Yeah
#help-36
obtuse heath
tender rampart
okay great, so then by that fact me putting that half angle identity down i have not yet integrated, so i still need to do that which is why they go back to sin, because that's the integration of that half angle identity, do I have that correct?
obtuse heath
So you're just confirming that integrating sin^2 x gives back an expression containing sin(2x)?
tender rampart
no
here
- convert starting trig identity to half angle identity
- (i still don't exactly know why we do this)
- half angle identity returns 1/2(1-cos(2x))
- we then integrate which returns the sine trig function
- plug in limits and finish
obtuse heath
Looks good to me
worldly vale
you do it so that you actually can integrate sin^2(x) using things you already know how to integrate
tender rampart
OH
so its really not possible to integrate 1-cos(2x)?
at my level of calculus 1?
worldly vale
? yes it is
tender rampart
then why are we doing this
tender rampart
you do it so that you actually can integrate sin^2(x) using things you already k...
^
worldly vale
you are trying to integrate sin^2(x)
we dont know how to do this
so we rewrite it as 1/2(1-cos(2x))
we know how to integrate this
so now we know how to integrate sin^2(x)
worldly vale
yes
tender rampart
but the 2x inside parentheses doesn't, right?
worldly vale
we simply replaced sin^2(x) with 1/2(1-cos(2x)) in the integral and then did the integration
doesnt what?
tender rampart
doesn't integrate
worldly vale
it does not change no
tender rampart
and that's because it's in parentheses right
its basically the "trig function of..." right
worldly vale
sure yeah
tender rampart
awesome
thank you very much my friend, that really helps a massive ton
have a good one
.close
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frosty ridge
how do i solve this
i get the answer 2/3(x1+x2)
final saddleBOT
how do i solve this
i get the answer 2/3(x1+x2)
@frosty ridge Has your question been resolved?
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@frosty ridge Has your question been resolved?
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@frosty ridge Has your question been resolved?
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@frosty ridge Has your question been resolved?
raven mason
facts about math
noble sorrel
yo can someone help me find the intervals
final saddleBOT
yo can someone help me find the intervals
final saddleBOT
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@noble sorrel Has your question been resolved?
shell compass
hi, can anyone helo me with this question? I don't know b, c, and d. I figured out question a.
final saddleBOT
hi, can anyone helo me with this question? I don't know b, c, and d. I figured o...
karmic kindle
This is systems of equations
Do you know how to solve those?
Here is an example of how you solve c
You could use elimination too
Which might be easier tbh
final saddleBOT
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hasty tide
The limit as x approaches 0 from both sides of f(x) is going to be nonexistent for this function, correct?
Because it's not continuous through the point x = 0?
final saddleBOT
The limit as x approaches 0 from both sides of f(x) is going to be nonexistent f...
azure trellis
the limit both ways exist, but they wont be the same
and that makes the function not continuous
hasty tide
Okay, so I know that the limit as x approaches 0 from the LEFT is -1, and the limit as x approaches 0 from the RIGHT is 1, but if it's the limit as x approaches 0 from BOTH sides, is the limit nonexistent?
marsh temple
the limit doesn't exist, yes, however
Because it's not continuous through the point x = 0?
is not the reason
worn knot
Okay, so I know that the limit as x approaches 0 from the LEFT is -1, and the li...
do you know when a limit from both sides won't exist? there are multiple reasons this can happen
hasty tide
do you know when a limit from both sides won't exist? there are multiple reasons...
I know if both points do not "converge" on the given x value, then the limit doesn't exist. That's, essentially the only one I know of.
worn knot
yes so in this case why doesn't the limit as x->0 exist?
does it converge on the same value as x->0 from both sides?
final saddleBOT
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grand birch
Chris uses IVT and concludes that the equation x3β5x^2β6=K has a solution over the interval [5,6]. Given their work is accurate, which of the following can be the value of K?
A. K=-7
B. K=186
C. K=-5
D. K=-8
E. K=36
F. None.
final saddleBOT
Chris uses IVT and concludes that the equation x3β5x^2β6=K has a solution over t...
grand birch
Im just confused how to even begin solving this
robust mulch
What does ivt state?
grand birch
IVT and the theorems in general are difficult for me
hearty solstice
Intermediate value theorem
robust mulch
Yes i know that
grand birch
π(x) is continuous on an interval [a,b]
π(π) β€ N β€ π(π)
π(π) = N
I believe are the rules for IVT
robust mulch
so over the interval [5,6], there exists a number c between 5 and 6 such that f(c) satisfies
f(a) < f(c) < f(b)
grand birch
Right
robust mulch
so over the interval [5,6], there exists a number c between 5 and 6 such that f(...
what are a and b here?
grand birch
a = 5, b = 6
grand birch
f(5) = (5)^3-5(5)^2-6 = -6
f(6) = (6)^3-5(6)^2-6 = 30
But neither of these fall between the interval
Unless theres more work to be done
robust mulch
Now look at your options
we know that for some c, f(c)=K
By ivt,
f(5) < f(c) < f(6)
grand birch
Correct
robust mulch
Evaluating everything out, we get
-6 < K < 30
robust mulch
Yeah
robust mulch
Sure
grand birch
The sqrt in A is all of it
robust mulch
so each one is sqrt of whats on the right of the equal sign?
grand birch
No no
I just mean the sqrt symbol you see in answer choice A is the whole answer
Hold on, this may help
π₯²
<@&286206848099549185>
Sad
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grand birch
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simple nebula
Can somebody please help with finding the resultant force vector and magnitude? Iβve been stuck for like 30min and donβt know what Iβm doing wrong. Thank you!
final saddleBOT
Can somebody please help with finding the resultant force vector and magnitude? ...
simple nebula
<@&286206848099549185>
wispy jay
Can somebody please help with finding the resultant force vector and magnitude? ...
This is physics?
simple nebula
yes, i was just trying this discord to see if anybody could help
final saddleBOT
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ashen tree
Hmmm, are the axises orthogonal?
I mean, it looks like the angle between the x and z axises is 45+60 degrees
Oh my bad, that assumes that F_2 lies on the xz plane
So, you want to find the components of F_1 and F_2. Have you managed to figure any of them out?
@simple nebula
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small ember
Is there a universal way to solve a system of 3 linear equations?
final saddleBOT
Is there a universal way to solve a system of 3 linear equations?
small ember
I find myself getting stuck on them more often than i'd like
tiny gorge
row reduction?
small ember
Well more just solving the equation for all of the variables. I'm not sure what method to use
3 seperate equations with 3 seperate variables
I've tried using both elimination and substitution to no avail
small ember
x - y + z = -6
3x - 4y - z = -4
-2x + 3y + 4z = 14
tiny gorge
ok, so what are you doing for your first step(s)?
with row reduction you would start by eliminating x from the 2nd and 3rd equations
then eliminate y from the 3rd equation
the idea being that when you're done, the 3rd equation will only depend on z, the 2nd will depend on y and z, and the 1st will depend on x y and z
then it's easy to find the values of xyz
small ember
with row reduction you would start by eliminating x from the 2nd and 3rd equatio...
Oh ok I'll give that a try, I was sort of just floundering, substituting into the second equation and ignoring the 3rd
tiny gorge
yea it's best to proceed methodically, this way you know you'll end up with something you can solve
small ember
I hadn't even been taught row reduction I'll need to try that
tiny gorge
you can check somewhere like khan academy for a tutorial
might also be called "gaussian elimination"
two names for the same thing
final saddleBOT
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eternal wharf
why do they add pie or substract?
final saddleBOT
why do they add pie or substract?
eternal wharf
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wet tundra
I need help with this problem: find a curve that passes through (1,-1) and has an arc length of (4.37653)
final saddleBOT
I need help with this problem: find a curve that passes through (1,-1) and has a...
wet tundra
i honestly have no clue what im doing
i had an integral in place of the arc length, but i already solved it
tired walrus
y = -1, for x from 1 to 5.37653
wet tundra
hm
tired walrus
can you show the entire problem exactly as stated?
wet tundra
im looking for more of an equation. and yea
i don't think i've learned with this tbh
tired walrus
you're supposed to recover the function from the stuff under the integral
@opaque juniper please open your own channel, #βhow-to-get-help
opaque juniper
okk
wet tundra
you're supposed to recover the function from the stuff under the integral
ah, so just take the sqrt and then do the integral
thanks
-6/x^2 + 5
thank you @tired walrus!
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tranquil pine
final saddleBOT
tranquil pine
what does (x,y) even mean in this case
does the (x,y) have any relation to the 3x+y
if i set (0,0) would that means 3*0+0=0?
marsh temple
(x,y) means an ordered pair, yes it has a relation to the 3x+y, yes it's 0
plush merlin
this set consists of all couples (x,y) that satisfy the given condition
you can rewrite the given condition as $y=-3x$ and so your set consists of all couples $(x,-3x)$ $\forall x\in \mathbb{R}$
soft zealotBOT
calculus is fun
plush merlin
every point in the xy plane that satisfies this condition is in the set
so you can think of (x,y) as the coordinates of a point in the xy plane
which satisfies the given relation between x and y
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paper copper
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ancient thistle
!status
final saddleBOT
What step are you on?
1. I don't know where to begin
2. I have begun but got stuck midway
3. I got an answer but I'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above
!show
final saddleBOT
Show your work, and if possible, explain where you are stuck.
ancient thistle
You can't just post a question
And the picture is also cut
What are you expecting
paper copper
so im trying to get the expected value of the values given in the table, would I use the formula (6x0.15)+(8x0.35)+(10x0.5) to get the expected value?
sour umbra
Yes.
runic storm
final saddleBOT
runic storm
What do I do when dy/dx is split up
I have the integrating factor which is e^x^2
I move -4e^-x^2 to the right side
sour umbra
Do you know how to solve exact equations?
runic storm
I just have to find general solution
vital crag
I just have to find general solution
k but that doesn't answer his question
runic storm
No I don't know then
vital crag
do you know how to solve first order linear differential equations
runic storm
Yes
tranquil pine
how do you get a rule in a sequence where the product of the next term is twice the number before it?
like in
2, 4, 8, 16, 32, ...
final saddleBOT
how do you get a rule in a sequence where the product of the next term is twice ...
tranquil pine
it's not making sense for me
cosmic warren
geometric sequence
dense pumice
$a_n = 2a_{n-1} \forall n \geq 2 ; a_1 = 2$
dry light
$a_n = 2a_n-1, \forall n \geq 2 ; a_1 = 2$
soft zealotBOT
Umbraleviathan
dry light
@dense pumice make sure to include the dollar signs and \forall lol
Otherwise the "for all" will turn into a jumbled word
tranquil pine
wait so is that the rule?
geometric sequences haaent been taught to us so i can't process this info atm
soft zealotBOT
Normed
dense pumice
wait so is that the rule?
Yes
tranquil pine
Sorry i forgot to reply my classmate said it's just
$a_n = 2^n$
soft zealotBOT
NickEman132
blazing ridge
Can someone help me with thid
final saddleBOT
Can someone help me with thid
blazing ridge
Which segment is longer ad or DC?
I've been looking at this for like 20 mins I don't get it
final saddleBOT
@blazing ridge Has your question been resolved?
fossil geyser
You have the side lengths of triangle ABE marked. Notice triangle ACD is similar to triangle ABE
final saddleBOT
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next merlin
how do i write an equation for this table?
final saddleBOT
how do i write an equation for this table?
next merlin
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silent cave
final saddleBOT
silent cave
find the value
I don't really remember many trigonometric properties so I don't know what to do first
severe reef
iβm just learning square roots for the first time. What would be the process for 5 a)?
echo sable
I wanna know of this is correct for the function e^2x+1
final saddleBOT
I wanna know of this is correct for the function e^2x+1
echo sable
End behavior = As the function approaches infinity the function is infinite, as the function approaches negative infinity it is 0
<@&286206848099549185>
final tangle
!15m
final saddleBOT
Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
grim badger
<@&286206848099549185>
Weren't you told in your last channel to wait before pinging?
echo sable
Nah
final tangle
seems to be a recurring offence
immediate ping, in many cases before even asking an actual question
if you intend to seek help here,its recommended that you try to abide by server guidelines
anyway your left end behaviour is wrong
final saddleBOT
@echo sable Has your question been resolved?
unborn pilot
not really sure what to do i havenβt even gotten past step 1
final saddleBOT
not really sure what to do i havenβt even gotten past step 1
tired walrus
do you know the triangle inequality
unborn pilot
do you know the triangle inequality
the sun of the lengths of any two sides of a triangle must be greater than the length of the third side
tired walrus
yes
to which triangle, and in what way, could you apply this inequality here in a useful fashion?
jolly sentinel
no JLK
since sum of any 2 side lengths is greater than the length of the 3rd, KJ + KL > JL
and since JL is congruent to LM, KJ + KL > LM
unborn pilot
oh thanks
got another question tho
it says determine all the possible values and i could just sit there testing all the values but is there any faster method?
help
close channel
ornate solstice
fossil geyser
close channel
i'm assuming you're trying to close this channel, you can do it like this
.close
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timber hearth
how do i prove that a function is monotonically increading without calculus?
timber hearth
ik you find the derivitive and see if its > the x axis, but i havent taken calculus yet so i dont know how to take it
warm wraith
well, if you're somehow able to prove that f(x+1) > f(x), that would do it
timber hearth
well, if you're somehow able to prove that f(x+1) > f(x), that would do it
why what about f(0) vs f(0.1)
warm wraith
not sure I understand you?
timber hearth
not sure I understand you?
why does it work?
warm wraith
well, if you prove that at any point x, the point x+1 is greater than it, that means the function is increasing
at least that's my understanding of it
timber hearth
oh ig i'll try that
static fractal
I don't think that quite works on the first unit interval on the domain (if the domain is unbounded below this isn't a problem)
for instance y = x for x>0, y = 10 for x = 0 is not increasing if we include the first point of the domain (and the function could be more generally decreasing in the first unit interval)
timber hearth
yeah that's what i was trying to say
ur gonna have to prove the interval [0,1) if you were to have x>0 domain
static fractal
if it's an infinite domain then it's not a problem
but otherwise, yeah, that'd require a separate proof
f(x+1) > f(x) still proves it for everything else though, so you can just do it in two parts if that's feasible
the other way is to prove the definition directly; that x > y implies f(x) > f(y)
final saddleBOT
@timber hearth Has your question been resolved?
trail mango
is there a specific function you had in mind?
it would be simple to do it by definition for e.g. linear functions
timber hearth
k ty
trail mango
yea i think this works
trail mango
i was trying to prove that for all x>0, x/(x+100) is monotonically increasing
this is equivalent to: for all x > 0, for all a >= 0, (x+a)/(x+100+a) >= x/(x+100)
timber hearth
yeah how am I supposed to prove that
trail mango
you can divide and multiply and add stuff etc to get an equivalent inequality thatβs clearly true
trail mango
changed the inequalities a bit but you can change them back to > if you want strictly increasing
final saddleBOT
trail mango
np ^_^
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sinful tide
I need help with this
final saddleBOT
I need help with this
abstract mango
What have you tried?
sinful tide
Plugging 3 into h
abstract mango
Have you got h yet?
sinful tide
No
abstract mango
Find the function h(x) from itβs definition
Essentially have you plug whatever information given above
soft zealotBOT
.doc
abstract mango
you reached here?
sinful tide
sinful tide
And I plug in 3?
sage nexus
idk where to put this
fossil geyser
go to one of the channels in the available section above where you found this channel
sage nexus
how do I get 14
abstract mango
sure you can plug x=3 into each equations and evaluate
I think the goal is to multiply them algebraically
then plugging in 3
First get the answer by inspection without algebra
then try the algebra
abstract mango
Without algebra, by this i mean, find f(3) then square it, find g(3)*3, take their difference
abstract mango
**.doc**
After you are done, try expanding all these above and write it into something nice
final saddleBOT
@sinful tide Has your question been resolved?
final saddleBOT
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fossil shuttle
is this right
final saddleBOT
is this right
grim badger
is this right
What's that blue dot at -3 on the y axis for?
grim badger
And no it's not right, because -7/3 can either mean down 7, right 3 or up 7 left 3
grim badger
While if it was 7/3 that can either mean up 7 right 3 or down 7 left 3
grim badger
And no it's not right, because -7/3 can either mean down 7, right 3 *or* up 7 le...
.
You can't go both down and left if you have a negative slope
grim badger
Negative sloped lines look like on the right
fossil shuttle
so it's not negative?
grim badger
is this right
If you connected the dots on what you drew here, would the line look like the left with a positive slope or the right with negative slope?
fossil shuttle
If you connected the dots on what you drew here, would the line look like the le...
left
grim badger
But your equation have a negative slope, so it should look like the right one
grim badger
And no it's not right, because -7/3 can either mean down 7, right 3 *or* up 7 le...
That's why I said this
grim badger
is this right
Start with your y intercept
fossil shuttle
okay
grim badger
And no it's not right, because -7/3 can either mean down 7, right 3 *or* up 7 le...
Apply one of these
So go down 7 right 3 or up 7 left 3
grim badger
Then you do the other one
grim badger
No
That's not the y intercept
Y intercept is where it crosses the y axis
What you did initially
You applied the slope wrong
fossil shuttle
grim badger
So go down 7 right 3 or up 7 left 3
Apply one of these
grim badger
If you can't go up 7 left 3, then do the other one, down 7 right 3
fossil shuttle
grim badger
but can't I also do the left?
No
fossil shuttle
oh
grim badger
And no it's not right, because -7/3 can either mean down 7, right 3 *or* up 7 le...
.
You can't go left 3 and down 7
fossil shuttle
Connect the dots
and then I'm finished?
grim badger
Yes
storm orchid
Would this not be the int from 0-1 of pi*(1-x^1/4)^2 dx?
final saddleBOT
Would this not be the int from 0-1 of pi*(1-x^1/4)^2 dx?
hybrid heath
why x^1/4?
storm orchid
Fourth root x.
storm orchid
The only thing new here is the introduction of the function revolving around the line y=1, did I incorporate this properly?
Using 1-r(x).
hybrid heath
yeah you did
storm orchid
Oh nvm, I figured it out. User error when entering the value. This was technically correct. I guess it just wanted the decimal approximation with a ton of numbers after the decimal.
Thank you!
.close
glad basalt
Can someone explain the significance
final saddleBOT
Can someone explain the significance
blissful meadow
You find where the parabola characterized by this equation touches the x-axis
glad basalt
Does that mean y, the solution is zero? Are you looking for the x intercepts?
glad basalt
You find where the parabola characterized by this equation touches the x-axis
Is that for everything
Any equation set to zero
blissful meadow
yeah
In particular it means you look for intersections with the line y=0
which is the x-axis
If you set your equation = 5
Then you find intersections with y=5
glad basalt
Hm ok
What does it mean when weβre looking for solutions to a quadratic?
Like factoring then getting x - 3 = 0
Are all solutions where the parabola touches the x axis?
blissful meadow
Yes
glad basalt
Ok
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hot hornet
Prove that the space $\mathbb{R}^{[0,1]}$ is infinite dimensional
final saddleBOT
Prove that the space $\mathbb{R}^{[0,1]}$ is infinite dimensional
soft zealotBOT
sadkid
strong mantle
Prove that the space $\mathbb{R}^{[0,1]}$ is infinite dimensional
You can do it by proving that any finite set of vectors cannot span the whole space
hot hornet
You can do it by proving that any finite set of vectors cannot span the whole sp...
I need a hint on how to do this
strong mantle
I need a hint on how to do this
Given any finite set of vectors, find another vector outside its span
strong mantle
Prove that the space $\mathbb{R}^{[0,1]}$ is infinite dimensional
That's all functions, right?
hot hornet
yeah it's for all functions
hot hornet
Given any finite set of vectors, find another vector outside its span
how should we find another vector outside of the span?
strong mantle
how should we find another vector outside of the span?
Consider the function outputting 1 iff its input is a certain real number r, and 0 otherwise
hot hornet
Consider the function outputting 1 iff its input is a certain real number r, and...
okay π
oh I think I know where this is going
nah I can't figure it out all the way π¦
@strong mantle how would you proceed from here?
strong mantle
<@559696948471398427> how would you proceed from here?
Maybe you can think about it later
strong mantle
<@559696948471398427> how would you proceed from here?
But there has to be at least one vector of this form that doesn't fit in the finite set
hot hornet
are we actually going to find the vector or just show that it exists?
strong mantle
nah I can't figure it out all the way π¦
What have you tried
strong mantle
Consider the function outputting 1 iff its input is a certain real number r, and...
I meant that the set of all these is linearly independent and spans the space
soft zealotBOT
sadkid
hot hornet
but I couldn't figure it out past this
Do you think I'm going in the right direction?
strong mantle
Consider the function outputting 1 iff its input is a certain real number r, and...
I've come up with another approach
Suppose that f_1, f_2, ..., f_n span the vector space
Then they must be different at n points
Or at least distinguishable
Then consider an n+1th point
Find a function which is the sum of all n functions at the previous n points, but different from it at the n+1th point
And conclude
hot hornet
Find a function which is the sum of all n functions at the previous n points, bu...
So basically you show that this new function can't be a linear combination of all the previous n functions but at n+1 it deviates by 1 past
strong mantle
So basically you show that this new function can't be a linear combination of al...
Yes
Up to you to fill out the details
final saddleBOT
@hot hornet Has your question been resolved?
hot hornet
<@478703800463523861> how is it not resolved?
oh sorry I was just working on it
Just kept it open to work on some details
in case I had any follow up questions
final saddleBOT
@hot hornet Has your question been resolved?
warm python
Find the number of solutions $1+ sin^4(x) =cos^3(3x)$ has in $[\frac{-5Ο}{2} , \frac{5Ο}{2}]$. I was thinking os solving it by substituting $sin(x)=u$, but I think that will be too long, what other method can I use
final saddleBOT
Find the number of solutions $1+ sin^4(x) =cos^3(3x)$ has in $[\frac{-5Ο}{2} , \...
soft zealotBOT
physicsrocks
final saddleBOT
limber scaffold
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rich talon
I could not get 545 x 25 in my head⦠any tips?
final saddleBOT
I could not get 545 x 25 in my head⦠any tips?
strong mantle
Try using paper
final tangle
consider 25 = 100/4
multiplication by 100 is trivial
and should be easier to do division by 4 mentally (compared to long mulitplication)
barren hound
think about 545 quarters and consider how much money that is
tired walrus
think about 545 quarters and consider how much money that is
ok american
limber scaffold
Reopen
barren hound
for all of these, you just think about each piece of it
so like for the park, it's made of a rectangle and a semicircle
figure out the area of each one and then add them up
barren hound
50 + 60? 
barren hound
why are you adding 50 and 60 is my question?
limber scaffold
The area is 110
limber scaffold
Yes
limber scaffold
Oh
barren hound
,tex .plane geom
soft zealotBOT
hayley!
barren hound
look at the rectangle (second one)
barren hound
yes
barren hound
where are you getting plus from
limber scaffold
It says 2b + 2h
barren hound
,rccw
soft zealotBOT
barren hound
the area of the rectangle part is 50m β’ 60m yes
limber scaffold
So I got it right
barren hound
well you probably want to actually multiply them
strong mantle
Can you find the radius of the semicircle?
barren hound
there's a diameter
limber scaffold
I thinks itβs 50
strong mantle
I thinks itβs 50
can you find the radius from the diameter?
limber scaffold
Nope
barren hound
,tex .plane geom
soft zealotBOT
hayley!
barren hound
look at the circle, the first one
see how the diameter goes all the way across, while the radius goes only halfway across?
barren hound
yes
barren hound
nope
final saddleBOT
@limber scaffold Has your question been resolved?
strong mantle
Can give me the answer for all three
have you completed 3 yet
shy tendon
Prove or disprove that function g(t,u)=a|u|+b(t^2)(u^2)+ct(u^3) satisfies the Lipschitz condition on R={(t,u)||t|β€a, |u-b|β€c}, where a, b, c are positive constants. Find Lipschitz constant(if exists).
final saddleBOT
Prove or disprove that function g(t,u)=a|u|+b(t^2)(u^2)+ct(u^3) satisfies the Li...
shy tendon
I applied the Lipschitz condition and proved that it satisfies it. Just have one doubt, if we have u1+u2 then can I say u1+u2β€2(b+c)
for finding the value of K(Lipschitz constant)
desert mantle
b and c are fixed while u1 and u2 are variables. you would have to show why u1+u2 cannot be bigger than the fixed value 2(b+c)
desert mantle
I am not saying it would be hard to show. but it needs justification
desert mantle
yes but why. from where are you getting that. be more precise
shy tendon
do I have to show them by solving the inequality?
desert mantle
if you wanna call it solving, then sure
torpid meteor
Hello
strong mantle
Hi
strong mantle
Whatβs the context?
torpid meteor
strong mantle
do you have more?
strong mantle
you could differentiate pir^2 wrt t
final saddleBOT
@torpid meteor Has your question been resolved?
tranquil pine
You have interpreted the question wrongly. "...with respect to radius" here simply means the corresponding change in Area of circle (per unit time) for some value of the initial radius r.
And dr/dt here denotes by how much radius is changing per unit time
final saddleBOT
@torpid meteor Has your question been resolved?
torpid meteor
Yes
jade cedar
did i wake up in a new universe today? because apparently according to the calculator, in 4^x=9, x=log3/log2, even though it's CLEARLY x=log9/log4
final saddleBOT
did i wake up in a new universe today? because apparently according to the calcu...
amber holly
Those are the same
$\frac{\log9}{\log4} = \frac{\log(3^2)}{\log(2^2)} = \frac{2\log3}{2\log2} = \frac{\log3}{\log2}$
soft zealotBOT
A Lonely Bean
foggy flower
final saddleBOT
foggy flower
How would you go about doing b)
<@&286206848099549185>
I need to find p(a) correct?
rustic wedge
Hmm if I remember correctly isn't P(A intersection B) = P(A).P(B)
rustic wedge
U do know what's P(A' )=?
rustic wedge
A' means everything that isn't a?
Yes mathematically 1-P(A)
foggy flower
I don't know the number
Maybe 0.6
rustic wedge
So
Let P(A)=x and P(B)=y
Can u find P(A intersection B) and P(A' intersection B)?
In terms of x and y
@foggy flower
rustic wedge
Ye sorry
foggy flower
All good, thanks for assisting
rustic wedge
I got x=1.6
Lemme see one sec..I get y = 0.8
foggy flower
Yea
foggy flower
How'd you get that if you don't mind me asking
rustic wedge
How'd you get that if you don't mind me asking
xy=0.2
(1-x)y =0.6
y-xy = 0.6
y-0.2=0.6
y=0.8
Then sub y in xy = 0.2
foggy flower
Oh is xy the same as p(a n b)
Answering myself
Yes it is
Okay
How'd u get to next step?
rustic wedge
How'd u get to next step?
Ah yes forgot to mention, I believe P(A' intersection B) = P((1-P(A) intersection B)
foggy flower
How is that so, is that just how it is
Like I'm just supposed to know that
Oh well know that I think about it
It makes sense
Yea
rustic wedge
U haven't been taught probability?
rustic wedge
foggy flower
Exactly that emoji
Wait let me write your working and I'll get back if I have any questions
rustic wedge
K
foggy flower
xy=0.2
(1-x)y =0.6
y-xy = 0.6
y-0.2=0.6
y=0.8
Why did you do p(b)-p(a n b)= p(a' n b)
rustic wedge
Why did you do p(b)-p(a n b)= p(a' n b)
I did (1-p(a))(p(b)) = p(a' n b)
foggy flower
I'm talking about Ur 3rd line of working btw
rustic wedge
I'm talking about Ur 3rd line of working btw
I distributed
rustic wedge
?
foggy flower
Then sub y in xy = 0.2
What would this achieve
rustic wedge
What would this achieve
the value of x
foggy flower
Wait a minute, we did all that working while we already had xy=0.8 I could've divided by y which we already had
Oh my days
rustic wedge
Wait a minute, we did all that working while we already had xy=0.8 I could've di...
xy=0.2 not 0.8
foggy flower
Oh I meant xy =0.2
I got p(a) now
P( a u b) should be here from know
Easy from here
rustic wedge
Yeah
chrome cove
Hi! π
I need help, I have to solve this statistics problem and for some reason my solution feels off
A fair coin is flipped. If it lands heads H we flip it again, if it lands tails T we stop. What is the average number of times the coin is flipped?
I tried to find the probability of the sequence (H, T)
The possible cases are: (T), (H, T), (H, H, ...)
I don't know how many cases the third sequence should be.
final saddleBOT
Hi! π
I need help, I have to solve this statistics problem and for some reason ...
strong mantle
Hi! π
I need help, I have to solve this statistics problem and for some reason ...
have you tried making a tree diagram
chrome cove
Nice, so the probability of (H, T) is 1/3, right?
strong mantle
Nice, so the probability of (H, T) is 1/3, right?
No, how did you get that?
desert mantle
you shouldnt write stuff that depends on order as sets
chrome cove
Here's what I did
chrome cove
you shouldnt write stuff that depends on order as sets
Yes my bad π
strong mantle
Here's what I did
then you can use this tree to calculate the expected values of each of the Tails
chrome cove
Nice, so the probability of (H, T) is 1/3, right?
But why is this not correct?
strong mantle
But why is this not correct?
You can see that by marking the probabilities on the tree
it takes 1/2 to go to H
and then another 1/2 to go to T
then when those are multiplied, you get 1/4
which is not 1/3
chrome cove
Ah I see you are right
final saddleBOT
@chrome cove Has your question been resolved?
chrome cove
Thank you @strong mantle!
final saddleBOT
final saddleBOT
Available help channel!
real juniper
"The nth hop lasts for a time t__n, which solves the equation.
Make an equation for the time t_n. how many jumps will the ball make in total? Is this answer realistic?"
final saddleBOT
"The nth hop lasts for a time t__n, which solves the equation.
Make an equation ...
real juniper
the equation at hand:
we know the v_0= 1m/s and it only keeps 1/9 of its energy per jump
so i got the velocity to be 1/3^n
however
When i do all of this, i get the limit as n goes to infinty to be 0 jumps?
final saddleBOT
@real juniper Has your question been resolved?
magic estuary
Working on #2
final saddleBOT
Working on #2
magic estuary
Honestly have no clue if Iβve done any right, but Iβve done 2 integrations by parts and donβt know how to continue
<@&286206848099549185>
final saddleBOT
@magic estuary Has your question been resolved?
final saddleBOT
@magic estuary Has your question been resolved?
jaunty portal
hello. The last integral you got is correct. That one can be solved directly. Also don't forget that when you have a definite integral and you use integrations by parts, you have to evaluate each "uv" in the limits (0 and inf)
Anyway, you can keep your work and evaluate the limits in each term you got
you should get a^2/s + 2ab/s^2 + 2b^2/s^3
Another way to approach this laplace transform is first expanding the square I think π€ (a+bt)^2 = a^2 + 2abt + b^2 t^2
magic estuary
I had 2b^2/s^3 as the third term but the first two I had as well
I can never tell when I can end the integration by parts on a laplace so I just do two of them and then start substituting the inf's and 0's
magic estuary
oh neat
jaunty portal
I can never tell when I can end the integration by parts on a laplace so I just ...
It ends when the integral can be directly solved, and you don't need int by parts anymore
magic estuary
So like in another example I've brought it down to sin(t)*e^-st
Since theres a laplace identity for sin(t) which is 1/s^2+1 I can sub that in and solve the rest of it?
jaunty portal
sure, as long as you're allowed to use that identity
There's a method based in int. by parts, it's the DI method. It's a way to do it less messy
magic estuary
In other cases such as with polynomials, can I just do by parts until I'm left with int(d e^-st) and then solve it using the upper and lower boundaries?
jaunty portal
yes
modest perch
working with lagrangians right now in classical mechanics but i think this is related enought to mathematics to ask the following question:
final saddleBOT
working with lagrangians right now in classical mechanics but i think this is re...
modest perch
I dont understand how we are using the chain rule here
Where q(a), as I understand it, is some function of the position and of time
oh it's actually possible im completely misunderstanding the chain rule
i figured it out hehehe
.close
final saddleBOT
final saddleBOT
Available help channel!
vivid birch
You are observing n rectangles from an origin (360O view) in arbitrary positions (distances and orientations) in cartesian coordinate system, as shown below (origin with red star).
Any rectangle can occlude any other rectangle, and a rectangle can occlude more than one rectangles; no rectangle can overlap with the origin or with any portion of other rectangles.
Calculate n random rectangles (with respect to position and orientation), with n being the user-defined input. Calculate the total occlusion angle over all rectangles with the lowest time complexity possible. Example of an occlusion angle between two rectangles is illustrated with the shaded area in the illustration below.
How would I do this?
final saddleBOT
You are observing n rectangles from an origin (360O view) in arbitrary positions...
vivid birch
Basically, how would I find the occlusion angles for this?
final saddleBOT
@vivid birch Has your question been resolved?
vivid birch
<@&286206848099549185>
final saddleBOT
@vivid birch Has your question been resolved?
vivid birch
.close
bleak granite
What does it mean by "Which summary statistics would you use to summarize the center and spread in this data? Why?"
final saddleBOT
What does it mean by "Which summary statistics would you use to summarize the ce...
bleak granite
well that but i dont understand this wording "summary statistics"??
robust mulch
Heres what wikipedia says
bleak granite
mhmm so I guess to measure spread you'd use mean and for spread, standard dev?
alright thanks @robust mulch! 
.close
stray stump
If $x$ is a real number, then either $x$ is an integer or there is an integer $n$ such that $n <x < n+1$
final saddleBOT
If $x$ is a real number, then either $x$ is an integer or there is an integer $n...
soft zealotBOT
KN
stray stump
Assume $x \notin \mathbb{Z}$. I considered 2 cases. The first case is when $x > 0$. Let $S = { n \in \mathbb{N} | n < x}$ and let $\alpha = \text{sup}(S)$. By construction $\alpha < x$.
soft zealotBOT
KN
stray stump
Now I want to argue why $x < \alpha + 1$.
soft zealotBOT
KN
stray stump
I am stuck here...
Since I want to show x < alpha + 1, can I assume that x > alpha + 1. But that is direct contradiction to how S is constructed. And this is done?
plush merlin
Why is $\alpha$ necessarily <x can't it just be x?
soft zealotBOT
calculus is fun
plush merlin
$\alpha$ might not and (is probably not) in S
soft zealotBOT
calculus is fun
stray stump
It has to be in S, no?
This is how I pictured it... S is natural numbers smaller than x
sup (S) is just the greatest natural number less than x.
which is definitely in S
plush merlin
It has to be in S, no?
Not that's not necessary
For example take the interval (0;1)
Sup((0;1))=1
stray stump
In that case, x would be like 1.1 and S = {1}
plush merlin
Sup((0;1))=1
And $1\notin (0;1)$
soft zealotBOT
calculus is fun
plush merlin
In that case, x would be like 1.1 and S = {1}
No If S={1} sup(S) doesn't exist
Because no matter which number you choose you can always choose another number closer but greater than 1
plush merlin
sup (S) is just the greatest natural number less than x.
No sup(S) is just the least upper bound of S so $sup(S)\leq x$
soft zealotBOT
calculus is fun
plush merlin
It can't be >x because it will contradict the fact that it is the least upper bound of S
consider the set S={0,1,2..,n} containing all natural numbers up to n
And let sup(S)=x
Then obviously n<x=<n+1
stray stump
But x is not an integer though
plush merlin
But x isn't an integer in this case because it is sup of set of some natural numbers
Thus if sup(S)=x then n<x<n+1
Now you have to prove for sup(S)<x
Because sup(S)>x can't happen
Oh wait
stray stump
Sorry, I still do not follow. If S = {1,2,3}, then sup(S) = 3 no?
plush merlin
You have to prove that you can always find a set of natural numbers up to some natural number n which has a sup=x
stray stump
3 is an upper bound, and for all other upper bounds, 3 < those upper bounds
plush merlin
Yea a supremum can or cannot be a part of the set
I missed that it can be
My mind is working such that it can't
But it can in fact
Thanks for notifying me
plush merlin
Sorry, I still do not follow. If S = {1,2,3}, then sup(S) = 3 no?
Yes
upper kite
No If S={1} sup(S) doesn't exist
no?
$\sup { 1 } = 1$, I think you will agree but what did you mean?
soft zealotBOT
M8732
plush merlin
My mind is working such that it can't
.
I was working under the assumption that a sup can't be in the set but it can be
upper kite
oh okay
upper kite
Proof Explanation by contradiction 
upper kite
yes and no
Ideally they are very explanatory
but in practice usually people take formal details and correctness as a priority.
Especially for statements like yours.
upper kite
Anyway, "calculus is fun" is definitely correct in that $\sup(N) \not\in N$ is possible generally.
soft zealotBOT
M8732
plush merlin
Sorry guys I gtg sleep I am brain farting it's around 1:17 am rn
upper kite
So I think this bit could use further explanation/proof/whatever you will call it. π
upper kite
Sorry guys I gtg sleep I am brain farting it's around 1:17 am rn
alright, good night
I will try help.
plush merlin
Have a nice day/night both of you cya both
stray stump
ty for help
upper kite
How did you define real numbers?
plush merlin
Np I didn't do anything I messed up alot
stray stump
How did you define real numbers?
we didnt
upper kite
Since this is a very fundamentals-based question there is of course the important question what we are allowed to use.
upper kite
Np I didn't do anything I messed up alot
Don't worry too much, happens to the best of us.
upper kite
"it fills in the gaps of the rationals"
πππππππππ
ha ha okay, let's see
stray stump
We did talk about the archimedean property of R
upper kite
Generally I like your approach with the supremum.
stray stump
yeah, that was the only tool we learned so far that I could think of.
upper kite
For the whether or not a wull be an integer ot not
there is a property of integers that any subset of the integers ahs the supremum and infimum as an element of it's set.
stray stump
axiom of completeness?
upper kite
However this is not godfgiven, so if you didn't proof it in class, I think it would need to be proved
upper kite
axiom of completeness?
well, I think you mean this for reals thoguh, right?
stray stump
upper kite
Any infiumum or supremum of a real ubset will be real (or infinity or negative infinity).
yes
The thing is though that we want a to be an integer.
We need to find a reason why a is an integer.
stray stump
which "a" are you referring to?
stray stump
Oh, we are keeping the idea of how I defined S earlier?
upper kite
I thought so?
soft zealotBOT
KN
upper kite
a = sup S
We have a lot to do now
- Why is a an integer
- Why is a < x
- Why is x < a + 1
have you at all heard about minimum/maximum vs supreumum/infimum?
stray stump
yes
upper kite
great
So I suggest proofing that our supremum is in fact a maximum
If we know this, than we get 1 & 2 for free
because IF it is a maximum, then a is in S
big IF
I think this is the thing "Calculus is fun" was tryingt to say.
upper kite
basically
stray stump
So is it even true that sup = max here?
upper kite
intuitively that is how i thought about supremums
if max exists, then always sup = max
However sometimes it is only a sup and not a max.
stray stump
well here the max does exist
upper kite
oh, why?
stray stump
Just take the biggest element in S, would that not be the maximum?
upper kite
Just take the biggest element in S, would that not be the maximum?
I am not convinced
we can't just take the biggest element of { x in real | 0 < x < 1 }
stray stump
right.
stray stump
we can't just take the biggest element of { x in real | 0 < x < 1 }
under which situation would this happen?
hmm
ok
maybe we dont split into cases
S = ${ n \in \mathbb{Z} | n < x }$
soft zealotBOT
KN
stray stump
so if x = 0.5, then S = {...,-2,-1,0}