you need help again but yeah the proof is also just on the wiki page
marsh citrusBOT
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wanton niche
marsh citrusBOT
wanton niche
Q42
Wants us to find the e area
The area
uh
Anyone
Do we need to use pythagorean
To find the leg
(Height)
sturdy condor
any other information given in the question?
marsh citrusBOT
@wanton niche Has your question been resolved?
wanton niche
but this seems kinda odd
since we have never seen a question with this many letters in our book
tight kite
2 letters?
wanton niche
What
no i mean like
when u write the equatiojs
And stuff
It looks like alot
which can confuse (us) ig
im at 9th grade
tight kite
try doing what you suggested
marsh citrusBOT
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modest blade
I've been trying to solve this maximal flow problem but I'm not sure if this is the optimal network or if i did something wrong
marsh citrusBOT
I've been trying to solve this maximal flow problem but I'm not sure if this is ...
sleek lake
i don;t understand :(
wow there are two numbers for every edge
is it like you can choose a direction, and that changes the weight?
modest blade
i used ford-fulkerson algorithm in this problem
marsh citrusBOT
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modest blade
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marsh citrusBOT
@modest blade Has your question been resolved?
marsh citrusBOT
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modest blade
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marsh citrusBOT
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tulip sundial
How would I prove that a jump discontinuity is not a removable discontinuity?
late geode
if you already know you have a jump, then by definition
tulip sundial
whats the definition of jump discontinuity
like lim x->a- f(x) =/= lim x->a+ f(x)?
late geode
one sided limits are different / exist
tulip sundial
ohh okok
and then for removable it would be like lim x -> a f(x) =/= f(a)
but for lim x -> f(x) to exist, that implies lim x->a- f(x) = lim x->a+ f(x)
which contradicts our definition of jump discontinuity?
is that a valid proof
late geode
yeh
marsh citrusBOT
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torn estuary
I understand that the factorization goes like this
And plugging in 6 for x yields 5-1 as the first time, making it very easy to show that it's divisible
but how do i get from the first step (i guess 6^n-1) to the factorization?
vital oracle
I understand that the factorization goes like this
are you saying you dont know where the factorization comes from?
torn estuary
are you saying you dont know where the factorization comes from?
i guess? i'm not quite sure if i can just slap that into my proof, and how to do so
torn estuary
https://math.iupui.edu/~ccowen/OldCoursePages/Math300F07/300F07Quiz4.pdf
it does hold true, though. (see attached link, not picture.)
vital oracle
youre not going to see where the factorization comes from until you expand it and see how its true
torn estuary
okay by RHS are we talking 6^n-1 or the image
vital oracle
it cant be this, becuase this is not an equation
also, you cannot expand something that is not factored
6^n - 1 is not factored
so you cannot expand it
do you understand
torn estuary
i guess but now i'm at a dead end because i don't see how to expand that giant mess
vital oracle
whats x^(m-1) times x
torn estuary
are you talking about this?
torn estuary
no, i don't see how i can implement it
torn estuary
no, it doesn't
vital oracle
whats x^(m-1) times x
?
consider the following about how belief works
and dont interrupt
in order to use something for a proof, you need to believe that its true first
now see here that, in their proof, they offer these lines to show that its true
however, learning this expansion is not part of your course
therefore its not required to be written
therefore you do not need to believe it to write it
torn estuary
i understand that
vital oracle
additionally
vital oracle
and dont interrupt
.
if you understood the proof, youd be able to see that one would need to lead to the other
since this is expected to be learned beforehand, you dont really have a choice in doing the same
so you just have to go from one side to the other without any requirement for additional lines to support it
vital oracle
i guess but now i'm at a dead end because i don't see how to expand that giant m...
however you are not doing this
which can only come from that you dont believe it enough to use it so tersely
torn estuary
no, it cannot only come from the fact that "i don't believe it enough to use it". it came from my belief that i might have to prove (that the factorization does go like that)
vital oracle
however, learning this expansion is _not_ part of your course
is learning to expand part of your course?
torn estuary
no
torn estuary
thank you
vital oracle
I told you this already
torn estuary
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marsh citrusBOT
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arctic crow
im so lost
marsh citrusBOT
im so lost
arctic crow
<@&286206848099549185>
guys pls help me ive been trying this problem for 30 minutes and i might crash out 😫 🙏
elfin berryBOT
JD
idle lagoon
that should be correct
marsh citrusBOT
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shy shard
Did I do the first problem correct?
marsh citrusBOT
Did I do the first problem correct?
quick moth
it says -infinity
marsh citrusBOT
still temple
Negative one then mb
No, answer is 1
But in the limit you write it as infinity instead of minus infinity
shy shard
But in the limit you write it as infinity instead of minus infinity
oh is it thank u
marsh citrusBOT
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lyric kelp
we found the dot product here to be 2.
marsh citrusBOT
we found the dot product here to be 2.
lyric kelp
what does the dot product stand for?
lyric kelp
Like fundamentally
i guess? cuz my textbook gave me a nice formula and a pat on the back
and how to find an angle but
never said what the dot product does
vital frost
the dot product essentially tells you how much two vectors are aligned
You can think of it in terms of forces
Say a train is held stationary onto a track
It can't go left or right
Only straight
And you apply a force at an angle 45° to the train
So how much of that force gets actually transferred to the train?
That's sort of the intuition here
lyric kelp
That's sort of the intuition here
but
is it normal my etxtbook isnt telling me anything about the use of it
at least
the
dot product itself
fervent rampart
they have to tell you what it is early on so they can use it in formulas throughout the book
vital frost
is it normal my etxtbook isnt telling me anything about the use of it
There's a lot of super cool uses of the dot product
lyric kelp
they have to tell you what it is early on so they can use it in formulas through...
yeah formulas, but not the result of the dot product itself
at least yet
vital frost
Of the top of my mind I can think of a lot of data science applications
fervent rampart
the dot product results in the dot product
there are many formulas which make use of it, but fundamentally the dot product is just an operation that you can perform on two vectors that gives a scalar
lyric kelp
ah okay okay
vital frost
No but do u get what the dot product is actually giving you
Try to get what that scalar is actually is
marsh citrusBOT
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bright sundial
The table below gives the selected values for a differentiable function h and its derivative h'. let g be the function with g(2)=0 and the derivative given by g'(x)=xcos(x^2) use the tale to answer question 6-9.
Q9) The point (2,1) lies on the curve in the xy plane given by the equation h(x)=g(x)^2 +xy. what is the value of dy/dx at the point (2,1)? Express your answer as a decimal approximation
marsh citrusBOT
The table below gives the selected values for a differentiable function h and it...
bright sundial
,rotate
elfin berryBOT
bright sundial
Does my work look good?
if anyone could double check
I am allowed a calculator but I didnt need one and i got the answer dy/dx= 1.5
marsh citrusBOT
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marsh citrusBOT
proud arch
Ur welcome
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rain hare
How do i approach this?
marsh citrusBOT
How do i approach this?
modern sedge
How do i approach this?
dy/dx = f
How do you get y?
marsh citrusBOT
@rain hare Has your question been resolved?
rain hare
How do i solve this?
modern sedge
How do i solve this?
if dy/dx = f
then integrate both sides with respect to x to find y
marsh citrusBOT
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marsh citrusBOT
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lyric kelp
for b)
lyric kelp
abgle between d and v, where is it exactly?
cuz they dont even have the same starting point
limber hearth
You can do translation
lyric kelp
translation?
still temple
show the relationship between d and v
lyric kelp
one is (1,1,0) and one is (1,1,1) ig?
thats the only way i think i can put it idk tho
still temple
so the angle would be when you draw the vectors from the same point
do you understand that?
lyric kelp
so the angle would be when you draw the vectors from the same point
oh
i need to assum
im starting at same point?
if so yeah i can just product dot
still temple
right
marsh citrusBOT
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fallow furnace
how to proove this? If the function ( f ) is neither bounded above nor bounded below on any punctured neighborhood of the point ( a ), then ( \lim_{x \rightarrow a} f(x) ) does not exist.
marsh citrusBOT
how to proove this? If the function \( f \) is neither bounded above nor bounded...
elfin berryBOT
Slowaq
night mica
you can construct a sequence for which f diverges to infinity or -infinity
fallow furnace
you can construct a sequence for which f diverges to infinity or -infinity
then f(x) must be in some neighbbut i need to prove that it doesnt exists not that it diverges to infinity
spark otter
then f(x) must be in some neighbbut i need to prove that it doesnt exists not th...
you can construct two sequences that for the first diverges to infinity and the other diverges to -infinity
then if by absurd lim(f(x)) when x-> a existed, the limit of all sequences that converge to a must be that limit
fallow furnace
you can construct two sequences that for the first diverges to infinity and the ...
can i just say that let limit exist and is equal to L iff for all epsilon>0 exist delta >0 that for all x in punctured delta neighbourhood of a f(x) in epsilon neighbourhood of L and that is in contradiction with my assumption that f(x) isnt bounded thus limit doesnt exist?
spark otter
can i just say that let limit exist and is equal to L iff for all epsilon>0 exi...
what is an epsilon neighborhood of L if L is an infinite limit
fallow furnace
well in that case the definition of limit is this
and that is also in contradiction with my assumption
marsh citrusBOT
@fallow furnace Has your question been resolved?
spark otter
and that is also in contradiction with my assumption
that's + infinity limit
you will also have -infinity limit with a different definition
so suppose by absurd the limit exists
show that it can neither be finite, + infinity nor -infinity
fallow furnace
that's + infinity limit
Let us prove by contradiction. Assume that ( \lim_{x\to a} f(x) = L \in \overline{\mathbb{R}} ) and ( a \in \overline{\mathbb{R}} ). Since ( f(x) ) is not bounded on any punctured neighborhood of ( a ), we can consider two sequences
[
{x_n} \text{ such that } x_n \rightarrow a \text{ and } f(x_n) \rightarrow +\infty
]
[
{y_n} \text{ such that } y_n \rightarrow a \text{ and } f(y_n) \rightarrow -\infty
]
For ( \lim_{x\to a} f(x) = L ) to exist, it must hold that for all sequences converging to ( a ), their function values under ( f ) must converge to ( L ). However, this contradicts the existence of the sequences ( {x_n} ) and ( {y_n} ).
elfin berryBOT
Slowaq
fallow furnace
is this valid?
spark otter
Let us prove by contradiction. Assume that \( \lim_{x\to a} f(x) = L \in \overli...
I think the closing argument, rather than "this contradicts the existence of ..." which is a little vague, would better be 'Then L = +infinity and L = -infinity, which is absurd'
marsh citrusBOT
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covert falcon
hi can someone help with this? I need to find AOB
marsh citrusBOT
hi can someone help with this? I need to find AOB
whole thorn
hi can someone help with this? I need to find AOB
can you find bcd
just find as many angles as you can
you can also find them in terms of x
covert falcon
thankss, i will try
marsh citrusBOT
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marsh citrusBOT
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cinder pilot
Let a and b be some constants satisfying 0<a<π/2<b<π. How can you show that the equation cos(x) = (x^4- a^4) (X²-b²) sin(x²) has at least two real solutions? Can you help me?
marsh citrusBOT
Let a and b be some constants satisfying 0<a<π/2<b<π. How can you show that the ...
junior copper
Let a and b be some constants satisfying 0<a<π/2<b<π. How can you show that the ...
what have you tried
marsh citrusBOT
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gray crystal
I solved
$$(4x^{7})^{8} \times 4x^{11}=$$
$$4^{8}x^{56}\times4x^{11}=$$
$$4^9x^{67}$$
In the answers in the book I use it says the answer is $2^{18}x^{67}$
I know my answer is correct but their is simplified into prime numbers
$$(2^{2})^9x^67 = 2^{18}x^{67}$$
I just want to ask is there any reason for me to simplify it into prime numbers if I haven't been asked to?
marsh citrusBOT
I solved
$$(4x^{7})^{8} \times 4x^{11}=$$
$$4^{8}x^{56}\times4x^{11}=$$
$$4^9x^{...
elfin berryBOT
𝕿𝖆𝖘𝖐
pliant willow
no ig
well it depends
ask your teacher in which form (s)he wants his/her answers
marsh citrusBOT
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quick raft
Hello, I am a 9th grade student on a fast paced math track (algebra 2), I enjoy more advanced level math (calculus) and often self study. My question is: Is there a limit definition for integrals?
marsh citrusBOT
Hello, I am a 9th grade student on a fast paced math track (algebra 2), I enjoy ...
quick raft
If so, how do I implement it for integrals
quick raft
it is called the riemann sum
thanks so much!
devout mauve
presumably you were introduced to integrals by taking rectangles and making them smaller and smaller
riemann sums are just that but precise
quick raft
I am solving derivates using the limit def, and getting into integrals but I found that there was no universal rule available
mellow crag
if you do even more maths, it'll turn out in the future that the definition of the Riemann integral is actually quite bad and doesn't satisfy a lot of the properties you want it to satisfy
once u study measure theory, you'll find that the Lebesgue integral is a better way to define the integral
anyway that's something u can also read about once ur comfortable with the defn of the Riemann integral
devout mauve
while its a classic exercise to compute derivative with the limit def, you very rarely do that for integrals
fervent rampart
strictly speaking for the riemann integral we need the limit of every possible riemann sum to converge to the same value. but if it does show up as an exercise in some calculus classes, usually you just calculate the limit of one riemann sum
quick raft
while its a classic exercise to compute derivative with the limit def, you very ...
thanks!
quick raft
strictly speaking for the riemann integral we need the limit of every possible r...
How is it possible for me to help other people in this server?
devout mauve
just go in their channel and help
quick raft
just go in their channel and help
👍
fervent rampart
look in the other help channels or #1021175428326633542 for questions that seem interesting, then just start helping. there is a bit of guidance in #helpers-info
inland hill
I'm trying to remember this, some case I had come across ...
That,
√(2)² can only be +2
And, √4 can be ±2
inland hill
Is that correct?
hidden dawn
no
hidden dawn
alr
inland hill
It was this lol just searched up
hidden dawn
well it can be explained by calculus
if u take the function f(x) = sqrt(x) youll that its positive
so it will have a positive outcome
x is a positive number, and sqrt(x) is also positive
junior copper
It was this lol just searched up
https://youtu.be/NBOlK_rSsm4?si=5MhJ9u8PqE5zw...
W bprp
inland hill
x is a positive number, and sqrt(x) is also positive
Kinda getting the feel of it
hidden dawn
x is positive not negative
inland hill
Thanks man later
hidden dawn
np
inland hill
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marsh citrusBOT
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inland hill
(speaking of real numbers ofc)
√x is a real function and x² isn't?
It's dumb question I get it but
uncut oasis
√x is a real function and x² isn't?
Depends on how you define them
Most of the time, you can assume both are real
inland hill
I was scouring the net, and "Saying √4 is ±2 wouldn't make sqrt a real function, as gives off more than one real numbers" statement popped up
What does that mean exactly
uncut oasis
I was scouring the net, and "Saying √4 is ±2 wouldn't make sqrt a real function,...
Welp, first of all
How would you define a real function?
uncut oasis
Domain is real and so is range
That's a real and real valued application, but you're still lacking something
uncut oasis
Pairs?
If you're saying what I think you're saying then yeah
You can imagine a function as something that relates a set to another
inland hill
Btw I like how you guys do it here, explaining through interactions and not just stating the right facts!
uncut oasis
Yup
There's one thing real functions always satisfy though
Assume we've got two sets A={a,b,c,d} and B={e,f,g,h}
A function f:A->B will relate each element of A with, at most, one element of B
In other words, for any element of A, a_i, f(a_i) will give only one output, which will be in B
For real and real valued functions(the most common type) the only difference is that both sets, A and B, are the reals
uncut oasis
Going back to the square root now
We know(or at least in most cases assume) it is a real valued function
So if a belongs to R, the square root of a can, at most, be one number
As to why it's the positive one, it's more of a definition thing, but there's why the square root of any real number(over the reals) is both unique and positive
uncut oasis
So what we say x² a kind of function is?
Wdym?
uncut oasis
What kind of function x² be
Idk if it's what you're asking but it's a non-surjective non-injective function
inland hill
Idk if it's what you're asking but it's a non-surjective non-injective function
Yeah what i was thinking was all over the place but I get it thanks
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maiden relic
I just need a fast explenation about something.
im studying logarithms and im encountering log3(1/9)
the "1/9" became 3^-2.
why? and how did it transform a fraction into a power?
marsh citrusBOT
I just need a fast explenation about something.
im studying logarithms and im en...
maiden relic
yes
nimble prairie
So what is 1/3^2?
nimble prairie
Yes
But we are trying to rewrite it
Do you recall how to rewrite a reciprocal like above into a power?
tight kite
List the powers of 3
3, 9, 27, 81, ...
notice that each term you're multiplying by 3, so 3×9 = 27, 3×27 = 81, etc
If you go backwards you can say that any term is a third of the next
so 81/3 = 27, 27/3 = 9, etc.
we know 3¹ = 3, how would we find 3⁰? well, to get the previous term, divide by 3, so 3⁰ = 3¹/3 = 1 so 3⁰ = 1.
What about 3 to the minus one? you know 3⁰ is 1 so if you divide by 3 again you get 1/3, so 3 to the minus one is 1/3, and this goes on, so 3 to the negative k is 1/3^k
maiden relic
hmm
nimble prairie
ye 1/x = x^-1
Yup!
maiden relic
oh so i just remove the 1, then 3^2 becomes 3^-2
nimble prairie
Yes
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fallow holly
Hi, could anyone give me some pointers for this problem? Our professor likes to take each of these roots and "substitute" for them (So it becomes An-Bn/Cn-Dn where n is like the index of the set).
He then went on to do lim An/n = 1, same for BCD and that is easy to understand. What isn't simple, though, is everything else.
I don't expect anyone to solve this with steps, neither do I need the answer (which is -1 for those interested), but I would really appreciate some help.
marsh citrusBOT
Hi, could anyone give me some pointers for this problem? Our professor likes to ...
still temple
fallow holly
ummmmm, I don't think I'm catching on? :d
My question might make more sense if I send the notes from my tablet I think
still temple
from green to blue you can factor out the n^x from under the radical
for example, take the first term in the numerator from the original function
you can factor out n^3 to get
cube-root(n^3(1+1/n^3))
then just take the cube root of n^3 to output n
the term becomes n*cube-root(1+1/n^3)
you can repeat this process for the remaining 3 terms to obtain what i wrote in the blue
does that make sense?
as a side note, i should really learn latex 😂
fallow holly
Yep, we get n from all the roots there, ye
still temple
or wtv bot this server uses
fallow holly
That far I understood
but after that, the professor just went rogue - I really think I need to send the SS so it makes sense
still temple
do you know how i went from green to blue though?
fallow holly
Yes
still temple
okay so from there, the n's cancel out right?
fallow holly
All of the roots behave "almost" like n for large enough n
fallow holly
okay so from there, the n's cancel out right?
yep
still temple
okay mb
fallow holly
This is what we've done. The top half makes sense, but the lim n to the fifth over n to the fourth A-B is where I am completely lost)
still temple
im blind bruh
from here im also a little stumped
but i can tell how i reasoned it
don't have a rigorous scientific explanation though
so for the numerator, for any n>0 the second term will always be greater than the first
in the denominator, for any n>0 the first term will always be greater than the second
thus the numerator evalutes to a tiny negative value, and the denominator a small positive value
and the magnitudes of the values are the same so as n->infinity it becomes -1
fallow holly
I appreciate your help - I'll look into it more and hopefully get that eureka moment.
<@&286206848099549185> would any of you understand the "6th line"? Where we take the limit of (A-B), but multiply it first?
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mellow frigate
Please I’m stuck on it for two hours
marsh citrusBOT
Please I’m stuck on it for two hours
main idol
i recommend reading up on concave/convex and when it changes https://tutorial.math.lamar.edu/Classes/CalcI/ShapeofGraphPtII.aspx
especially example 1
mellow frigate
i recommend reading up on concave/convex and when it changes https://tutorial.ma...
So when it opens up
So that’s what I did
It starts going up at -1
Then starts going down at 1
main idol
It starts going up at -1
you're not thinking of the definition of concavity correctly
mellow frigate
Wait wtf
It can concave up down going down
Okay so I know which is which but how does that tell me the inflection points
main idol
i recommend reading up on concave/convex and when it changes https://tutorial.ma...
read to find out
kind shadow
It can concave up down going down
A bowl can be facing up or down. In both cases u can either trace ur finger up or down the bowl
red nimbus
Unless I’m blind
mellow frigate
i recommend reading up on concave/convex and when it changes https://tutorial.ma...
Here
kind shadow
Im a bit rusty on vocab, is inflection point just the point where the bowl starts facing the otther way?
red nimbus
In the form he sent
On the side there is even the definition of inflection point
mellow frigate
Im a bit rusty on vocab, is inflection point just the point where the bowl start...
That’s what I’m trying to understand 😭
kind shadow
Im a bit rusty on vocab, is inflection point just the point where the bowl start...
Yep
main idol
It doesn’t talk about inflection points
can i introduce you to ctrl + f
mellow frigate
WAIT ITS THE IN BETWEEN THEM???
kind shadow
WAIT ITS THE IN BETWEEN THEM???
The huh? In between the bowls? Sure
mellow frigate
So 0,0???
kind shadow
Nah
U gotta find where the next “bowl” begins
This location is where the slope suddenly changes direction
kind shadow
The bowl is just an analogy for concavity
kind shadow
The first bowl still ends
The start of a bowl, if there is one, is the end of another
If u wana think of it like that
mellow frigate
My brain
I understand none of this
And reading doesn’t help me understand it
I understand by doing it and by seeing someone do it
I know you guys are trying to make me understand it myself but I don’t
It makes 0 sense
mellow frigate
Yep
This is the point between the two bowls
Here in my graph it’s 0,0
But it’s wrong
So I don’t get it
kind shadow
It’s infinite
Maybe this vid can help? https://youtu.be/dkoJWBhx8CA?feature=shared
kind shadow
But it’s wrong
Weird, id assume it would be that point as well based on ur graph
kind shadow
It’s infinite
Wdym its infinite?
mellow frigate
Wdym its infinite?
To the sides there is no other bump
mellow frigate
No sorry
Two bumps
One graph
But then the point in the middle is the thing you sent before
kind shadow
Seems like it
kind shadow
Weird
kind shadow
Whats the whole question? Can i see it plz?
kind shadow
The hell
mellow frigate
🥲🥲
kind shadow
Lmao yeah its the origin
No clue y its wrong
Try the peaks
Just to f around
Also
Maybe this is the “first bowl”
Check by seeing if the slope decides to change direction
In the red, the slope seems to get more negative and negative until it decides to turn around and start increasing to make the new bowl
marsh citrusBOT
@mellow frigate Has your question been resolved?
mellow frigate
😫😫😫
marsh citrusBOT
@mellow frigate Has your question been resolved?
kind shadow
Was it not this? @mellow frigate
lyric kelp
i would like to know
marsh citrusBOT
i would like to know
void elm
what would you like to know? 
fervent rampart
they are the same formula, just rewitten with trig identities
lyric kelp
i meanbt like the
left and right
they the matrix here combined
two orthogonal projections
fervent rampart
you will get the orthogonal projection of any vector $\begin{bmatrix} x \ y \end{bmatrix}$ by mulitplying by the given matrix
elfin berryBOT
cloud
lyric kelp
ok ty thats what i meant
fervent rampart
it's quite powerful because it means we can represent a pretty large class of transformations (what we like to call functions that turn vectors into other vectors) as just multiplying by a particular matrix
marsh citrusBOT
@lyric kelp Has your question been resolved?
lunar sedge
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lyric kelp
if i undertsnad
marsh citrusBOT
if i undertsnad
fervent rampart
we would get the same result for any scalar multiple of a, yes
honest creek
marsh citrusBOT
honest creek
we want to find the maximum area of BCDF
i’m confused what “maximum area” even means lol
didn’t they already define a trapezoid? there’s only one area
is there not?
crystal lintel
honest creek
<a:waves:880516410948853840>
heyy
honest creek
Is that the full question
yes
ornate heath
Maximum area of that thingo?
honest creek
the full question is to show that the maximum area > 37
but that part is irrelevant
so i was just talking about the maximum area of that thing
yes
honest creek
no
ornate heath
But usually for these question they give a maximum perimeter and want you to use a quadratic equation to find the maximum area for it
But this doesn’t have anything else so I just need to clarify
What about the FD line
honest creek
yeah cuz you have the bias that this is an optimization question
when it doesn’t have to be 😭
honest creek
anyway this is a geometry problem
now that i think about it
B, F, and D have no restrictions on them
well other than the most obvious ones
but they can move around in their respective line segments
and u have to maximize the area of whatever formed
crystal lintel
But usually for these question they give a maximum perimeter and want you to use...
the side lengths of the triangle can all be computed 
ornate heath
the side lengths of the triangle can all be computed <:hmmcat:831508801646755850...
Yeah so
Would that not just be
The area of the whole shape?
honest creek
what
crystal lintel
honest creek
B probably can’t be A
ornate heath
Wait
honest creek
for example
also if you make the trapezoid a triangle, you’re deforming the topology of the trapezoid
it’s not four sided anymore
and it says maximum area of BCDF so you can’t decay it into BCD or BCF, for example
ornate heath
So you can’t have a 3 sided shape?
crystal lintel
that doesn’t affect the topology of the trapezoid
ornate heath
Ok got it
ornate heath
that doesn’t affect the topology of the trapezoid
No I think I get what she’s tryna say
You can’t have a 3 sided shape right?
honest creek
that doesn’t affect the topology of the trapezoid
i mean the english definition of topology
not the math one 
ornate heath
Ok wait what’s the answer for this?
crystal lintel
i’m still not 100% sure what the question is. you can put B and D anywhere so long as BF and CD are parallel?
ornate heath
i’m still not 100% sure what the question is. you can put B and D anywhere so lo...
Yeah tbh I’m also confused abt that
honest creek
i’m still not 100% sure what the question is. you can put B and D anywhere so lo...
yes
i think
tbh i don’t know either
hence my question
crystal lintel
what about F? anywhere as well?
honest creek
yes
ornate heath
Bruh
ornate heath
not on A or E though
Is there a provided solution or answer
honest creek
the provided solution is something something rectangle maximizes the area
that makes no sense
honest creek
the annotation is a hint towards the actual solution
ornate heath
Wait
crystal lintel
you can just choose F and B close to A and D close to E and get arbitrarily close to the area of the whole triangle?
honest creek
you can just choose F and B close to A and D close to E and get arbitrarily clos...
😭that’s true
i’m lost as well
that’s literally all there is to the question
also wait
meh okay the angle 105
ornate heath
you can just choose F and B close to A and D close to E and get arbitrarily clos...
That’s exactly what I was thinking
honest creek
can be preserved too while doing that
ornate heath
But it can’t possibly be that
crystal lintel
so F is fixed
honest creek
so F is fixed
not that i know of
the question posted is all info there is
uhm the chances of the question being posed weirdly is also high
so you won’t be wrong in saying the question is bad or smth
crystal lintel
idfk. is it supposed to be whatever point makes that angle 105 degrees?
ornate heath
so you won’t be wrong in saying the question is bad or smth
ok well
the question is absoluetly terrible
honest creek
idfk. is it supposed to be whatever point makes that angle 105 degrees?
😭yesss i think so
the line segment connecting F and D
honest creek
has to be at a 105 degree angle (my guess)
ornate heath
or does it just say whatever you said back there
honest creek
numerical answer?
it’s greater than 37 always
ornate heath
bruh
crystal lintel
well ms mmmm7 idk what to say anymore
ornate heath
does it say less than any number?
it’s okay
crystal lintel
ornate heath
no wait we gotta get to the bottom of this
honest creek
does it say less than any number?
remember the exercise i posted above
u have to show the maximum area > 37
ornate heath
if we move the sines as close to the triangle's edges as possible then technically its always less than the triangle's area is it not?
ornate heath
(according to them)
bruh
honest creek
if we move the sines as close to the triangle's edges as possible then technical...
of course
ornate heath
is F the midpoint
honest creek
no
that’s an annotation
the printed stuff should say something
the one on red is just writing
clever spade
How would you notate a probability density function?
marsh citrusBOT
How would you notate a probability density function?
clever spade
Say I have a particular probability function p(x) with equations:
how would I notate these equations to represent the probability function p(x)? Note that I've used f(x) as a placeholder
still temple
U can use the notation of piece wise functions
clever spade
U can use the notation of piece wise functions
how would I do that exactly?
I'm not sure how to do piecewise notation in google docs
still temple
One sec I’ll show you
$$P(X = x) =
\begin{cases}
\frac{4}{143} x - \frac{92}{143} & \text{if } 23 \leq x \leq 28.5 \
-\frac{4}{195} x + \frac{48}{65} & \text{if } 28.5 < x \leq 36 \
0 & \text{otherwise}
\end{cases}
$$
clever spade
also, do I need to capitalise the X in any place considering this is a random variable?
elfin berryBOT
Edmund Cloudsley
still temple
also, do I need to capitalise the X in any place considering this is a random va...
Yeah
clever spade
alright, thanks! Just wondering if there's any way to notate this in google docs
still temple
alright, thanks! Just wondering if there's any way to notate this in google docs
Are u righting a maths paper or smth?
If so, I highly recommend Overleaf
clever spade
Are u righting a maths paper or smth?
similar
still temple
Over google docs
still temple
Okay then
It might be a good idea to use #latex-testing to create your screenshots using latex
And put that into google docs
Cause maths with google docs is really frustrating (one of the reasons I learnt overleaf :))
weak surge
overleaf is a good place to do latex
still temple
Also if you want more privacy, you want more privacy
Like if it’s a confidential paper or smth
You can create your own server
And put latex bot there
@elfin berry
This one ⬆️
clever spade
alright!
still temple
Ciao
marsh citrusBOT
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Available help channel!
runic crag
marsh citrusBOT
runic crag
how do we find x
without a calculator (this is supposed to be done quickly)
answer is (-2(1+i) , 1)
fathom ridge
do the matrix multiplication
runic crag
alright, let me try
elfin berryBOT
neon
runic crag
there's a trick to inverting a 2x2 matrix
the det = 0, i cant inverse it
fathom ridge
Yeah well
fathom ridge
Carry out the multiplication
on the left side
And then match the entries on either side
runic crag
?
fathom ridge
You can multiply the two matrices on the left
Normally
You'll get a column vector, equate each component of that vector to 0
marsh citrusBOT
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fallow furnace
how can i rigorouosly proove that $\lim_{x \to 0 }\sin\left(\frac{1}{x}\right)$ doesnt exist
marsh citrusBOT
how can i rigorouosly proove that $\lim_{x \to 0 }\sin\left(\frac{1}{x}\right)$...
elfin berryBOT
Slowaq
marsh citrusBOT
@fallow furnace Has your question been resolved?
spark siren
think about $lim_{y\rightarrow \infty}sin(y)$?
elfin berryBOT
ThM
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restive silo
hi
restive silo
Can we say y=2e^t+ce^(-t) or did I do smth wrong ?
cerulean prairie
No
late geode
you'd have take the reciprocal of the whole side
not whatever you want in any order you want
cerulean prairie
if z = x + y we can't say z-1 = x-1 + y-1
restive silo
if ``` z = x + y``` we can't say ```z-1 = x-1 + y-1```
we have 1/( 2e^(-t)+ce^t )
we can take stuff with the opposite pow, right
oh wait
we have to take the whole expression right ?
like ( )^-1
cerulean prairie
Yeah
marsh citrusBOT
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ruby matrix
hi i am confused by R ∘ R⁻¹
marsh citrusBOT
hi i am confused by R ∘ R⁻¹
ruby matrix
shouldnt aRb in R and bRa in R-1 lead to the final pair (a,a), thus (a, b) and (b, a) should not be included
marsh citrusBOT
@ruby matrix Has your question been resolved?
marsh citrusBOT
@ruby matrix Has your question been resolved?
marsh citrusBOT