oops
#help-28
nova basin
xn = 2pi
nova basin
can you prove it?
rugged raft
i cant😭 i guess it just bc i cant think of any counterexamples
nova basin
D is the hardest imo
rugged raft
b's example might involves some (-1)^n stuff
nova basin
.
since all functions are continuous, you might as well consider only constant sequences
rugged raft
Why consider constant seq if the functions are continuous?
nova basin
adjacent exercise: if you have a counterexample to one of these you have a constant one
rugged raft
B is -1 C is -1
nova basin
yes
rugged raft
Why consider constant seq if the functions are continuous?
But why? Actually i didn't expect that it'll be constant which made it so hard so me to think
nova basin
adjacent exercise: if you have a counterexample to one of these you have a const...
do it
rugged raft
adjacent exercise: if you have a counterexample to one of these you have a const...
You mean to prove it?
nova basin
yes
rugged raft
Need a hint
nova basin
xn converges
rugged raft
Suppose limxn=0, since sinx is continuous, so lim simxn = sinlimxn=sin0=0
Uhh what am i writing
So i guess as long as f(xn) is continuous, it'll never not equal to 0, so it has to be discrete
nova basin
if you assume xn -> 0, you'll go nowhere for this exercise
rugged raft
Ummmmmm
nova basin
assume xn -> a, some a but not necessarily 0
rugged raft
Suppose limxn=a, since sinx is continuous, so lim simxn = sinlimxn=sina
nova basin
you can do this for any continuous f, doing all 4 at once
rugged raft
Suppose limxn=a, since x+sqrt abs x is continuous, so lim x+sqrt abs xn = a+ sqrt abs a
For c, lim x+x^2 = a+ a^2
We can all get find the roots of these expressions=0
But for D
nova basin
you can do this for any continuous f, doing all 4 at once
if f(xn) -> 0, since f is continuous f(a) = 0
So the question boils down to: does f have a nonzero root?
rugged raft
Yes..?
nova basin
depends on f
rugged raft
Hmm depends on f continuous or not?
nova basin
it is for all 4 examples
but the answer to the question depends on the exact function
rugged raft
if f(xn) -> 0, since f is continuous f(a) = 0
So the question boils down to: doe...
So this is true for this question?
nova basin
you already solved abc
solve d
easy version: use the lemma
harder (intended version): don't use the lemma
rugged raft
What lemma
rugged raft
It doesn't have root so it's false
nova basin
if f(xn) -> 0, since f is continuous f(a) = 0
So the question boils down to: doe...
or thise
nova basin
It doesn't have root so it's false
well 0 sure is one
the question is is there another one?
prove your answer
nova basin
yeah you're doing D, you've been doing D for at least the last 5 minutes
nova basin
But idk how to prove it
well, you may take time
rugged raft
Ooh shooot
nova basin
but it can't
nova basin
right what
rugged raft
It's contradicting to the lemma cuz it only has a zero root
rugged raft
1-cosx=0 etc
I guess this is the idea but I'm not on pc so typing rigorous proof will be a pain that's why my msg is piece by piece
But I think I've got the idea, thanks for your patience!!
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somber hill
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somber hill
would it be incorrect to prove this using combinatorics
like if i find the number of n chain k elements long, and take the sum from 2 which is the shrotest to n+2 which is the longest
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torn jolt
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ruby wasp
think ive done right here but since its my first time wouldnt hurt to double check. Are the steps correct am i missing something?
ruby wasp
im supposed to find the limit value i think its called in english
languid junco
think ive done right here but since its my first time wouldnt hurt to double che...
that looks correct to me
ruby wasp
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naive sable
need help ASAP
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need help ASAP
naive sable
this is the matrixes i need to find the determinate first of A,B and C (edited)
[21:32]
then i have to find the reverse matrixes of A,B and C if they are reversable
[21:36]
and lastly we need to find the creation if the translation its right (A.B), (B.A) (A(t its upwards of A).D) and (C,D(T again upwards))
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feral urchin
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feral urchin
can anyone tell me what exactly this is asking?
stiff musk
which part is unclear?
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solar bison
Help
solar bison
Lets discuss this question paper plz
stable tulip
any question in particular?
solar bison
1.3
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solar bison
I got an answer of n=2,4,6,8.....
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buoyant saffron
could someone lend a hand
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could someone lend a hand
buoyant saffron
i have managed to do part a and b
but now with c i am not really sure what to do
placid oak
You need to calculate the transformed partial derivatives, and therefore show the laplacian in x', y' is the same as in x, y.
$$\Delta f = \pdv[2]{f}{x} + \pdv[2]{f}{y} = \pdv[2]{f}{x'} + \pdv[2]{f}{y'}$$
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Shuba
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teal nimbus
A one-dimensional line of N nodes indexed 0 to N-1. At each step there is a probability p = 1/8 of initiating a walk to a random destination. Once walking, one can move one node per step until destination reached with each step maintaining the p = 1/8 probability of selecting another random destination.
I believe I'm looking for the steady state distribution π(i) for each node i but I'm lost on how to get there and if this thought process breaks down in a 2d space. I initially made the assumption that this would be distributed the same as a 1/8 * 1/2 probability to move to any adjacent node in hopes that I could more easily model it as a graph to get an estimation but I'm doubting it is.
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rare dock
A one-dimensional line of N nodes indexed 0 to N-1. At each step there is a prob...
i would start by drawing the markov chain and writing the matrix. maybe just for an explicit small n if the matrix is overwhelming
but it shouldn’t be too bad. each row (other than the first and last) should just look like 0, …, 0, 1/8, 6/8, 1/8, 0, …, 0
teal nimbus
i wrote something quick and dirty earlier to get just a recorded frequency and like sure i could increase the iterations but i'm unsure how to approach it mathematically as it's not a problem I recognise how to even solve outside of bruteforcing like that
rare dock
steady state distribution is an eigenvalue problem
teal nimbus
At 0 0.875 0 0 0 0 0.025 0.025 0.025 0.025 0.025
At 1 0 0.875 0 0 0 0.025 0.025 0.025 0.025 0.025
At 2 0 0 0.875 0 0 0.025 0.025 0.025 0.025 0.025
At 3 0 0 0 0.875 0 0.025 0.025 0.025 0.025 0.025
At 4 0 0 0 0 0.875 0.025 0.025 0.025 0.025 0.025
Walking to 0 0.125 0 0 0 0 0.875 0 0 0 0
Walking to 1 0 0.125 0 0 0 0 0.875 0 0 0
Walking to 2 0 0 0.125 0 0 0 0 0.875 0 0
Walking to 3 0 0 0 0.125 0 0 0 0 0.875 0
Walking to 4 0 0 0 0 0.125 0 0 0 0 0.875```oh that looks shite
for n=4
rare dock
teal nimbus
wait no
actually no thats fine, made me second guess myself with walking to 0 from 0
but that can happen
or am i misunderstanding what you meant
rare dock
i can’t parse this table
teal nimbus
so its split into stationary and moving right
if you're stationary at node 0 theres a 7/8 to stay there
and a 1/8 to move to any other node, 1/8*1/5 to begin walking to any specific node
if you're already walking, there's a 1/8 that you arrive at the node you want and a 7/8 to be walking to the node you want
wait
guh
rare dock
i’m not sure i understand the prompt anymore actually
A one-dimensional line of N nodes indexed 0 to N-1. At each step there is a probability p = 1/8 of initiating a walk to a random destination. Once walking, one can move one node per step until destination reached with each step maintaining the p = 1/8 probability of selecting another random destination.
tf does that even mean
teal nimbus
N nodes, you can travel both left and right, every step of time you have a 1/8 chance to begin moving to any random node (1/N)
you con move between nodes, 1 per step
but you will continue to roll the 1/8 to begin walking to a different node even if already on a walk to a node
teal nimbus
me either tbh
rare dock
like as it is given, if i’m in the walking to 0 state i don’t just have a 0.125 chance to go to 0
the chance is 0 if i’m on anything other than 1
so not sure that ‘simplification’ is appropriate
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twin lotus
For a brute force solution, by choosing states (n,m) with n being the current node and m being the destination node, we have n^2 distinct states that we can use to make a transition matrix. We always have an eigenvalue of 1, so we would be able to find an eigenvector by solving a system of n^2 equations. We can obtain a steady state from here, but I’m not sure if this gives a necessary condition.
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teal nimbus
So for state (n,m) with probability p = 1/8:
We can transition to any state (n,m) where m ∈ [0,N-1] with probability p/N
With probability 1-p = 7/8:
If n < m: Move to state (n+1,m)
If n > m: Move to state (n-1,m)
If n = m: Stay in state (n,m)
and then we're making N^2 transformation matrices of N^2*N^2 size?
twin lotus
For the 7/8 case, that seems correct. For the 1/8 case, it depends on what happens when a new destination is chosen. If choosing a new destination uses up the step, what you wrote looks good. If there is movement in the step as well, you would have to have transition to (n+1,m) or (n-1,m) as well.
I believe we only need 1 transformation matrix of N^2 * N^2 size for this. For any state (n,m) we can determine the probability of transitioning to any state (n’,m’). Then we just need to put them in the N^2 * N^2 matrix.
teal nimbus
I believe the decision to move is independant of any movement so the latter
does that mean that in a 3d space this would be a matrix of N^3 * N^3?
twin lotus
Do we change destination before moving or after?
If for instance it is before, a new destination m’ ∈ {0, …, N-1} is chosen with probability p/N and we will move to (n+1, m’), (n, m’) or (n-1, m’) depending on whether n>m’, n=m’ or n<m’. We also have the 7/8 probability of retaining the original destination m, which is just the same as choosing m as a new destination.
By 3d space, do we mean a random walk in a 3d lattice?
It’s N^2 because we have N^2 states right now. If we had 3 nodes {0,1,2}, we will have 9 states.
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oak turret
I’m trying a something out where I’m trying to find a series for arcoshx in terms of x by integrating its derivative by parts a recognising repeating pattern in the uv section however I plugged in 2 into the 3 terms I have so far but it was 2.078 which is larger than the actual value of 1.3159 which means something has gone wrong because the summation can only get bigger as it’s addition and I can’t figure out what I did wrong
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I’m trying a something out where I’m trying to find a series for arcoshx in term...
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oak turret
<@&286206848099549185>
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spare fractal
I’m trying a something out where I’m trying to find a series for arcoshx in term...
I didn't read everything but are you sure that function is the arcosh?
I suppose you're talking about the inverse of hyperbolic cosine
oh nvm im dumb thats right
one sec ill take a better look
spare fractal
The reason is your series does not converge
for the most part it shoots into infinity
you can see this through the ratio test
let's generalize the terms you found:
$$
a_n = \frac{x^{2n - 1}}{2n - 1} \cdot (x^2 - 1)^{- \frac{2n - 1} 2}
$$
glossy valveBOT
Sires
spare fractal
so doing the ratio test,
$$
\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n}\right| = \lim_{n \to \infty} \frac{ \frac{x^{2n + 1}}{2n + 1} \cdot (x^2 - 1)^{- \frac{2n + 1} 2} } { \frac{x^{2n - 1}}{2n - 1} \cdot (x^2 - 1)^{- \frac{2n - 1} 2} }
$$
glossy valveBOT
Sires
spare fractal
so cutting a bunch of stuff there we get
$$
\lim_{n \to \infty} \frac{x^2}{x^2 - 1} \cdot \frac{2n - 1}{2n + 1}
$$
glossy valveBOT
Sires
spare fractal
keep in mind the limit is for variable n, so that fraction asymptotically goes to 1 and we're left with $\frac{x^2}{|x^2 - 1|}$
for the ratio test we want to know when that limit is less than 1 for it to converge
oh mb I forgot the absolute value
glossy valveBOT
Sires
spare fractal
so we have two cases, $x^2 - 1$ is positive or negative, if x is in (-1, 1) it is negative then we have:
$$ \frac{x^2}{1-x^2} < 1 \Rightarrow x^2 < \frac 1 2 \Rightarrow |x| < - \frac 1 \sqrt{2} $$
So x converges in (-1/sqrt(2), 1/sqrt(2))
For the positive case you can readily see the fraction gives more than 1 so it does not converge
glossy valveBOT
Sires
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spare fractal
so yeah that's that, the series you found only works in that small interval
oh lmao arccosh domain is x greater than 1 so yeah the series just doesn't work at all
spare fractal
@oak turret just pinging in case you forgot
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neat temple
I would like to know what the red answer was since I got it wrong.
This is part of my differential equations class.
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I would like to know what the red answer was since I got it wrong.
This is part...
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supple canopy
How do I view virtual keyboard in GeoGebra Classic 6?
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How do I view virtual keyboard in GeoGebra Classic 6?
supple canopy
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candid schooner
?
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grim skiff
How do I view virtual keyboard in GeoGebra Classic 6?
Google it, we're not google
candid schooner
Couldn't find any answer
Lemme see if I can help
supple canopy
Lemme see if I can help
Ty
I cant find anything in settings to turn it on
idk whats wrong, i had it earlier
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quasi forge
There should be a button in like the right corner
supple canopy
There should be a button in like the right corner
Yeah, i don't got it tho
weird that keyboard showed up on another GeoGebra project I had
idk I will close, maybe it got fixed
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fast peak
A+2I = A-(-2)I
storm flint
@topaz valley I have another question
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storm flint
if you have a good method for solving it
hidden harbor
post the question?
storm flint
is there an easier method other than computing the hessian?
storm flint
like this?
for example
or should there also be a hyperplane seperating b and the cone on the right
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bitter tree
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graceful glacier
\
slate violet
can i solve this with product rule?
yes
graceful glacier
slate violet
you could use logarithmic differentiation as well
graceful glacier
what is logarithmic differentiation?
slate violet
that's what they did
slate violet
what is logarithmic differentiation?
take the logarithm of both sides
then apply the chain rule
$\frac{d}{dt} (\ln f(t)) = \frac{1}{f(t)} \cdot f'(t)$
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south's secret twin brother
slate violet
for the right hand side you should of course simplify the logarithm
then differentiate
slate violet
there's an e^ in the function q(t) so it's more convenient
it's just a choice they made, maybe to show you another way you can use differentiation
also since you need q'(1) then logarithmic differentiation allows you to find q(1) directly, and multiply by something else to find q'(1)
(q(1) is the denominator of the LHS after subbing in t = 1)
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graceful glacier
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✅
slate violet
do you get $(5e^{-t/5} \cos 7t) (1/5 + 7t / \cos 7t)$?
glossy valveBOT
south's secret twin brother
slate violet
oh wait you are missing a minus sign
graceful glacier
ye
clever sluice
**south's secret twin brother**
this is practically unreadable south
slate violet
oh yeah and you didn't differentiate ln(cos 7t) correctly
clever sluice
true
graceful glacier
ohh okay i kinda get it
but this feels like a hassle compared to the product rule
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torn jolt
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torn jolt
I did it.
torn jolt
cold?
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white crown
I am a little confused with how can I know the dimension of a function's graph? For example <x,y,T(x,y)> from R^2->R
When T is a linear transformation.
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lavish sequoia
what does 2,7kg apples cost if 0,84 kg costs 201 kr?
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what does 2,7kg apples cost if 0,84 kg costs 201 kr?
vocal sonnet
Have you tried writing it out as an equation?
grave elm
,calc 2.7/0.84 * 201
glossy valveBOT
Result:
646.07142857143
seems correct
grave elm
hmm
grave elm
0,84 kg - 201 kr
2,7 kg - x
I think that this method is probably the easiest one
,calc 2.7 / 0.84
glossy valveBOT
Result:
3.2142857142857
I did
2.7a = ?
0.84a = 201
so a = 201 / .84
2.7(201/.84) = 646
grave elm
,calc 201 * 3.21
glossy valveBOT
Result:
645.21
yeah that works just rounded a lot (if you use all the decimals its the same for me)
grave elm
it's always gonna come down to something like
2.7 / .84 * 201
lavish sequoia
,calc 2.7 / .84 * 201
glossy valveBOT
Result:
646.07142857143
but that isnt a way to do that though
its just like if wanting to divide 158 by 2 using the writing division method vs. calculator
grave elm
you could calculate it by hand if you wanted to
im just too lazy to do that smh
but it's exactly what you'd get by the proportion thing
0,84 kg - 201 kr
2,7 kg - x
x/201 = 2.7 / 0.84
x = 2.7 / 0.84 * 201
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pure flare
Can someone tell if my answers are correct?
a) rank=3, nullity=2
b) {[-2 [-3
0 1
2 1
2 0
1], 0]}
c) {(1,0,3,0,-4),(0,1,-1,0,2),(0,0,0,1,-2)}
d) {[1 [2 [0
2 5 1
3 7 2
4], 9], -1]}
e) No they are linearly dependent
f) a1 a2 a4
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a) rank=3, nullity=2
b) {[-2 [-3
...
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pure flare
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vestal verge
Higher Order Existence and Uniqueness Theorem Confusion:
Can someone please explain the idea behind the uniqueness of a higher order differential equation? I dont understand how the theorem shows such is the case.
umbral dome
can you post the theorem?
vestal verge
can you post the theorem?
umbral dome
,rccw
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umbral dome
well the theorem isn't really trying to prove that the solutions to initial value problems exist and are unique, it just asserts that they do (because the proof is well above the level of the course)
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vestal verge
well the theorem isn't really trying to prove that the solutions to initial valu...
Can you give me a visual intuition though?
I want to be able to think of it intuitively
umbral dome
i mean "this differential equation has a solution which is unique" is pretty much something you would want to be true always
you can prove it directly for the first order case, then it's pretty much analogous for higher order cases
vestal verge
Yeah but not visually
You can see two solutions pass through a point for a first order slope field
How am I supposed to see that for higher order?
Also why is it unique if y=0 is a thing
umbral dome
i'm not sure that trying to get a visual picture is always helpful for these sorts of things
umbral dome
Also why is it unique if y=0 is a thing
every initial value problem for a given starting input has a unique solution. so y = 0 solves one, and only one, IVP (if it's homogeneous)
fast peak
higher order differential equations are basically first order differential equations in more than one dimension. and then you can still imagine this slope field visual
vestal verge
but I can’t even imagine how I could deal with asserting this won’t happen in higher order
vestal verge
higher order differential equations are basically first order differential equat...
Could you give an example for 2nd order?
I don’t quite understand what it would like look
umbral dome
i think it would be good to do an algebraic proof for the first order version. then you can imagine the situation as being analogous in higher orders
vestal verge
Don’t know where to begin with that
fast peak
see eg the examples here https://math.stackexchange.com/questions/70993/how-to-express-a-2nd-order-ode-as-1st-order-odes
fast peak
but I can’t even imagine how I could deal with asserting this won’t happen in hi...
how should that happen? the derivatives of all of those graphs at a are so different. your initial condition prevents that
umbral dome
using the integrating factor method you can write out an explicit solution to any arbitrary first order ode [in terms of integrals, etc]. then you can verify that the coefficients being continuous is enough for the solution you provide to exist and be unique
vestal verge
how should that happen? the derivatives of all of those graphs at a are so diffe...
What about this
fast peak
the second derivative at a will be different for those
vestal verge
That’s not relevant for second order though
vestal verge
But it’s not an initial condition
no where in the theorem does it assert the second derivatives must match
umbral dome
if you plug in your initial values of function & derivative into the ODE, then you can solve for the initial value of the second derivative (in the linear case it's fairly straightforward)
vestal verge
I don’t get it
So what if their second derivatives are different
The point is I have 3 different functions going through the same point with the same slope at that point showing the theorem wrong in my brain
umbral dome
the point being that it's impossible for two solutions of the same second-order linear IVP to have different initial values of the second derivative, provided that the IVP is "nice enough" as laid out in the theorem
fast peak
they still need to satisfy the ode
which gives a relationship between the second derivative and the rest
vestal verge
Oh
fast peak
if two of those are the same then the third one also has to be
vestal verge
how far away from the point does that hold
fast peak
well bla bla continuity bla bla close enough
vestal verge
hmmm
So if I find a second derivative, first derivative, and initial point where two functions pass through and posses I will have disproved the theorem for a second order right
fast peak
good luck trying to disprove a theorem
umbral dome
that would imply that the function you are talking about violated one of the assumptions of the theorem
in which case it would be worthwhile figuring out which one, exactly
vestal verge
Mhmm
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spring marsh
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hidden harbor
it's a sequence. See that the rods that you add are 6*n, where n is the current layer.
On the first layer, you have no rods (or 1, if you count the middle one, but the problem seems to be not doing that). Then you added 6 rods. Then you added 12 rods. Then you added 18 rods, and so on.
Does this help?
spring marsh
Ok
hidden harbor
the first red ring would be layer 1, and has 6*1 = 6 rods
the blue ring would be layer 2, and has 6*2 = 12 rods
the 2nd red ring would be layer 3, and has 6*3 = 18 rods
The total rods is the sum of all the layers, so sum from 1 to n of 6n
which is the same as 6 times the sum of the first n integers
does this help enough?
spring marsh
Yes
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hidden harbor
that is not correct
the sum of the first n integers is n(1+n)/2
you got 6 times that, and it has to be as close to, but under, 2000 as possible
solve for n
spring marsh
Can you show me how
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river patrol
This is probably such a simple question and im sorry, but what does the dot in the middle of the () stand for? I never had this in highschool, and now that im at uni this confuses me 😭 its for homework
fast peak
its a placeholder indicating that this is where you put your input
its useful when you dont want to give that input a name like x or t
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short crest
i'm having trouble understanding volume of revolution using integral...
short crest
why is it integral(y^2) not, (integral y)^2
it uses the formula of a circle right?
πr^2
isn't the integral of y = radius here?
rare pine
integral is a sum, and we're summing slices of the rotated area where y =f(x)
short crest
but why is the square inside the integral rather than.. outside
rare pine
short crest
Ah wait
I get it
Thanks
So you find the disk first
Then you integrate
Aaa
Thanks alot
❤️
rare pine
yw👍
short crest
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bitter tree
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bitter tree
can u help me with part a
slate violet
you have tan(alpha) = b/a
slate violet
Help me with my secant tangent theorem etc
I'm already fricking helping you
modest helm
that's hard
slate violet
you have tan(alpha) = b/a
if you want to express it as R cos(t - alpha) ofc
modest helm
I don't understand any
bitter tree
you have tan(alpha) = b/a
wut
modest helm
the 1 B
bitter tree
a = 28 and b = 45
i got a = 53
modest helm
HELP
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slate violet
are you a bot
communicate with the people you are helping
no that's a server command
modest helm
is this human I'm speaking to
modest helm
if we're all bots then don't talk to us
is there any mathematician
slate violet
how did u get a and b
it's in the form a cos x + b sin x which is the standard form
a = 28, b = 45, x = pi t/6
slate violet
is there any mathematician
pls talk to me in #geometry-and-trigonometry
modest helm
I didn't know I was tagging u sry
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daring spoke
what happend here? 0 divided by e to power of x?
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what happend here? 0 divided by e to power of x?
fast peak
a product is zero if at least one of the factors is zero
e^x is never zero, so 1+x has to be zero
daring spoke
oh. now i remember. thank you
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broken wind
hi
wraith garnet
decreasing means negative slope
you need to identify where the function is going downwards to the right
broken wind
yeah thats why I wrote 0
bc its decreasing at x = -1.5
and then the next thing is 0 no?
grim skiff
yeah thats why I wrote 0
It decreases more
It's the interval where the slope is negative
broken wind
hm
grim skiff
Not only just above the x axis, but below too
grim skiff
Not quite
wraith garnet
this might help
broken wind
OHHHH I GET IT
broken wind
this might help
helped a lot
if you guys are free can you guide me through inflection point in these type of graphs
ok so for this one now right
late kite
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broken wind
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steel patio
can someone please explain to me why the exponents in the denominator switch from positive to negative and vice versa
umbral dome
we have that $\frac{a^b}{a^c} = a^{b-c}$
glossy valveBOT
cloud
frigid garden
yes, that law says that we can express the number $\frac{1}{a^{c}} ; \text{as} ; a^{-c}$
glossy valveBOT
Jhohn
frigid garden
that is a properties of exponents and fractions! when you "switch" the position of "top" to "bottom" you also switch the sign of the exponent
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umbral dome
,, \frac{-9\pm\sqrt{121}}{2}
glossy valveBOT
cloud
umbral dome
a good start would be to factor the part inside square root
where did this come from?
ignoring the square root, can you just factor 121?
remote zealot
no one knows
umbral dome
ok so if we have 121 = 11 * 11, what does that tell us about sqrt(121)?
hearty ivy
a square root is asking a question. The square root of B is asking what two numbers multiplied together give you B
so if 11 times 11 is 121, the square root of 121 is what?
yes 11
so
$\frac{-9\pm\sqrt{121}}{2}=\frac{-9\pm 11}{2}$
glossy valveBOT
Nacho Boi
hearty ivy
how can we further simplify this?
where did what go
remember we said $\sqrt{121} = 11$
glossy valveBOT
Nacho Boi
hearty ivy
well this would just be a constant number so im not sure why u want to graph it
the quadratic formula finds the 0s (aka x-intercepts aka roots) of a parabola
you used the quadratic formula, right?
yes
$\frac{-b \pm \sqrt{b^2-4ac}}{2a}$
glossy valveBOT
Nacho Boi
hearty ivy
yes
so what does that tell you?
remember it solves for x whenever you have an equation in the form $a x^2 + b x + c = 0$
glossy valveBOT
Nacho Boi
hearty ivy
so it’s finding the x value when y is 0
and an x intercept is when y=0
complete the square?
that’s not properly factored
fouil that out and you’ll see it’s wrong
foil*
no
u got this
foil is this:
$(a + b)(c + d) = ac + ad + bc + bc$
glossy valveBOT
Nacho Boi
hearty ivy
the acronym is i think
first
inner
outer
last
u multiply the first terms, the two middle terms, the two outer terms, and the last two terms
yes
you’ve gotta study a lot
just do it
no you’re not
take a break for 15 mins
you’ve got to force yourself to
get it into the form $y=(x-h)^2 + k$
glossy valveBOT
Nacho Boi
glossy valveBOT
Nacho Boi
hearty ivy
to get it into the form $y=(x-h)^2+k)$, we need to have a $64$ since that is $(-8)^2$. so note that
$x^2-8x+23 = x^2 -8x + 64 - 64 + 23$ since $64-64 = 0$, we’re not actually changing at all
glossy valveBOT
Nacho Boi
hearty ivy
because then you foil out $(x-h)^2$ you get an $(-h)^2$ term
glossy valveBOT
Nacho Boi
hearty ivy
then we can factor:
\
$x^2-8x+64-64+23 = (x-8)^2 -64 + 23 = (x-8)^2 -41$, with vertex (8, -41)
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Nacho Boi
warm canopy
factorize: a^3 - 9b^3 + (a+b)^3
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factorize: a^3 - 9b^3 + (a+b)^3
limpid moat
factorize: a^3 - 9b^3 + (a+b)^3
you are a prime, aren't you?
warm canopy
kinda confused, is the answer supposed to be -> (a-2)(a^2 + 2a + 4)(a+b)^3 ??
warm canopy
you are a prime, aren't you?
didn understand what you meant
warm canopy
yes
limpid moat
cool
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gray heath
They asked me to demonstrate if the function is bijective or not and to draw its reciprocal function
Okay all that is easy, but the last question I have no idea how to solve it
Why is that equality is true and what can I deduce from it graphically?
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gray heath
Vous ne peut pas la calculer
Je sais que c'est cos-¹(πx) je pense
Mais la methode utilisé dans la correction est avec la derivé et j'ai rien compris de ce qu'il essaye de faire
Le but de cet exercice est de démontrer cette propriété sans la réciproque
gray heath
Ah d'accord
Merci en tout cas
For anyone wondering what the correction said
This is it
And I have no idea why he chose this approach
If you have another way of solving this or any other ideas I'm all ears
<@&286206848099549185>
Thank you everyone for any help or ideas/explanation
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cedar flint
@full forum
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<@805217272087838750>
cedar flint
Can someone help me with this question:
<@&286206848099549185>
<@&286206848099549185>
cedar flint
nvm i solved it already
cedar flint
aops
cedar flint
it's just extra things to make doing math more "interesting" and "fun" in a way
they don't really affect anything
knotty grail
no just that ive never seen it
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raw pendant
Can i be ayanokoji kyotaka
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rare dock
spiral vigil
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peak sun
primal hull
xD
glad surge
Lol
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lime crescent
I need help setting up this equation
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I need help setting up this equation
lime crescent
I can't add exponents, so is this the way it wants to be set up?
<@&286206848099549185>
azure eagle
haughty saddle
@lime crescent https://youtu.be/HL2fDIOMLJ0?si=1yed8mkuAsy1w-mk
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torn bane
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torn bane
Im a bit confused on c), why multiply P(X=1) by P(Y=0)? and what formula is Y=0 using? also binomial?
ivory cairn
Yes, technically Y is using the binomial, but the only values are Y=0 or Y=1, so there isn't much to do there.
You multiply by Y because that event is (X=1 and Y=0), and those events are independent.
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ivory cairn
yes
devout valley
One trial for Y in both cases, and that single one is either a second or not
torn bane
thank you!
bitter smelt
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gritty flax
You're welcome
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native barn
how many solutions does the equation have?
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how many solutions does the equation have?
rare dock
what values can x,y,z take
native barn
integer
rare dock
hm. so we want the number of ways to write 100 as a sum of 3 squares basically
native barn
yes
rare dock
do you have to do this by hand?
native barn
yes
rare dock
native barn
the solution is 30 but i only got 9
gleaming prairie
rare dock
hi lily ily
gleaming prairie
rare dock
let’s ask @granite torrent he knows everything
we could take cases on z
we have |z| <= 10
and 100 - z^2 has no two square representations for many z’s
computing number of two square representations isn’t bad
granite torrent
how many solutions does the equation have?
a^2 + b^2 + c^2 = 100
a, b, c must be some variant of (0, 0, 10) (6, 8, 0) (does not take too long to verify if you already had the time to figure out the answer's at least 9)
permuting (0, 0, 10), (6, 8, 0) and taking ± due to the squares gives you 30 which aligns with the answer you have
rare dock
native barn
permuting (0, 0, 10), (6, 8, 0) and taking ± due to the squares gives you 30 whi...
i think i didnt take negative values
i did take negative values
for a^2 to be 6^2 there is 2 values and 2 for b^2 to be 8^2 and 1 for c so I get 4
for a^2 to be 8^2 there is 2 values and 2 for b^2 to be 6^2 and 1 for c so I get 4
only 1 solution for c^2 and its 10^2 so I get 1
i still dont know
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native barn
ha its me again.
ill be back in 30mins though
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ha its me again.
ill be back in 30mins though
@native barn Has your question been resolved?
drifting flame
You could find the value of AY. You should also find the value of AX/XY. You know YZ//BC <=> AY/YB=AZ/ZB=AX/XY (the second equality is due to XZ//YC).
This should allow you to find YB
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fickle adder
right now i am thinking along the lines of "all of the ways to pay the toll 10 cents cheaper, plus a dime, PLUS all of the ways to pay the toll 5 cents cheaper, plus a nickel"
and i think this is on the right track, except that "plus a dime" and "plus a nickel" are super ambiguous here and i dont know how to fix that. in theory, if you laid out every combination of ways to pay a toll of n-5 cents, you could place the nickel in any position before or after any of the coins in that solution, *for every solution for n-5 * .
it's hard enough trying to think of a way to extract whatever the "number of positions" would mean for even one solution, but all of them? i have no clue.
fickle adder
@nimble ridge @wild sleet @rare dock sorry for ping, you three were very helpful before, am i on the right track here ?
rare dock
this problem radiates generating function
fickle adder
we had a unit on those and i did terribly i still dont understand them 😭
green merlin
this looks like gen func
fickle adder
oh god
glad surge
The function can't be defined for n cents
green merlin
but you probably dont need to use gen func
glad surge
It has to be some multiple of 5
fickle adder
it can, its just 0 for non multiples of 5 no?
green merlin
is 0
wild sleet
you should just do that instead
glad surge
it can, its just 0 for non multiples of 5 no?
I guess that make more sense
rare dock
anyway yea you’re on the right track
uneven hornet
we had a unit on those and i did terribly i still dont understand them 😭
if I may ask, what class is this?
wild sleet
i'm not sure that helps, you shouldn't insert coins inbetween
fickle adder
if I may ask, what class is this?
discrete math
uneven hornet
Interesting
fickle adder
well, specifically "discrete structures for computing"
rare dock
this problem is a lot easier than the bit string one honestly
fickle adder
a comp sci course
uneven hornet
Is this really a gen func question? I'm really bad at these sorts of things but can't you just consider the last coin
fickle adder
Is this really a gen func question? I'm really bad at these sorts of things but ...
the order matters
rare dock
you can have a nickel or a dime last
uneven hornet
Well yeah but if you have a nickel last
then the problem reduces to n - 5
right...?
so I'm assuming that's where you would build your recurrence relation
fickle adder
imagine you have 1 specific way to pay 25 being dime nickel dime
then to pay thirty, you can add a nickel to that with any of:
nickel dime nickel dime
dime nickel nickel dime
dime nickel dime nickel
thats +3 ways no?
for just that one solution
now consider all of the ways to pay 25..
wild sleet
the order matters
the order doesn't matter between the coins of the same denomination, you can't even account for this
fickle adder
yes but its not "all dimes" or "all nickels"
its both
thats why it says "order matters"
fickle adder
imagine you have 1 specific way to pay 25 being dime nickel dime
then to pay thi...
basically this is whats confusing me
rare dock
what’s the issue
wild sleet
you get massive overcounting this way
fickle adder
🤔
fickle adder
how many payments are there that end with a dime? how about a nickel?
whatever the amount of payments is for the amount 10 cents lower? all of those + ending with a dime
uneven hornet
whatever the amount of payments is for the amount 10 cents lower? all of those +...
exactly
uneven hornet
I see
rare dock
uneven hornet
That's a bit hard to explain but
It's like we're forcing it to go to the end to count all the possibilities easily
As long as we still count all possibilities it doesn't matter if we force the dime to go to a specific position
rare dock
a payment either ends with a dime or a nickel
fickle adder
oh wait
uneven hornet
Sorry if that's a bit unclear but it's basically just cleverly splitting up cases
fickle adder
all of the ways where its not at the end would be counted by the a_(n-k)?
uneven hornet
Is it possible to apply generating functions to this problem?
uneven hornet
all of the ways where its not at the end would be counted by the a_(n-k)?
exactly
so we don't really have to worry about what comes before
wild sleet
dime nickel dime + nickel makes one way
dime dime nickel with nickel inserted second makes the same way
uneven hornet
dime nickel dime + nickel makes one way
dime dime nickel with nickel inserted se...
are you talkign about part a?
wild sleet
but if you insert last you will not miss out on any sequence and you will not count anything twice
fickle adder
i see
uneven hornet
but if you insert last you will not miss out on any sequence and you will not co...
Oh I see what you mean now, you are justifying why we are focusing on the last coin?
fickle adder
so is it really just as simple as a_n = a_(n-5) + a_(n-10)?
uneven hornet
so is it really just as simple as `a_n = a_(n-5) + a_(n-10)`?
I would think so, yes
wild sleet
it is
fickle adder
oh
good news and bad news, good news is that problem is easy, bad news the next problem i just looked at might be the hardest one ive seen all semester 💀
this is due in two hours i legit might have to skip it
fickle adder
fickle adder
yeah im uh.. i'm gonna come back to that one, maybe.
rare dock
dynamic programming is trivial
uneven hornet
dynamic programming is trivial
we're not all as smart as you ._.
part (a) and (b) don't seem too bad though and I would assume (c) is just impelmentation
fickle adder
i dont have to do (a), just b-e
uneven hornet
Hmmm
For part (b), I think it's basically just whether you want to keep adding onto your current sum or start a new sum, if that makes sense
I'm not too sure though I suck at coding
fickle adder
i mean "programming" isnt really a thing you can suck at, if you can do the earlier problem you can program. its just problem solving
serene yarrow
dynamic programming is trivial
bold claim
rare dock
b is basically the answer to (c). you just go through the sequence and update the max with each element
serene yarrow
its linear time for e
you do 1 add and 1 max compare at each time step
thats O(1)
uneven hornet
and you need to go through the entire list once, so O(n)?
serene yarrow
ye so its linear
fickle adder
thx
wild totem
If a person hold a poll with 10 choice in which a person can only voted once. If there's 200 voters, what is the chance to that the 2 topmost candidstes will tie?
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If a person hold a poll with 10 choice in which a person can only voted once. If...
@wild totem Has your question been resolved?
tepid slate
,tex \text{Let X} \subset R^m;
\text{ K} \subset R^p \text{ compact set;
f:XxK ->R continuos function.}
\text{Let g:X->R} g(x)=max{a \in K} f(x,a).
\text{Show that } \mid g(x)-g(x{0}) \mid \leq max_{a \in K} \mid f(x,a)-f(x_0,a) \mid
tepid slate
,tex \text{Let X} \subset R^m;
\text{ K} \subset R^p \text{ compact set;
f:XxK ->R continuos function.}
\text{Let g:X->R} g(x)=max_{a \in K} f(x,a).
\text{Show that } \mid g(x)-g(x{0}) \mid \leq max_{a \in K} \mid f(x,a)-f(x_0,a) \mid
glossy valveBOT
spygenio_
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)
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tepid slate
nio
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@tepid slate Has your question been resolved?
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muted lodge
probably would be easier to do f(g(x)), yeah?
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probably would be easier to do f(g(x)), yeah?
glad surge
Hi @muted lodge
grave elm
the definition pretty much is f(g(x)) = x
glad surge
Then they are inverse of each other
muted lodge
Hmm
Making sure i have this down right
How does 9/5 and 5x-160/9 multiply into each other?
glad surge
Yup
austere cove
You can do either f(g(x)) or g(f(x)) and from what I'm seeing it's probably easier to do f(g(x)) yeah
if that was what your question was
muted lodge
Hmm
Making sure i have this down right
How does 9/5 and 5x-160/9 multiply into ...
I wonder if the easiest way would be to multiply by denominator lcm, which is 45
muted lodge
if that was what your question was
Yes
austere cove
you just multiply the 9/5 by each term in the expression, you separate the fraction of g(x)
glad surge
See for9/5 * 5x/ 9 it's just x
austere cove
!occupied
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glad surge
Yup
austere cove
\begin{align*}
f(g(x)) &= \frac{9}{5} \cdot g(x) + 32 \
&= \frac{9}{5} \cdot \frac{5x - 160}{9} + 32 \
&= \frac{9}{5} \qty(\frac{5x}{9} - \frac{160}{9}) + 32 \
&= \frac{9}{5} \cdot \frac{5x}{9} - \frac{9}{5} \cdot \frac{160}{9} + 32
\end{align*}
and so on
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OmnipotentEntity
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woeful anvil
What did I do wrong for problem 69? Sorry bad handwrite
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What did I do wrong for problem 69? Sorry bad handwrite
leaden ermine
What did I do wrong for problem 69? Sorry bad handwrite
dividing by 1/3 is the same as multiplying by 3
timid iris
BAD HANDWRITING?
leaden ermine
So instead of 1/6 it's 3/2
woeful anvil
Oh
timid iris
I wish I could write like that 💀
woeful anvil
Ok skull emoji
leaden ermine
Ok skull emoji
thank
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woeful anvil
.reopen
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✅
leaden ermine
I make a^2 -x^2 u
thank
leaden ermine
du/(2a-2x) = dx?
a is constant 🙄
woeful anvil
a is constant 🙄
Oh so just du/2x
leaden ermine
ya 🙄
woeful anvil
How do I solve 70(?) <:blood:1159168459972808885>
Is this correct am idk for 70 sorry bad hand write
leaden ermine
Is this correct am idk for 70 sorry bad hand write
where did the 1/2 go
woeful anvil
god questiond
i added
thanksafnaksfnkasnfkanfkansfknasf
thiaknkanf
kan
ngnakfnakfnas
fnaks
f
c.oseo
c.oose
.close
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woeful anvil
timid iris
dundundun?
leaden ermine
😂 outro type shit
gritty rose
do you work in MKBHD's studio
