i will apply my anwer to the previous question i answered earlier
#help-27
zenith prism
balmy quest
$Area=36y^2+60yz+25z^2=(6y)^2+2\times6y\times 5 z+(5 z)^2=(6y+5z)^2$
woven radishBOT
秋水
devout snowBOT
hybrid snow
Nice
devout snowBOT
Available help channel!
bleak edge
What is the remainder when $31^{70} + 211^{49}$ is divided by 342
woven radishBOT
Damnatio Odiosis
bleak edge
I have tried a bit of fermats little theorem and a basic number theory approach, but that doesnt seem to be working
stone stump
hmm maybe try it first mod the factors of 342 and then CRT?
bleak edge
hm, one sec
what i notice is 341 (predecessor) is exactly 31 x 11 and also a pseudoprime
maybe that can help in connecting the remainders with 342
stone stump
I dont know any theorems which connects those to each other
bleak edge
(a+1)^n always gives a remainder of 1 when divided by a
maybe this
there are two more with (a-1)
stone stump
well yes ok I meant more a theorem of the form mod a tells us something about mod a+1. cause it seemed like you wanted to use that
surreal jungle
Wait whats 1+1
devout snowBOT
Wait whats 1+1
neon folio
about 35
surreal jungle
Ok thanks
neon folio
np
main gull
Wait whats 1+1
Don't troll
restive river
Can I take {4,4} as a subset of A={1,2,3,4}?
devout snowBOT
Can I take {4,4} as a subset of A={1,2,3,4}?
sonic smelt
Yeah, {4} is a subset of {1, 2, 3, 4}
restive river
yee
{4,4} is just the set of {4} because sets are unique collections
idk in what circc umstance you would even start with {4,4} tbh
devout snowBOT
@restive river Has your question been resolved?
prisma frost
At the B part. What does it mean
devout snowBOT
At the B part. What does it mean
devout snowBOT
@prisma frost Has your question been resolved?
tame palm
,rotate
woven radishBOT
tame palm
And why do they do what they do
They are explaining how to solve the problem given the condition that the Imaginary part of the solution is 0i.
If
z^2 + 2z = a + 0i
(z^2 + 2z is real, therefore the imaginary part is 0i.)
then
(8a + 8)i
can be set equal to zero to solve for a.
8a + 8 = 0
a = -1.
rare roost
devout snowBOT
rare roost
Could someone explain to me what the second part of the question is asking?
topaz axle
Could someone explain to me what the second part of the question is asking?
got it?
devout snowBOT
@rare roost Has your question been resolved?
topaz axle
I don't know the solution for a but b) is certainly that times 5c3
rare roost
c++ vector of strings constructor
c++ check if value in vector
Ohhhhhhhh
Sorry XD I'm in the wrong chat window
umbral roost
How would you go about solving this?
devout snowBOT
How would you go about solving this?
supple knot
How would you go about solving this?
Write your vector as a linear combination of e1 and e2
pseudo basin
you are given enough info to reconstruct what T does to any vector in R^2
and in fact you will be able to do so very easily since {e1, e2} is in fact the standard basis of R^2
scenic ether
@supple knotwhat is ur pfp?
supple knot
<@261933205387477002>what is ur pfp?
Don't spam other people's help channels
scenic ether
Don't spam other people's help channels
oh ok
so that is what its called
@supple knotno seriously what is it
supple knot
<@261933205387477002>no seriously what is it
Seriously stop spamming Stefan's help channel
umbral roost
you are given enough info to reconstruct what T does to **any** vector in R^2
I am still unsure.
pseudo basin
I am still unsure.
do you know what a linear transformation is?
umbral roost
yes
pseudo basin
if you do, then it should be clear that once you know $T\bd{e}_1$ and $T\bd{e}_2$ (which you do) you also know $T(\alpha \bd{e}_1 + \beta \bd{e}_2)$ for any scalars $\alpha$ and $\beta$
woven radishBOT
Ann
pseudo basin
and every vector in $\bR^2$ can be put into the form $\alpha \bd{e}_1 + \beta \bd{e}_2$, and in fact such a decomposition is always unique
woven radishBOT
Ann
pseudo basin
(this last message is me spelling out explicitly what in the jargon of linear algebra is concisely described by the word "basis")
pseudo basin
for some definition of "finished", i did
what's with the "jesus" interjection though? did i go too hard?
umbral roost
i said jesus because I feel like I should be getting this quicker
and it concerns me
pseudo basin
is this your first time doing linalg
umbral roost
I mean we started 2 weeks ago, and I couldnt follow well in class because I missed some lectures. So now I am revising everything we did. Exam is in 2 weeks.
pseudo basin
you started two weeks ago and the exam is in two weeks?
you're being given an exam over what amounts to like, a month's worth of material?
pseudo basin
in any case, for a first course in linear algebra it's normal if you have trouble getting things down quickly
i did try to lubricate it a little but it can still be a lot to take in
pseudo basin
wdym
as in do i offer tutoring services
it's kind of complicated and for personal reasons i don't do that at the moment, but probably in the near future i might
vivid barn
devout snowBOT
vivid barn
I have found the equations and the three lines. The y goes from 3x to 9. The x starts from 0 but im not sure what the top limit is.
sonic smelt
Clearly x goes from 0 to 3
vivid barn
Clearly x goes from 0 to 3
Dont I have to use the equation to determine the point of intersection with the y=3x line?
I already tried 3 and it didnt work
sonic smelt
Not really, you either have x going from 0 to 3 and y going from 3x to 9 or y going from 0 to 9 and x going from y/3 to 3
vivid barn
Not really, you either have x going from 0 to 3 and y going from 3x to 9 or y go...
The equation is x+y correct? because I usually just had y = ? or f(x) = ?
sonic smelt
Here you have a double integral and the integrand is (x + y)
vivid barn
Here you have a double integral and the integrand is (x + y)
Ill try it again. Thank you
.close
plain furnace
Chang invested his savings in two investment funds. The $3000 that he invested in Fund A returned a 2% profit. The amount that he invested in Fund B returned a 7% profit. How much did he invest in Fund B, if both funds together returned a 4% profit?
devout snowBOT
Chang invested his savings in two investment funds. The $3000 that he invested i...
plain furnace
how do i solve this
long kettle
What have you tried
plain furnace
nothing i dont know how to solve it
long kettle
Then let's start by naming a variable
Let's say x is the amount he invested in Fund B
How much money did he invest in total?
plain furnace
3000
long kettle
Just 3000?
That's how much he invested in Fund A
I'm talking about both funds together
plain furnace
well we dont know how much he invested in b
long kettle
If I had 2 apples and then I added x apples, I'd have 2 + x apples
If he invested 3000 dollars in Fund A and also invested x dollars in Fund B, how much did he invest in total
plain furnace
3000x?
long kettle
Why are you multiplying them
plain furnace
i dont know x how did you find x
long kettle
We aren't finding x yet
long kettle
If I had 2 apples and then I added x apples, I'd have 2 + x apples
We didn't find x here
plain furnace
3000 + x?
long kettle
What happened to the x?
plain furnace
3120x
long kettle
If you invested y dollars, and you made a 4% profit, how much money did you make in terms of y?
long kettle
We're doing a bit of a detour to help you understand what we're supposed to do
Let's say I invested y dollars in a fund, and that fund made a 4% profit
How much money did I make
plain furnace
youd make a 4% profit of whatever you invested being y
long kettle
And what is that as an expression?
plain furnace
y + 4%?
long kettle
So if I invested 100 dollars with a 4% profit, you're saying I made 100 + 4 = 104 dollars?
plain furnace
ye
long kettle
No
If I invested 3000 dollars, and I made a 4% profit, I'm not making 3004 dollars
What's 4% of y
plain furnace
what was y again?
long kettle
Just a variable
plain furnace
y + .04?
long kettle
So 4% of 100 is 100 + 0.04 = 100.04?
plain furnace
prolly not
plain furnace
5?
plain furnace
125 . 0.04
plain furnace
y . 0.04?
long kettle
Yep. You can also write it 0.04y
Now back to the original problem
We invested 3000 + x dollars and made a 4% profit
Whats 4% of 3000 + x?
plain furnace
120x?
long kettle
Why?
long kettle
To find 4% of a number, we multiply by 0.04, right?
plain furnace
yes
long kettle
What's 0.04(3000 + x)?
plain furnace
120 + 0.04x?
long kettle
Yes
That's the total profits
Remember that expression, we'll need it later
Now for a slight detour
He invested 3000 dollars into Fund A and made a 2% profit
How much profit did he make from Fund A alone?
plain furnace
60$?
long kettle
,calc 0.02*3000
woven radishBOT
Result:
60
Yep
In addition, he invested x dollars in Fund B and returned a 7% profit
How much profit did he make from Fund B alone?
plain furnace
0.07 . X
long kettle
Yes
And if he made a profit of 60 from Fund A and a profit of 0.07x from Fund B, how much profit did he make in total?
plain furnace
270$?
long kettle
Where'd you get that number from?
plain furnace
i did 0.07 . 3000 then got 210 then added 210 +60
long kettle
Whyd you find 7% of 3000?
We don't need that
Fund A made a profit of 60
Fund B made a profit of 0.07x
How much did they make combined
plain furnace
idk
long kettle
If I have 60 apples, and then I add 0.07x apples, how many apples do I have in total
plain furnace
64
long kettle
Why 64?
plain furnace
60 . 0.07
long kettle
If I have 3 apples and I add 4 apples, how many apples do I have
plain furnace
7
long kettle
How'd you find that?
plain furnace
3 + 4 = 7
long kettle
Yes, you add them
So if you have 60 apples and you add 0.07x apples, how many apples do you have in total
plain furnace
60.07
plain furnace
60.07x
plain furnace
...
long kettle
That's the total profits
But the total profits is also 120 + 0.04x
So they must be equal
What do you get when you set the expressions equal to each other
plain furnace
i dont know how to do that
plain furnace
x is 2000?
agile spire
Am I on the right track ? I’m stumped on integrated tanx^2
devout snowBOT
Am I on the right track ? I’m stumped on integrated tanx^2
agile spire
lone ravine
So you need to integrate tan^2 x? Do you know the derivative of tan x?
lone ravine
No the derivative
lone ravine
Or if you write it differently: tan^2 x +1
lone ravine
Ok, if you still need help, feel free to ask
agile spire
I think I got it
Hard to remember to keep those identities in mind throughout entire problem!
.close
lone ravine
I think I got it
Isn't the identity tan^2 = sec^2 - 1
devout snowBOT
Isn't the identity tan^2 = sec^2 - 1
lone ravine
.close
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Available help channel!
restive river
the question is ''which is not equivalent to this proposition?''
restive river
restive river
the question is ''which is not equivalent to this proposition?''
i couldnt even understand what should i do lmao
gleaming socket
Do you know what is equivalent
restive river
yes
gleaming socket
Ok do you know about truth tables
restive river
yep but we didnt learn abt <=>'s (idk what its called in english) equivalent
restive river
aaah the <=> symbol's equivalent? like opposite ver. of it 
i cant explain oms
gleaming socket
What is the symbol of equivalent according to you?
restive river
i didnt mean there is a symbol of equivalent
im trying to say idk that symbol's opposite version
arent these symbols equivalent of each other? and im trying to say idk if theres an equivalent of <=>, like these
restive river
Fun fact: <=> is the symbol of equivalence!
ooo i didnt know abt it 
/srs
restive river
Oh I don't think there is
i see
gleaming socket
I am not sure but crossing it with "/" might mean the opposite. The same way "=/=" is opposite of "="
But that's not relevant here
Sorry I'm new to logic
I think brackets help, like p<=>q means they are equivalent and (p<=>q)' not equivalent
Makes sense?
restive river
Makes sense?
mm yeah
gleaming socket
Do you know what the V underlined symbol means?
I'm seeing that for the first time
restive river
Do you know what the V underlined symbol means?
yes
gleaming socket
Interesting
restive river
it has the opposite truth table w/ <=> but idk
gleaming socket
And I'm assuming p' means "not p"?
gleaming socket
Because I am familiar with ~p
restive river
i didnt see it be4 :O
gleaming socket
There's another symbol also which means the same thing
But it's irrelevant here
Ok so you need to find [(p<=>q) <=> which of A, B, C or D]'
gleaming socket
Yeah
So create a one giant truth table and basically do all of them lol
That's the only way I know
restive river
oh i'll try wait
restive river
the question is ''which is not equivalent to this proposition?''
<@&286206848099549185> how can i solve it faster, like instead of making a truth table
pls ping when answering. im afk.
help
devout snowBOT
@restive river Has your question been resolved?
devout snowBOT
@restive river Has your question been resolved?
devout snowBOT
@restive river Has your question been resolved?
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Available help channel!
winter hound
If $log_{9}15 = m, express log_{5}9 in terms of m.
devout snowBOT
If $log_{9}15 = m, express log_{5}9 in terms of m.
winter hound
Express log_5 of 9 in terms of m.
I got my working until
$\frac{1}{m} log_{5} 15 = log_{5} 9$
woven radishBOT
notzeussz
devout snowBOT
@winter hound Has your question been resolved?
winter hound
<@&286206848099549185>
jolly leaf
devout snowBOT
jolly leaf
The question is the first line
umbral creek
solving?
jolly leaf
The answer key shows 4 answers but i only got pi/2
jolly leaf
Thats not aloud?
umbral creek
when solving trig eqns don’t cancel it
you’ll lose answers
instead just factor out
like you know if you have
(X+1)(X-1)=0
x+1 =0
or x-1 =0
jolly leaf
oh okay let me try that
umbral creek
okay
red quartz
devout snowBOT
wild creek
yeah
red quartz
I’m not sure about after
wild creek
heres the first step
do you see how to apply the equivalence between square root and power 1/2?
red quartz
No I dont
wild creek
doo you get the first line with the sqrt(x * sqrt(x))?
wild creek
its also true
but you can also replace that big square root
if you call b whats inside the square root
you have sqrt(b)
which is just b^1/2 right?
red quartz
Yes
wild creek
thats what i did here roughly
red quartz
Mmm some
wild creek
what would that give you?
devout snowBOT
devout snowBOT
Available help channel!
jovial mauve
Let X $\sim \text{Normal}(\mu, \sigma^2)$. What will be the distribution of $aX+b$ where a and b are constants?
devout snowBOT
Let X $\sim \text{Normal}(\mu, \sigma^2)$. What will be the distribution of $aX+...
woven radishBOT
QuantumBee
jovial mauve
I needed your help
Thanks for coming the right time
I'm only sure the distribution of Y=aX+b will also be a normal distribution
Normal distributions are not monotonic, so I also cannot use the formula I learnt
unkempt tree
when to use a sub n and s sub n
pseudo basin
@unkempt tree don't intrude in other ppl's channels
pseudo basin
Normal distributions are not monotonic, so I also cannot use the formula I learn...
what do you mean by a "monotonic distribution"?
jovial mauve
not montonic distributions.
pseudo basin
"Normal distributions are not monotonic"
jovial mauve
functions which either strictly increases or strictly decreases
pseudo basin
it appears that i couldn't communicate to you the issue in your wording, but okay...
do you claim that the function x ↦ ax + b fails to be monotonic?
jovial mauve
The graph of any normal distribution increases and peaks at the mean value and starts to decrease, so any normal distribution is not monotonic.
jovial mauve
**QuantumBee**
If only they were monotonic, I could use a theorem or formula I learnt back then to get the distribution.
pseudo basin
can you state the formula again?
i think you are misremembering it
or rather, misremembering its condition of applicability
jovial mauve
For any function g(x) being monotonic for $x \in supp(X)$ with derivative $g'(x)$. Then, the PDF of Y=g(X) will be, $$f_{Y}(y)=\frac{f_{X}(g^{-1}(y))}{|g'(g^{-1}(y))|}$$
woven radishBOT
QuantumBee
pseudo basin
you do realize that this DOESN'T require the PDF of your random variable to be monotonic, right? only the function by which you transform the variable itself is required to be monotonic
in fact, it is impossible to have a function be monotonic on the entire number line and also be the PDF of a random variable...
jovial mauve
Alright, my bad
pseudo basin
that and also you do not even need this formula
jovial mauve
Sorry
pseudo basin
a gaussian random variable's distribution is described entirely by its mean and stdev, and both of those transform in straightforward ways when you pass from X to aX+b
woven radishBOT
QuantumBee
jovial mauve
OK I have no idea what I'm doing
pseudo basin
$X \sim N(\mu, \sigma^2)$. what are the mean and stdev of $aX$?
woven radishBOT
Ann
jovial mauve
Mean=$a \mu$ and stdev=$|a| \sigma$
pseudo basin
one of these is correct and the other is not
and i realize i would've done better to ask for variance, but still...
woven radishBOT
QuantumBee
jovial mauve
Var(X)=$a^2 \cdot \sigma^2$
woven radishBOT
QuantumBee
woven radishBOT
Ann
pseudo basin
can you now tell me the mean and variance of aX + b?
woven radishBOT
QuantumBee
devout snowBOT
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Available help channel!
coral wraith
@pseudo basin what do i do next?
devout snowBOT
<@183668144404037632> what do i do next?
pseudo basin
undo the screwups you just did
you have either misread or ignored my directions in which i explicitly stated the value of c
pseudo basin
there are still more values you can calculate by applying similar logic to what i described to different choices of known products
coral wraith
if so, i dont know which values
pseudo basin
think a little about what known products i made use of
coral wraith
think a little about what known products i made use of
ab=1/2, de=4, gh=1/2
pseudo basin
no, that is not what i meant.
those products were not known at the beginning of the problem; they were derived.
coral wraith
oh that each product of 2x2 square = 2 and products of row and column = 1
pseudo basin
now you are being a bit too unspecific
coral wraith
now you are being a bit too unspecific
The product of numbers in each row and each column is equal to 1, and product of numbers in each 2x2 square is equal to 2.
pseudo basin
you repeated the last thing you said
coral wraith
what did u mean by unspecific
pseudo basin
there are four 2×2 squares and 6 rows/columns in the grid
i did not use all of them
perhaps it would do you some good to look at how the ones i did use are positioned on the grid
pseudo basin
among other ones, yes.
coral wraith
abde/degh=2/2=1
ab/gh=1
pseudo basin
try seeing what happens when you use other pairs positioned like this
maybe something good will come of it
or maybe, yknow, the setup is symmetric so you won't have to do the same calculations with different letters a thousand times
coral wraith
or maybe, yknow, the setup is _symmetric_ so you won't have to do the same calcu...
are you saying a=2, d=1/2, etc? im not suree...
maybe...
pseudo basin
ok sure now you got bhe^2 = 8
this is a good step towards finding the value of e, which is what you were asked for
coral wraith
are you saying a=2, d=1/2, etc? im not suree...
is this true?
bc im not sure how to continue
i hate to be annoying, but u there @pseudo basin ?
pseudo basin
are you unable to continue without presenting every single assertion of yours for validation by me?
i don't want to keep pointing out the obvious to you, but you are given that
the product of each column is 1
nimble hemlock
hello
devout snowBOT
@nimble hemlock Has your question been resolved?
devout snowBOT
@nimble hemlock Has your question been resolved?
devout snowBOT
@nimble hemlock Has your question been resolved?
devout snowBOT
@nimble hemlock Has your question been resolved?
restive river
I have a question. This is regarding the equation 6d-11/2=2d-13/2. My original answer was ¼ as I gave 13/2 giving -11/2 giving me 1. After this I would take 6d from the 2d on the other side giving me -4d. I divide both sides giving me d=4. Khan Academy disagrees and does the equation in a different order giving 11/2 to the 13/2 instead. Changing the final answer to -¼ instead. I am confused by the order you are supposed to do problems like this.
devout snowBOT
I have a question. This is regarding the equation 6d-11/2=2d-13/2. My original a...
restive river
winter patrol
order doesn't matter as long as the manipulation is valid
supposedly you had reached
1 = -4d
right?
restive river
yes
winter patrol
what exactly did you divide by after that
restive river
Oh dang just figured it out. I was dividing the wrong side. Oops. Thanks anyways.
.close
inland seal
Does anyone know a tool that can solve a system of equations with 2 parameters? (4 equations, 4 unknowns)
devout snowBOT
Does anyone know a tool that can solve a system of equations with 2 parameters? ...
inland seal
Wolfram alpha doesn't seem to work
inland seal
I typed the equations, separated by a comma
Wait
x + ay + az + (a-b)w = b+1, x + (a+1)y + (a+b)z + (2a-b)w = a+b+1, 3x + 3ay + (3a+b)z + (3a-b)w = 4b+3, x + ay + az = 2b
main gull
And what are you solving for?
inland seal
x, y, z, w
main gull
Let's see if this works
,w NSolve[{x + ay + az + (a-b)w = b+1, x + (a+1)y + (a+b)z + (2a-b)w = a+b+1 , 3x + 3ay + (3a+b)z + (3a-b)w = 4b+3, x + ay + az = 2b}, {x, y, z, w}]
woven radishBOT
main gull
@inland seal
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Available help channel!
vapid marsh
how do i find the answer to 5x8 and what is the most efficient way to figure it out
devout snowBOT
how do i find the answer to 5x8 and what is the most efficient way to figure it ...
restive river
Help anyone? Either one.
vapid marsh
.close
errant whale
devout snowBOT
errant whale
help
devout snowBOT
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devout snowBOT
errant whale
.reopen
devout snowBOT
✅
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Available help channel!
umbral creek
hi
buoyant heron
equals ye
umbral creek
so the answer would look something like
k= something
and k= something
it’s not a range right
umbral creek
oh nono
sorry I forgot to mention
it’s a carry on from a previous part
it gets to $(x-k)(x^2+kx+k^2)=(x-k)(2x-1)$
woven radishBOT
duckiescute!
arctic field
still
it will be a range of values of k
because its a cubic
there will always be at least 1 real root
woven radishBOT
arctic field
i.e. when b^2 - 4ac < 0
devout snowBOT
@umbral creek Has your question been resolved?
umbral creek
i.e. when b^2 - 4ac < 0
ohh
devout snowBOT
@umbral creek Has your question been resolved?
jade fox
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restive river
My fault didn’t know
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polar apex
A minivan puts on the brakes and comes to
a stop in 12 seconds. If it took 200 meters
to stop, and decelerates at 10 meters per
second2
, how fast was it originally going, in
meters per second?
polar apex
Help
restive river
I don't understand, but at the same time I feel that the equation is wrong.
devout snowBOT
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autumn grove
devout snowBOT
sonic smelt
That's not what you'd get using quotient rule nor power rule properly
quaint citrus
Check sign
autumn grove
Check sign
14/x^3 + 1
autumn grove
14+x^3 / x^3
quaint citrus
Ye
Now u gotta find where that equals 0 or where it’s undefined
When is it undefined
Like what value does the denominator take for it to be undefined
autumn grove
0
winter patrol
misconception
quaint citrus
?
winter patrol
the original function would need to be defined there for there to be a critical point there
winter patrol
pretty much just solve
14+x^3 = 0
quaint citrus
So setting x^3 = 0 wouldn’t work because 0 isn’t in the domain of g? @winter patrol
winter patrol
So setting x^3 = 0 wouldn’t work because 0 isn’t in the domain of g?
yeh
autumn grove
so that's DNE?
quaint citrus
Ye ig there’s only 1 crit number
autumn grove
not sure why DNE was wrong then
quaint citrus
So u have cbrt-14 and DNE
Maybe it wanted it in the 2nd spot?
Webassign weird sometimes
autumn grove
i feel like the cbrt can't be a negative number
quaint citrus
It can
What’s cbrt(-1)
What happens if u multiply a negative number by itself 3 times? U get a negative
autumn grove
so is sqrt the only one that cant be a negative
autumn grove
o
quaint citrus
What abt 4th root
What happens when u multiply a negative number by itself 4 times
Or 6 times
Or 8 times
Always positive
quaint citrus
Sure
quaint citrus
first find g’ then set it = 0
If g’=0 somewhere in the [0, 2pi) interval, then it’s an answer
If g’=0 but it’s outside the interval, then it’s not an answer
autumn grove
yeah bit stuck here
quaint citrus
Well u wanna isolate the cos term so divide both sides by 10
autumn grove
Got it
quaint citrus
Now what do u get
autumn grove
10x = pi
autumn grove
x = pi/10
autumn grove
nope
quaint citrus
U put in “1”?
autumn grove
quaint citrus
Gimme a sec Lemme double check
Alr so I found this online for a problem with 8x instead of 10x, take a look
So we did the derivative right and everything
But our answers shd be (pi+2pi*n)/10
That’s how the function touches the x-axis 10 times for 10 solutions
autumn grove
that's so weird
quaint citrus
Hold on 1 sec
So looks like since we have cos10x
We get 10 solutions
Take a look
They all fall in the interval because they are divided by 6
Do u see now
@autumn grove
so arccos(-1) is pi, 3pi, 5pi, etc
But since we divide by 10, the answers fall in our interval
Does that make sense
Well I gtg just tag if u have questions
cunning stream
need help finding the derivative of y=9e^-x + e^4x. I think I know how to use the product rule to differentiate e^4x, but idk how to do 9e^-x
devout snowBOT
need help finding the derivative of y=9e^-x + e^4x. I think I know how to use th...
hybrid snow
Chain rule
Recall that the derivative of e^x is itself
But you have to use chainrule
"Slap slap slap clap clap clap" - my man Bolbi
cunning stream
"Slap slap slap clap clap clap" - my man Bolbi
glad to see another bolbi fan
stone stump
-x = (-1)*x. so its really the same thing as 4*x or any other multiple of x
cunning stream
could you explain in further detail how to use the chain rule here? my teacher didn't actually teach it despite it being on my homework
hot thicket
d/dx (9e^-x)
9 d/dx (e^-x)
we know the der of e^x
let e^-x be e^x
this becomes
9e^-x d/dx (-x)
which gives us -9e^-x
cunning stream
let e^-x be e^x
so are we breaking it down into d/dx(e^x) and d/dx(-x)?
hot thicket
yes
cunning stream
so why does the power of e remain -x?
okay i looked stuff up and i think i kinda get the chain rule. the derivative of e^x is e^x, and then you find the derivative of whatever expression x is and multiply e^x by that?
.close
devout snowBOT
devout snowBOT
Available help channel!
uneven quest
how do i calculate the area of a triangle while knowing all 3 of it's sides when they all are different?
devout snowBOT
how do i calculate the area of a triangle while knowing all 3 of it's sides when...
uneven quest
(please ping me if you are talking to me)
tropic dune
how do i calculate the area of a triangle while knowing all 3 of it's sides when...
Heron’s formula?
heavy blaze
hello, I am working on fixed point iteration method and I am trying get my g(x). I know to get g(x) i have to get my f(x) which is tan(𝜋𝑥) − 𝑥 − 6. As x by itself. I have been trying to get x by itself however I have been having trouble of where to start
devout snowBOT
hello, I am working on fixed point iteration method and I am trying get my g(x)....
heavy blaze
I've tried doing the inverse of tan and got 𝑥 = tan^-1(x+6)/𝜋 but I'm not sure if that is right
wary ruin
A∩C = ∅⊆A∪C?
devout snowBOT
A∩C = ∅⊆A∪C?
sonic smelt
That's an invalid equation, you have a set on the left and a statement on the right
wary ruin
That's an invalid equation, you have a set on the left and a statement on the ri...
How would you write it?
∅⊆A∩C?
sonic smelt
Depends on what you intended to mean in the first place
wary ruin
The intersection of set A and C
Well, is the empty set a subset of, or equal to, A intersection C
sonic smelt
The empty set is a subset to any set, so yes
devout snowBOT
@wary ruin Has your question been resolved?
wary ruin
@sonic smelt ⊆ or ⊂?
sonic smelt
⊆
wary ruin
Ok thanks
hallow goblet
hey, have a question involving Set Theory
devout snowBOT
hey, have a question involving Set Theory
hallow goblet
Find a bijection between all integers and all positive integers
hollow pollen
did you work one out?
hallow goblet
Find a bijection between ℤ and ℤ+
hallow goblet
did you work one out?
not yet,, but i know how to find one in words
i just dont know how to actually put it into form
hollow pollen
thats a good start
restive river
one tip, map 0 to 1, and then jump around between negative and positive
hallow goblet
j = i+1 where i is an element of ℤ and j is an element of ℤ+?
restive river
thats doesnt work
hallow goblet
im so lost
restive river
do you know what a bijection is?
hallow goblet
yea, a one to one correspondence. everything gets hit at most once, and nothing is left unhit
and i fully understand that ℤ and ℤ+ are countably infinite and therefore have a bijection
i just dont know how to find one
and put it into proper form
restive river
good, now the problem is that Z is infinite to "both sides", while Z+ is only infinite to "one side"
hallow goblet
ok hang on
restive river
thats why you have to go back and forth
restive river
think less about formulas first, the formula isnt that easy
lets think about how a function might look like
what could you map f(0) to?
hallow goblet
what could you map f(0) to?
1?
restive river
and f(1)?
hallow goblet
0?
restive river
well 0 isnt in Z+ right?
restive river
we are going from Z to Z+
hallow goblet
ah so 2
hallow goblet
wait but dosnt it not matter which way we go since its a bij?
restive river
now, instead if looking at f(2), f(3), f(4), we instead look at f(-1) now
what could f(-1) be?
restive river
yes
hallow goblet
f(-1) -> 0?
restive river
well 0 isnt in Z+ right?
hallow goblet
wait wtf how lol
restive river
its a legal choice, -1 is in Z and 3 is in Z+
hallow goblet
f(2) -> 4?
restive river
yes, and f(-2)?
restive river
f(0) = 1
f(1) = 2
f(-1) = 3
f(2) = 4
f(-2) = 5
f(3) = 6
f(-3) = 7
see how we can slowly fill both of these sets?
restive river
hmm, maybe look at what 1, 2, 3, 4 goes to, and then look at what 0, -1, -2, -3, -4 go to
make a case handling for positive and non positive
hallow goblet
ok give my smooth brain a sec
btw tysm
f: ℤ -> ℤ+
Let i be an integer
Letj = (i+1) * -1 for all i > 0
Then there exists a bijection between ℤ and ℤ+?
please god tell me im at least getting closer
restive river
this is a piecewise function for example
hallow goblet
ok wait so
restive river
instead of x lets call it z
just as a reminder
f(1) = 2
f(2) = 4
f(3) = 6
f(4) = 8
restive river
f(0) = 1
f(-1) = 3
f(-2) = 5
f(-3) = 7
f(-4) = 9
hallow goblet
ok so f(-z) will map to all odds and f(z) maps to all evens? and 0 just maps to 0?
o wait
i mean 1
restive river
0 to 1 yeah
hallow goblet
shit
hallow goblet
right
restive river
not even or odd (for z)
you just need to make sure the function puts out even for positive and odd for negative z
hallow goblet
f(z) = z+1 for all x / 2?
restive river
what is x?
hallow goblet
f(z) = z for all z / 2?
restive river
make something like
f(z) =
..., if z>0
..., if z≤0
hallow goblet
like just give a few iterations you mean?
restive river
just look at
f(1) = 2
f(2) = 4
f(3) = 6
f(4) = 8
so how do you get f(z) = even?
hallow goblet
f(z) = z + 1 where z > 0
and
f(z) = z where z <= 0?
or is that just what i had before?
restive river
thats not quite it, it would give you
f(1) = 2
f(2) = 3
f(3) = 4
f(4) = 5
hallow goblet
z+2?
restive river
then its
f(1) = 3
f(2) = 4
f(3) = 5
f(4) = 6
you want
f(1) = 2
f(2) = 4
f(3) = 6
f(4) = 8
hallow goblet
im a moron
restive river
f(600) = 1200
hallow goblet
ok it makes sense now lol
tysm, seriously. im sorry to have bothered you with something so trivial
restive river
wait, so you got f(z) = 2z now?
restive river
we still got the negatives left
hallow goblet
fuuuuuck my class is abt to start
restive river
f(0) = 1
f(-1) = 3
f(-2) = 5
f(-3) = 7
restive river
f(z) =
2z , if z>0
-2z+1, if z≤0
hallow goblet
tys fucking m
restive river
np
hallow goblet
gonna sprint to class now like a freak. can i trouble you with more questions later tonight possibly?
restive river
just ask in one of the help channels
hallow goblet
will do. thank you again
devout snowBOT
@hallow goblet Has your question been resolved?
lofty crater
can someone check my work please
devout snowBOT
can someone check my work please
vivid cliff
aye aye
seems good
but idk bro
let’s
wait for someone
more credible
to check
restive river
for this question are there 4 equivalence classes, {-x, y}, {x, y}, {x, -y}, {-x, -y} since the relation just says |x| = |a| and |y| = |b| ?
so then for any number between - 10 and 10 if -x = y, x = -y, or -x = -y they would all satisfy the relation
i have no idea if im going ab this the right way
devout snowBOT
for this question are there 4 equivalence classes, {-x, y}, {x, y}, {x, -y}, {-x...
@restive river Has your question been resolved?
restive river
<@&286206848099549185>
devout snowBOT
@restive river Has your question been resolved?
low holly
for this question are there 4 equivalence classes, {-x, y}, {x, y}, {x, -y}, {-x...
so W has 441 elements
and you are correct that (-x, y), (x, y), (x, -y), (-x, -y) are the members of the equivalence class of (x, y) when x and y are nonzero
so you can count the number of elements in W with both elements nonzero and divide by four to get the number of equivalence classes in that subset
then you're left with considering the equivalence classes of pairs with at least one element zero
i think you can make some progress with that
restive river
so you can count the number of elements in W with both elements nonzero and divi...
wouldnt that just be 100
low holly
wouldnt that just be 100
yep
restive river
but then including 0 it would be 11 * 11 so 121 rather than 100 right?
low holly
but then including 0 it would be 11 * 11 so 121 rather than 100 right?
not sure i follow
you know you have at least 100 equivalence classes by restricting to one quadrant e.g. the equivalence classes of (x,y) with x and y both positive
10 choices for each gives you 100 equivalence classes
there's a single equivalence class with both elements 0
now you're left with counting the number of equivalence classes with one element 0
you can do this by noting that the equivalence classes in such case always have 2 elements, so again if you can count the number of elements with one element 0 you can divide by 2
restive river
yea so it would be 1 more class for each pair, 20 more classes not including the one with both elements 0
so then including the one with 0 its 21 then i get 100 + 21
restive river
lol sorry was just visualizing it graphically and was getting the number of points in the first quadrant
but to follow up, would there be 121 classes then?
low holly
but to follow up, would there be 121 classes then?
yeah
fairly sure
(x,y), (x,0), (0,y), (0,0) for 1 <= x,y <= 10
cardinality of the set on the left is 100 + 10 + 10 + 1
if you're struggling to see this you can see that the absolute value of each element in a pair must be between 1 and 10 for the first type of eq. class, and the latter three are cases on it being 0
restive river
so the logic behind this then would be since for each pair (x, y) where x, y are between -10 and 10 and non-zero, we have 100 different possibilities for (x,y), making 100 equivalence classes. taking account (x, y) pairs that include 0 between -10 and 10, we get 21 elements, making the total number of equivalence classes 121?
low holly
so the logic behind this then would be since for each pair (x, y) where x, y are...
100 different possibilities for the absolute values of x and y
1 through 10 for x and 1 through 10 for y
we don't have 21 elements anywhere
(0,10) is equivalent to (0,-10)
oh i overcomplicated it LOL
there's a very easy explanation for why 121
11 different absolute values for the first element, 11 different absolute values for the second element
i think that's what you were saying
restive river
o yeah
low holly
there's an equivalence class for each absolute value of each element