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#help-27
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magic ivy
how do i prove this
devout snowBOT
how do i prove this
misty crest
magic ivy
i tried to write it in delta epsilon form and manipulate it somehow but i cant find exactly how
magic ivy
can i show you what working?
magic ivy
i tried squaring the 0<|x|<delta part for the first limit but i have no idea where to go from that
i also tried equating the first limit with lim_(x->0) f(|x|) but i cant figure that out either
misty crest
well let's look at the statement for lim x -> 0+ f(x) = L. it says that for every eps > 0 there exists delta > 0 such that for all x satisfying 0 < x < delta, |f(x) - L| < eps. so all we need to show is that L = lim x-> 0 f(x^2) yes?
magic ivy
yes
magic ivy
epsilon>0 then delta>0 where 0<|x|<delta and |f(x^2)-L|<epsilon
misty crest
right so we should probably try and match this statement with the first statement we had with an appropriate substitution
magic ivy
with x to u^2?
misty crest
with x to u^2?
no i mean like for finding a suitable delta in our new definition
misty crest
from the first one we have $\dots 0 < x < \delta \implies |f(x) - \ell| < \varepsilon$
so if we just get $0 < x^2 < \delta$ then we know $|f(x^2) - \ell| < \varepsilon$
magic ivy
the seocnd one is sqrt delta
misty crest
what new choice of delta do we need in our statement for lim x-> 0 f(x^2)
magic ivy
for the first?
misty crest
the seocnd one is sqrt delta
yes
take idk delta' = sqrt(delta) then for every eps > 0 you get 0 < |x| < delta' implies 0 < x^2 < delta'^2 = delta so |f(x^2) - L| < eps
as per our first limit statement
magic ivy
why would delta'=sqrt(delta) change the f(x) to f(x^2)
misty crest
in words, any positive number within distance delta of 0 maps to a point within distance epsilon of L under f
misty crest
why would delta'=sqrt(delta) change the f(x) to f(x^2)
well look at our original statement
let me color
$\dots 0 < \mathcolor{blue}{x} < \mathcolor{red}{\delta} \implies |f(\mathcolor{blue}{x}) - \ell| < \varepsilon$
woven radishBOT
knief
misty crest
the "x" is just a placeholder for any positive number within distance delta from 0
misty crest
take idk delta' = sqrt(delta) then for every eps > 0 you get 0 < |x| < delta' im...
here with our choice of delta' we arrived at the inequality [0 < \mathcolor{blue}{x^2} < \mathcolor{red}{\delta}]
woven radishBOT
knief
magic ivy
0h so if x^2 can be a distance delta' then the x in f(x) can also be x^2
misty crest
x^2 is just some positive number within distance delta from 0
misty crest
**knief**
thus it maps to a point within distance eps from L according to our first limit statement here
magic ivy
ok that makes sense
misty crest
**knief**
like all this says is that the image of any point within distance delta from 0 is within a distance eps from L. the second inequality involving x^2 satisfies exactly that
misty crest
like all this says is that the image of any point within distance delta from 0 i...
positive point
magic ivy
alr
misty crest
0h so if x^2 can be a distance delta' then the x in f(x) can also be x^2
yea so we chose delta' so that we get x^2 to be within distance delta from 0
magic ivy
and that means f(x^2) is a distance epsilon from l
misty crest
and because of our first statement we know that f(x^2) must necessarily be less than eps from L because x^2 is less than delta away from 0
magic ivy
ok '
also is it normal to define a new delta and work from there for problems like these or is that case specific
magic ivy
ok alr thanks
misty crest
no worries
magic ivy
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fervent terrace
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cedar garnet
hi?
fervent terrace
how come for the third last sentence, they say to reject the null hypotheses for that interval of p?
i don't understand when to reject or accept null hypothesis
sometimes when they solve for the test statistic and look at the appropriate table sometime they accept or reject the null hypothesis for the same interval of p
for example
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@fervent terrace Has your question been resolved?
lunar harbor
Recall you reject the null hypothesis if the p-value is less than or equal to the significance level
And that you don’t reject the null hypothesis if the p-value is greater than the significance level
lunar harbor
lunar harbor
fervent terrace
so here p value is equal to the significance level?
i don't really understand what significance level is
i sped through the content honestly and crammed
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charred eagle
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charred eagle
is this right track?
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@charred eagle Has your question been resolved?
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@charred eagle Has your question been resolved?
tender cobalt
you made a mistake here
instead you should do
x^3 - x^2 + kx + 1
= x^3 + x - x - x^2 - 1 + 1 + kx + 1
= (x^3 + x) - x + (-x^2 - 1) + 1 + kx + 1
= x(x^2 + 1) - (x^2 + 1) + kx - x + 2
= x(x^2 + 1) - (x^2 + 1) + (k - 1)x + 2
remember, you have to keep fk(x) the exact same while still getting the convenient stuff you want
so any time you add something, you subtract it after
here I add and subtract x, add and subtract -1
charred eagle
hm
tender cobalt
think about if I want to write out 123 in 100s, 10s, and 1s
I wouldnt just do 100 + 10 + 1
Id have to do 123 = 100 + 2 * 10 + 3 * 1
charred eagle
ohh. i see
tender cobalt
so if you want x^2 + 1 stuff, add and subtract
charred eagle
i tihnk i understand... i added x^2 instead of x?
tender cobalt
oh I see
thats a messier way of going about the same business
because then youre doing mental math when you dont need to
also, yea you turned -x^2 to x^2 - 2x^2 when theres nothing good with changing the x^2
you should be - x + x
you also changed the x^2 to an x
- x^2 -> + 1 (which is really an x)
alr now for the next part
lets say you finish the division and you get some sort of Ax + B + (Cx + D)/(x^2 + 1)
charred eagle
hmm
tender cobalt
A, B, C, D are just standins for numbers
charred eagle
alr now for the next part
wait
isnt this immediately wrong
cz i separated the fraction wrong first
tender cobalt
janos think in parts
we cant talk about what happens next?
maybe the work you did here was right, its just that the part before it was wrong
think in terms of what you did, not what you got
tender cobalt
now what you did here is the right idea, but you dont have to do it this way
we know to intersect the asymptote, the remainder part Cx + D has to be 0 somewhere, right
charred eagle
yes
tender cobalt
so that usually means we do Cx + D = 0
but keep in mind, we dont need to care about the x
charred eagle
lets say you finish the division and you get some sort of Ax + B + (Cx + D)/(x^2...
wait
this is talking about
f_k (x) right
and ur cx + d part is talkign about
tender cobalt
yea, it looks like it
charred eagle
(if it was this)
tender cobalt
the Cx + D part of the f_k(x), yea
thats (Cx + D)/(x^2 + 1), not Cx + D
the Cx + D is the numerator
charred eagle
but keep in mind, we dont need to care about the x
what why
tender cobalt
well what were you going to do with it
we need to find a k where fk doesnt intersect the asymptote at all
getting an x in the first place already means we have the wrong k, somehow
take a look at Cx + D = 0
Cx = -D
x = -D/C
charred eagle
ok wait i see.
charred eagle
getting an x in the first place already means we have the wrong k, somehow
ohh
so i shld not be getting x in the first place?
tender cobalt
yep
charred eagle
ok hold on lemme try again
tender cobalt
go do the first part again, so that youre looking at the right C and D this time when you try this
(this is also because if we try it with this current C and D, you might get the wrong idea)
charred eagle
((k-1)x + 2) /(x^2+1)
charred eagle
k-1
tender cobalt
and D is 2, yes
charred eagle
yes
charred eagle
uhhh
tender cobalt
x is something that either will have a solution or wont have a solution
(k - 1) x + 2 = 0
tender cobalt
yep
tender cobalt
you see any other values other than k = 1?
tender cobalt
thats correct
charred eagle
what abt D
tender cobalt
if D was 0, then we'd always get a solution and we'd be screwed
charred eagle
if d is zero, this q wouldnt work?
tender cobalt
this is a bit of a cool question as you can see
its like theyre throwing a new word at you, thats just glued together portions of previous words
you just gotta know the prefix and suffix
the prefix here is you simplifying the fraction to get the asymptote and remainder, that you know you gotta do because it makes the question easier
charred eagle
hmmm i see
tender cobalt
the suffix would be thinking "how would I know what the right answer is?"
and maybe then seeing you need to set C to 0
to solve for k
tender cobalt
so prefix and suffix are a way you can think of solving it
youre thinking forwards, or youre thinking backwards from the solution
thinking backwards is the main reason algebra is useful, you can directly put letters to things you dont know and still do math on them
see here we didnt need to graph anything to figure out k = 1, we figured on a "win condition" first then found it
charred eagle
see here we didnt need to graph anything to figure out k = 1, we figured on a "w...
is this risky?
or optimal
tender cobalt
well lets see
how would we know if we won or not? we'd have to put a variable to it
itd be more risk to not know when we're done
sometimes geometry is faster, but most of the time youll be dealing with heavy algebra problems
this is the optimal way to go about this kind of problem
if its not this, youll eventually stumble upon requiring C = 0 some other way
@charred eagle do you need anything else
charred eagle
this is the optimal way to go about this kind of problem
ohh okay. i see
charred eagle
if its not this, youll eventually stumble upon requiring C = 0 some other way
yea i’ll watch out for like constants
charred eagle
<@538904028709519393> do you need anything else
no, that was acc kinda perfect explanation TYSM
thank u
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tender cobalt
np
charred eagle
i tihnk i dropped a negative symbol somewher ethere, cant find tho
covert root
Have you tried using an integration calculator
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tribal fog
Question B. I need help solving the problem in cas (computer algebra system(
flat ibex
Question B. I need help solving the problem in cas (computer algebra system(
what does au equal in vector form
tribal fog
flat ibex
noo i mean scalar a times vector u
tribal fog
idk
last parrot
what is vector u, rina?
tribal fog
(1,-2)?
last parrot
Now, multiply it with a scalar a, what does it gives?
tribal fog
a,-2a
last parrot
Now do the same for bv?
last parrot
No, not that complicate 🙂
tribal fog
okay
last parrot
2 vectors are parallel if they are linearly dependent.
Whereas, having 2 vectors, namely u and v, such that u = kv
last parrot
Question B. I need help solving the problem in cas (computer algebra system(
now substitute them in the given statement? check if they makes 0
uhh no
tribal fog
okay
last parrot
substitute qy and qx in the statement pxqy-pyqx
tribal fog
'
last parrot
ehh
last parrot
now you have to check the second side :)
last parrot
if pxqy-pyqx=0 then p and q parallel
last parrot
because that is an if and only if statement
tribal fog
zero
last parrot
nahhh thats one side, if p and q parallel then pxqy-pyqx=0
tribal fog
so we have to prove
its not parallel
?
idk
soyr
can u explain what we're going to do
flat ibex
you proved that if p = kq, then that expression is 0
now you need to show that if the expression is 0, p = kq
since its asking you to show if and only if
last parrot
You are going to see it more in the future
if pxqy-pyqx=0 then you can rewrite as pxqy=pyqx
last parrot
a-b=0, then a=b
tribal fog
take it back
last parrot
😅
then you can just write qy/py=qx/px = k where k is a constant
ad=bc -> a/c = b/d
tribal fog
ad=bc -> a/c = b/d
wdym
tribal fog
then you can just write qy/py=qx/px = k where k is a constant
and what is the solution going to be for all these tsteps
last parrot
well then if you have qx/px = k
then you can say qx = kpx
similarly, qy/py=k, then qy=kpy
right?
tribal fog
yes
last parrot
then q = kp, which means q and p are parallel
last parrot
i did write it down bru
last parrot
wut
last parrot
ill js ask chat for a complete solution
This server does not allow for a complete solution btw
I give you hint, and you figure it out by yourself
last parrot
Which one?
tribal fog
after we
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desert seal
kinda confused midway of this question
devout snowBOT
kinda confused midway of this question
desert seal
heres how much i have done so far, lmk if you cant ready anything
kindred cedar
yeah
kindred cedar
you forgot k in the right side
west ridge
You'll then get only 1 term with sqrt
desert seal
you forgot k in the right side
wym theres only sqrt(k+1) on rhs no?
kindred cedar
you got $\sqrt{k}\sqrt{k+1}\geq k \Leftrightarrow k(k+1) \geq k^2$
woven radishBOT
q-analog
desert seal
ah how did i not see that
kindred cedar
and now you can simplify
desert seal
finally?
desert seal
yeah fr
kindred cedar
$\sqrt{k+1}>\sqrt{k}$ is trivially true
woven radishBOT
q-analog
desert seal
what does it mean trivially true
kindred cedar
we can prove it easily
kindred cedar
youre welcome
desert seal
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potent compass
Can u help me w number 7 and 8
devout snowBOT
Can u help me w number 7 and 8
kindred cedar
what you did in 7 is correct i think
potent compass
But the book says -2/x
kindred cedar
yeah you just need to multiply both sides by (-1)
because in the left side you have -f(x), you should put -g(x) tho, because its a new function
potent compass
you should have g(x) = 3 f(x), right?
Yeah but why
kindred cedar
because youre strechting it by a factor of 3, so if f(1) = 1
then after the stretch gonna be 3
f(1) = 3
vestal surge
I don’t understand
its streched like a rubber band so every value is affected
kindred cedar
think of 1 meter, if you have 1 meter and stretch it by a factor of 3
its gonna be 3
if you stretch 0 meters gonna be 0 meters
kindred cedar
youre welcome
potent compass
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potent compass
Question 4 pls
devout snowBOT
Question 4 pls
pearl relic
Question 4 pls
just draw the graph shifted 2 units to the right
faint gorge
Your point (2,0) is wrong
2^x>0 always positive so 2^{x-2}>0 as well 2^{x-2}+1>0
You should observe that p goes to infinity as x goes to infinity
pearl relic
also 2^0=1
faint gorge
and that it's strictly increasing
potent compass
Wha
pearl relic
Wha
the graph goes up and to the right
it does not go towards the x axis
so there's no x intercept
potent compass
I don’t understand what ur saying
pearl relic
I don’t understand what ur saying
,w graph 2^(x-2)+1
woven radishBOT
pearl relic
do you see how the graph does not touch the x-axis
potent compass
I think u mean x
pearl relic
that's because 0=2^(x-2)+1 has no solutoin
potent compass
that's because 0=2^(x-2)+1 has no solutoin
How do u tell this pls
faint gorge
How do u tell this pls
What does 2^x mean to you
pearl relic
How do u tell this pls
its a positive number to some exponent
thats always >0
sorry ill leave it to @faint gorge but you can ping if you need my explanation
faint gorge
potent compass
I still don’t get it
uncut crow
faint gorge
I still don’t get it
What do you get when you multiply two positive numbers? You get a positive number
2^x means really you are multiplying powers of 2
potent compass
What do you get when you multiply two positive numbers? You get a positive numbe...
Oh
potent compass
Which intercept are we doing pls
faint gorge
you can compute the y-intecept
And strictly increasing means that p is always rising
If you know 2^x then really p is just two transformations that you did with 2^x, which is a vertical (+1) and horizontal (x-2) translation
potent compass
I rlly don’t get u
faint gorge
<a:ablobsadpats:771538687284346883>
help your daughters out pls
uncut crow
i don't want to help i'm just here for moral support
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@potent compass Has your question been resolved?
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grave idol
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dull jasper
yes
dull jasper
nah
grave idol
dull jasper
bro i got a calc 2 quiz tmrw 🥀
already got 70/100 im low cortisol
it was like
function
yk
f(x) shit
but i forgot
dull jasper
f(5) i mean
put numbers on x
like
yk u undersatan
it was a kind of like
what am i even explainig
grave idol
dont make me summon mahoraga
devout snowBOT
teal tree
i dont think any x's should be in your limits once you swap the order of integration
maybe check up on that
grave idol
i dont think any x's should be in your limits once you swap the order of integra...
mmm i can't seem to find anything better to put in it tho
wait nvm i think I can just turn do some swap shenanigans
There
Are there still any errors?
also, unrelated but how do I ping mods (in case I got troll or smth)
teal tree
Are there still any errors?
dont think so
grave idol
dont think so<:catthumbsup:614540188747563008>
Alright thanks a lot, appreciate it
grave idol
also, unrelated but how do I ping mods (in case I got troll or smth)
Do i need like, certain role or smth?
teal tree
also, unrelated but how do I ping mods (in case I got troll or smth)
type the at symbol and then type moderator
use this ping wisely
they dont take it too kindly when you misuse it
grave idol
they dont take it too kindly when you misuse it
ohh i see
nah i wont use it randomly lol
just asking cause i got trolled
(before i reopen this post again)
u can scroll up
anyways, thanks again
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clever sphinx
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clever sphinx
Idk I m lost
sand quarry
Idk I m lost
It is unintuitive but try to work with C + iS where S is the same sum but with sin instead of cos
Then apply euler's formula ans binomial theorem
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@clever sphinx Has your question been resolved?
devout snowBOT
untold ravine
a "more intuitive" way might have been to directly use euler's formula and compare them, but this is cleaner
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hexed berry
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lunar harbor
Try looking for a pattern
So find the remainders when $2^1$, $2^2$, $\dots$, are divided by $6$
woven radishBOT
Civil Service Pigeon
hexed berry
Why? ;-;
Sometimes they ask you to use fermat, euler and a^n= b^n
lunar harbor
Why? ;-;
Sometimes they ask you to use fermat, euler and a^n= b^n
oh 6 is quite small
hexed berry
there's no such a thing as an universal tool and that sucks.
lunar harbor
if you know fermat/euler then that's probably easier since it's basically a sledgehammer on this (well, kinda)
consider what that tells you about powers of 2 mod 6
lunar harbor
there's no such a thing as an universal tool and that sucks.
eh discovering new things is what it's all abt
hexed berry
Let me try...
hexed berry
Yeah, I tried
the reminder should be 4
because the reminder is 2 when a^(2k+1)= and 4 when 2k
supple knot
,w 2^210 mod 6
lunar harbor
remainders of big powers -> use theorems to reduce big powers like fermat/euler
sounds sarcastic but I promise I'm being genuine
but yeah there's a sense of fucking around and finding out and there's also a sense of intuition at play
hexed berry
I guess in this exercise I can just use the numbers without exponent?
@lunar harbor
I think it's something related to fermat's numbers
lunar harbor
I think it's something related to fermat's numbers
fermat only works if you're doing modulo primes
and 9 doesn't decompose into a product of distinct primes
there's a theorem that fermat is a weaker case of that you can use here though
well, for most of it at least.
Hello?
lunar harbor
that's fine, what I sent above still applies
lunar harbor
You mentioned Fermat and Euler earlier
my point was that Fermat is a weaker case of Euler
because $\varphi(p)=p-1$ for all primes $p$, so if $\gcd(a,p)=1$, then
$$a^{p-1} = \underbrace{a^{\varphi(p)} \equiv 1 \pmod{p}}_{\text{Euler's theorem}}$$
woven radishBOT
Civil Service Pigeon
lunar harbor
tldr use Euler's theorem
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keen sundial
how do we go about this question?
devout snowBOT
how do we go about this question?
devout snowBOT
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high charm
yo , my u-sub is looking like crap, anybody got tips on u sub? (calculus)
spiral raptor
Do you have an example?
bronze void
yo , my u-sub is looking like crap, anybody got tips on u sub? (calculus)
i like how I set mine up, its nice and simple
but an example of your work would be great!
-# a general rule of thumb would be to follow LIATE for u-subs
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high charm
i like how I set mine up, its nice and simple
how
bronze void
could you send a pic of how you do yours?
bronze void
how
for, $$\int_{a}^{b} f(x) dx$$ where f(x) is a composite function of the form h(g(x)), we can follow LIATE for our U-subs. So, first find a good u via LIATE. $$u=K(x) \implies du = \frac{d}{dx} (K(x)) dx \implies dx = \frac{du}{K'(x)}$$ and adjust bounds accordingly. Then integrate
and I sub everything in directly, rather than finding: oh yea, this part is apart of my du so im going to combine then from the start etc.
woven radishBOT
･ﾟ✧ 𝓀ℬ ✧ﾟ･
bronze void
in theory, the numerator will cancel with parts of K'(x) leaving a simplified integrand
that or give us a form which can be integrated via known antiderivatives
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kindred mauve
can someone explain how we sketch level curves
devout snowBOT
can someone explain how we sketch level curves
kindred mauve
aafter finding what y is equal to
sand quarry
aafter finding what y is equal to
you basically pick different values of c
y_0 = -1, y_1 = e^x - 1, y_2 = 2e^x - 1, ...
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kindred mauve
y_0 = -1, y_1 = e^x - 1, y_2 = 2e^x - 1, ...
and then sub these in?
sand quarry
and then sub these in?
ye
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hybrid maple
Given the fields F3[x]/f and F3[x]/g where f = X^3-x+1 and g = x^3-x-1
i need to construct an isomorphism between them
i found generators in both (F3[x]/f)^x and (F3[x]/g)^x and i want to consider the isomorphism that maps (x-1)^n to (x+1)^n and 0 to 0
my issue is showing that its an isomorphism
how do i show for two arbitrary elements that phi(a+b) = phi(a) + phi(b)
devout snowBOT
Given the fields F3[x]/f and F3[x]/g where f = X^3-x+1 and g = x^3-x-1
i need to...
@hybrid maple Has your question been resolved?
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@hybrid maple Has your question been resolved?
last parrot
A wire has linear density k g/m if 1 metre of the wire has mass k grams. Suppose that you have a 10 metre-long wire whose density ρ varies continuously along the length of the wire. Suppose that the density is proportional to the length of the wire. Use upper and lower sums to bound the total mass of the wire.
How would I continute? I have done fair bit.
devout snowBOT
A wire has linear density k g/m if 1 metre of the wire has mass k grams. Suppose...
last parrot
We are given that the density $\rho$ is proportional to the length of the wire. Now, we let $x$ be the length of the line, then $\rho(x)$ is the density of the wire, and we can rewrite as:
\begin{gather*}
\rho(x)=cx \text{ (Where c is a positive proportional constant)}
\end{gather*}
We are also given a piece of wire that is $10$ metre-long, we can divide the wire into $n$ sections such that each section has the with $\Delta x$:
\begin{gather*}
\Delta x = \frac{10-0}{n} =\frac{10}{n}
\end{gather*}
From this, we can also have the gridpoint $x_i=i\cdot\Delta x$ for $i={0,1,2,..,n}$
woven radishBOT
Nefer
last parrot
Do not give me answers! I just want a hint on where to do next
<@&286206848099549185>
stable wraith
have you tried setting up the sums?
last parrot
I am doing it
Ehh I know it is strictly increasing function
So max value is at right end point and min at left and point
$L(f,P)= \sum^n_{i=1}M_i(t_i-t_{i-1})=\sum^n_{i=1}?$
woven radishBOT
Nefer
last parrot
But what is M_i here?
pure stone
max value of f(x) in (x_i-1, x_i) subinterval
last parrot
Mhm, true, so that for any interval $[x_{i-1},x_i]$, its length is always $\Delta x$.
woven radishBOT
Nefer
last parrot
also this is the formula for the upper sum not for lower sum
$L(f,P)= \sum^n_{i=1}m_i\Delta x=\sum^n_{i=1}\rho(x_{i-1})\Delta x=\sum^n_{i=1}cx_{i-1}\frac{10}{n}$
How about now?
pure stone
are you sure M_i = p(x_i-1)?
last parrot
Wait
woven radishBOT
Nefer
last parrot
Good now
timber violet
now just explicitly sum it
last parrot
And similarly, $U(f,P)= \sum^n_{i=1}M_i\Delta x=\sum^n_{i=1}\rho(x_i)\Delta x=\sum^n_{i=1}cx_i\Delta x$
woven radishBOT
Nefer
last parrot
now just explicitly sum it
Add them together?
timber violet
and take limit as n goes to infty
last parrot
Wdym explicitly sum
timber violet
okay so substitute back x_i and delta x with their n-dependent expressions
last parrot
I substituted back
timber violet
where
last parrot
It would just be x_n-1?
last parrot
Ah then it will be (i-1)deltax
mb was tripping
$L(f,P)= \sum^n_{i=1}m_i\Delta x=\sum^n_{i=1}\rho(x_{i-1})\Delta x=\sum^n_{i=1}cx_{i-1}\Delta x=\sum^n_{i=1}(c\frac{10(i-1)}{n})\frac{10}{n}\
U(f,P)= \sum^n_{i=1}M_i\Delta x=\sum^n_{i=1}\rho(x_i)\Delta x=\sum^n_{i=1}cx_i\Delta x=\sum^n_{i=1}(c\frac{10i}{n})\frac{10}{n}\$
woven radishBOT
Nefer
last parrot
Okay these are the 2 sums
timber violet
clean it up
leave only the i-dependent term inside the sum and take the constants out
last parrot
Oge
last parrot
only i
For the upper sum:
\begin{gather*}
U(f,P)= \sum^n_{i=1}M_i\Delta x\
=\sum^n_{i=1}\rho(x_i)\Delta x\
=\sum^n_{i=1}cx_i\Delta x\
=\sum^n_{i=1}(c\frac{10i}{n})\frac{10}{n}\
=\frac{100c}{n^2}\sum^n_{i=1}i
\end{gather*}
For the lower sum:
\begin{gather*}
L(f,P)= \sum^n_{i=1}m_i\Delta x\
=\sum^n_{i=1}\rho(x_{i-1})\Delta x\
=\sum^n_{i=1}cx_{i-1}\Delta x\
=\sum^n_{i=1}(c\frac{10(i-1)}{n})\frac{10}{n}\
=\frac{10c}{n^2}\sum^n_{i=1}10(i-1)\
=\frac{100c}{n^2}\sum^n_{i=1}i-1
\end{gather*}
This is what I got
woven radishBOT
Nefer
timber violet
ye
last parrot
mk
\begin{gather*}
U(f,P)= \sum^n_{i=1}M_i\Delta x\
=\sum^n_{i=1}\rho(x_i)\Delta x\
=\sum^n_{i=1}cx_i\Delta x\
=\sum^n_{i=1}(c\frac{10i}{n})\frac{10}{n}\
=\frac{100c}{n^2}\sum^n_{i=1}i\
=\frac{100c}{n^2}n\
=\frac{100c}{n}
\end{gather*}
For the lower sum:
\begin{gather*}
L(f,P)= \sum^n_{i=1}m_i\Delta x\
=\sum^n_{i=1}\rho(x_{i-1})\Delta x\
=\sum^n_{i=1}cx_{i-1}\Delta x\
=\sum^n_{i=1}(c\frac{10(i-1)}{n})\frac{10}{n}\
=\frac{10c}{n^2}\sum^n_{i=1}10(i-1)\
=\frac{100c}{n^2}\sum^n_{i=1}i-1\
=\frac{100c}{n^2}\sum^n_{i=1} \text{ (Let $k=i-1$)}
=\frac{100c}{n^2}nk\
=\frac{100ck}{n}\=\frac{100ck}{n}
\end{gather*}
Like this
timber violet
wtf?
last parrot
wut
pure stone
you messed up badly
timber violet
use wolfram if u dont know how to evaluate the sum
pure stone
remember how you would calculate the sum of consecutive natural numbers
last parrot
fack
timber violet
,w sum i , i=0..n-1
timber violet
still wtf
last parrot
Which one ;-;
timber violet
both
last parrot
isnt sum^n_i=1 of i = n?
last parrot
remember how you would calculate the sum of consecutive natural numbers
ahh i see
pure stone
isnt sum^n_i=1 of i = n?
are you sure that 1 + 2 + 3 + ... + 100 = 100?
last parrot
I was miscalculating for sum^n_i=1 of 1
nvm let me fix
For the upper sum:
\begin{gather*}
U(f,P)= \sum^n_{i=1}M_i\Delta x\
=\sum^n_{i=1}\rho(x_i)\Delta x\
=\sum^n_{i=1}cx_i\Delta x\
=\sum^n_{i=1}(c\frac{10i}{n})\frac{10}{n}\
=\frac{100c}{n^2}\sum^n_{i=1}i\
=\frac{100c}{n^2}\frac{n(n+1)}{2}\
=\frac{50c(n+1)}{n}
\end{gather*}
For the lower sum:
\begin{gather*}
L(f,P)= \sum^n_{i=1}m_i\Delta x\
=\sum^n_{i=1}\rho(x_{i-1})\Delta x\
=\sum^n_{i=1}cx_{i-1}\Delta x\
=\sum^n_{i=1}(c\frac{10(i-1)}{n})\frac{10}{n}\
=\frac{10c}{n^2}\sum^n_{i=1}10(i-1)\
=\frac{100c}{n^2}\sum^n_{i=1}i-1\
=\frac{100c}{n^2}\sum^{n-1}_{k=0}k \text{ (Let $k=i-1$)}\
=\frac{100c}{n^2}\frac{n(n-1)}{2}\
=\frac{50c(n-1)}{n}
\end{gather*}
Okay good now
woven radishBOT
Nefer
last parrot
Somehow i fking forgot the sum identities
last parrot
Alr one sec
pure stone
or you can leave it there if you're happy with it
because the problem only asks you to bound the sum and not calculating the limit itself
last parrot
I can just make it for example: $\frac{50c(n-1)}{n}$ into $50c(n^2-n)$
pure stone
check your maths again
woven radishBOT
Nefer
last parrot
This for upper sum
pure stone
good
last parrot
The other lower sum is $50c(1-\frac{1}{n})$
woven radishBOT
Nefer
last parrot
So i take U-L for min bound and U+L for max bound?
pure stone
no
last parrot
That is just deviation right
pure stone
remember, L is your lower bound for your sum and U is your upper bound
last parrot
Wait isnt those lower and upper súm already bounds
bruh
Okay, let me take note that
So for the mass M, it would just be $50c(1-\frac{1}{n})\leq M \leq 50c(1+\frac{1}{n})$
woven radishBOT
Nefer
last parrot
perfect
pure stone
exactly
last parrot
alright i will write that down, i will ask other questions later
devout snowBOT
@last parrot Has your question been resolved?
last parrot
Okay, how would I write down an integral which computes the mass of the wire? And what is the relationship between this integral and the answer in a)?
pure stone
Okay, how would I write down an integral which computes the mass of the wire? An...
alright so again, imagine diving the wire into really small units
call the length of 1 unit dx
last parrot
Yes
pure stone
and now you calculate the mass dm of each unit
last parrot
do i just calculate L and U again?
pure stone
no
last parrot
mass is basically just density times length
Then for an arbitrary section , then dm = p(x)dx
last parrot
cx
pure stone
excellent
last parrot
Then i would just take int_0^10 of cxdx
pure stone
yep
last parrot
Alr one sec, let me type these down
total mass is just 50c surprisingly
But what are their relationships?
pure stone
tbh the wording of the question is weird
pure stone
the gist of the Darboux integral is: if the integral exists, when you take lim L and lim U as n -> inf then the 2 limits are equal
and the result of the integral should be equal to that common limit
last parrot
So the upper sum and lower sum in (a) is the approximation of area under p(x), while integration in (b) gives the exact area
pure stone
yea sort of
and tbh this question is a really weird way to teach about integral sums like darboux or riemann
last parrot
mostly focus in darboux
lwk need to practice calculus and english again, delved into cs too much
pure stone
arent you 1st year
last parrot
yes
pure stone
same
last parrot
I am lwk shit in Calc 1, both in VN and aus
pure stone
australia?
pure stone
I am lwk shit in Calc 1, both in VN and aus
oh ho ho wait until you see calc 2
last parrot
consequences of skipping calc1 lectures in bk
pure stone
i did kinda shit in first sem but i am much more focused on my studying now
i can grasp pretty much all i've learned
just awful execution in exams
last parrot
I am lwk cooked first sem (currently), trying best to achieve a 70% for a distinction
pure stone
oh well best wishes
pure stone
so you have exam in 2 weeks yet you're still studying these sums?
or are the curriculums wildly different
last parrot
Mhm
We have 2 seperate parts, LA mainly on basis, change of bases, eigen, and differential
last parrot
lin alg
pure stone
We have 2 seperate parts, LA mainly on basis, change of bases, eigen, and differ...
what is differential you mean here?
last parrot
Calc is darboux, riemann, trigonometry, deriv,integration, lim, cont, and diff
Solving system of linear equations for differential equations
pure stone
odes wtf
pure stone
i know but that in linear algebra is weird
last parrot
Yea just studied that this morning
pure stone
especially when you havent done with calc 1 yet
last parrot
My course is basically semester 1: Calc 1 and a bit of 2, semester 2: Calc 2 and a bit of 3
pure stone
For example, given y''=-y, y(0)=1, y'1(0)=0
i think the only part in linear alg that is even remotely related to this is the Wronsky determinants
last parrot
i think the only part in linear alg that is even remotely related to this is the...
Haven't heard of that, if it is something determinant then LaPlace
Alr peace out, its quite late here, i will ask for LA questions tmr
thanks!
pure stone
i think the only part in linear alg that is even remotely related to this is the...
ohh laplace transform
havent gotten to that yet
pure stone
yea
last parrot
Giải định thức bằng cách xóa phần tử
pure stone
chưa nghe khái niệm đó bao giờ
pure stone
học trường me tây nó chán thế ạ
last parrot
me tây là gì
pure stone
phép khai triển laplace
uk nhưng cái đó liên quan gì đến giải pt vi phân cấp 2
nghe rất lạ
last parrot
cái đó là khác, laplace là học từ đầu năm, giờ thì giải bằng khử gauss và eigen với pt vi phân cấp 2
xài vecto riêng
pure stone
nghe lạ thật
last parrot
tutu để tìm bài giảng sáng nay
pure stone
đóng kênh vs gửi DM đi
devout snowBOT
pure stone
ok
hasty cargo
no way blud switched languages mid conversation
shell hare
~~no way blud switched languages mid conversation~~
Get out
shell hare
why? they finished the convo and im just remarking?
I was making it in a joke-y manner not seriously :<
hasty cargo
I was making it in a joke-y manner not seriously :<
shell hare
<:hmmcatfone:849254980812013639>
devout snowBOT
Available help channel!
potent compass
Can u help me pls
devout snowBOT
Can u help me pls
rain summit
Can u help me pls
try drawing the heights from D to AB
for instance DH is the height of AB
now notice for me that $\angle ADH$ is $\frac{\beta}{2}$ and $AH$ is $\frac{AB}{2}$
woven radishBOT
1 divided by 0 equals Infinity
rain summit
according to trig then $AH = x \cdot \sin \left(\frac{\beta}{2}\right)$
woven radishBOT
1 divided by 0 equals Infinity
potent compass
No
potent compass
Everything u said
potent compass
Ok
rain summit
so there's an important property of isosceles triangle
potent compass
Ok!
rain summit
here i have an isosceles triangle ABC where AB = AC
if you draw the height AD
or AD such that D is the midpoint of BC
or AD such that BAC splits in half
then AD would satisfy all those 3 properties
so for example, if i drew the height AD, i can show that BAD = ADC through triangle ABD congruent to ACD
@potent compass
potent compass
Ok
rain summit
**1 divided by 0 equals Infinity**
so if you used the property above, in this case with triangle ADB, you get this result
potent compass
I don’t get it
rain summit
since AD = DB, then triangle ADB is isosceles
so i can draw a height DH of the triangle
potent compass
Can u start again pls
rain summit
draw the height DH of the triangle
potent compass
Ok done
rain summit
if you can prove that triangle DHA is congruent to triangle DHB
then you get AH = HB and angle ADH = HDB
potent compass
Ok
wintry socket
if you can prove that triangle DHA is congruent to triangle DHB
Hint:use RHS
rain summit
**1 divided by 0 equals Infinity**
so if you use trigonometry, you get this
so the only thing left to do is to prove $2x \cdot \sin\left(\frac{\beta}{2}\right) = RHS$
woven radishBOT
1 divided by 0 equals Infinity
rain summit
**1 divided by 0 equals Infinity**
which i think you can use the property $\sin^2 \theta + \cos^2 \theta = 1$
woven radishBOT
1 divided by 0 equals Infinity
rain summit
I want to break
good luck!
potent compass
.close
devout snowBOT
devout snowBOT
Available help channel!
polar nebula
i have this question and im kinda confused
devout snowBOT
i have this question and im kinda confused
polar nebula
when x go through 2, the positive sign doesnt change into negative sign
soooo
do i have to do like (-2x+6)²=2
summer summit
could you translate this please?
polar nebula
the question just asks us to find monotonicity of the function like increasin decreasin
y = g(x) = f(-2x+6)
that variation table is from f(x)
im confused when finding f'(-2x+6)=0
dapper tiger
you meant f(-2x+6)?
dapper tiger
the derivative of g(x) is -2f'(-2x+6), so you need to study the sign of f'(-2x+6), but you already have the sign study of f'(x)
what's left is to relate the two
polar nebula
so just find f'(-2x+6)=0 to find x and make another variation table for y=g(x) to find the monotonicity, right
ive got another problem
i cant find where f'(x) = 0
only know f'(1) = f'(3) = f'(5) = -1
still the same question, find the monotonicity of g(x) = f(x²+x+1)
the graph is from f'(x)
supple knot
i cant find where f'(x) = 0
why do you need to know these x values
again you need to translate the problem
summer summit
the question just asks us to find monotonicity of the function like increasin de...
.pin
devout snowBOT
the question just asks us to find monotonicity of the function like increasin de...
polar nebula
why do you need to know these x values
so i can find when g'(x) = 0
to make the variation table
we only learn to solve the problem by that way
supple knot
why do you need g'(x) = 0
supple knot
again you need to translate the problem
.
polar nebula
Given a function y = f(x) with a continuous derivative on R. The graph of the function y = f'(x) is as shown in the figure.
Find the monotonicity of the function g(x)=f(x²+x+1)
polar nebula
why do you need g'(x) = 0
so i can make a variation table for g(x)
pure stone
Find the monotonicity of the function g(x)=f(x²+x+1)
by that you mean find the intervals where g(x) increases/decreases?
supple knot
the question and/or graph is malformed. you don't have enough information to find f'(x^2 + x + 1) = 0, so you only have enough information to find just the one point you already found g'(x) = (2x + 1) * (...) = 0 where x=-1/2
bring it up to your teacher and move on
polar nebula
oh
pure stone
not sure if there's a workaround with this
polar nebula
i guess the question is wrong
pure stone
yea nvm you cant find the intervals at all
pure stone
yea nvm you cant find the intervals at all
... or the graph wants you to assume that f'(x) is a cubic function?
polar nebula
so the only way g'(x) = 0 is when f'(x) = -1 i guess
soooo g'(x) = f'(x) + 1
or g(x) = f(x) + x + 1, which match the function in the question
pure stone
the only thing i can get at this is that g'(x) = 0 has 7 solutions
pure stone
so the only way g'(x) = 0 is when f'(x) = -1 i guess
are you sure?
polar nebula
thats just my thinking, im not sure 100%
pure stone
yeah i think the question is wrong
pure stone
... or the graph wants you to assume that f'(x) is a cubic function?
even with this assumption the solutions for f'(x^2+x+1) look awful
hasty scarab
!help
devout snowBOT
To ask for mathematics help on this server, please open your own help channel or help thread. See #❓how-to-get-help for instructions.
devout snowBOT
@polar nebula Has your question been resolved?
devout snowBOT
Available help channel!
hasty scarab
!help
devout snowBOT
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hasty scarab
hey
did i break it lol
ok im bacak
yo
so
this is the things
and my problem is
i dont know how to solve them
the formulas are writen but
i cant meorize them
and idk how to solve them step by step
@lethal wraith
true coral
supple knot
can you make a equation and solve it
you should have a textbook or notes to look for actual problems
hasty scarab
lets say the first one P and V are ?