we have dt/dx and dy/dt
#help-26
frank lodge
yea
buoyant horizon
$\frac{\dd{y}}{\dd{x}}=\frac{\dd{t}}{\dd{x}}\times\frac{\dd{y}}{\dd{t}}=\frac{1}{2\sqrt{x}}\times-\sin{\sqrt{x}}$
ee
thorny flameBOT
@buoyant horizon
buoyant horizon
right?
frank lodge
yess
buoyant horizon
this gives $\frac{-\sin{\sqrt{x}}}{2\sqrt{x}}$
thorny flameBOT
@buoyant horizon
buoyant horizon
Now in theory, you should've been done
but the book goes an extra step further
Do you recall:
$\sin^2\theta+\cos^2\theta=1$
thorny flameBOT
@buoyant horizon
buoyant horizon
but wait for this why did u not take -1/sinsqrtx
why reciprocal?
we got dy/dt as -sin(root x), right?
we just multiply the two diffs we did
to get the answer
frank lodge
ah okay
frank lodge
<@951897314333700186>
yes
buoyant horizon
alright
now
can i do this:
$\sin^2\theta=1-\cos^2\theta\implies \sin\theta=\sqrt{1-\cos^2\theta}$
thorny flameBOT
@buoyant horizon
buoyant horizon
which?
frank lodge
i just asked u abt it and checked my book but i really can’t seem to get it
wait
here for arccos they took -1/sint but u didn’t 😭 why so (i’m slow as hell)
buoyant horizon
here for arccos they took -1/sint but u didn’t 😭 why so (i’m slow as hell)
Uh
let's go back to the very first step
When we did this, did you see that we get a cos on the right, not arccos
We changed the function we want to diff into cos
Because well x = (arccos y)² is harder to differentiate when written in that way
buoyant horizon
We got y = cos(√x)
frank lodge
We changed the function we want to diff into cos
yeah i always do this step but still take the reciprocal 😭
buoyant horizon
Y is a function of x with a cos, not an arccos
dy/dx means "differentiation of y with respect to x"
after you isolate y
you need to see what the new function is
in this case it's cos
that's why I used d(cos(t))/dt = -sin t
does this clear it up?
frank lodge
that's why I used d(cos(t))/dt = -sin t
okay 
tysm bro 😭😭😭
buoyant horizon
np
buoyant horizon
<@951897314333700186>
so, did you understand this?
buoyant horizon
<@951897314333700186>
from here,
$-\sin\sqrt{x}=-\sqrt{1-\cos^2\sqrt{x}}$
thorny flameBOT
@buoyant horizon
buoyant horizon
right?
frank lodge
yeah right
buoyant horizon
<@951897314333700186>
so put that instead of -sin√x in this and you should get the expression the book's asking
frank lodge
yeah it’s the same expression 
buoyant horizon
Because well x = (arccos y)² is harder to differentiate when written in that way
If you are confused about this "isolating y as function of x", we can still try diffing this
using the arccos formula in ur book
frank lodge
would appreciate it 🥲
buoyant horizon
alright
frank lodge
im sorry bro i’m actually very slow at understanding
buoyant horizon
now notice that you havex on the left
frank lodge
yea
buoyant horizon
so if you differentiate both sides, you write dx/dy, not dy/dx right?
frank lodge
yeah right
buoyant horizon
alright, so
frank lodge
OHHHHHHH
buoyant horizon
going by the book method
Since we have a function squared, we can let t = arccos (y)
frank lodge
wait…i just realized sometjing
frank lodge
or maybe i’m wrong…wait continue maybe you’ll mention it while explaining
buoyant horizon
lol
frank lodge
buoyant horizon
so
buoyant horizon
Since we have a function squared, we can let t = arccos (y)
this works right?
frank lodge
Since we have a function squared, we can let t = arccos (y)
i don’t understand 😭
frank lodge
like we let the squared function be t?
buoyant horizon
You let the "inside" thing as t, right?
buoyant horizon
like we let the squared function be t?
you let the thing that's "inside" a bigger thing as t
buoyant horizon
yes
yes, now
we have arccos y "inside" a square
so we let t = arccos(y)
$\frac{\dd{t}}{\dd{y}}=-\frac{1}{\sqrt{1-y^2}}$
thorny flameBOT
@buoyant horizon
buoyant horizon
right?
frank lodge
yea
buoyant horizon
Now we have dt/dy
We will need to find dx/dt
Now, $x=t^2\implies \frac{\dd{x}}{\dd{t}}=2t=2\arccos{y}$
frank lodge
alr
thorny flameBOT
@buoyant horizon
buoyant horizon
Now we multiply these two
$\frac{\dd{x}}{\dd{y}}=\frac{\dd{x}}{\dd{t}}\times\frac{\dd{t}}{\dd{y}}=2\arccos y \cdot \frac{-1}{\sqrt{1-y^2}}$
thorny flameBOT
@buoyant horizon
buoyant horizon
right?
buoyant horizon
Now, we have dx/dy, but we need dy/dx
buoyant horizon
why not just 2t tho
t is a different variable we took
we gotta get rid of it in the answer
frank lodge
alr
buoyant horizon
when you solved the example, you substituted t back to x^2, right?
thats what i did here
frank lodge
oh yeah
buoyant horizon
Now, we have dx/dy, but we need dy/dx
So, we need to reciprocal this
This gives
$\frac{\dd{y}}{\dd{x}}=-\frac{\sqrt{1-y^2}}{2\arccos y}$
thorny flameBOT
@buoyant horizon
buoyant horizon
Right?
frank lodge
yeah
buoyant horizon
Now, we need to get rid of the y's
since the answer expects us to have everything in terms of x
to get rid of the $y^2$:
$x=(\arccos y)^2\implies \sqrt{x}=\arccos y\implies y=\cos\sqrt{x}\implies y^2=\cos^2\sqrt{x}$
thorny flameBOT
@buoyant horizon
buoyant horizon
right?
buoyant horizon
hmm
we got rid of y^2
we still have an arccosy on the bottom
$x=(\arccos y)^2\implies \sqrt{x}=\arccos y$
thorny flameBOT
@buoyant horizon
buoyant horizon
right?
frank lodge
yess
buoyant horizon
so we got rid of y's
buoyant horizon
<@951897314333700186>
put that back into this fraction
you get the same expression
buoyant horizon
real
frank lodge
😭😭 neat work
i wanna save this explanation somewhere so i can comeback to it 😭 i’ll copy the link to one of ur messages and save it 💀
frank lodge
yesss
buoyant horizon
i think this channel gets dissolved after you mark your question as done
im not sure
frank lodge
it gets used by other ppl but the previous msgs will still remain in it
buoyant horizon
ee
frank lodge
i’ll just screenrecord
buoyant horizon
alr
topaz sinewBOT
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earnest apex
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earnest apex
yo how do I do this now 😭
topaz sinewBOT
yo how do I do this now 😭
acoustic tangle
Consider substituting for x+y
glossy nymph
Let u = x+y+2
then u' = y'-1 then u' = u^2+1
earnest apex
Consider substituting for x+y
Was y = Y - 2 the wrong move
acoustic tangle
No
acoustic tangle
Right
topaz sinewBOT
earnest apex
.reopen
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earnest apex
@acoustic tangle sorry to disturb
will this be the final answer
it's given that for x = 0 y = -2
so const = 0
,w √3 - pi/3 - 2
thorny flameBOT
acoustic tangle
You could check by just plugging your solution into the original equation
earnest apex
You could check by just plugging your solution into the original equation
oh yes!
i think it fits
thank you all!
.close
white grotto
i have a question about the jabcobian and why is it used when we change variables for example from x to u when dealing with a double integral, i really want to know why because the teacher is just spoon feeding us
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i have a question about the jabcobian and why is it used when we change variable...
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eager coral
what can cosx-sinx be rewritten as?
topaz sinewBOT
what can cosx-sinx be rewritten as?
rapid lake
proof?
eager coral
proof?
eager coral
yeah
rapid lake
yeah
i'd work with the right hand side
just rewrite cotg in terms of sin and cos
unless you have special instructions to work with the left hand side
eager coral
I guess I could try
rapid lake
left hand side is doable as well its just cosx-sinx isnt really a known identity
which side would u prefer to work with
eager coral
left
eager coral
i was thinking multiply with the denominator but im honestly not entirely sure how to do that or if I can
rapid lake
we can divide the numerator and denominator by the same thing and the identity stays equal
lets start with the numerator
what comes to mind so that it becomes as the right hand side's numerator
eager coral
somehow make it cosx/sinx
eager coral
/sinx
rapid lake
yea
and if we divide the numerator by sinx, we have to do the same for the denominator
eager coral
so the numerator is fine
denominator is cosx-sinx/sinx
rapid lake
use brackets
(cosx-sin)/sinx
but yeah
can we separate that fraction as cosx/sinx - sinx/sinx ?
verbal shadow
noice
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lusty dock
I’m afraid of failing math
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I’m afraid of failing math
lusty dock
What?
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@lusty dock Has your question been resolved?
red relic
could you technically call a cylinder a square just because it could be turned into a square and two circles as 2D shapes? my friend is arguing for this and it doesn't make sense to me.
topaz sinewBOT
could you technically call a cylinder a square just because it could be turned i...
@red relic Has your question been resolved?
graceful wolf
could you technically call a cylinder a square just because it could be turned i...
cylinder is 3-dimensional
square is 2-dimensional
you can't call a 2-dimensjonal figure a 3-dimensjonal figure
red relic
cylinder is 3-dimensional
square is 2-dimensional
you can't call a 2-dimensjon...
she's saying that because a cylinder (or pipe)'s "original form" is a square, it is technically a circular square.
or that because a square can be folded into a cylinder, a cylinder is a circular square
graceful wolf
well then all kinds of other 3-dimensional figures would be squares
red relic
well then all kinds of other 3-dimensional figures would be squares
I said this, but the examples I gave would be composed of multiple squares, whereas a cylinder only one
she shot it down on that basis
she used a square piece of paper as an example. if you rolled it up you could make it a cylinder, but by letting it go it would revert to a square
topaz sinewBOT
@red relic Has your question been resolved?
opal vault
in P(n) the sum goes to n+1
so supposing P(k), you know something about the sum that goes from 1 to k+1
ocean pier
so supposing P(***k***), you know something about the sum that goes from 1 to **...
i know that 1+1=2
opal vault
i know that 1+1=2
yes, so if you're trying to prove P(k+1) using P(k), you want to know something about the sum that goes from 1 to k+2
to do that, put the "k+2" term aside
now you got the sum that goes from 1 to k+1
opal vault
so supposing P(***k***), you know something about the sum that goes from 1 to **...
that's given by P(k)
we use it
so now k*2^(k+2) + 2
- the "k+2" term we set aside
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@ocean pier Has your question been resolved?
sweet gale
Can someone please give me tips on how to be better at maths specifically geometry
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dense plover
Let $f : X \rightarrow Y$ be a morphism of algebraic curves (one-dimensional integral separated schemes of finite type over a field $k$, not necessarily algebraically closed). How do we construct regular projective curves $X'$ and $Y'$, and a morphism $f' : X' \rightarrow Y'$ such that $f'$ and $f$ are birationally equivalent?
topaz sinewBOT
Let $f : X \rightarrow Y$ be a morphism of algebraic curves (one-dimensional int...
thorny flameBOT
DavidL1450
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wet holly
Am I correct with this one?
topaz sinewBOT
Am I correct with this one?
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wet holly
Another question
I went from that to h'(x)=f'(2x+1) (2)
then I have legit no idea how to get f'(3)
<@&286206848099549185>
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wet holly
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wet holly
I went from that to h'(x)=f'(2x+1) (2)
then I have legit no idea how to get f'(3)
topaz sinewBOT
@wet holly Has your question been resolved?
plucky elbow
to find h'(x) we can use chain rule
let u = g(x) and let y = f(g(x)) = f(u)
h'(x) = du/dx * dy/du
= g'(x) * f'(g(x))
so h'(1) = g'(1)f'(g(1))
= 2f'(3)
= and to find f'(3) you need to find the gradient of the line that connects (1,4) and (4,-2)
neon iron
Its visible that 2 components of Fg are Fgx and Fgy (In the book the axis are diffrent from this pic where x is along the ramp) but to find Fgx is tricky how the trig works because its not similar to the normal cirlce trig.
topaz sinewBOT
@neon iron Has your question been resolved?
neon iron
<@&286206848099549185>
neon iron
Its visible that 2 components of Fg are Fgx and Fgy (In the book the axis are diffrent from this pic where x is along the ramp) but to find Fgx is tricky how the trig works because its not similar to the normal cirlce trig.
arctic hawk
hmm
arctic hawk
Its visible that 2 components of Fg are Fgx and Fgy (In the book the axis are d...
oh it's mgcos(theta)
it nicely checks out 😄 (mgcos(theta)^2 + mgsin(theta)^2)) =mg
neon iron
Yeah but visualy how to understand it?
Its cos(60grade)
Why bot sin60 grade?)
Whos stops me?
velvet hound
it's just how the angles add up, we have two related right angle triangles being made from the forces, one of Fg and the other from Fgy
so when you're calculating Fgy, we need to reverse which trig function we use because now our "y" is adjecent rather than opposite to our theta
topaz sinewBOT
@neon iron Has your question been resolved?
upbeat minnow
What am I doing wrong while integrating this? The cos(4x) doesn't cancel out
agile harness
poor choice of u
upbeat minnow
poor choice of u
what would have been better for u?
agile harness
the sin(4x) should make it obvious outside
upbeat minnow
the sin(4x) should make it obvious outside
so u=sin(4x)? im a little confused, what made you think that was the logical pick? I just took the biggest mess and set that as u
agile harness
no
notice how the sin(4x) cancels with your original u sub, but it brings along the unwanted cos
any ideas how to kick cos out like a social reject
upbeat minnow
any ideas how to kick cos out like a social reject
honestly, not really besides setting 2sin(4x) as u, and im not sure if htat would work
upbeat minnow
yes
agile harness
so maybe try using cos somewhere in your u sub
but not the entire 1 + asd;lfjdsal'fgasjdg'lidsuga['sop
upbeat minnow
ok i will try that
agile harness
good man
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signal stirrup
what is the answer
topaz sinewBOT
what is the answer
raven sparrow
The one you get when you compute the sum
signal stirrup
how to do it
agile harness
review summation notation
you add each term with each terms x value increasing by 1 each time
starting from 1 and ending at 5
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devout fable
in the standard form of Ax+By+C=0, is there any convention that A has to be positive?
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in the standard form of Ax+By+C=0, is there any convention that A has to be posi...
@devout fable Has your question been resolved?
devout fable
.close
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mystic siren
how tf do i draw these curves
topaz sinewBOT
how tf do i draw these curves
mystic siren
i cant do it
prime flame
how tf do i draw these curves
What do you mean by draw? Haven't you already drawn it
mystic siren
What do you mean by draw? Haven't you already drawn it
thats the prof's
i cant draw it on paper
it's wrong
prime flame
i cant draw it on paper
So you are giving a piece-wise function and have to draw it on paper?
mystic siren
So you are giving a piece-wise function and have to draw it on paper?
yes
i cant draw the damn curves
like how do i even do that
prime flame
Just choose some point on the interval and then plug it into the function and then connect the point
mystic siren
Just choose some point on the interval and then plug it into the function and th...
it dont work bro 😭
like i know what to do
it dont work
prime flame
it dont work
Why? For these kinds of sketch questions, you only need to maintain the general shape of a function and be careful about endpoints (whether the interval is open or closed)
vernal matrix
(I shall also ask what the context behind this question, if any, is? Have you been asked something in particular?)
topaz sinewBOT
@mystic siren Has your question been resolved?
slow tusk
when you work out the brackets, how does the minus in front of Beta 1 hat, turn into a plus sign
topaz sinewBOT
when you work out the brackets, how does the minus in front of Beta 1 hat, turn ...
verbal shadow
when you work out the brackets, how does the minus in front of Beta 1 hat, turn ...
- * - = +
slow tusk
\- * - = +
What minus, don’t you just multiply with the epsilon Xi thing
verbal shadow
-(-b hat)
verbal shadow
i think so
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near laurel
This probably makes me look stupid but I really can’t get my head around this
near laurel
So it’s just b-a??
near laurel
Hang on I’m trying to get my brain around this
near laurel
since u_1=3
So for the next one, if n=2 so I’d substitute the last term into the equation? (B-3a)
buoyant horizon
yes
near laurel
buoyant horizon
yep
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near laurel
For this question, I’m not sure what my next step is. I tried getting all into one side and I got a x^4 so unsure of what to do
topaz sinewBOT
For this question, I’m not sure what my next step is. I tried getting all into o...
eager compass
For this question, I’m not sure what my next step is. I tried getting all into o...
Can you show what you mean by you got x^4
near laurel
Can you show what you mean by you got x^4
It’s only 3 marks so there must be something I should do
near laurel
the function and the inverse function must meet at line y = x
I was about to say that, it’s in my notes when the question was gone through
But I’m not sure how to use that
finite storm
x >= 0
finite storm
oops
finite storm
near laurel
I can find out the intersection
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pliant kelp
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opal vault
what does #2 mean?
if you take a divisor d, it generates the subgroup {d,2d,3d,...}
the exercise is asking you to take one of those subgroups
(so one of those divisors)
and write the operation table
of the generated subgroup
pliant kelp
if you take a divisor d, it generates the subgroup {d,2d,3d,...}
so just pick a number and add it over and over till you hit the max that is under 60?
Z_60 = <30>
{ 0, 30 }
smoky sparrow
and then write the operation table
but yes these are the only two elements in that subgroup
https://math.stackexchange.com/questions/2168385/cayley-table-of-mathbbz-5
this is what you need to do
pliant kelp
and then write the operation table
smoky sparrow
cool looks good
opal vault
the multiplicative group of integers mod 60
is the subset of Z_60
with elements that are invertible (by multiplication)
59 is invertible because 59 * 59 = 1
20 is not invertible because 20 * x is never 1 in Z_60
pliant kelp
59 * 59 = 1?
opal vault
59 * 59 = 1?
in Z_60
pliant kelp
until you are in the 0-59 range?
opal vault
until you are in the 0-59 range?
yes
there are easier ways to see why 59 * 59 in Z_60 but
59 is invertible
while for example 20 isn't
maybe definition 11.4.17 would be worth looking at if the question asks for it
opal vault
is there some other part?
opal vault
so i just use numbers divisible by 2,3,5
no, you have to get rid of those numbers actually
the elements of the multiplicative group are precisely those that share no common factor with 60
those that are coprime
smoky sparrow
yeah if you have taken elementary number theory this will be super familiar
that say there is no solution to $10k \equiv 1 \pmod {60}$
cause that implies $10k = 60m + 1$ for $k, m \in \mathbb Z$ and now look at divisibility by $10$ on both sides
thorny flameBOT
south's secret twin brother
opal vault
in 0 - 59
smoky sparrow
that's how you reason that you don't want the divisors of 60
and anything else is possible
(by the Euclidean algorithm, cause the gcd will be 1)
pliant kelp
ok
{ 1, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59 }
i gtg to class
opal vault
all prime numbers + 1 and 49, nice
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@pliant kelp Has your question been resolved?
azure dune
Hey I'm working through some elementary number theory problems about divisibility and want to prove (or disprove) the following:
$a | (b +c) \implies a | b \lor a | c$
topaz sinewBOT
Hey I'm working through some elementary number theory problems about divisibilit...
thorny flameBOT
texaspb
azure dune
i don't see how I can go from b + c = ax for some integer x
azure dune
I mean i tried to find counter examples but idk
for ex
5 | 15 + 5 => 5 | 15 or 5|5
but
5 | 13 + 7
but 5 does not divide neither 13 or 7
long stirrup
yeah or 2 | 7+ 9
azure dune
yeah
long stirrup
you disproved it
azure dune
o
oh
lol
true iguess
found a counterexample dur
sorry i'm kinda slow today
thansk!! ❤️
.close
neon iron
prove that if (not iff, i'll do it later) n>9, then 2^n > n^3
topaz sinewBOT
prove that if (not iff, i'll do it later) n>9, then 2^n > n^3
silent siren
!status
topaz sinewBOT
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
compact zodiac
induction?
limber prairie
prove that if (not iff, i'll do it later) n>9, then 2^n > n^3
induction?
neon iron
yes
compact zodiac
so where you stuck at?
neon iron
Assume it to be true for n = k
2^k > k^3
2^(k+1) > 2k^3
limber prairie
lemme think
fresh ridge
is using derivatives to prove that allowed for you @neon iron ?
limber prairie
k>9
1/9 > 1/k
10/9 > (1 + 1/k)
(10/9)^3 > (1+ 1/k)^3
limber prairie
its a roundabout way
limber prairie
k>9
1/9 > 1/k
10/9 > (1 + 1/k)
(10/9)^3 > (1+ 1/k)^3
But I imagine there to be a more direct way
fresh ridge
hold on, is the original correct
odd forge
Assume k >=10
limber prairie
k>9
1/9 > 1/k
10/9 > (1 + 1/k)
(10/9)^3 > (1+ 1/k)^3
2> (10/9)^3 > ( (k+1)/k )^3
limber prairie
because from what i know 2^x>x^3 till 1.37 (approximately)
(10/9)^3 is approx 1.37
neon iron
2^k > k^3
2^(k+1) > 2k^3
Now let's compare 2k^3 and (k+1)^3
Find the answer to this ineq, -> 2k^3 > (k+1)^3
neon iron
wolfram alpha says 2k^3 > (k+1)^3 for k> 3.84 (approx)
since this will be true for k>9
odd forge
k(k^2)>9k^2
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neon iron
.reopen
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odd forge
natural numbers?
fresh ridge
then why is it not valid for 9.01?
it is valid
neon iron
happens
topaz sinewBOT
@neon iron Has your question been resolved?
prisma coral
prove that if (not iff, i'll do it later) n>9, then 2^n > n^3
what all is allowed
for you
prisma coral
happens
assume 2^n > n^3 ( base case 9) 2^(n+1) = 2^n *2 > 2 x n^3 , so all you have to do is prove that 2x n^3 > (n+1)^3 => 2> (1+1/n)^3 => (1+1/n)< cuberoot(2) , cube root of (2) ~1.26, ( 1+1/9) = 1.1111 , so for n>= 9 we have (1+1/n) <= (1+1/9) <cube root(3) and u are done
topaz sinewBOT
assume 2^n > n^3 ( base case 9) 2^(n+1) = 2^n *2 > 2 x n^3 , so all you have to ...
prisma coral
try that
neon iron
i've done it
topaz sinewBOT
topaz sinewBOT
Available help channel!
hollow pumice
Can I please have some guidance on this
topaz sinewBOT
Can I please have some guidance on this
ionic oar
do you know how to invert functions
hollow pumice
I dont
ionic oar
Alright
if you have y = f(x) all you need to do is replace the x and y, so x = f(y), and then solve for y
hollow pumice
so the X's turn into Y's?
ionic oar
yes and vice versa
night imp
Let's take an example
y = x²+2
Now swap x and y
x = y²+2
Find y in terms of x
x-2 = y²
√(x-2) = y
So f`(x) = √(x-2) - this is inverse of f(x)```
Yes?
y = x²+2
If we swap x and y it will become
x = y²+2
I am working as a software engineer, my school days are over
Good luck
You can proceed with your original question now
I hope
Nooooo, that was an example I had given
😅
night imp
Nooooo, that was an example I had given
To help you understand inverse
topaz sinewBOT
Available help channel!
distant totem
I don't really understand the point of limits, as example this one here. I understand that to solve it I would need to refactor it and in that way find out the answer, but what is really the difference between a normal polynomial and one where we have lim before? Is the only difference that when we say P(x) = y that y is the actual answer meanwhile that when x goes towards 2 that the output y goes towards that number?
topaz sinewBOT
I don't really understand the point of limits, as example this one here. I under...
cedar wagon
Do you know what is the definition of derivative ?
distant totem
Do you know what is the definition of derivative ?
not really, in the book we first get to learn about limits before deriviation
raven field
I don't really understand the point of limits, as example this one here. I under...
for this one you can’t directly make x=2
cedar wagon
not really, in the book we first get to learn about limits before deriviation
Ic
Do you know l'hôpital ?
distant totem
Do you know l'hôpital ?
i just quickly checked it out and i think so, we didn't have that name in the book but it told us that it can decide values for unlimited equations like 0/0 and infinity/infinity
distant totem
so im really new to limits, but from what i can tell in the example i gave you is that p(2) is a factor in both the denominator and numerator making it 3(x-2)(x+2)/(x-2)
raven field
so im really new to limits, but from what i can tell in the example i gave you i...
then just simplify
distant totem
I did so eventually i came to
3(x+2)
raven field
we still got x->2
fallow torrent
replace x with 2
raven field
replace x with 2
Not always tho
distant totem
12 is the correct answer, but my question is really what the big difference is between simplifying this and a normal polynomial? Is the key difference that limits show you an indication of what y could be if x goes towards 2 in this case?
so if x -> 2, y goes towards 12, but is not 12
cedar wagon
Wdym by normal polynomial ?
distant totem
for example P(x) = x^2+4x-4, we put in P(2) and get a value for y
Don't know if that is a correct second degree equation, but to get the idea, we put in x, get out y
cedar wagon
Yes
distant totem
However limits
fallow torrent
Not always tho
yes , it depends if it contains an indeterminate form after doing the calculations
distant totem
what is the point, what does it tell us/do?
cedar wagon
That limits is a trouble iff there is no continuity
raven field
what is the point, what does it tell us/do?
Limits are used to check the behavior of a function at a certain point
distant totem
so as I said in that case, in the example i gave when x -> 2, y -> 12, but is not 12?
raven field
so as I said in that case, in the example i gave when x -> 2, y -> 12, but is **...
For the limit definition we check the behavior of the function as it gets closer and closer to 2
but when we write x=2 the function is not defined at x=2
so we use limits to see how a function behaves at some point
distant totem
so in that case, does that mean that the domain is Dv = R \ {2}
raven field
so in that case, does that mean that the domain is Dv = R \ {2}
Yep
But using limits we could see how the function will behave at x=2
distant totem
Alright so, I tried to visualise it, but here comes some confusion with, as I see that when x = 2, y = 12. However when x = 2 that means the fraction would become 0/0
raven field
Alright so, I tried to visualise it, but here comes some confusion with, as I se...
yep but here it says discontinuity
so the function is not defined at x=2
cedar wagon
In the original form you can't calculate x = 2 but you can transform it such that x = 2 became something doable
Removing discontinuity
distant totem
hm alright, so with a 'normal' polynomial we can say that when input is x, y is the output. but with limits we can check the behavior of a function when it gets close to x, but it can't be x thus Dv = R \ {x}
did i get that correct?
raven field
hm alright, so with a 'normal' polynomial we can say that when input is x, y is ...
yep but another definition for limits check values closer to what value x approaches like
x->1.99
x->1.9999
x->1.999999 so on
Or
x->2.000001
x->2.0001
x->2.01
as checking these values gets closer and closer to 2 and we can understand that the y values are approaching 12
distant totem
yep but another definition for limits check values closer to what value x approa...
right, so limits are basically used to check the behavior of a function when it closes towards x, so in my example indeed we can understand that when x goes towards 2 that y goes towards 12, but wont ever become 12 as that would become infinity
raven field
right, so limits are basically used to check the behavior of a function when it ...
you can’t say infinity but yea
distant totem
you can’t say infinity but yea
if we would put in 2 as x in the function i gave as example that would become 0/0, basically infinity right?
or did i get that wrong
raven field
if we would put in 2 as x in the function i gave as example that would become 0/...
not really
0/0 is an indeterminate form meaning the value of it exists it’s just we can’t determine what value is it
It could be 4 could be 35.72 could be infinity
But we can’t determine a specific value for it
distant totem
ah, kind of like imaginary numbers?
raven field
ah, kind of like imaginary numbers?
at least imaginary are specified as i but not for 0/0
You check it using another function you could see y approach a completely different value
distant totem
i see, that clears a lot up! so to summarize, limits basically help us tell what value y approaches when x goes towards 2 (in my example), but when it becomes 0/0 we know that Dv = R \ {2} since it would become an indeterminate form
raven field
i see, that clears a lot up! so to summarize, limits basically help us tell what...
Yep can I give you another limit with same idea/process
?
distant totem
yeah if you'd have time for it, i'd appreciate it
raven field
lim x^2-9/(x-3)
x->3
distant totem
so that would simplify to (x-3)(x+3)/(x-3) = x+3 = 3+3 = 6
raven field
Yep but ofc the original function doesn’t have 3 in the domain
as you go on it’ll get harder either using other algebraic processes like difference of cubes multiplying by conjugate so on
distant totem
I see, so to solve them we would have to find out what value of x makes the function 0/0?
Because in class today we also got an example of when the numerator was above 0, and the denominator was 0, that became quite hard
raven field
I see, so to solve them we would have to find out what value of x makes the func...
Sometimes some function won’t yield this indeterminate forms
Like 0^0 for example
Or sometimes won’t give an indeterminate form like
1/x when x approaches 0
But you’ll come to these cases later in limits
distant totem
I understand, well thanks for at least making me understand what really the function of limits are ^^
.close
topaz sinewBOT
topaz sinewBOT
Available help channel!
river trail
Help! (Calc 3) My professor asked us to use differentials to estimate the maximum error of Kirchhoff’s law.
topaz sinewBOT
Help! (Calc 3) My professor asked us to use differentials to estimate the maximu...
@river trail Has your question been resolved?
Available help channel!
mortal bone
Hi, i need to find the lowest common divisor between 3(X+1), (X+1) and (X-1)(X+1).
Wolfram alpha isn't really helping with it and i'm unsure about remembering how to do it, i've gotten 3(x+1)(x-1) but i've got the feeling it is wrong, if someone could also help me understand Lowest Common Divisor again it'd be really helpful for future reference
topaz sinewBOT
Hi, i need to find the lowest common divisor between `3(X+1)`, `(X+1)` and `(X-1...
wary tulip
that’s not a divisor of X+1?
craggy haven
i think you're a bit confused, lowest common divisor isn't a thing
mortal bone
oh, is it lowest common denominator maybe? i knew the term in another language and tried looking it up
lean pebble
you mean greatest common divisor
lean pebble
the lowest would always be a constant
wary tulip
are you sure you don’t mean lowest common multiple
mortal bone
context should help, let me get the problem i'm solving
trying to find a denominator to use for all 3 of these
checking my notes rn and it actually is the Greatest Common Divisor, i'm struggling to apply it with this
agile harness
common denominator
craggy haven
hmm ok well the terminology is a bit confusing but 3(x+1)(x-1) is indeed what you want for a common denominator
mortal bone
i see, i know i wasn't very helpful in bringing up my issue, will try to present it better next time and i just wanted to make sure i was on the right track
wary tulip
checking my notes rn and it actually is the Greatest Common Divisor, i'm struggl...
i don’t think that’s what you want?
uneven grotto
I am having trouble reading the handwriting. What is the generic name for this Unit in Calculus?
topaz sinewBOT
I am having trouble reading the handwriting. What is the generic name for this U...
sweet shard
loud oasis
the specific topic covered in the picture seems to be quadric surfaces
topaz sinewBOT
@uneven grotto Has your question been resolved?
ornate trellis
what is the terminal point of 17 pi
topaz sinewBOT
what is the terminal point of 17 pi
ocean terrace
what in
ornate trellis
unit circle
ornate trellis
what’s the terminal point of pi?
-1,0
which is what i put
but google is telling me i’m wrong
ocean terrace
what did u google
agile harness
but google is telling me i’m wrong
googles stupid
agile harness
was it gemini
ocean terrace
google and gemini is a bit different
agile harness
don’t trust ai sir
agile harness
google and gemini is a bit different
gemini always pops up on my google searches
agile harness
told ya
ornate trellis
at the very top
ocean terrace
gemini always pops up on my google searches
that occasionally trend on twitter
agile harness
POKE
ornate trellis
so it is -1,0?
ornate trellis
lemme seee what the ai is saying again
maybe i’m dyslexic
wouldn’t be the first time
ocean terrace
lemme seee what the ai is saying again
dont do ittttttt
agile harness
don blam me
yo what starter pokémon did you pick back in the day
ocean terrace
yo what starter pokémon did you pick back in the day
mew
agile harness
nah i’m talking charmander, squirtle, or bulbasaur
you seem like the bulbasaur type
charmander ftw
ocean terrace
dissing me
agile harness
indeed
ornate trellis
oh so uhhh
ocean terrace
maybe i’m dyslexic
same bro
ocean terrace
lol
ornate trellis
am i wrong
agile harness
why do you trust ai
ocean terrace
no
ornate trellis
why do you trust ai
i dont TRUST ai
agile harness
then why do you question whether you’re wrong
ocean terrace
why do you trust ai
why do u trust this guy
ornate trellis
i just have a lot of self doubt so id prob second guess myself when ai says im wrong
ocean terrace
its ok
agile harness
why do u trust this guy
cj and i go way back
ornate trellis
cj and i go way back
do we...?
ocean terrace
dont trust @agile harness but trust me. id say you are right
ornate trellis
"Since a whole number of rotations always end at the starting point (1, 0), the terminal point of 17π is also (1, 0)."
the correct thing to say would be
ocean terrace
lol
agile harness
yo cj
ornate trellis
hm
agile harness
what was your starter pokémon
ornate trellis
what was your starter pokémon
ughhh
agile harness
you were too young huh
ornate trellis
it always stressed me out because they were all so cute
agile harness
idk what was it red?
ornate trellis
i think i usually picked charmander
agile harness
i was like 4-5
ornate trellis
or Squirtle
agile harness
pokémon red
ornate trellis
idk what that is
agile harness
hmm
ornate trellis
hmm
ornate trellis
well it was some interactive thing where you could type "up" or "down"
and the whole community eventually beat it
agile harness
used to play at daycare back in the early 2010s
ornate trellis
used to play at daycare back in the early 2010s
how old are you?
agile harness
18
ocean terrace
this is clearly not math anymore
agile harness
i’m taking 4 math classes
^
ornate trellis
i’m taking 4 math classes
which ones?
anything fun like real analysis or linear algebra?
agile harness
differential equations, intro to higher math which is like a proof writing course, multivariable calc, and a probability course
agile harness
anything fun like real analysis or linear algebra?
i already took linear algebra over the summer
i take real analysis in the spring
agile harness
indeed
ornate trellis
lucky lucky boi
agile harness
how
ornate trellis
real analysis fun?
Yes
agile harness
you’re in alegbra 2 or some shit
ornate trellis
looks very fun imo
agile harness
looks very fun imo
have you ever seen it
ornate trellis
have you ever seen it
yeah
looks like a new language
anywho
i dont wanna be in precalc rn
agile harness
you can’t skip fundamentals
ornate trellis
like half the kids in my class dont know how to factor
agile harness
yea the basics
ornate trellis
There are like 3 important things in precalc everything else is just review of alg2
agile harness
you’ll be lost in further math classes if you haven’t mastered this stuff
ornate trellis
you’ll be lost in further math classes if you haven’t mastered this stuff
such as?
Limits?
agile harness
math is cumulative
ornate trellis
math is cumulative
to a certain extent
Was it wrong of me to skip pre algebra because i tested out of it?
I wouldnt say so
agile harness
but you don’t know precalc that well
ornate trellis
but you don’t know precalc that well
Well really, what exactly IS precalc?
agile harness
if the higher stuff is appealing to you you should self study
ornate trellis
if the higher stuff is appealing to you you should self study
Im working on it
ive been liking essence of calc and pauls online notes
agile harness
Well really, what exactly IS precalc?
basically all the algebra you need for calculus
ornate trellis
basically all the algebra you need for calculus
mhm... All algebra which is either trivial and or already taught?
agile harness
not exactly
ornate trellis
Precalc just seems like a lame excuse to give students a year before calc
agile harness
it’s needed
ornate trellis
it’s needed
But you havent really given any specific examples of what is important in pre calc
ornate trellis
you go over limits
trivial
agile harness
have you ever read a precalc book
ornate trellis
you asked what the terminal point of 17 pi was
Limits?
agile harness
you’re calling these things trivial
ornate trellis
What does that have to do with AI disagreeing with me and asking for reassurance
ornate trellis
you’re calling these things trivial
I would argue limits are pretty trivial
agile harness
yet you were confused about the terminal point of 17pi
agile harness
I would argue limits are pretty trivial
i think so, but i’m not sure you can say that atm
agile harness
What does that have to do with AI disagreeing with me and asking for reassurance
why would you need ai if it was trivial
🤔
ornate trellis
why would you need ai if it was trivial
Did i say the unit circle was trivial?
I didnt mean to if so
agile harness
a concept which is easier
ornate trellis
a concept which is easier
According to you..?
agile harness
the unit circle is taught in early algebra for a reason
ornate trellis
the unit circle is taught in early algebra for a reason
It isnt where i live
probably due to how fucked up the state is but oh well
W florida am i right
anyways
the point is
i feel like im kinda snoozing
I wish i could take like half a year where they taught the really important stuff instead of spending weeks on how to factor again
agile harness
so self study
agile harness
you just said you’re not using any books
ornate trellis
I cant afford a textbook
agile harness
essence of calc isn’t self studying
that’s just a nice video series
there’s no theory to it really
just gives some intuition
ornate trellis
essence of calc isn’t self studying
Its prob the best im gonna get other then pauls online notes which ive also been reading
ornate trellis
not by reading notes like a novel
Some notes do both
Start with a question which begs the iddea of a certain concept
walk through it
be able to apply it later
agile harness
them providing examples of solving problems isn’t the same
ornate trellis
them providing examples of solving problems isn’t the same
Why not?
Part of math is patern recognition
agile harness
you learn math by solving problems
^
ornate trellis
Watch a couple of problems
agile harness
math isn’t a spectator sport
agile harness
you have to solve the problem
ornate trellis
I think i might just uhhh
Ima ask one of my rich friends to buy me the textbook for my b day
ornate trellis
*you* have to solve the problem
Also
I also think you get some benefit in first understanding why something works
once you understand the WHY then you can learn to apply it
agile harness
I also think you get some benefit in first understanding why something works
who said you don’t
there’s a reason why textbooks put the exercises at the end of the chapter
ocean terrace
dang knief