like
#help-13
crimson sedge
fallen moat
i might have to google it
crimson sedge
the intercepted arc is 38 degrees right
so like arc formed by chord?
i think
am not sure
fallen moat
maybe we should google it
crimson sedge
yeah
fallen moat
"what is measure of arc"
crimson sedge
38 degrees
fallen moat
"what is measure of arc"
i mean google that
silent finch
You can use that angle segment theorem that you talked about in another channel right
fallen moat
You can use that angle segment theorem that you talked about in another channel ...
yep, but you will have to first extend the angle at centre to angle at circumference
fallen moat
yep, but you will have to first extend the angle at centre to angle at circumfer...
using another theorem
silent finch
Well a simpler solution is to notice that the triangle is isosceles, so the other 2 angles are 71. 71 + the desired angle = 90 because the radius and tangent lines form a 90 degree angle. So our answer is 19
silent finch
fallen moat
nice!
fallen moat
tangent chord angle theorem
oh, I don't know this 😛
silent finch
ye me neither lol
crimson sedge
it is like
that angle created by tangent line and chord
is half of the intercepted arc
silent finch
oh that makes sense
silent finch
Well a simpler solution is to notice that the triangle is isosceles, so the othe...
Basically just generalizing this
crimson sedge
yeaaa
uneven chasm
cedar kilnBOT
uneven chasm
Oh
dull oxide
oh wait
dull oxide
no we're both wrong
buoyant latch
,w 33.110.52pi + pi10.5^2 + 4pi10.5^2/2
uneven chasm
What about it?
wraith daggerBOT
buoyant latch
that's in mm^3 btw
uneven chasm
SA = 2πr^2+2πrh
= SA = 2π(10.5)^2+2π(10.5)(33.1)
= SA = 220.5π + 695.1π
= SA = 915.6 * 3.14 (π)
= SA = 2875
What did I do wrong?
@buoyant latch
buoyant latch
Whats 2pi r^2
uneven chasm
692.7
buoyant latch
The bottom?
uneven chasm
Yeah thats what I put
buoyant latch
What shape is the bottom
buoyant latch
What is the formula of the area of a circle
uneven chasm
a = nr2
buoyant latch
So why do you have 2pi r^2
cedar kilnBOT
@uneven chasm Has your question been resolved?
frosty ocean
<@&268886789983436800>
celest seal
crimson sedge
help pls
cedar kilnBOT
cedar kilnBOT
Available help channel!
tranquil harbor
the argument of complex number states that tan-1 (y/x)
cedar kilnBOT
the argument of complex number states that tan-1 (y/x)
tranquil harbor
if my question shows only 3i, how should i input it into the formula
since 0 real number will only result in math error if i sub it into x
crystal raptor
Well think about what angle θ makes tan(θ) undefined
tropic oxide
arg(i) is geometrically obvious
tranquil harbor
hm? i dont quite understand
tropic oxide
you know the geometric meaning of arg(z) right?
tranquil harbor
not very, mind explaining?
quite new to this topic
oh is it this?
arg (z) = arg (x+iy) = tan-1(y/x)
tropic oxide
arctan(y/x) only works as is in the first quadrant.
tranquil harbor
yes
tropic oxide
arg(z) is the angle, measured CCW, from the positive real axis to the line joining z to the origin
lyric jungle
(You might be interested in https://en.wikipedia.org/wiki/Atan2)
tropic oxide
arg(i) is geometrically obvious
what's the angle between the positive real axis and the line from 0 to i?
tranquil harbor
90 ?
tranquil harbor
(You might be interested in https://en.wikipedia.org/wiki/Atan2)
thanks ill have a read in a moment
languid hill
how has this got 5 answers
cedar kilnBOT
how has this got 5 answers
languid hill
cedar kilnBOT
@languid hill Has your question been resolved?
spice pier
<@&286206848099549185> got it wrong but still wanna understand it
It is generally a good idea to draw a graph to figure out how many solutions a trigonometric equation might have.
languid hill
alr what do you do with the sin n tan though since theyre both different
spice pier
The question can be put this way, how many times does the graph of tan(2x) and 5sin(2x) intersect.
spice pier
And we also need to find the values in this case.
Usually with such questions, we want to create factors to solve such trig equations. This is what the question does by taking 5sin(2x) to the other side and creating factors.
languid hill
alr ty
normal pawn
Your input seem right to me, or is it the solution itself ?
languid hill
oh i found the answers online, its bcos i didnt round one to 1dp
normal pawn
Lmao
cedar kilnBOT
cedar kilnBOT
Available help channel!
bright shadow
cedar kilnBOT
bright shadow
I confused the approach again, how can I get it right? 😭
south tundra
The one-sided limits are equal though
Oh unless there's a modulus
What's the original function?
(x^2 + x - 6)/(2 - x), right?
bright shadow
As x approached 2 from the right, the the absolute values remain the same right? But the answer is the opposite
south tundra
What function are we looking at?
bright shadow
The one sided right hand limit is -5 and the other one is 5
limit of (x^2+x-6)/(|x -2|) as x goes to 2
south tundra
Then the limit doesn't exist, yeah
south tundra
The one sided right hand limit is -5 and the other one is 5
The left* hand limit is -5 and the other is 5 though
bright shadow
But my one sided limits, are opposite, I’m smh still taking the wrong approach
south tundra
Yeah they're opposites because of the modulus
It's correct
(x^2 + x - 6)/(|x - 2|) can be rewritten as (x + 3) * (x - 2)/|x - 2|
or (x + 3)sgn(x - 2)
As x approaches 2 from the right, sgn(x - 2) = 1
So the limit is 5 there
And as x approaches 2 from the left, sgn(x - 2) = 1
So it's -5 from that side
Thus the limit at 2 doesn't really exist
mellow loom
I confused the approach again, how can I get it right? 😭
Your answer is correct.
bright shadow
Your answer is correct.
It’s wrong
south tundra
How do you know?
mellow loom
Why it is wrong?
bright shadow
Wolfram alpha, the graph was the opposite way
It’s supposed to be -5 for left approaching, i got it as 5
south tundra
Ah you're talking about just the graph?
bright shadow
And the other way around
bright shadow
Ah you're talking about just the graph?
I got it wrong directly from the limits part
south tundra
You got 5 from the right and -5 from the left though, wdym
x -> 2^+ means approaching from the right and x -> 2^- means from the left
bright shadow
south tundra
You got those values too 
south tundra
The function can be simplified as -x - 3 when x < 3
So you shouldn't have it increasing overthere
bright shadow
I took the points wrong i guess
bright shadow
Ahh so the limits were correct
So I’m finally starting to think correctly about approaching limits
Noo I’m not
Ahhhh yess
I need a break
Sorry for confusing everyone 😭
Thankyou!
.close
sonic hedge
Let L = {f,p,q} be a language in which f is a unary function symbol and p and q are unary relation symbols. Let I be the interpretation for L defined by D^I = {0,1,2,3,4,5,6,7}; p^I = {0,3,7}; q^I = {7};
f^I(0) = 4; f^I(1) = 5; f^I(2)=2; f^I(3)=6; f^I(4)=7; f^I(5)=2; f^I(6)=7; f^I(7)=1.
Define a congruence relation ∼ on I that has four equivalence classes, justifying your answer. Describe the I/∼ quotient interpretation.
SOLUTION
We have to partition D^I into four sets. Observing p^I and q^I we note that 0 and 3 are the only elements that belong to the first and not to the second; they can therefore be congruent with each other, but not with the other elements of D^I. it is also evident that 7 cannot be congruent to any element other than itself. Now taking f^I into consideration we note that f^I(4) = f^I(6) = 7 and therefore these elements cannot be congruent with 1, 2 and 5, which are mapped together.
These observations lead us to conclude that the four congruence classes can only be {0, 3} {1, 2, 5}, {4, 6} and {7}.
Furthermore ∼ thus defined also verifies the condition concerning f, because f^I(0) ∼ f^I(3), f^I(1) ∼ f^I(2) ∼ f^I(5) and f^I (4) ∼ f^I(6).
You have then
D^I /∼ = {[0], [1], [4], [7]};
f^I/∼([0]) = [4], f^I/∼([1]) = [1], f^I/∼([4]) = [7], f^I/∼ ([7]) = [1];
p^I/∼ = {[0], [7]}, q^I/∼ = {[7]}.
sonic hedge
I didn't understand this part "Furthermore ∼ thus defined also verifies the condition concerning f, because f^I(0) ∼ f^I(3), f^I(1) ∼ f^I(2) ∼ f^I(5) and f^I (4) ∼ f^I(6)." and also this "f^I/∼([1]) = [1]".
cedar kilnBOT
@sonic hedge Has your question been resolved?
cedar kilnBOT
@sonic hedge Has your question been resolved?
sonic hedge
<@&286206848099549185>
sonic hedge
maybe I understand
cedar kilnBOT
@sonic hedge Has your question been resolved?
cedar kilnBOT
@sonic hedge Has your question been resolved?
crimson sedge
you need help
cedar kilnBOT
@sonic hedge Has your question been resolved?
twin tinsel
How can we prove its limit is 1/3 using delta epsilon?
cedar kilnBOT
How can we prove its limit is 1/3 using delta epsilon?
copper crest
cedar kilnBOT
@twin tinsel Has your question been resolved?
cedar kilnBOT
@twin tinsel Has your question been resolved?
gilded elm
I only know the single variable analysis version but
you kind of just start manipulating backwards until you get delta in terms of epsilon no?
lethal jackal
yeah it's the same thing
limits can be defined in arbitrary metric spaces
so you're basically just using some vector norm instead
cedar kilnBOT
@twin tinsel Has your question been resolved?
gilded elm
ok where are you stuck
to prove this you work in reverse no?
|x^2 / (x+y) - 1/3 | < epsilon
then just start manipulating it
until you get some delta for (x, y)
twin tinsel
yeah I was trying to do that but did not work
i mean i was not able to find delta
could you show me the manipulation
lethal jackal
okay you can probably show that for 0 < epsilon < 1, epsilon/1000 works for delta, and otherwise 0.00001 works
like it doesn't have to be tight
cedar kilnBOT
@twin tinsel Has your question been resolved?
cedar kilnBOT
@twin tinsel Has your question been resolved?
crimson sedge
hi all i need help for assiments anyone can help ?
cedar kilnBOT
cedar kilnBOT
Available help channel!
mint fable
rewrite an equation
cedar kilnBOT
rewrite an equation
mint fable
I have to rewrite f(x)= (4/5x) to a form like this: f(x)= a*x^n
Could someone help me and explain how to get to the answer?
tropic oxide
do you mean $\frac{4}{5x}$?
wraith daggerBOT
Ann
clear berry
or $\frac{4x}{5}$
wraith daggerBOT
numbpy
rotund fox
do you mean
$\frac{4x^-1}{5}$ ?
wraith daggerBOT
Shockshwat
wraith daggerBOT
Shockshwat
rotund fox
yea this
tropic oxide
kind of spoilery
mint fable
In the answer book it says (4/5)*x^-1
rotund fox
See , $\frac{1}{x^n}$ can be written as $x^{-n}$
wraith daggerBOT
Shockshwat
mint fable
The thing I don't get is how you can isolate 4/5 out of the fraction
rotund fox
Here we have $\frac{4}{5}$ constant so we can take that out which gives us $\frac{4}{5} \cdot \frac{1}{x}$
wraith daggerBOT
Shockshwat
mint fable
ah, thanks
rotund fox
Now we know $\frac{1}{x}$ is actually $\frac{1}{x^1}$ And knowing that $\frac{1}{x^n}$ can be written as $x^{-n}$ we get ${x^{-1}}$ and then we get $\frac{4}{5} \cdot {x^{-1}}$
wraith daggerBOT
Shockshwat
mint fable
thank you for your help!
rotund fox
and multiplying it we finally get $\frac{4x^{-1}}{5}$
wraith daggerBOT
Shockshwat
rotund fox
thank you for your help!
if your problem was solved then please type .close
mint fable
.close
cedar kilnBOT
cedar kilnBOT
Available help channel!
crimson sedge
what is vertical assymptote in this case?
cedar kilnBOT
what is vertical assymptote in this case?
crimson sedge
what is vertical assymptote in this case?
,rotate
wraith daggerBOT
crimson sedge
<@&286206848099549185>
cedar kilnBOT
@crimson sedge Has your question been resolved?
timber violet
What is the greatest integer function of 0.58x
timber violet
Is it possible to find this
crystal raptor
What do you mean by "find"?
cedar kilnBOT
@timber violet Has your question been resolved?
timber violet
What do you mean by "find"?
The greatest integer function of 0.58x
For any real number x
crystal raptor
Well calculate 0.58x and then find the greatest integer less than it?
Really not clear what you're asking
fervent radish
how can i prove this is divergent or convergent?
cedar kilnBOT
how can i prove this is divergent or convergent?
fervent radish
can someone help me pls?
crimson sedge
Nth term test should work
worn patio
You can prove this is not convergent by proving that the limit of the general term is not 0
crimson sedge
Because I have a feeling this is just the limit definition of e with extra steps
Yeah it is basically that
crystal raptor
It is zero so that doesn't help
worn patio
It is zero so that doesn't help
Care to elaborate?
cosmic steppe
Nth term test is inconclusive if the output is 0
worn patio
It is zero so that doesn't help
Nvm you're right
worn patio
maybe Cauchy's Criteria?
crimson sedge
Ratio test probably will work after a lot of painful algebra
Was thinking Cauchy too
wraith daggerBOT
draganb
fervent radish
sry guys im jst in calc 1, where can i learn cauchys criterias? never heard of it
worn patio
sry guys im jst in calc 1, where can i learn cauchys criterias? never heard of i...
Maybe this helps https://www.youtube.com/watch?v=wTq6HI9w4n8
fervent radish
oh this...
oh jesus now i see
everytime i use cauchy there is only n, not a multiplme of that, but now i see can be anything
jst divide by n and find the limit
gotcha guys, thank u so much
thx @worn patio,
thx @crimson sedge ,
thx @crimson sedge and
thx @cosmic steppe
.close
crimson sedge
Can someone help with this?
crimson sedge
cedar kilnBOT
@crimson sedge Has your question been resolved?
crimson sedge
This isn't a test,
it's a practise thing my teacher set out for us before an exam as practise questions and I don't know how to do this
and my teacher isn't replying
stuck granite
After his retirement, Peter decides he wants to receieve a regular payment at the start of each month for 25 years. He invests his lump sum at an annual rate of 4.2%. What is the monthly payment?
His lump sum = $644,037
I'm having trouble interepreting the question
How can he withdraw some amount from the account every month and try to get an interest on it?
crimson obsidian
I'm confused that if he is retired he should get pension, why is he getting payment
stuck granite
I'm interpreting it as - every month, the interest he earns + $644,037/(25*12)$
wraith daggerBOT
Awesome_Ruler_007
stuck granite
I'm confused that if he is retired he should get pension, why is he getting paym...
he apparently invested his pension
crimson obsidian
he apparently invested his pension
I see
crimson obsidian
Principal amount, yeah
crimson obsidian
How can he withdraw some amount from the account every month *and* try to get an...
I am just guessing his annual interest is divided by 12 to make it monthly interest?
stuck granite
I am just guessing his annual interest is divided by 12 to make it monthly inter...
No, its compound interest
crimson obsidian
No, its compound interest
I see mb
stuck granite
$\sqrt[12]{1 + 0.042} - 1$
crimson obsidian
The question didn't mentioned anywhere so I thought it's simple interest
wraith daggerBOT
Awesome_Ruler_007
cedar kilnBOT
@stuck granite Has your question been resolved?
cedar kilnBOT
@stuck granite Has your question been resolved?
rotund dagger
hi I am having trouble with this word question. Do I just multiply 30ft and 10m? I don’t really understand 😅
wind glade
So you either want to convert 10m to ft or 30 ft to meters
The table gives ft to meters directly so I think its easier to do 30 ft to meters
swift sky
Hello all! I'm learning Multi-Var Calc, and am trying to solve this problem
Use polar coordinates to find the volume the given solid: above the cone $z=\sqrt{x^2+y^2}$ and below the sphere $x^2+y^2+z^2=1$
wraith daggerBOT
Anti-Clownfish
swift sky
Using double integrals*, so:
- Convert into polar
wraith daggerBOT
Anti-Clownfish
swift sky
Then set equal to find r value, which would be sqrt{2}/2
Then the double integrals will be: integral from 0 to 2pi and integral from 0 to sqrt{2}/2
But I'm not sure what will be the "inside" the integral (what will be integrated). Would it be sqrt{r^2} or sqrt{1-r^2}? And why?
cedar kilnBOT
@swift sky Has your question been resolved?
swift sky
<@&286206848099549185>
cedar kilnBOT
@swift sky Has your question been resolved?
short magnet
You are trying to compute one or two double integrals? Your problem are not stated well enough so that I can understand what is missing @swift sky
cedar kilnBOT
@swift sky Has your question been resolved?
swift sky
Its not that IDK how to compute the integral, I don't know how to set it up
Basically it's:
Int from 0 to 2pi and int from 0 to sqrt2/2 (I think) but I don't know the expression to integrate: is it
Sqrt(r^2) or sqrt(1-r^2)?
cedar kilnBOT
@swift sky Has your question been resolved?
floral topaz
.close
verbal seal
Its not that IDK how to compute the integral, I don't know how to set it up
Bas...
since this is in polar and a double integral it's a little bit of a pain but I'll attempt to help and explain it in a way that makes sense.
Your bounds look good, your integral should be (sqrt(1-r^2)-r)r because sqrt(r^2) can be similifed down to r since bounds can tell us above or below the x,y plane. Since we want the space between the sphere and the cone, we're going to take the equation of the sphere, then subtract that by the cone and set the bounds so that the integral only collects information within the snowcone like shape . Then we multply that all by r because its been convertee to polar (integrals converted to polar need an extra r mutlplied in bc of reasoms you will learn later in class)
im about to go to sleep so i hope that made sense, if not my apologizes, and anyone feel free to jump in if I missed anything
swift sky
Your bounds look good, your integral should be (sqrt(1-r^2)-r)r because sqrt(r^2...
i know what you mean -- im going to math it out and check it with the answer key rn
verbal seal
okay cool, sphereical is usually much better for this kinda integral so if its off some I probably converted something wrong in my head😅
swift sky
okay cool, sphereical is usually much better for this kinda integral so if its o...
I think the problem is with sqrt(1-r ^2)r
I use u sub, but when you plug in 2 (from the integration limits) into the 2/3 (1-r^2)^3/2, you get an imaginary answer, and that is most likely not the answer
verbal seal
did you mean sqrt(2)/2 or 2pi?
ur correct in that u shouldnt get an imaginary answer though
swift sky
$\int^2_0 x^2$
wraith daggerBOT
Anti-Clownfish
wraith daggerBOT
Anti-Clownfish
swift sky
u get r * sqrt{1-r^2) if you factor/whatever terminolgoy that is
u subbing it and then when i do the integral out its -3^(3/2)
verbal seal
I believe the upper bound should be sqrt(2)/2 but lemme just check something real quick
swift sky
Oh wait yeah
It's cleaner, but still not as clean as the solution
Then if I integrate theta, thats just theta^2/2 (plug 2pi) and thats 2pi^2, which would not get the same solution as the above
verbal seal
you should just have to multiply what you get the first time you integrate by 2pi since there is no theta in the integral
swift sky
Yeah I meant multiply by 2pi* oops, it still won't result in the answer
verbal seal
so depending what you get, it might look different but be the same bc of factoring
what did you end up getting as an answer
swift sky
I'm going to redo it in case
verbal seal
okie dokie
swift sky
.. and I got the same answer as the answer key now!
Thank you very much, so essentially the two shapes subtracted from each other which I get
lol..., now this?
verbal seal
yeah, and just make sure the bounds only include what you want
swift sky
This was the one where it was
$\int^2_0 \int^{2pi}_0$
wraith daggerBOT
Anti-Clownfish
verbal seal
i believe so
I gotta head to bed but if no one has gotten to this by tomorrow i might be able to help you out mote
swift sky
Alright, tmrw my teacher is going to teach this unit so I'll see if he covers something I missed in the readings
Thanks!
crimson sedge
WHat is a difference of two squares ?
cedar kilnBOT
WHat is a difference of two squares ?
river lichen
When an expression can be viewed as the difference of two perfect squares, i.e. a²-b², then we can factor it as (a+b)(a-b). For example, x²-25 can be factored as (x+5)(x-5).
river lichen
WHat is a difference of two squares ?
In this the expression we are given is 3x^2-5y^2
To find the difference of two squares both numbers must be perfect squares
crimson sedge
oh ok that is familiar
river lichen
Yes and since, both terms are not perfect squares so
we cannot factor it further.
crimson sedge
so it cant be factored as two different squares because there is a x and a y ?
river lichen
Hmm not quiet. To find the difference of two squares both of the numbers. (The coefficients of x must be perfect squares. Perfect squares are 4, 9, 16, 25, 36, 49 so on. Both terms are not perfect squares so we cannot factor it further.
Does that make sense?
crimson sedge
Does that make sense?
oh so a perfect square is a number that can be created by 1 number multiplied by itself like 7*7 = 49
river lichen
Yup
And u need a perfect square to find the difference of two squares
And both of the numbers in ur equation are not perfect squares
Meaning u cant find the difference of the two squares
cedar kilnBOT
@crimson sedge Has your question been resolved?
untold locust
help
untold locust
i cant see
untold locust
ok
crimson sedge
So the problem is how 3x - x = 2x
untold locust
yes
crimson sedge
Close your eyes
untold locust
ok
crimson sedge
Ok don't
untold locust
ok
crimson sedge
Look at what I'm typing
untold locust
alr
crimson sedge
Imagine a small square
untold locust
ok
crimson sedge
Think of that as x
untold locust
ok
crimson sedge
Now in the problem, we have 3x being subtracted by an x
untold locust
yes
untold locust
ohhhhhh
crimson sedge
How many do you see left?
untold locust
2
crimson sedge
Very good
crimson sedge
yes
untold locust
crimson sedge
The reason why you can subtract them
is because
they are like terms
Both are x
untold locust
ohhhhh
crimson sedge
They both have the same shape
untold locust
ohhh yeaaaa
crimson sedge
c:
crimson sedge
Nope
untold locust
ohhhhh
crimson sedge
y has a rectangular shape
crimson sedge
NP
untold locust
goodbye
crimson sedge
Conclusion:
You can't combine unlike terms or unlike shapes
brazen spade
hi
latent bloom
Hey
brazen spade
latent bloom
How can we help you?
brazen spade
the x²=4ay one?
latent bloom
Yeah
Express t as a function of y
And plug it into x's equation
Actually this now looks like a vertical one
wraith daggerBOT
VulcanOne
latent bloom
$x = \left( \frac{y-1}{2} \right)^2 +1$
wraith daggerBOT
VulcanOne
latent bloom
$x-1 = \frac{1}{4} (y-1)^2$
wraith daggerBOT
VulcanOne
latent bloom
$4(x-1) = (y-1)^2$
wraith daggerBOT
VulcanOne
brazen spade
i am sorry the wifi got discconected
latent bloom
Y^2 = 4aX
latent bloom
i am sorry the wifi got discconected
No worries :)
I got my wifi disconnections today morning too
🥲
latent bloom
I plugged into x=t^2 + 1
brazen spade
i am sorry but i am having problems understanding. you plugged in the x and y in which equation and what about the "a"
latent bloom
$y = 2t+1$
wraith daggerBOT
VulcanOne
latent bloom
Right?
brazen spade
yes
brazen spade
i get this
latent bloom
Okay that's t = function of y
brazen spade
yes
wraith daggerBOT
VulcanOne
brazen spade
oh
latent bloom
brazen spade
oh i get it now
latent bloom
brazen spade
so the equation of directrix is x-1=-1 right?
latent bloom
The directrix is usually how many a?
2 or 1?
1 a
And since the parabola is positive, so it is pointing to the positive x direction
And it's shifted towards the right by 1 unit
So (x-1) = -a
And a = 1
brazen spade
yes
brazen spade
thank you
latent bloom
brazen spade
.close
cedar kilnBOT
brazen spade
one more question
cedar kilnBOT
Available help channel!
scenic ermine
Hey,im doing Grade 10 Quadratics and i'm stuck at one part of converting a vertex form equation to a standard form one. It would be appreciated if someone could help me learn what to do with fractions at the point i'm stuck at
vital jay
I don't recall all my grade 10 terminology, can you give an example?
scenic ermine
Hey, sorry i just did a little bit of digging and found exactly what i was looking for online, thanks anyways.
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cedar kilnBOT
scenic ermine
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cedar kilnBOT
✅
scenic ermine
I am stuck at the final part of converting the equation to standard form
I have 3 different fractions so i do not have like terms to add
crimson sedge
fractions can be added
scenic ermine
should i send a picture of where i am at right now?
crimson sedge
sure
scenic ermine
crimson sedge
just to clarify the picture says $y = \frac{2}{43}x^2-\frac{20}{43} \frac{100}{43}x-4$ right?
wraith daggerBOT
starlight
scenic ermine
yes
frosty ocean
There is no x in the second term 
crimson sedge
the exact same thing though
scenic ermine
the tutorial video i watched had 2 of the same fractions at the point i'm at so i dont know what to do if they are different
frosty ocean
$-\frac{20}{43}x\frac{100}{43}$ makes me uncomfortable
wraith daggerBOT
tropic oxide
better to write $-\frac{20}{43}x \cdot \frac{100}{43}$ js
wraith daggerBOT
Ann
frosty ocean
or $-\frac{20}{43}\frac{100}{43}x$
wraith daggerBOT
scenic ermine
ok so what would i do with those two fraction at that point
is it possible to gather them as like terms?
or is that the final answer in vertex form?
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dull kayak
Confused as to how i end up with this graph even tho my calculator shows this
still barn
What course is this for that is teaching you polar coordinates but giving you a worksheet that looks like you are in junior high school
dull kayak
Honors pre calc
still barn
At a glance you need to evaluate your points more carefully. When theta = pi/6, r = 4sin(pi/2) = 4. Not 2. If you calculated it all wrong your points are going to be all over the place
dull kayak
I dont understand how when theta=pi/6, r=4
The rest of the problems match up with my graphs fine
still barn
Its right off the unit circle, at an angle pi/2, it points straight upwards and intersects the circle at (0,1) so sin(pi/2) = 1 and cos(pi/2) = 0.
dull kayak
But how would r=4 at pi/6
Wait
Ok im looking at your previous statement and when theta=pi/6, r=4sin3(pi/6)
Which would simplify to r=4sin(pi/2) correct?
<@&286206848099549185>
cedar kilnBOT
@dull kayak Has your question been resolved?
cedar kilnBOT
cedar kilnBOT
Available help channel!
deep light
Hi
fair pebble
1-H
2-B
3-A
4-D
5-F
6-C
7-E
8-G
https://en.wikipedia.org/wiki/Differentiation_rules
fair pebble
yep
deep light
fair pebble
There is an error in the condition, but I think the correct answer is
1-A
2-B
3-C
4-E
5-D
E should be -10
deep light
Thank you
crimson sedge
fr don't cheat on a test
also confused how derivatives and basic level factoring is on the same test
cedar kilnBOT
Available help channel!
surreal aurora
where's the error in my work? correct radius of conv = $k^k$ and not $1/k^k$
wraith daggerBOT
drsleep
surreal aurora
lol....
earnest socket
in the first line
nova snow
so im currently learning chain rule but i have a couple questions; im given an example of y = (5x+7) and the steps show as;
Let u = 5x + 4
Then du/dx = 5
Why is du/dx made to be equal to 5 here
stray otter
cuz of coefficient
long swan
I don't see how you'd use chain rule here
stray otter
the coefficient of 5x is 5
tropic oxide
well the derivative of 5x+4 is 5.
stray otter
I don't see how you'd use chain rule here
yes
nova snow
I don't see how you'd use chain rule here
its an example im pulling from my textbook
that makes sense now
crimson sedge
this isn't related to chain rule-
nova snow
well i didnt state everything
stray otter
if it's y = (5x+7)^2 then that's a chain rule
nova snow
ill send the screenshot
oops
wait
i mistyoed
i meant to say
y = (5x+4)^7
i forgot the power of 7
tropic oxide
well technically nothing would have stopped you from applying the chain rule with the identity function as your outer function.
it still works, you're just subjecting yourself to a bit of extra work and bookkeeping.
stray otter
and then we use the chain rule
dy/dx=(dy/du)(du/dx)
so we know that y = (5x+4)^7, then y=u^7
so dy/du=7u^6
and u=5x+3
nova snow
stray otter
yes
nova snow
where does the y = u^7 come from
stray otter
by substituting u=5x+4
nova snow
so because you took the 5x+4 from y = (5x+4)^7 youre just left alone with the power of 7?
stray otter
that's when the u comes in
stray otter
yes
nova snow
so you technically use a bit of the power rule in the chain rule then?
livid hound
differentiating without expanding the binomial involves a combo of power and chain rule
nova snow
you get the 1/2 from the reciprocal or something?
stray otter
oh ok
nova snow
trying to atleast
stray otter
seems right enough
nova snow
dont know how im doing though lol
nova snow
nice
what kind of calc are you doing right now
also
it shows du/dx = -1
where do you get -1?
stray otter
what kind of calc are you doing right now
both derivatives and intergration
nova snow
nice
stray otter
where do you get -1?
-x
nova snow
and the 3 is gone because of the power rule right
stray otter
yes
nova snow
how long did it take you to learn calc to where you are at currently
where does the (3-x) come from
stray otter
u
stray otter
how long did it take you to learn calc to where you are at currently
Started at 7th grade, derivatives for around 3 weeks and integration around 1 month, preparation for ~6weeks
nova snow
nice
stray otter
how does the final answer become a sqrt
cuz the power is -1/2
preparation includes:
Graphing
Trig
Exponents
Geometry(a bit)
you need to be quite good at these stuff
nova snow
functions?
nova snow
im fine with most of that stuff except for trig and geometry
stray otter
and also trig identities
nova snow
is that the same as trigonometric functions
or different
also
for using chain rule on trinomials
does anything change?
du/dx = 6x + 2?
stray otter
cuz 3x^2+2x
stray otter
actually, a better answer would be 18(3x+1)(3x^2+2x-1)^8
nova snow
isnt 3x^2 + 2x = 9x^2 + 2x
stray otter
power rule
stray otter
stray otter
np
nova snow
.close
crystal mason
Let $\mathcal{V}$ be a finite-dimensional unitary vector space and let $f \in \mathcal{L}(\mathcal{V},\mathcal{V})$ be a normal endomorphism, i.e. $f \circ f^{ad} = f^{ad} \circ f$ holds. Show that $f$ is unitarily diagonalisable.
wraith daggerBOT
Levens
crystal mason
How would one approach this?
cedar kilnBOT
@crystal mason Has your question been resolved?
prime karma
can someone help me with this?
cedar kilnBOT
can someone help me with this?
prime karma
my work
livid hound
how are you getting that second last line
also writing dy/dx = stuff in that first line is very inappropriate
prime karma
how are you getting that second last line
oh ok
earnest socket
Check how you are going from line 4 to line 5
cerulean sail
prime karma
wait doesnt it cancel out? because it'd be 1/2 times 2?
cerulean sail
If you want to cancel it out you'd have to also change that 6x^5 and 6y^5 terms
prime karma
ohh ok
so then for this probelm should i be factoring some constant out from both 1/2 and 6?
cerulean sail
Well that would be an idea, or multiply both top and bottom by 2
humble cipher
is this space all of R^2?
cedar kilnBOT
is this space all of R^2?
clear berry
If you mean if they span R^2 then yes they do
In fact they form the basis called the standard basis for R^2
humble cipher
In fact they form the basis called the standard basis for R^2
yeah exactly
so does that mean if thats the image, the image is R^2?
crimson delta
be careful. what you showed is just a set with two vectors. that cannot be R^2. it does span R^2, yes
clear berry
Image is generally used when we are talking about functions
What I mean is that, linear combinations of the given set with field R is R^2
humble cipher
be careful. what you showed is just a set with two vectors. that cannot be R^2. ...
earlier i had smth that said if the dimension of the image is 2, then the image is all of R^2
crimson delta
if you are in R^2, yes
humble cipher
T : R^3 -> R^2
crimson delta
then yes
jovial eagle
okay i found this, the leading coefiecient is positive in my example but why the calculator graphs this function that is located in 2nd and 4th quadrant
cedar kilnBOT
okay i found this, the leading coefiecient is positive in my example but why the...
@jovial eagle Has your question been resolved?
cedar kilnBOT
@jovial eagle Has your question been resolved?
neat sluice
Hi I cannot really understand why do we use conditional operator with the universal quantifier in logic
neat sluice
i mean why don't we just use the "and" operator?
the reason they give in my textbook is that by using the "and" operator we imply that every object in the universe should have the predicates we are testing
however, by looking at the truth table of the conditional operator, it returns true even when the antecedent is false...
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junior iron
Why does the determinant of a nxn-matrix stay the same if I expand it with a 1 along the diagonal?
crimson delta
expand along the last row or column
junior iron
I don’t know how to compute the determinant. We have been given some properties and some definitions (of alternating groups, etc). We were told the above holds for all nxn matrices.
Well I do know the formulas for 2x2 and 3x3 but thats not really useful for my understanding of the general nxn case
crimson delta
you will hopefully learn a property that allows you to compute an nxn determinant by expanding along a row or column and computing n determinants of size (n-1)x(n-1)
not sure rn how to prove this particular case without that tho. should probably be possible tho
do you know that you can row reduce to compute determinants?
junior iron
I don’t think that we will proof the above that way. We always do things like this shortly after introducing them. I’ve seen similar things on stackexchange that argued with the properties of alternating maps, etc but I cant find one that explains the above
crimson delta
and do you know that for a triangular matrix the determinant is the product of the diagonal entries?
junior iron
I dont know anything about computation
Well
I know the long way
With permutations and the Sn group
With sums that have like a thousand terms
crimson delta
I mean you could force your way through this and see that a lot of products are 0 I guess
except for those were you fix the last element in the permutation
that allows you to go from Sn to S(n-1)
junior iron
Argument using induction?
crimson delta
I mean if you don't know anything about general nxn determinants except the leibniz formula then whatever you do will be painful
junior iron
Whats the leibniz formula?
crimson delta
the one with the sum over all permutations
junior iron
the one with the sum over all permutations
Just googled a way of nit forgetting a permutation: is rule of sarrus something that I use with the leibniz formula?
crimson delta
rule of sarrus is only applicable for 3x3 matrices
junior iron
:(
crimson delta
you never use the leibniz formula to actually compute the determinant
it's just there for theoretical purposes
what you can try to show for yourself: show that row operations play nicely with the determinant
more specifically, switching rows flips the sign of the determinant, scaling a row by c also scales the determinant by c and adding a multiple of one row to another does not change the determinant at all
I am hoping that you know that the determinant is multilinear and alternating in the rows
crimson delta
ok great
this gives you a way to calculate determinants by row-reducing a matrix
row reduce until you have a triangular matrix and then the determinant is the product of the diagonal entires
which you can show by completing row reducing a triangular matrix to the identity and using that the determinant of the identity is 1
junior iron
Will that help me for general nxn matrices
With variables for every entry
The row reducing would be pretty ugly I feel like
crimson delta
so now lets say you have a matrix $A$ which you can row reduce to a triangular matrix $T$. so you know that $\det A = \det T$. now lets think about the matrix $\begin{pmatrix} A & 0 \ 0 & 1 \end{pmatrix}$ which is just $A$ with a $1$ at the bottom right. you can just row reduce the first rows like previously and then you will get the matrix $\begin{pmatrix} T & 0 \ 0 & 1 \end{pmatrix}$. but this is again just a triangular matrix, so the determinant is the product of the diagonal entries. which are just the entries of $T$ and the $1$. so $$\det \begin{pmatrix} A & 0 \ 0 & 1 \end{pmatrix} = \det \begin{pmatrix} T & 0 \ 0 & 1 \end{pmatrix} = \det T = \det A$$
wraith daggerBOT
Denascite
crimson delta
when I say row-reduce here, I don't mean actually doing it. I mean that you could in theory do it and it has the properties i described above
junior iron
Does this have a name or is it just basic application of determinant properties?
crimson delta
what exactly should have a name here?
junior iron
The reducing to apply leibniz easier
crimson delta
well, row-reducing
and yes just basic knowledge that this plays well with the determinant
junior iron
Should I just take note of all the row operations that multiply something?
When doing it on a real matrix later
crimson delta
I find that in practice I generally never actually multiply a row by something
I only ever add a multiple of some row to another
or rarely switch two rows
but yes if you were to multiply them then you have to account for that
I mean it's just pulling a factor out
cedar kilnBOT
@junior iron Has your question been resolved?
junior iron
except for those were you fix the last element in the permutation
I tried to do it by splitting it up into two sums
One sum where you have permutations that dont give you the (n+1)-element in the first n columns and then a sum where the last column gets the 1 in the bottom right
Just realized that I can make the conditions better I dont need two conditions one is enough
Just one sum where the last column gets the n+1 element and one where it doesnt
junior iron
Just one sum where the last column gets the n+1 element and one where it doesnt
This should be enough I think
.close
grave beacon
I am confused with complex numbers. How do I solve log(-i), if i stands for imaginary part?
dire geode
,calc log(-i)
wraith daggerBOT
Result:
-1.5707963267949i
dire geode
$-i = \cos(3\pi/2) + i \sin(3\pi/2)$
wraith daggerBOT
riemann
dire geode
now use euler's identity
cedar kilnBOT
@grave beacon Has your question been resolved?
light pine
cedar kilnBOT
tropic oxide
no, because you have missed an important part of the defn of an algebraic number.
under your definition, every complex number is algebraic.
tropic oxide
do not call me sir.
tropic oxide
i cannot help but analyze "yessir" as "yes sir".
light pine
ok
tropic oxide
anyway
light pine
bruh
tropic oxide
would you like to try and recall the defn of an algebraic number yourself or do you want me to remind you
grizzled oasis
Stop being transphobic they have clearly told you not to refer to them using masculine terms
light pine
would you like to try and recall the defn of an algebraic number yourself or do ...
A number α in C is algebraic if there exists a polynomial such that f(α) = 0.
tropic oxide
A number α in C is algebraic if there exists a polynomial such that f(α) = 0.
no, incorrect. you are still missing the most important part.
wide plume
sorry if it offended you ann
tropic oxide
right now, with your definition, pi is algebraic because it is the root of the polynomial (x - pi).
@light pine
light pine
right now, with your definition, pi is algebraic because it is the root of the p...
right
ok what am i missing then
tropic oxide
you are missing that the polynomial has to have integer coefficients.
(rational coefficients are good enough too, in the sense that a number is annihilated by some polynomial in Z[x] if and only if it's annihilated by a polynomial in Q[x] -- it's a matter of preference which one is chosen as The Definition™️)
light pine
you are missing that the polynomial has to have **integer coefficients**.
so all coeffs. appearing in f are integers?
tropic oxide
yes
light pine
yes
so
back to question 5
what do i need to change
ive changed my dfn in question 4 now
tropic oxide
well your work for question 5 is unsalvageable
given a polynomial f ∈ Z[t] which annihilates x (i.e. for which x is a root), how do you construct a polynomial g ∈ Z[t] such that g(sqrt(x)) = 0?
cedar kilnBOT
@light pine Has your question been resolved?
crimson elm
You invest $400 in a magazine stand. You buy each magazine at $3.00 and you are planning to sell them at $7.00.
crimson elm
Write a linear function to represent the situation. Choose your own variables.
cedar kilnBOT
@crimson elm Has your question been resolved?
crimson elm
<@&286206848099549185>
crimson sedge
Do you mean get a function to represent profit?
crimson elm
Yes
crimson elm
3
crimson sedge
Yep
And if we’ve sold 0 magazines and spent 400 on the stand, what’s the current net profit
crimson elm
-400
crimson sedge
Yep
So this gives a y-intercept of -400
And if every magazine sold gives a profit of 3, then the line will have a gradient of 3
Leaving y=3x-400
crimson sedge
The 7 is in there cause 3=7-4
jaunty mural
Should be guiding ppl to answer, not giving it
crimson elm
The 7 is in there cause 3=7-4
Ok thank you
I also need help with another question
If that’s ok with you
What is the minimum number of magazines you have to sell to break even?
When it says break even, does it mean that the y must be zero?
crimson sedge
Yep
wary summit
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