so if they're the same thing, they get canceled out?
#help-0
earnest bridge
wary stream
If it's in the numerator and denominator
earnest bridge
oh okay mhm
wary stream
Like this, because a is in the numerator and denominator, a gets canceled out
But that's not what we want, correct?
earnest bridge
correct
wary stream
As I stated b is the obvious one
If the final expression is a/d, should c still exist?
earnest bridge
oops hold on
earnest bridge
If the final expression is a/d, should c still exist?
im not understanding
1sec made a mistake
wary stream
In my example, the final expression should be a/d. In the first fraction, you can see that the b's cancel out. That means now for the second fraction, certain things must exist to cancel out stuff from the first fraction
earnest bridge
mhm
wary stream
If the second fraction contains a in the denominator, it gets canceled out, because as I mentioned, if the same stuff appears in the numerator and the denominator, they get canceled out
earnest bridge
yup
wary stream
So knowing that we have (ab)/(cb) and the b's cancel and the a needs to exist since it's part of the final expression, what in this expression should be canceled out?
earnest bridge
c, because it doesn't have a place in there second fraction
wary stream
Good, so we need c to get canceled. Should c be in the numerator or the denominator of the second fraction?
earnest bridge
numerator, because if there's 2 of it, it can get canceled out as u said
earnest bridge
mhm
wary stream
We still have a in the numerator, because it never got canceled
earnest bridge
mhm
wary stream
In the 2 fractions, what is missing that the final answer has?
earnest bridge
d
wary stream
So d needs to exist in the second fraction, should d be in the numerator or denominator?
earnest bridge
denom, so it won't get canceled out?
wary stream
You're right about the denominator but that's not the proper reason
earnest bridge
oh
wary stream
It's because d exists in the denominator in the final answer
It doesn't make sense to have d in the numerator when the final answer has d in the denominator
earnest bridge
ah
wary stream
So overall, this is an answer, the b's cancel, and c's, and all that is left is a and d
earnest bridge
ah
wary stream
So in your problem here, the x + 5 will cancel out. What needs to exist from this fraction because it exists in the final answer?
earnest bridge
x-6
wary stream
Not exactly. I'm just asking about the stuff in that fraction only
What in that fraction alone, should still be there because it's in the final answer?
wary stream
And what in that fraction needs to get canceled?
earnest bridge
x+5
wary stream
Besides that
earnest bridge
x-4
wary stream
So the x - 4 needs to get canceled because it doesn't exist in the final answer
earnest bridge
mhm
wary stream
So in the second fraction, x - 4 needs to exist, where? Numerator or denominator?
earnest bridge
numerator, to get canceled out
wary stream
Good
In the second fraction, something needs to exist to match the final answer, what is that something?
earnest bridge
x-6
wary stream
And should it be in the numerator or denominator?
earnest bridge
denominator
wary stream
Good
Now another thing to note, the question given, the second fraction has a power of two, meaning it would need to be factored
earnest bridge
mhm
wary stream
So when you factor, you'll have two set of parentheses in the numerator and two in the denominator
Like the first fraction
Because you're going to have factors, and they don't exist in the final answer, that means that they got canceled out
earnest bridge
mhm
wary stream
So that means, the numerator and denominator of the second fraction should have the same factor so it can get canceled
earnest bridge
yup
wary stream
So what's a factor that comes to mind?
earnest bridge
6?
wary stream
Factor as in (x + something) what's that's something you want
wary stream
So then x + 6 needs to exist in the numerator and denominator of the second fraction to cancel out
earnest bridge
yes
wary stream
So that's what it should look like
The x + 5 cancel, the x - 4 cancel and the x + 6 cancel
The last step, expand that second fraction
wary stream
This is the last step, all you need to do is expand the second fraction, applying distribution, then plug in the values into your problem
earnest bridge
So that's what it should look like
where did the extra x+6 come from
im a bit lost
wary stream
It's what you picked
earnest bridge
oh yeah
wary stream
Because the initial question had a quadratic, you needed a second factor, that gets canceled out
earnest bridge
This is the last step, all you need to do is expand the second fraction, applyin...
apply distribution? how can i use distribution on this part
wary stream
Do you know FOIL?
earnest bridge
I've heard of it
wary stream
You should apply it
earnest bridge
i mean i've "heard" of it but ive never used it before
wary stream
I suggest looking into that, so you can properly get the answer
earnest bridge
alright I'll do that, one sec
lone heartBOT
@earnest bridge Has your question been resolved?
earnest bridge
im back
just figured out the foil method
it's pretty simple
now let me try applying it
@wary stream
lone heartBOT
@earnest bridge Has your question been resolved?
earnest bridge
how am i to expand it now?
i dont know what that means
i applied the foil method
wait
a
damn
minute
is it supposed to be
x^2 + 12x + 12?
instead of 12x^2?
or am i combining like terms all wrong
i think im doing the whole thing all wrong
<@&286206848099549185> does anyone understand how i can go about doing this
earnest bridge
The x + 5 cancel, the x - 4 cancel and the x + 6 cancel
this is how im understanding this part
wary stream
this is how im understanding this part
You don't cancel stuff out
That second fraction, expand that stuff
earnest bridge
what does that mean? expand how?
wary stream
You were working backwards to form this
wary stream
<@485903210272391178>
Like this, expand the numerator and the denominator
earnest bridge
ah but without canceling?
wary stream
Yes, without canceling
wary stream
Yes
earnest bridge
okay
wary stream
Numerator and denominator, separately
earnest bridge
x^2 + 6x -4x -24
wary stream
And that equals?
earnest bridge
x^2 + 2x -24
wary stream
Good
wary stream
Same thing with (x-6)(x+6)
wary stream
Yes
earnest bridge
thank you
very very very much
you are so intelligent
i appreciate you
.close
quasi rover
Working on a larger problem, I got this as a part of an equation. Wasn't sure how to approach, so plugged numbers in. Got the Bernoulli Triangle sums [(1), 1, 3, 8, 20, 48]. I worked out that the sum permutation on the left -- if isolated -- form the fibonacci numbers, and the isolated sum of the permutation on the right give triangular/tetrahedral/etc. numbers.
Anyways, would like to prove whether or not the summation will give the n-1th Bernoulli sum.
quasi rover
this thing
lone heartBOT
@quasi rover Has your question been resolved?
sharp glacier
a question on my homework which i need help with, I remember doing these earlier, but completely blanked out now
last ether
-
Critical points are when $y' = 0$
-
If $a>0$, then increasing intervals will be $\left{x\in\bR | x\geq \frac{-b}{2a}\right}$. The opposite is true for decreasing. This is just knowing how parabolas work with their vertices.
-
Inflection point is when $y'' = 0$. Use sign analysis to prove it does change concavity.
-
Concave down is when $y'' < 0$, concave yo when $y'' > 0$.
ocean sealBOT
Umbraleviathan
last ether
@sharp glacier
sharp glacier
wym easiest
last ether
Finding increasing decreasing
last ether
Like on that list I gave out
last ether
Yeah
sharp glacier
and 2ax + b = 0 for CV
last ether
Yeah
sharp glacier
then u solve for x?
last ether
Yup
sharp glacier
but theres a and b?
last ether
You solve for x in terms of a and b
sharp glacier
ah i see
last ether
I mean that's how they get the formula for the abscissa of the vertex
sharp glacier
-b/2ax
sharp glacier
I mean that's how they get the formula for the abscissa of the vertex
whats that
oh wait
no x
mb
-b/2a
last ether
If $(h, k)$ is the vertex of parabola $ax^2 + bx + c$, then $h = \frac{-b}{2a}$
ocean sealBOT
Umbraleviathan
sharp glacier
i think you plug in values greater than and less than the CV
to get increasing and decreasing intervals?
last ether
Save yourself some time
Can you mentally graph what a parabola would look like if a > 0
Like a sketch
sharp glacier
concave up?
last ether
sharp glacier
ahk
last ether
From tehre, it's just looking at a graph: no calculus needed
sharp glacier
k ty
increasing intervals (-infty, +infty)
decreasing intervals: n/a
concave up (-infty, +infty)
but inflection point?
@last ether
last ether
No
last ether
There are decreasing intervals
sharp glacier
???
last ether
Like if you look at the drawing
sharp glacier
what
last ether
There are clearly decreasing parts
last ether
And it's even funnier considering I gave you the answer
sharp glacier
And it's even funnier considering **I gave you the answer**
if im smart i wouldnt be here
sharp glacier
ok
last ether
You can somehow plug in a value less than -b/2a. It's really hard to, and I wouldn't
sharp glacier
but thats without calc?
last ether
You should instead know how the parabola would look
sharp glacier
yes kk
last ether
but thats without calc?
Sign analysis if f'(x) is Calc
sharp glacier
so inc from (-b/2a, infty) dec: (-infty, -b/2a)
last ether
Yes
last ether
Concave up for $x\in\bR$
ocean sealBOT
Umbraleviathan
last ether
Write it like that
last ether
Saves you ink
last ether
ah i can just calculate that
set $y'' = 0$
ocean sealBOT
Umbraleviathan
last ether
But if I'm gonna be honest
You don't even need to do that
It's concave up for all real values of x
So it never changes concavity, so there's no inflection point
last ether
Np
lone heartBOT
@sharp glacier Has your question been resolved?
iron mulch
.close
light scarab
How do I solve this? I'm only able to mark up the triangle right now. I don't know how I get the heights and bases.https://uploadi.ng/
normal ingot
toxic wasp
what is a curly bracket next to 2 equations mean?
lone heartBOT
what is a curly bracket next to 2 equations mean?
worn fox
Show an example
worn fox
Just a notational thing like grouping the two equations into a system
olive oar
Is it possible to define the integral $$\int_{a}^{b} f = \lim_{n\to \infty} s_n,$$ where ${ s_n}$ is a sequence of step functions converging uniformly to $f$? (idk the answer to this im just thinking it is after reading a bit about sequences and series)
ocean sealBOT
ohNoiAmHere
olive oar
The main thing im thinking is that a continuous function is a regulated function, so this should work right?
fallow vale
In a sense yes and this is another useful way to think about integrating functions
Are you familiar with Banach spaces or only with spaces like $\mathbb{R}^{n}$?
ocean sealBOT
ninjahuman
fallow vale
Because this is what you can do
Let $[a,b]$ be a closed (and bounded) interval of $\mathbb{R}$. Let $E$ be a Banach space (or just take this as $\mathbb{R}^{n}$ if you don't know what a Banach space is) and you can define a step function $f:[a,b]\to E$ as a function such that you can partition the interval $[a,b]$ with some points $\alpha_{0}, \alpha_{1}, \cdots ,\alpha_{n}$ with the property that $i<j$ implies $\alpha_{i}<\alpha_{j}$, $\alpha_{0}=a$ and $\alpha_{n}=b$, and that $f$ is constant on $(\alpha_{j},\alpha_{j+1}$ for each $0\leq j<n$
ocean sealBOT
ninjahuman
fallow vale
Now there are a finite number of values each step function can take, so it is also a bounded function
But if you extend these step functions by adding in all the bounded functions that have a sequence of step functions converging uniformly to them, you get the space of jump continuous functions on $[a,b]$
ocean sealBOT
ninjahuman
fallow vale
These are the functions such that if $a<c<b$, then $f(c)$ has both a left limit and a right limit
ocean sealBOT
ninjahuman
fallow vale
And in addition, $f(a)$ has a right limit and $f(b)$ has a left limit
ocean sealBOT
ninjahuman
fallow vale
Now to integrate the step functions, there is a natural way to do this. Take a partition $(\alpha_{0},\cdots ,\alpha_{n})$. For every $j$ with $1\leq j\leq n$, $f$ is constant on $(\alpha_{j-1}, \alpha_{j})$. For $x\in (\alpha_{j-1}, \alpha_{j})$, let $f(x)=e_{j}$ define the integral of the step function $f$ as $\int f=\sum_{j=1}^{n}(\alpha_{j}-\alpha_{j-1})e_{j}$
ocean sealBOT
ninjahuman
fallow vale
You can show that this value is independent of the partition chosen and so the integral of this particular function is well-defined
Now if you have a jump continuous function, this means that there is a sequence of step functions converging uniformly to it
So if $(f_{n}){n\in\mathbb{N}}$ is a sequence of functions converging uniformly to a jump continuous function $f$, you can show that the sequence $\left(\int {f{n}}\right)_{n\in\mathbb{N}}$ forms a Cauchy sequence in $E$
ocean sealBOT
ninjahuman
fallow vale
And as $E$ is complete, it converges to some unique value
ocean sealBOT
ninjahuman
fallow vale
This value is independent of the sequence of step functions you choose to converge uniformly to $f$, and so this value is also well-defined
ocean sealBOT
ninjahuman
fallow vale
This is the general idea of how you can do this
You can look at the book "Analysis II" by Amann and Escher which has a very nice presentation of this topic if you would like to learn more about this
lone heartBOT
@olive oar Has your question been resolved?
olive oar
Are you familiar with Banach spaces or only with spaces like $\mathbb{R}^{n}$?
only $\mathbb{R}^n$
ocean sealBOT
ohNoiAmHere
fallow vale
Ok then take everything I said above with $E=\mathbb{R}^{n}$
ocean sealBOT
ninjahuman
fallow vale
Basically you can do what you said but it takes a bit of work to formalise it
olive oar
so i basically show that for $s_n$ for every $\epsilon > 0$, for $m, n > N$ we have that $|s_n(x) - s_m(x)| < \epsilon$ for all $x\in [a,b]$
ocean sealBOT
ohNoiAmHere
olive oar
then i find a sequence {t_n}
with for n > N, we have that $|s_n(x) - t_n(x)| < \epsilon|$ for $x\in [a,b]$
ocean sealBOT
ohNoiAmHere
olive oar
and then i can show that $$\lim_{n\to \infty} \int_{a}^{b} s_n = \lim_{n\to \infty} \int_{a}^{b} t_n$$
ocean sealBOT
ohNoiAmHere
olive oar
using the definition of the integral you used above
fallow vale
You can do that, but if you are wanting to show that value is unique, you have to take the sequences from the start. You only chose the $(t_{n})$ sequence later on so that it would satisfy that condition instead of taking it from the start
ocean sealBOT
ninjahuman
fallow vale
There is not much to change to fix it though
lone heartBOT
fallow vale
You're welcome
lone heartBOT
Available help channel!
hidden carbon
Hey, I need to calculate the following:
lone heartBOT
Hey, I need to calculate the following:
hidden carbon
What I did was to convert it to the exponential form and then using the De Moivre forumla
I got 2^91 * e ^ (ipi91/3). Now here's where I'm in trouble and I'd really love some help:
vale wigeon
so far so good
hidden carbon
How do I convert it back? I can't seem to understand what that 91/3 equals to
vale wigeon
well 91/3 is 91/3
it's just a fraction like any other
do you mean that you are struggling to convert $e^{91\pi i/3}$ into something that looks like $e^{i\theta}$ for $\theta \in [0, 2\pi)$?
ocean sealBOT
Ann
hidden carbon
**Ann**
I believe I need to go through this step first
So I guess yes too? 😄
is there a "quick" trick to convert the 91/3pi into tetha between 0 and 2pi?
vale wigeon
theta, not "tetha"
but anyway, you might want to subtract a suitably chosen integer multiple of 2pi from 91pi/3
hidden carbon
I see I see
Are you aware of any trick to find that integer? Or it's all trial and error?
It's 15 in this case isn't it
91/3 - 15*2pi = pi/3
Got it. Thanks!
.close
keen quarry
I am stuck at a problem in my book
if x+1/x = root3
Then what is the value of
x²⁴+x¹⁸+x⁶+1
tacit arch
Multiply through by x and use quadratic formula
There's probably a more clever way involving raising x + 1/x to a power
tacit arch
,w solve x + 1/x = sqrt(3)
ocean sealBOT
lone heartBOT
@keen quarry Has your question been resolved?
meager terrace
lone heartBOT
meager terrace
Hello, I have a related rates of change question with respect to a conical object.
Basically, I know I have to solve for the height (using volume) before I take the derivative and then use implicit differentiation to find dh/dt
but how I do I get the height value, which ratio or formula can I use with the given variables of height, diameter and volume?
echo socket
You mean how do you find "the rate at which the water level is rising when there's 8pi m^3 of water"?
meager terrace
yeah i believe i need the height at that point in time
echo socket
You mean how do you find "the rate at which the water level is rising when there...
Here it asks for dh/dt at that point in time though*
You just need to find value of t such that V(t) = 8pi and then plug that value into dh/dt
Assuming you've found an expression for dh/dt
meager terrace
yeah we need to find dh/dt
sorry
but i believe we need to do some algebra to solve for the other variables somehow (h, r) as we are only given volume at that point in time
all the text with the "steps" i wrote so could be wrong
echo socket
Btw I think it should be dV/dt = 1.2 instead of 5.2
5.2 is the height of the cone and 1.5 is the rate at which water is being pumped into it
meager terrace
yeah ur right ty
like we need to basically get the value of r somehow... because it just gives us it in general
know what i mean?
can we use a ratio of 2.5/5.2?
so its dynamic
and then use it solve with the volume
echo socket
I think I'd start by rewriting $\derivative{V}{t} = 1.2$ into $\frac{\pi}{3}(2rh\derivative{r}{t} + r^2\derivative{h}{t}) = 1.2$
ocean sealBOT
A Lonely Bean
meager terrace
i'll show you how i did another one if ur interested but its a bit different
but basically we need to do some algebra before the differentiation i think
because we don't have r or h when the volume = 8pi
here's one example they give me in the unit's work
basicall
echo socket
Don't really want to work with decimal points
meager terrace
i ask because i saw someone else do that b4
alright, i think i got it now thanks for the help though, for some reason i thought 3:8 was a general ratio but it's specific. so thanks
echo socket
Let's define T to be the point in time when V(T) = 8
Oh alright
I was gonna continue solving lol
meager terrace
ok go ahead
echo socket
$V = \frac{\pi}{3}r^2h = \frac{\pi}{3}(\frac{25}{52})^2h^3$
ocean sealBOT
A Lonely Bean
echo socket
So this should be true
This is the first equation that we need, the other we'll get by rewriting V(T) = 8
echo socket
From here you can solve for h(T), then plug in value of h(T) into here
And solve for dh/dt at T
Which is what the question asked you for
echo socket
Alright
meager terrace
i think the above works too maybe? and then take the derivative when the volume = 8pi
echo socket
Yeah that's pretty much what I wrote so far
echo socket
That's why I didn't bother expanding (25/52)^2 lol
You can just define some variable to represent that coefficient of h^3
meager terrace
is that equivalent though
echo socket
For everything to look simpler
Yeah I mean if I were to define k = 625pi/8112 I'd write this as V = kh^3
Looks simpler to me
And then plug in 625pi/8112 into k when you're done solving the problem
meager terrace
alright ill take down ur notes and save it for later
lets solve it and see if we get same answer
then ill call it
we need to solve for h using the ratio
i think
meager terrace
yes i think we need still need the height at that point in time
or we can solve it using the ratio itself...
echo socket
Oh damn I wrote 8 instead of 8pi up there mb
meager terrace
its super confusing question man im surprised
i gotta call it soon
i gotta go to bed
but we are missing an h in there
echo socket
Yeah from here you'll solve for h(T) and then be able to solve for dh/dt at T after rewriting dV/dt = 8pi
echo socket
but we are missing an h in there
Doesn't seem to be the case
coarse current
lone heartBOT
coarse current
Can anyone tell me if I did this properly
lone heartBOT
@coarse current Has your question been resolved?
lone heartBOT
@coarse current Has your question been resolved?
lone heartBOT
sorry my last channel closed <#486844829209198613>
gilded vessel
i basically just want to integrate cosine and sine functions to find the area under my "rollercoaster"
@mortal trellis @chrome plank
gilded vessel
it shouldnt be that hard to learn since i can just use u substitution right and antiderivative of cos and sine is easy
but not sure
a*f(bx + c) + d
probably things in this form
where f is sine or cosine
honestly i think i might have enough knowledge from a short video i watched to learn u subtitution
gilded vessel
a*f(bx + c) + d
i can split the d into a separate integral, take the constant a out, and let u = bx + c
does this sound correct?
keen socket
Maybe send the equation
chrome plank
send your work
lone heartBOT
@gilded vessel Has your question been resolved?
chrome plank
Yes looks correct, good job
chrome plank
Another thing, sin([something] + pi/2) is just cos([something]), but you can leave it however you like
It's like you translate sin() to get cos()
gilded vessel
Another thing, sin([something] + pi/2) is just cos([something]), but you can lea...
yeah i know and what i have is -sin x
i know its kinda pointless there
but i just wanted to have a number there idk why i chose that
thanks so much for the help :D
chrome plank
yeah i know and what i have is -sin x
what do you mean by what i have is -sin x
gilded vessel
well cos(x + pi/2) = -sin x
gilded vessel
ah gotcha
just a side effect of defining the initial function using cosine instead of -sine i guess
chrome plank
Ok, if you have more questions feel free to ask, otherwise .close the channel
lone heartBOT
chrome plank
lone heartBOT
Available help channel!
keen python
how do I calc this ?
lone heartBOT
how do I calc this ?
vale wigeon
are you having wolfram alpha do it
if so then wolfram alpha doesn't know that you want the real root and not the principal root
keen python
no i wonder how I would do that in exams
vale wigeon
-9 has a real cube root
keen python
is that only for -9 ?
vale wigeon
are you asking if -9 is the only real number that has a real cube root?
if so then the answer is "of course not"
keen python
but how does one know if -x has a real cube root
vale wigeon
all real numbers have a real cube root...
keen python
but how do I calc them ?
with a simple calculator ?
mortal trellis
well how would you calculate the cube root of 9?
cbrt(-9)=-cbrt(9)
because (-x)^3=-x^3
chrome plank
(because 3 is odd)
lone heartBOT
@keen python Has your question been resolved?
lone heartBOT
lone heartBOT
Available help channel!