Channel closed
Closed by @echo wharf
Use .reopen if this was a mistake.
#help-10
idle spade
obtuse pebbleBOT
idle spade
hello, i'm not sure how to do d
obtuse pebbleBOT
@idle spade Has your question been resolved?
idle spade
<@&286206848099549185>
timid silo
hello, i'm not sure how to do d
factorize the x^3 from the numerator
then evaluate the limit
$\frac{1 - 2x^3 + ... +3^4 x^8 -1}{x^3}$
warm shaleBOT
hannibal
timid silo
$\frac{- 2x^3 + ... +3^4 x^8}{x^3}$
warm shaleBOT
hannibal
idle spade
$\frac{- 2x^3 + ... +3^4 x^8}{x^3}$
so the limit of that?
or simplify then the limit
even then, wouldn't i just get infinity?
timid silo
even then, wouldn't i just get infinity?
no, factorize the x^3 first
idle spade
ohhhhh
timid silo
it's a common factor in that sequence
idle spade
obtuse pebbleBOT
obtuse pebbleBOT
@idle spade Has your question been resolved?
idle spade
<@&286206848099549185>
idle spade
.close
true dust
Can system of two linear equations in three variables can have an unique solution
obtuse pebbleBOT
Can system of two linear equations in three variables can have an unique solutio...
true dust
?
stoic salmon
I dont think so
lethal cloud
no
stoic salmon
is it possible for 2 planes to intersect in 3d space in just one point? it'll be either a line(infinite solutions) or not intersecting (no solutions)
true dust
Ohk
obtuse pebbleBOT
Available help channel!
silent cove
Is E[X-Y] = E[X] - E[Y] in statistics?
obtuse pebbleBOT
Is E[X-Y] = E[X] - E[Y] in statistics?
gilded needle
Is E[X-Y] = E[X] - E[Y] in statistics?
this is true as long as E[X] and E[Y] exist
by linearity of the integral that defines E[]
silent cove
Ah so this would be true in most cases?
gilded needle
yea
silent cove
How can
So this is E[X] but mathematically how can subtracting this from E[Y] be true
gilded needle
yes that's the definition for a discrete random variable
probably easiest to show if you start with the joint distribution, say that's called p(x,y) by abuse of notation
then:
$$\begin{aligned}
E[X-Y] &= \sum_{i,j} (x_i - y_j) p(x_i, y_j) \
&= \sum_{i,j} x_i p(x_i, y_j) - \sum_{i,j} y_j p(x_i, y_j) \
&= \sum_i x_i \sum_j p(x_i, y_j) - \sum_j y_j \sum_i p(x_i, y_j) \
&= \sum_i x_i p(x_i) - \sum_j y_j p(y_j) \
&= E[X] - E[Y]\end{aligned}$$
warm shaleBOT
Bungo
obtuse pebbleBOT
@silent cove Has your question been resolved?
clever pier
How do I calculate the period of this?
obtuse pebbleBOT
How do I calculate the period of this?
clever pier
$\sin{x}+\cos{\frac{x}{2}}$
warm shaleBOT
Raage
obtuse pebbleBOT
@clever pier Has your question been resolved?
clever pier
<@&286206848099549185>
obtuse pebbleBOT
@clever pier Has your question been resolved?
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
rapid karma
Hi, i'm just wondering what the point of this question is. And furthermore, how to answer the following statement: "The subspace of matrices that commute with the shift S has dimension__". What does commute mean?
obtuse pebbleBOT
Hi, i'm just wondering what the point of this question is. And furthermore, how ...
royal basin
we say two matrices commute if the commutative property holds for them
obtuse pebbleBOT
@rapid karma Has your question been resolved?
rapid karma
Oic. So in this case only a 3 by 3 would commute with S?
rapid karma
Which makes it 3 dimensions?
obtuse pebbleBOT
@rapid karma Has your question been resolved?
obtuse pebbleBOT
@rapid karma Has your question been resolved?
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
elder tulip
,rotate
warm shaleBOT
stoic salmon
looks like you'll need to use sin2x = 2sinxcosx
try it using this
then you'll also need to use double angle identity for cos
obtuse pebbleBOT
@elder tulip Has your question been resolved?
grizzled bramble
hello
grizzled bramble
i need help
i just read proof of weierstrass criteria and im confused
is that true or not, that (series is convergent <=> series is uniform convergent)?
haughty coyote
Uniform convergence is for series of functions
Do you mean absolutely convergent?
grizzled bramble
Uniform convergence is for series of functions
Do you mean absolutely convergent...
nope, uniform convergent
haughty coyote
Yes for functions
grizzled bramble
Uniform convergence is for series of functions
Do you mean absolutely convergent...
u can apply it to partial sums of series
haughty coyote
We talk about uniformity as opposed to nothing for a reason
They're not equivalent
haughty coyote
https://en.wikipedia.org/wiki/Uniform_convergence
The very start of the examples already answers your question
slim leaf
If series of functions is uniformly convergent then obviously the series is convergent so one direction is obvious. (Series of functions converges uniformly if the sequence of partial sums converges uniformly).
However if series of functions is pointwise convergent (This happens when sequence of partial sums converges pointwise) then we don't necessarily have uniform convergence.
If you are talking about Weierstrass' M-test, then yes the series is uniformly convergent if it satisfies the hypotheses in the test
grizzled bramble
If series of functions is uniformly convergent then obviously the series is conv...
why? if series is convergent, then for any eps>0 theres n0, that for every m>n>n0 theres sigma(m, i=n+1) ai < eps
so cauchys condition is fullfiled
slim leaf
This is Cauchy's condition for uniform convergence of series
$\sum_{n=0}^{\infty} x^{n}$ converges pointwise to $\frac{1}{1-x}$ on $(-1,1)$.
grizzled bramble
yeah
grizzled bramble
This is Cauchy's condition for uniform convergence of series
ok but every convergent series fullfill cauchys condition
warm shaleBOT
Sup?
slim leaf
ok but every convergent series fullfill cauchys condition
How are you getting this?
grizzled bramble
How are you getting this?
ok so, suppose we have convergent series Sigma(i=1, inf) Ai
right?
get some eps > 0
then, there must be some natural n0 number
that, for every m>n>n0
we have Sigma(i = n, m) Ai < eps
do u agree?
slim leaf
Can you prove this?
Okay nvm, I see what this is, Cauchy criterion for series
@grizzled bramble Okay now what?
@grizzled bramble Okay so I get what your confusion is. Suppose we have a sequence of functions (f_i) : A ---> R.
If for all e > 0, there exists N s.t |sum (i=m+1 to n) f_i (x) | < e, for all x in A and all n > m > N, then we say the series (sum i=1 to infinity f_i (x)) converges uniformly.
Now what you're doing is using the Cauchy criterion for series of fixed real numbers
So what you're doing now is fixing some x in A, say x = x_0
Which means that f_i (x_0) is now a sequence of real numbers
And if this series f_i (x_0) is convergent, then yes for all e > 0, there exists N s.t |sum (i=m+1 to n) f_i (x_0) | < e, for all n > m > N.
But notice that this only happens for one "x" and that "x" is "x_0".
slim leaf
ok so, suppose we have convergent series Sigma(i=1, inf) Ai
Whatever you said is true but only when A_i is a fixed number, not a function
grizzled bramble
Whatever you said is true but only when A_i is a fixed number, not a function
but does it work when function depends only from iterator, not some external x?
like sigma(n=1, inf) 1/n^2 ?
slim leaf
Yes obviously because then you are just talking about a series of real numbers
and not series of functions (dependent on x, and as x changes, the series changes)
grizzled bramble
ah, so when i would say: "for every series, where An depends only on n value, series is covergent <=> is uniform covergent" thats true then, right?
slim leaf
When you have this series then there is no notion of uniform convergence
Just convergence, absolute convergence, conditional convergence
When we introduce functions then the notion of uniform convergence is brought up
grizzled bramble
but why it makes difference for, idk An(x)? the base of the proof is that lim (n->inf) n rest of series = 0
slim leaf
When talking about normal series (of numbers), then we have that lim n to infinity must be 0
But when talking about series of functions, we don't talk about lim n to infinity must be 0
Because then the limit can be a function
grizzled bramble
Just convergence, absolute convergence, conditional convergence
ah, cause we dont have x, so we dont have to use diffrentations and stuff
grizzled bramble
Because then the limit can be a function
hm, but what if we are chcecking limit for every x from domain of functions
like, for every x the limes is zero
slim leaf
limes?
slim leaf
you mean limit?
grizzled bramble
mhm
slim leaf
Ah okay so you're saying we fix x, now we get a series of numbers and so lim n to infinity (series) = 0?
grizzled bramble
well, nvmd
thats another problem, gonna think bout it later
well rn i get why what i know is true and what i dont know still, so thank you all for help
/close
eh, how to close session?
.close
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
timid silo
This a good approach?
timid silo
.close
tawdry meteor
when calculating the comutator of these operators i usually use the method where you do [A,B] f = AB f - BA f
obtuse pebbleBOT
when calculating the comutator of these operators i usually use the method where...
tawdry meteor
but seeing how the these operators have derivatives to different variables
how do i effectively take the derivatives?
do i just decide that f is a function of x?
do that d/dy (f) = 0?
obtuse pebbleBOT
@tawdry meteor Has your question been resolved?
obtuse pebbleBOT
@tawdry meteor Has your question been resolved?
obtuse pebbleBOT
@tawdry meteor Has your question been resolved?
plush cedar
so using taylor series how do i solve this
obtuse pebbleBOT
so using taylor series how do i solve this
plush cedar
ah
royal basin
x^3/6 - x^3/3 can also be simplified into -x^3/6
so that's the denominator taken care of
now do the numerator
you know you only need terms up to x^3
plush cedar
no i dont , this is the first time using taylor series and I just learn about them today XD
so how can i know that?
is it because the first term is of power 3?
royal basin
it's because the denominator behaves as x^3
plush cedar
does O absorb negitive numbers too?
royal basin
so that's the level of "precision" you want
plush cedar
ah I see
royal basin
yes of course it does
plush cedar
ok
timid silo
just wondering, are u required to solve this using taylor anyways?
because i do see an alternative, potentially less excruciating method
plush cedar
so the numerator would be (x-1)(1 - x^2/2! + O(x^4)) - (1 + x^2/2+...+O(x^4) + 2
royal basin
because i do see an alternative, potentially less excruciating method
don't suggest lhop
royal basin
so the numerator would be (x-1)(1 - x^2/2! + O(x^4)) - (1 + x^2/2+...+O(x^4) + 2
missed a term for e^x
e^x = 1 + x + x^2/2 + x^3/3 + O(x^4)
plush cedar
yea I used the dots on porpuse couldnt bother writing it out XD
in any case now what
plush cedar
just wondering, are u required to solve this using taylor anyways?
no I am not requried and if there is a better method I am all for it, just that it has to be general purpose for these types of questions, training for an exam...
royal basin
@timid silo so what is the substitution you're talking about
timid silo
<@456226577798135808> so what is the substitution you're talking about
u = tan(x)
royal basin
does that help things
plush cedar
so what do I do with this?
timid silo
i believe so but i will verify with what i think the outcome will be
obtuse pebbleBOT
@plush cedar Has your question been resolved?
plush cedar
side question, is O(x^5) + O(x^4) = O(x^5) because it is way bigger?
plush cedar
then what is the result of this?
timid silo
xO(x^4) = O(x^5)
fathom bridge
is -ln(e) the same as ln(e)
obtuse pebbleBOT
is -ln(e) the same as ln(e)
fathom bridge
other than they return a -1 rather than 1
fathom bridge
as in do they undo the same way
minor cypress
wdym undo?
fathom bridge
bro ln(e) undo
minor cypress
ln(e) will undo irrespective of the scalar you multiply it with right
here you're multiplying by -1
minor cypress
yep
tame narwhal
yes
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
rough cove
A rope is tied to a heavy box (m = 95 kg) and pulled along the floor. If the rope makes an angle of 41.2° above the horizontal, and the coefficient of kinetic friction between the box and the floor is μ = 0.15, what applied force is needed in order for the box to maintain a constant speed?
please help
obtuse pebbleBOT
A rope is tied to a heavy box (m = 95 kg) and pulled along the floor. If the rop...
daring rock
obtuse pebbleBOT
@rough cove Has your question been resolved?
timid silo
how it works? like, 2 * (2*1.41) is like 5.., how it went to that, and what if it is right, why isnt 8 below? if simplify this, how come to root(2)
obtuse pebbleBOT
how it works? like, 2 * (2*1.41) is like 5.., how it went to that, and what if i...
slim leaf
$2 = \sqrt{2} \cdot \sqrt{2}$
warm shaleBOT
Sup?
slim leaf
That's it
slim leaf
$\frac{2}{2 \sqrt{2}} = \frac{\sqrt{2} \sqrt{2}}{2 \sqrt{2}} = \frac{\sqrt{2}}{2}$
warm shaleBOT
Sup?
slim leaf
If you want to simplify further
slim leaf
?
tough hawk
obtuse pebbleBOT
tough hawk
For c
Am I going in the right direction???
Not sure if I’m doing it richt?
Right
obtuse pebbleBOT
@tough hawk Has your question been resolved?
tough hawk
<@&286206848099549185>
tough hawk
.close
smoky tusk
Hello, this is part of my chemistry class. I’ve provided the question the Professor gave as well as their answer. And then my work, I got -312kJ and they got -314kJ. Did I plug into calculator wrong or something? I’ll provide a screenshot of how I plugged it into calculator if needed. Help?
obtuse pebbleBOT
Hello, this is part of my chemistry class. I’ve provided the question the Profes...
smoky tusk
I figure it might be a typo? But just need extra clarification because if I did go wrong somewhere I have no idea where
slow mango
i get the same answer as u, i think its a typo
mellow compass
I have a jar containing 30 red marbles, 30 green marbles and 30 blue marbles. I reach in
and pull out five marbles. How many ways are there to do this?
obtuse pebbleBOT
I have a jar containing 30 red marbles, 30 green marbles and 30 blue marbles. I ...
grizzled shore
90! Is much bigger than 43 million
mellow compass
but is the combination correct?
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
regal geyser
hi im doing my mock math ia on statistics, and im modelling temperature values in my city, so ive used r^2 value to check for whether a model is a good fit for simple inear, quadratic, cubic and quartic, but i also want to check sinusoidal but since i cant use r^2 value for it so is there a good way to check for whether its a good model based on a certain data set?
obtuse pebbleBOT
hi im doing my mock math ia on statistics, and im modelling temperature values i...
@regal geyser Has your question been resolved?
regal geyser
<@&286206848099549185>
obtuse pebbleBOT
@regal geyser Has your question been resolved?
runic thunder
obtuse pebbleBOT
runic thunder
Where does the “2” come from?
runic thunder
When taking the integral of e^.5t when t is 10 would it be e^5 * .5
fierce lagoon
Where did your t go
e^(0.5t) requires a u-sub (although that can be mentally done)
Start with identifying your antiderivative
stoic salmon
When taking the integral of e^.5t when t is 10 would it be e^5 * .5
limits are from 0 to 10, you gotta plug in both also you're using the chain rule which isn't used like that in integration
$\int f(ax+b) dx = \frac{1}{a}F(ax+b) + c$
warm shaleBOT
roxyit
runic thunder
Ya so
e^(u)
u’e^u
Ok i see ty
fierce lagoon
Latex hurts sometimes 
runic thunder
I just solved the integral by using the e derivative rule instead of the integral one
fierce lagoon
Well it's just a chain rule, although you'll realize it's not much of a hard one
Considering the inner function is a linear function
stoic salmon
yeah so the antiderivate is F(x)=10*2e^(x/2)
now you gotta evaluate the limits so F(10)-F(0)
runic thunder
I used that to solve for b (average number) but what would i use to solve average growth (rate of change)
Wait is this even correct for b
I did this average integral stuff was i meant to do y’ = y(10)-y(0)
obtuse pebbleBOT
@runic thunder Has your question been resolved?
gusty heron
.close
obtuse pebbleBOT
Available help channel!
timber zephyr
obtuse pebbleBOT
timber zephyr
w1 is correct but I can't get w2
royal basin
wdym by "can't get"?
timber zephyr
the value isn't correct
royal basin
is the output depicted yours or the answer key?
timber zephyr
the output
royal basin
is it YOURS or is it not yours
timber zephyr
is it YOURS or is it not yours
mine
royal basin
,calc 0.408^2 + 0.408^2 + 0.816^2
warm shaleBOT
Result:
0.998784
it looks correct to me. what's the issue?
timber zephyr
might be code related and not math related
yeah but maybe w2 is calculated incorrectly
idk the method to get w2
i don't know gram schmidt process that much
timber zephyr
https://gist.github.com/iizukak/1287876/edad3c337844fac34f7e56ec09f9cb27d4907cc7
I tried didn't work
leaden ginkgo
hmm
frosty spoke
why are your vectors row vectors
timber zephyr
why are your vectors row vectors
are they not supposed to be row vectors?
frosty spoke
most people would be much more familiar with column vectors
timber island
Well i don't think numpy has column vectors
frosty spoke
also not sure what your code is actually doing for the projection
like what is all of this tolist() and [0][0] nonsense
timber zephyr
Also i think i found the issue here
what is it
timber island
what is it
Can u send me the alg? Haven't used it yet sooo idk about it..
$$ v2 - \frac{a}{b} $$
warm shaleBOT
ItzKraken2
timber zephyr
Can u send me the alg? Haven't used it yet sooo idk about it..
wdym
timber island
**ItzKraken2**
R u supposed to do this to calc w2? a and b is the dot product and normalised vector in w2 calc
frosty spoke
the w1 and the w2 are correct
timber zephyr
yea
frosty spoke
my guess the problem is in this mess
timber zephyr
the w1 and the w2 are correct
so is the problem for the projection code related?
timber zephyr
my guess the problem is in this mess
how would this be solved
timber island
W is a Matrix right?
timber island
In this
frosty spoke
can you at least print the value of Pmv?
frosty spoke
why do you have Pwv in the second cell as opposed to Pmv?
the value for Pmv is what I'd expect
timber zephyr
why do you have Pwv in the second cell as opposed to Pmv?
?
it's for the check answer
frosty spoke
this is what sympy gives
frosty spoke
it's for the check answer
yeah why are you putting something else into checkanswer than the thing you computed
why does checkanswer say it's testing [-1, 1, 0]
timber zephyr
LMAO
i got
it
I need help on this part
w1, w2 E W
how do you show that
frosty spoke
well, you need to show that w1 can be written as a linear combination of v1, v2
and same with w2
timber zephyr
how would that be done
frosty spoke
the Gram-Schmidt process gives you those coefficients if you look carefully enough
or if you're lazy, just as the computer to solve for those, though you probably will need to use some least-squares thing and show that the distance is very low
because you're doing things with imprecise arithmetic
timber zephyr
or if you're lazy, just as the computer to solve for those, though you probably ...
so how would that be done
frosty spoke
pretty sure you can look that up
timber zephyr
pretty sure you can look that up
could u help me on this
it's related to the previous one
timber zephyr
do you not know the geometric meaning?
nah
timber zephyr
the Gram-Schmidt process gives you those coefficients if you look carefully enou...
about the coefficients how do u find them?
frosty spoke
for example, if you say that w1 = v1/|v1|, then isn't w1 = 1/|v1| * v1 + 0 * v2?
timber zephyr
for example, if you say that w1 = v1/|v1|, then isn't w1 = 1/|v1| * v1 + 0 * v2?
yea
how do u do this
this what the prof said to do
for that problem
frosty spoke
likewise, if you say that unnormalized w2 = (v2 - (v2 dot w1)w1), then you can find the coefficients for that too
basically you need to prove that w1 = some linear combination of v1, v2 and same with w2
that's assuming W is defined as span {v1, v2} and not something else, but it's up to you to figure that out
spice plank
determine the domain and range of y=2- √(2x+3)
obtuse pebbleBOT
determine the domain and range of y=2- √(2x+3)
prisma hatch
here expression under root must not be negative if we want real solution
warm shaleBOT
Pro_Hecker
prisma hatch
so the domain of the function is
{ x | x ≥ -3/2 }
here √(2x+3) is being subtracted from 2 ; there is no real way in which √(2x+3) will increase value of y
so, when √(2x+3) = 0
we get y = 2;
so the range of the function is
{ y | y ≤ -3/2 }
gray nest
obtuse pebbleBOT
gray nest
How am I wrong?
obtuse pebbleBOT
@gray nest Has your question been resolved?
gray nest
<@&286206848099549185>
timber island
ur application of tan(theta/2) is wrong
gray nest
Why?
gray nest
How can it have 2 diff values?
ashen nebula
what is the answer?
gray nest
I was solving some other questions which involved computations and the answers differed
ashen nebula
oh wait i see
gray nest
what is the answer?
The proof?
ashen nebula
The proof?
the answer you concluded is just a restatement of the answer given at the top
gray nest
Its a negative reciprocal
ashen nebula
3/(sqrt(7)+4), multiply both numerator and denominator by 4-sqrt(7)
the final answer has a rationalised denominator, yours doesn't
gray nest
Yea
ashen nebula
i guess you better rationalise denominators next time
prisma hatch
no rationalising doesnt work(actually it does)
gray nest
Yea got it
ashen nebula
no rationalising doesnt work(actually it does)
it gives the final expression that is desired
gray nest
ashen nebula
yes, that is indeed correct
prisma hatch
my bad i didnt calc properlu
obtuse pebbleBOT
Available help channel!
silver glen
I have an equation like this: x’’ +cx’ + 4225x = 0 and also I know x(0)=x(pi/56)=0. How can I solve this?
obtuse pebbleBOT
I have an equation like this: x’’ +cx’ + 4225x = 0 and also I know x(0)=x(pi/56)...
silver glen
I already know the characteristic polynom. And oh btw, I gotta solve c
The task is to find what c is
And between 0 and pi/56, the answer is positive
I ended up that c is +-130 but its wrong..
<@&286206848099549185> help appreciated
obtuse pebbleBOT
@silver glen Has your question been resolved?
silver glen
The english term for this I found was second order differential equation
obtuse pebbleBOT
@silver glen Has your question been resolved?
strong ermine
Jeez Louise
obtuse pebbleBOT
@fluid oak Has your question been resolved?
crystal solstice
<@&268886789983436800> seems like cheating in exam?
<@&268886789983436800>
Why isn't it pinging 
glossy yew
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
fathom bridge
for a) would this be every 1/3 of a year the radioactive substance decreases by 4%
obtuse pebbleBOT
for a) would this be every 1/3 of a year the radioactive substance decreases by ...
glacial obsidian
for a) would this be every 1/3 of a year the radioactive substance decreases by ...
Not exactly
After 1/3 years Q = 20·(0.96)^(1/9)
It decreases by 4% after more time
fathom bridge
so every 3 years
glacial obsidian
Yes
fathom bridge
okay, that makes sense
glacial obsidian
You can also mention in the answer that there are 20 grams at the beginning
fathom bridge
how would i incorporate that
every 3 years starting from 20 grams of radioactive substance it will decrease by 4%
glacial obsidian
Yes
fathom bridge
.close
lofty whale
obtuse pebbleBOT
slim leaf
@lofty whale I'm thinking H_0 : M_p > M_a and H_A : M_p <= M_a
The null hypothesis is what we want to check, right?
obtuse pebbleBOT
@lofty whale Has your question been resolved?
severe mica
i have a question
obtuse pebbleBOT
i have a question
lost tree
what is it
clear condor
severe mica
can i ask help for anything
clear condor
math
severe mica
can i have help on indices
spring trail
<:JustAsk:371662163489718273>
.
lost tree
yes send ur question
fierce lagoon
...
severe mica
5^5
lost tree
is this trolling
clear condor
uh
severe mica
is this trolling
me
lost tree
well what help do u need with 5^5
severe mica
its for a question and if i dont get it and if i dont know what it means i wont be able to join this groupe
hello
need to go
how do i leve
this chat
lost tree
5^5 means multiply 5 by itself 5 times
lost tree
to close this channel type .close
opaque galleon
How can I show that $0 \le \cos^2(x) \le 1$, I tried something but obviously that' s wrong. So how do I do it properly?
obtuse pebbleBOT
How can I show that $0 \le \cos^2(x) \le 1$, I tried something but obviously tha...
warm shaleBOT
steamhahasteamhaha
alpine raven
cos^2 x is always positive so its obviously greater than 0
cosx <= 1
so cos²x <= 1² = 1
in conclusion
0 <= cos²x <= 1
opaque galleon
but what about $1 \le \cos^2(x)$
warm shaleBOT
steamhahasteamhaha
glacial obsidian
How can I show that $0 \le \cos^2(x) \le 1$, I tried something but obviously tha...
f(x) = x^2 is not an increasing function, so you can't do that
However, f restricted to the positive numbers is an increasing function, so if cos(x) >= 0, then
0 <= cos(x) <= 1 and therefore 0^2 <= cos(x)^2 <= 1^2
On the other hand, f restricted to the negative numbers is a decreasing function, so if cos(x) <= 0, then
0 >= cos(x) >= -1 and therefore (-0)^2 <= cos(x)^2 <= (-1)^2
And in every case you get 0 <= cos(x)^2 <= 1
opaque galleon
,w graph x^2
warm shaleBOT
alpine raven
-1 <= cos(x) <= 1
|cos(x)| <= 1
|cos(x)|² <= |cos²x| = cos²x <= 1
half silo
you can use definition of cos = Base/ hypotenuse,
and just constraint base <= hypotenuse, achieving equality when the triangle ->steiaght line
clear condor
cah
half silo
using the unit circle works too
alpine raven
cos is a decreasing function on (-1,1)
so you can never have cos²(x) greater than 1
opaque galleon
Can I cherry pick and just do:
We know that:
$\cos(x) \le 1$
and
$\cos^2(x) \le 1$
$\cos^2(x)$ will never be negative so $\cos^2(x) \ge 0$
thus we have
$0 \le \cos^2(x) \le 1$
bruh
clear condor
there
warm shaleBOT
steamhahasteamhaha
alpine raven
why do you want to prove that a value squared is greater than 0
I mean its common sense
0 <= cos²(x)
no need to prove it already
opaque galleon
why do you want to prove that a value squared is greater than 0
I mean its commo...
we are kind of required to do it
i don't know why
but we always have to do like proving before doing the tests
half silo
**steamhahasteamhaha**
You might also have to show where the equalities are attained
opaque galleon
You might also have to show where the equalities are attained
you mean like: for all n $\ge 1$?
warm shaleBOT
steamhahasteamhaha
half silo
Yeah you need to show n=1 for some "n"
alpine raven
In two sentences :
cos²(x) is always positive so 0<= cos²(x).
since -1<= cos(x) <= 1 and cos is a decreasing function on [-1, 1], we have cos²(x) <= 1
In conclusion : 0 <= cos²(x) <= 1
opaque galleon
nice
alpine raven
now you have :
$\frac{\cos²(n)}{3^n} \leq \frac{1}{3^n}$
warm shaleBOT
Ｈｅｒｅｌｓ
opaque galleon
In two sentences :
cos²(x) is always positive so 0<= cos²(x).
since -1<= cos(x)...
why do I have to say that it's decreasing
alpine raven
its not really required but it can indicated that cos²(x) < cos(x)
so it will never be greater than 1
the fact that -1 <= cos(x) <= 1 is enough to say that cos²(x) <= 1
opaque galleon
okok thank you
dry lily
Could I have someone check my work? There is an error somewhere making the initial conditions not check out
obtuse pebbleBOT
Could I have someone check my work? There is an error somewhere making the initi...
dry lily
This is solving initial value problem with Laplace transform
putting it into a calculator and using a different method shows that the e^-2t sint shouldnt be there but not sure how I have it
obtuse pebbleBOT
@dry lily Has your question been resolved?
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
timid silo
obtuse pebbleBOT
timid silo
wsh
I tried expanding 2nd, 3rd, 4th term
was completely wrong
for 2nd term I got
8C1 * (1/2)^7 * (-2/3) (x^2)
which should be 8 / 128
then-16 / 384
completely wrong
nevermind I got it
.close
woven creek
need help where i went wrong for directional derivitive
obtuse pebbleBOT
need help where i went wrong for directional derivitive
woven creek
,rotate
woven creek
,rotate
warm shaleBOT
warm shaleBOT
woven creek
obtuse pebbleBOT
@woven creek Has your question been resolved?
woven creek
<@&286206848099549185>
obtuse pebbleBOT
@woven creek Has your question been resolved?
frosty spoke
have you checked all of your derivatives with an online calculator?
obtuse pebbleBOT
@woven creek Has your question been resolved?
woven creek
.close
timid silo
Hi, could someone help me with this integral
obtuse pebbleBOT
Hi, could someone help me with this integral
warm shaleBOT
CWolf
timid silo
Or undefined
gilded needle
that doesn't converge
or you could say it's +infinity if working in the extended reals
timid silo
I just started the definite/improper integral chapter , so not too familiar with the rules yet
if it doesn't converge, does that mean it's undefined or non existant?
For that integral between those bounds
gilded needle
it means that the limit which defines the improper Riemann integral is +infinity, in this case
$$\int_a^{+\infty}e^{-x}\ dx$$ exists and is finite
warm shaleBOT
Bungo
timid silo
And not a finite value
gilded needle
but if you take the limit as $a \to -\infty$, you get $+\infty$
warm shaleBOT
Bungo
gilded needle
yea, it's either "does not exist" or "+infinity" depending on the context
timid silo
For a bit of context, the exercises do have a bit of text saying "integrate if they exist"
gilded needle
yea, "does not exist" is what you should say in this case
haughty geyser
Hi, how do you figure out whether this series converges?
obtuse pebbleBOT
Hi, how do you figure out whether this series converges?
haughty geyser
I was suggested to use comparison test with 2/k^2 but i dont really understand how they got that in the first place
Ratio test for this is inconclusive i believe
obtuse pebbleBOT
@haughty geyser Has your question been resolved?
haughty geyser
<@&286206848099549185>
haughty geyser
im getting infinity but that’s wrong because that series is convergent
according to wolfram alpha
slim leaf
I'm getting 0
warm shaleBOT
Sup?
haughty geyser
yeah that’s what i got in the beginning too but when i simplify it something goes wrong
$\frac{x^x}{(x+1)^x}
slim leaf
After some simplification, you get $\frac{(x+1) x^x}{(x+1) \cdot (x+1)^{x}}$
warm shaleBOT
Sup?
haughty geyser
ok so how do you go from there
slim leaf
np
haughty geyser
.close
kindred lichen
anyone know how to solve this?
obtuse pebbleBOT
anyone know how to solve this?
@kindred lichen Has your question been resolved?
kindred lichen
.close
hard citrus
Can someone help with question 1?
obtuse pebbleBOT
Can someone help with question 1?
hard citrus
is it the gradient of a point
timid silo
oh do you not know derivatives?
timid silo
like, are those questions meant to be related to differentiation
hard citrus
yeah
timid silo
yeah okay
hard citrus
geometrical applications of differentiation
timid silo
then we are good
anyways, you do kinda need to understand what a derivative is
to understand how those questions are meant to be solved
hard citrus
i kinda get it
timid silo
yeah so what would u say is it
hard citrus
the rate of change between x and y
timid silo
err okay, that's not particularly the full explanation
like
[
\f{\Delta y}{\Delta x}
] is also the rate of change between $y$ and $x$, but it is not a derivative
warm shaleBOT
hard citrus
oh
timid silo
yes okay, it is just the gradient of the tangent line
hard citrus
so for this we have to derive the first equation?
timid silo
okay, so for part a) of your question, let's start by figuring out when the gradient is equal to 0
timid silo
so for this we have to derive the first equation?
yes
hard citrus
2x - 2 = 0
hard citrus
x=1
hard citrus
oh do we sub 1 back into the original equatoin
timid silo
yes
hard citrus
thanks 
timid silo
to write the equation of the tangent line, what is the equation of the line
like
just any line
hard citrus
y = x^2 - 2x + 4
hard citrus
y = f(x)
timid silo
no
timid silo
y = mx + b
timid silo
okay so what is the point slope form of the equation of the line
hard citrus
switch the x and y
hard citrus
o
warm shaleBOT
timid silo
so
timid silo
the rate of change
remember when u said this
hard citrus
yes
timid silo
yes so m is the gradient
and we are talking about the gradient of the tangent line
what does the question tell you it is
timid silo
so y2 - y1 = 0
yes
so (x_1,y_1) are the points u got
from ur calculations earlier
what are they
hard citrus
3, 1
timid silo
other way around haha but yes
just plug it in
and thats ur equation of the tangent line
timid silo
yes
hard citrus
2
timid silo
right
so we want our tangent line to have a slope of 2
so for the tangent line, what is the slope again?
timid silo
so the slope is 0?
no
thats for the previous part
the gradient is not zer0 anymore
just remember that the derivative is the slope
hard citrus
y is 4
timid silo
are u sure
hard citrus
alr
timid silo
okay so i want u to try the same idea now, from this
timid silo
yep
hard citrus
y-4 = 2(x-2)
timid silo
yep you got it
obtuse pebbleBOT
timid silo
aye have a great day
obtuse pebbleBOT
Available help channel!
dry dawn
could someone give me a hint as to how i should approach this problem?
obtuse pebbleBOT
could someone give me a hint as to how i should approach this problem?
weary barn
obtuse pebbleBOT
weary barn
having some problems with average velocity and is flat out confused on what to do here
stone spoke
do you know the definition for average velocity?
weary barn
yes
stone spoke
okay can you tell me what it is?
weary barn
basically how an object changes velocity over a period of time
stone spoke
yes, but can you give it in a mathematical equation?
I think if you write out the definition it'll become much clearer
stone spoke
yup, what do we let f(t) equal here?
weary barn
F(3)=-102
stone spoke
yeah that's right you went ahead of me on this step
weary barn
so we would than do F(.01) ... correct?
weary barn
wouldn't it be .01-3?
weary barn
im lost here based on the question if t=3 and its lasting for .01 seconds wouldn't that make t2=3 and t1=.01 ??
stone spoke
You're confused because you aren't interpreting what t1 and t2 represent.y(t) represents the location of the object at any time t. t1 is the time at the beginning of the motion, and t2 is the time that the motion ends. Try finding t1 and t2 again with this
obtuse pebbleBOT
@weary barn Has your question been resolved?
stone spoke
I had to go for a bit...
weary barn
no worries
stone spoke
did you figure it out?
weary barn
so would it be (3,3.01)
stone spoke
yeah
weary barn
ok so than i would go as normal find F(3.01) and do the formula
stone spoke
yup you got it
weary barn
thanks alot i was really overthinking this
weary barn
thanks alot
stone spoke
if you want to check for the other parts you can just plug t2 and t1 into this https://www.desmos.com/calculator/fy325axrof
stone spoke
thanks alot
np good luck!
obtuse pebbleBOT
@weary barn Has your question been resolved?
strange sphinx
HELP I am freaking desperate right now
obtuse pebbleBOT
HELP I am freaking desperate right now
@strange sphinx Has your question been resolved?
chilly narwhal
How would I simplify (x+8)(x-4)? Like turn it back into an equation?
obtuse pebbleBOT
How would I simplify (x+8)(x-4)? Like turn it back into an equation?
high lily
you can't turn that back into an equation
royal basin
you cannot "turn it into an equation"
chilly narwhal
Then how did the stupid bot do it 😭
high lily
do what
leaden ginkgo
what are you trying to do with it
chilly narwhal
It took that factor and like
high lily
can you show the full original problem and context
high lily
(x+8)(x-4) isn't equal to x^2 + 3x - 4
leaden ginkgo
How would I simplify (x+8)(x-4)? Like turn it back into an equation?
this would become x^2 -4x + 8x - 32
if you were to expand it
chilly narwhal
Update it refreshed and I lost the question