cinder pilot
#help-19
weary pelican
Seems all good
mystic saffron
odd edgeBOT
mystic saffron
not quite sure how to evaluate
maybe a checklist or a video might be useful
.close
gleaming trout
odd edgeBOT
bleak shard
I'd say you already did by writing it expanded out.
odd edgeBOT
odd edgeBOT
Available help channel!
gritty ore
hello! i'm having trouble figuring out where to start for this
brittle beacon
Assume that you have $a\mathbf{u} + b\mathbf{v} + c\mathbf{w} = \mathbf{0}$, and use the fact that the inner product of any two is zero
clever fjordBOT
@brittle beacon
torpid owl
hmm, I think proof by contradiction might work
gritty ore
what do you mean by "inner product of any two"?
brittle beacon
E.g. $\mathbf{u}\cdot \mathbf{v} = 0$ and any other pair you pick (except where they're the same - in which case, e.g. $\mathbf{u}\cdot \mathbf{u} > 0$ by $\mathbf{u}$ being nonzero)
clever fjordBOT
@brittle beacon
brittle beacon
so like u.w is zero, as is v.w etc etc
gritty ore
ohh ok and that's bc they're orthogonal
brittle beacon
Yep, the definition (assuming you're using the dot product, but it's similar for any other inner product)
gritty ore
<@788085606483361802>
the definition of linear independence is a=b=c=0 right? how do you use the inner products to prove that
brittle beacon
the definition of linear independence is a=b=c=0 right? how do you use the inner...
Yep - you must have all of those constants being zero! and what properties do you know about inner products?
brittle beacon
Not quite - a,b and c are scalars, while u, v and w are vectors
I meant more along the lines of general properties that inner/the dot product(/s) satisfy
Can you think of any way to apply an inner product to $a\mathbf{u} + b\mathbf{v} + c\mathbf{w} = \mathbf{0}$?
clever fjordBOT
@brittle beacon
brittle beacon
You can use that, yep 
Inconvenient I used a,b,c as the coefficients, peak 
But you have ${\color{green} a\mathbf{u} + b\mathbf{v} + c\mathbf{w}} = {\color{green} \mathbf{0} }$, those are two equal vector quantities...
clever fjordBOT
@brittle beacon
gritty ore
OHH so au, bv, and cw all = 0, and since u/v/w are nonzero the constants all have to be 0
brittle beacon
you need to kind of show that-
Because ${\color{green} a\mathbf{u} + b\mathbf{v} + c\mathbf{w}} = {\color{green} \mathbf{0} }$, you know that for any other vector ${\color{orange} \mathbf{x} }$, you have ${\color{green} ( a\mathbf{u} + b\mathbf{v} + c\mathbf{w} )} \cdot {\color{orange} \mathbf{x} } = {\color{green} \mathbf{0} } \cdot {\color{orange} \mathbf{x} }$, so you may want to choose particular choices...
clever fjordBOT
@brittle beacon
gritty ore
so if i choose u as x i get au.u + bv.u + cw.u = 0.u
v.u and w.u are 0 so au.u = 0
and u.u is nonzero so a must be zero
then repeat that with v and w to prove they're both 0 too
brittle beacon
then repeat that with v and w to prove they're both 0 too
Yep, it's the same idea, you could just say that "similarly b=c=0"
brittle beacon
hmm, I think proof by contradiction might work
You could also assume that one of them is nonzero, and find a contradiction that way 
brittle beacon
A pleasure 
gritty ore
.close
pastel loom
odd edgeBOT
pastel loom
Is this right?
dense glade
open knot
Is this right?
yop
odd edgeBOT
Available help channel!
storm cedar
Can somebody please walk me through the process of factoring this polynomial?
-x^3+7x^2+6x-112=0
storm cedar
I never learned the unit root theorem, and am struggling to grasp the online material
storm cedar
yes
storm cedar
ok
fair vine
Can you read it?
I will explain it still
Ok after you separate into two binomials you can swap the two inner numbers
Than factor
Actually I’m confused
fair vine
I got there two but something is wrong
The numbers inside the parenthesis should be the same
storm cedar
this is the way it is explained online:
fair vine
I don’t know 😕
storm cedar
no worries
storm cedar
the main goal here is finding eigenvalues for a 3x3 matrix
after taking the determinate of (A - Iƛ) I get that polynomial
no worries though I will figure it out
.close
mild matrix
odd edgeBOT
mild matrix
hello
I am not sure where I went wrong here I keep on getting the wrong answer
what I did was I moved -z over to the right side and -13 to the left side to get z = x - 5y -13
then I plugged in z into both the top and bottom equations to isolate for the values x and y
once I did that I multiplied each top and bottom appropriately such that the x values would cancel out leaving me with only y
but the y value I got was incorrect so I dont know if I went about this the right way
odd edgeBOT
@mild matrix Has your question been resolved?
odd edgeBOT
@mild matrix Has your question been resolved?
odd edgeBOT
@mild matrix Has your question been resolved?
odd edgeBOT
@mild matrix Has your question been resolved?
spice brook
@mild matrix it's much easier to express the system using a combined matrix
do you know how to do that?
and reduce that matrix to it's row-echalon form
languid drum
odd edgeBOT
languid drum
how did they get from the 2nd line to the 3rd?
fervent hound
They factored 2x(x-2)³
$f'(x)=4x^2(x-2)^3+2x(x-2)^4\ =2x\times 2x(x-2)^3+2x(x-2)^3(x-2)$
clever fjordBOT
Adam Chebil
languid drum
**Adam Chebil**
what happened after this? where did the 2x go😭
odd edgeBOT
@languid drum Has your question been resolved?
odd edgeBOT
@languid drum Has your question been resolved?
spring knot
division of 2 differential operators, how did they get 6tD^2 and 6D
odd edgeBOT
division of 2 differential operators, how did they get 6tD^2 and 6D
@spring knot Has your question been resolved?
odd edgeBOT
@spring knot Has your question been resolved?
zinc cairn
odd edgeBOT
zinc cairn
I dont know how to do any of it
tardy lagoon
the ~ symbol means similar and similar triangles have same corresponding angles
odd edgeBOT
@zinc cairn Has your question been resolved?
zinc cairn
the ~ symbol means similar and similar triangles have same corresponding angles
How do i find it tho
prime basalt
Let's look at angle P. It corresponds to angle J due to the similarity of the triangles. Therefore P and J both are 60 degrees. On the other hand, J = x + 10 degrees. Thus x = ? Try using similar arguments to deduce x, y, z and find the final answer
odd edgeBOT
@zinc cairn Has your question been resolved?
zinc cairn
Let's look at angle P. It corresponds to angle J due to the similarity of the tr...
So wjat do i do to it
prime basalt
Let's see what you have:
P = 60
P = J
J = 10 + x
Can you find x using this information?
regal chasm
the somme of all angles inside a triangle is 180°
so (x+10) + ( y/2 ) + (0,3*z -20) = 180
mystic saffron
I don't understand the factoring on the first blue line
toxic monolith
I don't understand the factoring on the first blue line
$\binom{n}{k}+\binom{n}{k+1}=\frac{n!}{k!\left( n-k \right)!}+\frac{n!}{\left( k+1 \right)!\left( n-k-1 \right)!}=\\=\frac{n!\left( k+1 \right)+n!\left( n-k \right)}{\left( k+1 \right)!\left( n-k \right)!}=_{\cdots }$
clever fjordBOT
Joanna Angel
odd edgeBOT
@mystic saffron Has your question been resolved?
still osprey
The exchange… as you can see Icelandic Krona and Polish Zlotych, is it best for Poland to have their PLN as less valuable for Isk?
So if 100 ISK is 2,87pln, is it profitable if 100 ISK is <2,87 (e.g 2,5)?
If it’s less zlotych, is the currency more powerful?
wooden python
what do you mean by "best"...?
wooden python
good for whom? good in what sense?
odd edgeBOT
@still osprey Has your question been resolved?
still osprey
good for whom? good in what sense?
Change rate
This ma’am doesn’t understand <@&286206848099549185>
odd edgeBOT
@still osprey Has your question been resolved?
wooden python
you're right, i don't understand.
from the viewpoint of getting more of the local currency for their own,
an Icelander visiting Poland wants a smaller ISK/PLN rate,
but a Pole visiting Iceland wants a higher ISK/PLN rate.
odd edgeBOT
@still osprey Has your question been resolved?
mystic saffron
The exchange… as you can see Icelandic Krona and Polish Zlotych, is it best for ...
The less złoty one krona is the more worth the złoty is compared to the krona
@still osprey
silver crag
For this question, I am struggling to get the correct graph. I saw that r should be .5 at an angle of 0 and pi, and 2 at an angle of pi/2 and 3pi/2, so I drew an oval, but the correct graph is more of a peanut shape. I am wondering how I can recognize that the shape will be like that for future questions?
silver crag
Correct graph
sullen ferry
there's some general rules for polar curves. You know it won't hit the origin for example because r is never 0
you need only look at a period (when it returns to where it was) to get the full picture of the curve (for just a pure trig function with a shift)
silver crag
This was my attempt
sullen ferry
well, you should notice it starts increasing from r>.5
sullen ferry
try plotting particular points
silver crag
But I guess it would decrease and the increase
Ok I think that makes sense
Thanks
.close
tardy moss
ember oak
snow hemlock
just wodnering in this question does the ' next to the "i" mean anything
snow hemlock
or is it just a mistake
snow hemlock
a misplaced comma I guess
ahh okayy thanks very much
i was just wodnering becuase its in like my offical textbook and idk if they would like make a mistake
nocturne belfry
i was just wodnering becuase its in like my offical textbook and idk if they wou...
it happens all the time
try searching for your book name here "errata"
nocturne belfry
yea
snow hemlock
its like an Alevel txtbook from Uk so dik if it would work
nocturne belfry
whATS the title of the book
snow hemlock
errata Pearson A Level Further Mathamatics Core Pure Book 1
snow hemlock
hrm what year is it
il check rq
nocturne belfry
either way, yea, people make errors all the time
there are errors in every textbook
snow hemlock
hrm what year is it
2017 it was made
is tyhere just like a list of all the listed books?
nocturne belfry
what?
snow hemlock
yk the errata command, is there a way too see all of the books that they have listed
nocturne belfry
errata command?
errata is just a part of a book that lists all the errors
its usually published online
sometimes if its just some shitty book or a newer book it doesnt exist
it requires the publisher or the author maintaining it so if its just some random book sometimes they dont bother
,w errata
clever fjordBOT
nocturne belfry
close enough wolfram lol
snow hemlock
it requires the publisher or the author maintaining it so if its just some rando...
ooooh shi yea
sorry im supid
thatnks
.close
nocturne belfry
sorry im supid
nah youre good nw
odd edgeBOT
nah youre good nw
nocturne belfry
.close
odd edgeBOT
snow hemlock
nah youre good nw
thanks again
odd edgeBOT
Available help channel!
modest badge
there is a boy thats in a moving train from the train he saw a girl he knew the girl was walking twords the train 8 seconds after the girl was at the window the boy got out of the train and followd the girl in what time will the boy ctch up with the girl if the boy is 2 times faster then the girl and the train in 5 times faster then the boy
ebon yarrow
there is a boy thats in a moving train from the train he saw a girl he knew the ...
You're gonna need some punctuation
And it's very ambiguous
odd edgeBOT
@modest badge Has your question been resolved?
mortal hinge
odd edgeBOT
mortal hinge
hi can someone help pls ? idk why they're wrong
sick wasp
what the frick, is this SAT?
mortal hinge
wdym ?
sick wasp
the test
mortal hinge
it's my homework 😭
sick wasp
ohh dayum
mortal hinge
you know how to do it ?
clever fjordBOT
lifefuel
gentle venture
moving square roots to numerators
sick wasp
right completely did not spot that
mortal hinge
$WZ=\frac{3\sqrt{2}}{2}$
still marked it wrong
sick wasp
how that Jamais vu sound?
gentle venture
still marked it wrong
does this work after all, then?
mortal hinge
yes
gentle venture
ok it's right, do i do that to the rest of them ?
probably
mortal hinge
okay let me try
gentle venture
all except the last one though, that one passes anyway for some reason
mortal hinge
what about the third one ?
mortal hinge
how that Jamais vu sound?
really good
gentle venture
what about the third one ?
rationalize the denominator
the numerator will have a radical one way or another
gentle venture
rationalize the denominator
you can do this in general by multiplying numerator and denominator by something
odd edgeBOT
odd edgeBOT
Available help channel!
austere hull
odd edgeBOT
odd edgeBOT
@austere hull Has your question been resolved?
leaden quiver
Hi @austere hull I can probably help. Are you just looking to show that the inequality is true?
austere hull
i must find x
ocean knot
Use logaritmic propperties to put x in one side
odd edgeBOT
austere hull
.reopen
odd edgeBOT
✅
austere hull
I think the last step was unnecessary
but I still don't know how to solve it
-49 * 3^x < -9*7^x
austere hull
can it be done without log?
ocean knot
Idk
leaden quiver
Ah gotcah, I will see if I can figure it out without logs
toxic monolith
can it be done without log?
this way:
toxic monolith
can it be done without log?
$3^{x+2}+7^{x}<4\cdot 7^{x-1}+34\cdot 3^{x-1}\Leftrightarrow \frac{3}{7}\cdot 7^{x}<\frac{7}{3}\cdot 3^{x}\Leftrightarrow \\\left( \frac{7}{3} \right)^{x}<\frac{49}{9}=\left( \frac{7}{3} \right)^{2}\Leftrightarrow x<2$
clever fjordBOT
Joanna Angel
austere hull
Oh i see
leaden quiver
haha yeah the key is to divide by 21 instead of multiplying by 21 in your work
bright iris
odd edgeBOT
bright iris
i have found the maxima and minima of the function when x<3
i know that the local minimum is -15
when x=3
and I got the minima of the function when x<3
and when I substitute the minima into the equation I'm taking it > -15 so for all x f(x) >-15
but I'm not getting anywhere after that
toxic monolith
but I'm not getting anywhere after that
the linear function is rigidly fixed at the point x = 3 with a value of -15, hence the polynomial on the left side of x = 3 must have a value at least equal to -15 so that there is a local minimum at the point x = 3
and this means that the inequality is satisfied:
$\lim_{x \to 3^{-}} f\left( x \right)\ge f\left( 3 \right)\Leftrightarrow \ln\left( a^{2}-3a+3 \right)\ge 0$
clever fjordBOT
Joanna Angel
odd edgeBOT
@bright iris Has your question been resolved?
marble vale
hello, how do I show that the norm function: $||.|| : (\mathbb{R}^n, ||.||) \rightarrow (\mathbb{R}, |.|)$ is continuous on $\mathbb{R}^n$?
clever fjordBOT
lilisworld
marble vale
$\forall n > 0, \exists \varepsilon > 0 , | x - a | < \varepsilon \implies | |x| - |a| | < \varepsilon$
brittle beacon
(while the definition of continuity wants the last term to be epsilon, what you've said is technically true(!))
clever fjordBOT
lilisworld
marble vale
what do i do now?
brittle beacon
do you know any properties that norms must satisfy?
marble vale
triangle inequality
brittle beacon
That's a good one, do you know what it/they state?
(there is a form that would be very useful here(!))
marble vale
wait
marble vale
(there is a form that would be very useful here(!))
idk the form
brittle beacon
idk the form
do you know the "standard" triangle inequality?
(there's a hint as to what the alternate form might be called, somewhere 
)
brittle beacon
Not quite that one 
brittle beacon
Generalised?
brittle beacon
Yep 
that's the one 
marble vale
so, $\left| | \mathbf{x} | - | \mathbf{a} | \right| \leq |\mathbf{x} - \mathbf{a} |$
clever fjordBOT
lilisworld
zenith talon
I have tan=-1
how do i find the rest of the angles?
0>angle>360
wicked kestrel
what do you mean "rest of the angles"?
zenith talon
what do you mean "rest of the angles"?
I dont even know...that was the question
stupid ah question
wicked kestrel
can you post the question's exact wording?
zenith talon
Determine the measue of the angle in standard position, where 0>angle>360, when tan angle = -1
zenith talon
can you post the question's exact wording?
@wicked kestrel
wicked kestrel
much better
next time, please start with the exact wording (a screenshot would be even better)
zenith talon
next time, please start with the exact wording (a screenshot would be even bette...
mb
wicked kestrel
!status
odd edgeBOT
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
tan -1 would be 359
wicked kestrel
why are you computing tan(-1)?
for one, that value isn't defined. On top of that, the question doesn't ask you to do so
zenith talon
so how would i getg the rest?
angle is standard position right? so where does tan fit in with that?
wicked kestrel
"the rest" implies you've already found one (or more)
zenith talon
"the rest" implies you've already found one (or more)
ok how would i find one??
wicked kestrel
generally when working with tan, you're often best going back to sin and cos (as those have values that should be easier to derive)
for example, if you know tan(x) = 1, what can you say about sin(x) and cos(x)?
odd edgeBOT
@zenith talon Has your question been resolved?
distant tartan
odd edgeBOT
distant tartan
shouldn't the k be raised to n where n is the number of rows?
(for the second statement)
or does it not matter if A and B have the same number of rows
Like this
brittle beacon
shouldn't the k be raised to n where n is the number of rows?
As a motivation for why not, note that kA is basically multiplying all of the n rows by k, so apply (2) n times
simple ether
runic swift
odd edgeBOT
Available help channel!
thin field
Fernando must deliver a market study for a client. Based on field work determines that at a price of S/46, his client would only offer 300 t-shirts, but if the price rises by S/10, the quantity to be offered would increase by 400 t-shirts; Likewise, it is known that the other market equation is determined by the equation 10p+3q=800
thin field
They asked me for the market equation and the equation of demand and offer, just that I'm not sure how to get them with the point of equilibrium
odd edgeBOT
@thin field Has your question been resolved?
thin field
<@&286206848099549185>
odd edgeBOT
@thin field Has your question been resolved?
odd edgeBOT
@thin field Has your question been resolved?
clear tree
odd edgeBOT
clear tree
Second line how did we get 1/4 outside the integral
forest sky
multiplied the entire thing by 1/4 * 4, then took the 1/4 out
forest sky
we didn't exactly divide by 1/4, instead we multiplied the inside by 4 and multiplied the outside by 1/4 (in total multiplying by 1), to get rid of the denominators inside
odd edgeBOT
@clear tree Has your question been resolved?
clear tree
factor out 1/4?
wicked temple
why doesnt my third derivative sol. match any option?
wooden python
what's the derivative of x-9?
also your handwriting (or mousewriting?) is kinda hard to read.
@wicked temple
wicked temple
its x
wooden python
no, the derivative of x - 9 isn't x.
wicked temple
i think im trippin
wooden python
you are
wooden python
yes, the derivative of x-9 is 1
by the way you do not actually need the product rule here.
wicked temple
yeah i figured
i just need to expand
and then find derivative
btw when i do product rule whyy is the ans on my 4th derivative?
winged halo
wicked temple
its all g thx
odd edgeBOT
@wicked temple Has your question been resolved?
odd edgeBOT
odd edgeBOT
Available help channel!
haughty blaze
Alright so
odd edgeBOT
Alright so
hard field
for the left, i would be nice to use x limits and for the 'height' a function of y with respect to x, really just the positive root of the circle, and for the right, maybe instead y limits, with the width x as a function of y.
chrome lynx
How do yk which one to use to find surface area?
it depends on what your exercise says
haughty blaze
It doesn't say anything I just made this problem up lol because Idk how to use the formula
$$\int_{a}^{b}2\pi xds$$
clever fjordBOT
quan
haughty blaze
I'm just wanting to know how do I know to use this for right or left
honest turtle
Surface area is: $$2\pi \int\limits_a^b f(x) \sqrt{1+(f'(x))^2} ; dx$$
haughty blaze
Surface area is: $$2\pi \int\limits_a^b f(x) \sqrt{1+(f'(x))^2} \; dx$$
for which one?
honest turtle
This is the general formula for surface area.
haughty blaze
Fr? I was taught something completely different
honest turtle
Fr? I was taught something completely different
This should be correct now.
haughty blaze
How would you apply that tho for each solid on the image I sent
Is what I don't understand tho
clever fjordBOT
Good
honest turtle
Is what I don't understand tho
You find a function that represents f(x).
honest turtle
Been a while I did something like this.
honest turtle
This do indeed have f(x) = sqrt(1-x^2). But the other one should be different.
I'm not sure how you do to get the other area, you're going to divide the upper function with the lower function to get the new function you're going to use.
haughty blaze
I'm not sure how you do to get the other area, you're going to divide the upper ...
Unless we could maybe reframe it as $f\left(x\right)=-\sqrt{1-x^{2}}+2$ for $x\in \left[0,1\right]$ and rotate about x-axis
clever fjordBOT
quan
honest turtle
You sure that those semicircles are sqrt(1-x^2)?
And again, it's been a while I did something like this, so if you want an answer, you might want to wait for a other person.
odd edgeBOT
@haughty blaze Has your question been resolved?
restive widget
trans: if matrix A ... and B ..., |A| = 5, then what's |B| = ?
restive widget
i don't even know if it's allowed to do this😭
the -10 is incomplete
i still don't know the value of 2bd - 2ac
latent scaffold
You didn't distribute the multiplications properly in the first line of your computation for |B|
But you have the right idea. You should expand all of that and group terms so that you can express |B| in terms of |A|.
brittle beacon
Alt: if you remember how row operations change the determinant of a matrix, then figure out what operations turn A into B and note how they'd have changed the determinant
restive widget
Alt: if you remember how row operations change the determinant of a matrix, then...
wait, how— english isn't my mother tongue so i have difficulty comprehending this😭
brittle beacon
Depending on whether you've been taught it, but for example swapping two rows in a matrix means you multiply the determinant by -1
Similarly multiplying the row of a matrix by a scalar, say k, means you multiply the determinant by k, and adding a scalar multiple of one row to the other doesn't change the determinant
restive widget
Depending on whether you've been taught it, but for example swapping two rows in...
you mean by changing the matrix arrangement from abcd to cdab?
brittle beacon
That matrix B is gotten from A from each of those operations being applied
brittle beacon
you mean by changing the matrix arrangement from abcd to cdab?
No, as in changing $\pmqty{ a & b \ c & d}$ to $\pmqty{c & d \ a & b}$
clever fjordBOT
@brittle beacon
restive widget
right, the unedited one was column
brittle beacon
But if you haven't learned about that, don't worry
restive widget
i just realized it
restive widget
But if you haven't learned about that, don't worry
yeaa i just knew it after you told me😅
i just taught these kekw
restive widget
You didn't distribute the multiplications properly in the first line of your com...
i just realized this💀💀💀💀💀 it's the second line i mistaken
cancel the 2cd and then -10
thank you for the help! also for the new matrix property that's just been told, i'll add it to my notes
.close
odd edgeBOT
odd edgeBOT
Available help channel!
ashen beacon
hey guys, this is my question, i haave tried solving it using different identities but am not able to, what is the answer?
undone mauve
lost helm
Hint: 2025=(45)^2
lost helm
Try to find some common factor ig
lost helm
Try to complete the square ig, that should help
You already have 2019^2. Try to get 2×2019 from 4035
jolly niche
Maybe try this
4035 = 2016 + 2019
So we have
2019^2 + 2016 + 2019
Extract the 2019
2019(2019 + 2016 + 1 )
2019(4036)
lost helm
That's much better ig
jolly niche
4036 is divisible by 2018
lost helm
Actually, wait. You can simplify it to 2018×2022. They directly cancel out from denominator
2×2019=4038, so we can write 2019^2+4038 as 2019^2+2×2019+1-4, which is (2019+1)^2-2^2, hence 2018×2022
odd edgeBOT
Available help channel!
paper ivy
absolutely no clue 😭
odd edgeBOT
absolutely no clue 😭
nocturne belfry
,rotate ccw
clever fjordBOT
nocturne belfry
what'd you try?
nocturne belfry
try turning it into a line instead
paper ivy
a line?
nocturne belfry
general rule is that we hate working with fraction
and we want to get rid of fractions if we can
multiply by x
nocturne belfry
you need to multiply everything by x
paper ivy
of
paper ivy
o
clever fjordBOT
jan Niku
nocturne belfry
$x \cdot \frac 2x + x \cdot 9 = x \cdot 16$
clever fjordBOT
jan Niku
nocturne belfry
yea
paper ivy
oh
nocturne belfry
i mean, what were really doing is this
paper ivy
i thought 2/x would be left alone cos it alr has x
nocturne belfry
$x \qty( \frac 2x + 9 ) = x \qty( 16)$
clever fjordBOT
jan Niku
nocturne belfry
were multiplyng both sides of the eqn by x
you have to do the same thing to both sides
paper ivy
okok wait
nocturne belfry
if you dont, they might not be equal any more
paper ivy
nocturne belfry
theres just one x, yea
so you just need to distribute here
its not x/x, just x
nocturne belfry
there is nothing to do here but to distribute
so we start here
then we distribute
nocturne belfry
$\frac{2x}{x} + 9x = 16x$
clever fjordBOT
jan Niku
nocturne belfry
i mean thats true in general but its not applicable here
nocturne belfry
we dont have any place where x is times x
i think youre maybe confused about the first term?
$\frac 2x$
clever fjordBOT
jan Niku
nocturne belfry
what do you get if you multiply this term by x
paper ivy
2x/x^2 no?
nocturne belfry
why?
paper ivy
cos youre multiplying the whole fraction by x
nocturne belfry
$a \times \frac bc \neq \frac{ab}{ac}$
clever fjordBOT
jan Niku
clever fjordBOT
jan Niku
nocturne belfry
then we know from fraction multiplication, we can rewrite this
$\frac{x \times 2}{1 \times x}$
clever fjordBOT
jan Niku
nocturne belfry
yea?
nocturne belfry
lets work with real numbers
paper ivy
ok
nocturne belfry
$\frac 14$
clever fjordBOT
jan Niku
nocturne belfry
what happens if you multiply this by 2
nocturne belfry
actually, $\frac 28 = \frac 14$
clever fjordBOT
jan Niku
nocturne belfry
its HALF
yea
and i mean we can write this out
$2 \times \frac 14 = \frac{2 \times 1}{4} = \frac 24 = \frac 1 2$
clever fjordBOT
jan Niku
nocturne belfry
feel comfortable with that?
nocturne belfry
why is it like that though
im not sure what answer would be satisfying here
youd maybe have to explain why youd expect it to work differently
this seems reasonable to me, like, if you have a quarter of a pie, and someone else has twice as much as you, they have half of a pie
nocturne belfry
**jan Niku**
if you feel comfortable with this
we can just switch the pieces around
$x \times \frac 2x = \frac{x \times 2}{x} = \frac{2x}{x} = \frac 2 1$
clever fjordBOT
jan Niku
paper ivy
okay
nocturne belfry
same steps:
- multiply
- combine
- simplify
- cancel common factors
nocturne belfry
by combine do u mean
id say lets just make sure we get this one term right
by combine, i mean use what we know about fraction multiplication
$\frac ab \times \frac cd = \frac{a \times c}{b \times d}$
clever fjordBOT
jan Niku
nocturne belfry
that we can 'combine' the two fractions being multiplied into a single fraction where the top and bottom are multiplied instead
paper ivy
yes
nocturne belfry
again, were just thinking of like, 2 as 2/1, and x as x/1 here
paper ivy
uhuh
clever fjordBOT
jan Niku
paper ivy
okay
nocturne belfry
lets just look at this $x \times \frac 1x = 1$
clever fjordBOT
jan Niku
nocturne belfry
say you have 1 pie
you divide it into ... 5 pieces
so each piece is $\frac 15$ of a pie
clever fjordBOT
jan Niku
nocturne belfry
how much of a pie do you have if you have 5 of these pieces?
paper ivy
1
nocturne belfry
yea
so we could write this out, our combining these pieces
$\frac 15 + \frac 15 + \frac 15+ \frac 15+\frac 15 = 1$
clever fjordBOT
jan Niku
nocturne belfry
right, 5 pieces, each one is 1/5 of a pie
paper ivy
yes
nocturne belfry
we add them all up, it totals one pie
paper ivy
yes
nocturne belfry
we can write this a different way
$5 \times \frac 15 = \frac 15 + \frac 15 + \frac 15+ \frac 15+\frac 15 = 1$
clever fjordBOT
jan Niku
nocturne belfry
because repeated addition is just multiplication, right?
paper ivy
uh
clever fjordBOT
jan Niku
nocturne belfry
seems reasonable right?
what would happen if we divided it into 7 pieces instead?
$\frac 17 +\frac 17 +\frac 17 +\frac 17 +\frac 17 +\frac 17 +\frac 17 = 1$
clever fjordBOT
jan Niku
nocturne belfry
agreed?
if we have 7 pieces of a pie thats been divided into 7 pieces, we have the whole pie
nocturne belfry
so, the same thing happens
$7 \times \frac 17 = \frac 17 +\frac 17 +\frac 17 +\frac 17 +\frac 17 +\frac 17 +\frac 17 = 1$
clever fjordBOT
jan Niku
nocturne belfry
and $7 \times \frac 17 = 1$
clever fjordBOT
jan Niku
nocturne belfry
can you see where this is going?
paper ivy
not long
nocturne belfry
it seems like you have all the necessary pieces floating around in your head
paper ivy
maybe
nocturne belfry
you are just having a hard time seeing them together
paper ivy
an hour n half
i sobbed over an angle question earlier
do u think im gonna pass this test
nocturne belfry
well IDK you can do as well as you can do
pushing yourself past your limit isnt going to help
paper ivy
yeah youre right
nocturne belfry
its good to take breaks and stuff it helps your head digest information
paper ivy
i'll pause for a bit
nocturne belfry
not you know 2 hours but a 10 minute break drink some water go outside
can really help
paper ivy
i'll have supper
then try again
thank you for trying to help
ive kept u for nearly an hour
paper ivy
sorry i couldn't get it
thank you for your time
if it's not too much of a bother, do u reckon u could try helping me again later if i don't piece it together
royal steppe
whats the best way to solvethese types of fractions (2x^2-5x-6)/(3x+1) = (x^2-3x+3)/(5x) ?
sterile blaze
first go with the assumption that the denominators are non zero and multiply both sides by both denominators to get rid of them
distant tartan
.close
odd edgeBOT
solid moth
wouldnt B be 3/5
it says that the first ball drawn isn't put back, so, supposedly, you have 5 balls left with 2 being red and 3 being green
latent scaffold
it says that the first ball drawn isn't put back, so, supposedly, you have 5 bal...
Granted the first one was red though. If it's green you need to adjust.
solid moth
Granted the first one was red though. If it's green you need to adjust.
balls numbered 1-3 are red
it says it has the number 1 on it?..
latent scaffold
They're two different events.
If the second ball is green, then it means that either the first is red and the second is green or the first is green and the second is green as well.
So all in all it's equivalent to 1/2.
clever fjordBOT
latent scaffold
It's P(2nd green) = P(first red)*P(green | first red) + P(first green)*P(green | first green)
There are two ways to get a green ball on your second draw.
Either you picked red on the first, or you picked green on the first.
Those are mutually exclusive, so you can add the probabilities to get the probability of getting a green ball second.
Both events alter the next draw differently though.
For the first event (you pick red first, then green), you get 3/6 to pick red, and then 3/5 to get the green on once the red ball has been picked.
So for that event, you get 3/6 * 3/5 = 3/10
Okok
Oh sorry I didn't really see you wanted that probability.
Let me check
Well P(AnB) = P(B) * P(A | B). P(B) is 1/2 as we figured out. Now P(A|B) represents the probability of picking the ball #1 first granted we picked a green ball second. I think it should be 1/6 since picking a green ball second doesn't alter the probability of the A event.
I just rewrote this equation to isolate P(AnB), You can compute P(B|A) still using logic. I think it's easier going the P(A) route actually.
P(A) = 1/6
P(B | A) is the probability of picking a green ball second granted we picked the #1 ball first, which is 3/5
In probability, AnB is the interfacing of events happening in both. This is not empty, because you can certainly pick the first ball to be #1 while picking a green ball second.
I think so yeah.
P(A | B) is actually hard to decide because A happens before B
So it's weird to think about.
P(B | A) is easier in that sense.
P(AuB) = P(A) + P(B) - P(AnB)
You can only add them when they are mutually exclusive (P(AnB) = 0 )
Yeah
You can compute it because the formula for P(AnB) is symmetric, so you can switch the point of view
P(AnB) = P(A)*P(B | A) = P(B)* P(A | B)
So P(A | B) = P(A) * P(B | A) / P(B)
clever fjordBOT
latent scaffold
Let me redo this in my head one sec
I think P(AnB) = 1/10
Going from the beginning, we have P(A) = 1/6, P(B) = 1/2.
Then, P(B | A) = 3/5, so we use P(A n B) = P(A) * P(B | A) = 1/6 * 3/5 = 1/10
Then we have P(A | B) = P(A n B) / P(B) = (1/10) / (1/2) = 1/5
It's not 3 ?
Do you mean why P(B | A) = 3/5?
This one is just computed from context, since if you pick a #1 ball first (event A) , then the second ball has 3/5 chances of being green (event B).
latent scaffold
Then, P(B | A) = 3/5, so we use P(A n B) = P(A) * P(B | A) = 1/6 * 3/5 = 1/10
That was computed using the formula for P(B | A)
Hahahah okay
If you look at the formulas I sent, they're just a way of "linking" together P(AnB), P(A|B) and P(B|A).
So you can use those to compute, but it doesn't mean that you can't compute them on your own from context.
P(B | A) is the probabiility of B happening given we know A happened.
In this context, A happening means that we picked #1 first.
Yes exactly!
Then you can go and use the formulas to find the values you don't know.
It's especially useful because sometimes the conditional probabilities are rather hard to think about.
See, for instance, in this case, it's not immediately clear why P(A | B) = 1/5
That's because P(A | B) asks what is the probability of picking ball #1 first given we picked a green ball second.
But this is hard to do because in reality A happens before we pick the second ball.
But it has meaning, it's still the probability of picking #1 first given we KNOW we're going to pick another green ball after.
It's just hard to think about.
But in contrast, P(B | A) is much easier to do
So you can go this path and then use the formulas to compute P(A | B) in a much easier way./
My pleasure, all the best for your studies.
clever fjordBOT
Umbraleviathan
reef sandal
The equation of a tangent line
You know that (3,8) must exist on this line, so
L(3) = 8
$$8 = f(x_0) + f'(x_0)(3-x_0)$$
clever fjordBOT
Umbraleviathan
reef sandal
You just gotta find x0 and f(x0)
Wdym first term
No
Well i mean if you're going from point slope form to the equation I showed
You'd be subtracting -f(x0) which is just adding f(x0)...
reef sandal
**Umbraleviathan**
_ _
You know that f(x) = x^2
Which implies f'(x) = 2x
Just replace f(x0) and f'(x0) with appropriate substitutions with the info i gave you
brittle beacon
Because the "x" in that "2x" you wrote, and the "x" for the line equation needn't be the same
In particular you created a quadratic equation for a general point on the line
brittle beacon
**Umbraleviathan**
Hence umbra did it like this - you are tangent to y = x^2 at the point (x0, f(x0)), and you want to find what that x0 is
You are given that you're tangent to y = x^2 at only one "guaranteed" point, not necessarily all of them (and in fact not all of them at all, but anyways-)
You just know at best, there's some point, call it (x0, x0^2) where you have the line tangent to, and that said tangent also passes through (3,8)
Only when x is at the point where you're tangent, not at all points
reef sandal
Because (3,8) is not on the parabola
brittle beacon
You need to be careful about distinguishing a general point on the line, and the point of tangency
Because otherwise you'd imply the gradient is not constant - but it is
The 2x only is for one point - you'd otherwise be saying that at (3, 8), the gradient is 6, but that does not need to be true
And as per before, you're basically implying a line is a parabola (which they're not)
(True, but you don't need to meet the parabola everywhere, just once (and it happens only once))
Take another point on the line - you cannot also represent it with x and y, because you used (x,y) to represent the point you're tangent at
One way is to say that the point you're tangent is, say, $(x_0, y_0)$, to empathise that it's a fixed point and doesn't vary (and as you know it's on the parabola $y = x^2$, you know that $y_0 = x_0^2$ too)
clever fjordBOT
@brittle beacon
brittle beacon
**Umbraleviathan**
Then as per before, this formula allows you to figure out a relation (with f(x0) replaced with x0^2) between points)
You're basically using that you're passing through two points and that one of them forms the gradient
It is - but you're noting that a genreal point on the line needn't form the gradient for it
Either you note that the x and y there are the point of tangency, or otherwise make sure you're aware
Yep at said tangency point!
Basically yea - what you have is fine but make sure you bear in mind that (x,y) is a fixed point, and doesn’t vary
You can’t e.g. go elsewhere and use x and y again (I kinda see you’re using it for one point, thought you were using x and y for a general point on the line)
I mean you don’t seem to reuse it - but I’d for example state that instead of “solv[ing] for [the] intersection”, that it is the point of intersection
odd edgeBOT
@mystic saffron Has your question been resolved?
odd edgeBOT
@mystic saffron Has your question been resolved?
brittle beacon
Having a quadratic in constant values is fine (and necessitated by a point being on the curve y = x^2)
(As long as you don’t imply that a general point on the curve will satisfy a quadratic equation - which was the comment before)
odd edgeBOT
@mystic saffron Has your question been resolved?
round jolt
quick question: If I have a pareto chart and I am trying to find the largest contributers to percentage with the 80/20 rule and it inersects a column in half not fully. Do I include that column or not?
odd edgeBOT
quick question: If I have a pareto chart and I am trying to find the largest con...
@round jolt Has your question been resolved?
round jolt
.close
old whale
Can you use trig sub to do this?
odd edgeBOT
Can you use trig sub to do this?
old whale
sullen ferry
Can you use trig sub to do this?
ye :)
but a u-sub approach would probably work out nicer
old whale
How would u sub work
old whale
but a u-sub approach would probably work out nicer
Can you walk me through it I am not sure what could be u subbed here
Oh wait
Nvm
Wait
No I’m still
Yeah
What can be u subbed her
sullen ferry
x=
old whale
What
sullen ferry
???
old whale
Cause u is a constant
sullen ferry
no..
sullen ferry
ok, so you have
old whale
Yes
sullen ferry
$du=\f{x}{\sqrt{x^2+16}}dx$
clever fjordBOT
Moosey
old whale
When do people usually learn this lol
sullen ferry
When do people usually learn this lol
depends
old whale
Ok
sullen ferry
it's mainly in calc 1-2
old whale
HEY REIMENN
ok
Interesting
Wait what even do people learn in calculus two
Integrals?
old whale
more about integrals, taylor series, etc.
Oh gosh
clever fjordBOT
Moosey
sullen ferry
$u=\sqrt{x^{2}+16}$
clever fjordBOT
Moosey
sullen ferry
$x=\sqrt{u^{2}-16}$