#help-28
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@coarse tundra Has your question been resolved?
coarse tundra
<@&286206848099549185>
rough bolt
What's the matter?
misty flare
Hey guys, I need help with this. I'm struggling on where to start.
I have to determine the area of the triangle, but as you can see, all I have is the angles and the bisector as information
(on the right side are the answer options )
open igloo
Hey guys, I need help with this. I'm struggling on where to start.
I have to d...
Are you aware of the sine rule?
misty flare
yes, but I can't use it when I don't have enough information to calculate with
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misty flare
<@&286206848099549185>
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fervent anchor
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<@&286206848099549185>
fervent anchor
this proof seems kinda odd and maybe unintuitive but it makes sense to me
can anyone follow this?
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long ridge
ok so by bezouts identity gcd(12, 25)=1 so there exist such a and b, so that 12a+25b=1
then you can just scale them
brb
fervent anchor
ok so by bezouts identity gcd(12, 25)=1 so there exist such a and b, so that 12a...
We never learned this in class so we aren’t allowed to use it on our homework’s
long ridge
the step is the problem im guessing
you know that f(n)<2-1/n
you basically want to prove f(n+1)<2-1/(n+1)
you can expand lhs and get f(n)+1/(n+1)^2 < 2-1/(n+1)
so you can subtract the hypothesis and get 1/(n+1)^2 < 2-1/(n+1)-2+1/n
which after some manipulation should reduce to what you want to prove
long ridge
i think theres a typo in the end, theres a really nice idea but they explain it really poorly
btw ping me if you need any more help
fervent anchor
ok so find such a and b so that 12a+25b=1
isnt that wat I did? 12(-2n) + 25(n)= n
long ridge
isnt that wat I did? 12(-2n) + 25(n)= n
yes basically
fervent anchor
the step is the problem im guessing
you know that f(n)<2-1/n
you basically want ...
what exactly do you mean by guessing?
long ridge
i mean im guessing thats the part you are unsure abt ig
fervent anchor
i think theres a typo in the end, theres a really nice idea but they explain it ...
to be honest I understand the idea but wording it is hard for me. any idea which parts come across as unclear?
fervent anchor
the step is the problem im guessing
you know that f(n)<2-1/n
you basically want ...
doesnt what I have already reduce to what we want to prove? is this just a more concise way to do it?
long ridge
to be honest I understand the idea but wording it is hard for me. any idea which...
im really bad at understanding purely mathematical explanations and stuff like that in general so for my case it would help to say it informally
granite shore
Hello @fervent anchor I might be able to help with logic. I'll stay quiet to allow @long ridge to help first though.
fervent anchor
Hello <@278629163915673600> I might be able to help with logic. I'll stay quiet ...
if u could look at theorem 4 i'd appreciate that
also, does theorem 3 look good? I think it holds but Clovk makes me think it might not hold
granite shore
Sure let me take a look
You will want this to be strictly less than instead of $\leq$
glossy valveBOT
granite shore
You may wish to state that $k \in \mathbb{N}$ and fix up the right side so that you have $2 - \frac{1}{k}$
glossy valveBOT
granite shore
This is not obvious to me and would need more explanation
Why not this?
In red is where I have applied the induction hypothesis
Let me know if my comments make sense
I was able to simplify the RHS to get $2 - \frac{1}{k+1}$ and can show you, if interested
glossy valveBOT
granite shore
Looking at (4) now
Base case looks good, and induction hypothesis good
(again you may wish to state that $k \in \mathbb{N}$)
glossy valveBOT
granite shore
May wish to state: Let $Y$ be a set such that $Y = X \cup \left{ m\right}$, where $m$ is an additional element that was not previously in $X$.
glossy valveBOT
granite shore
The notation here is a bit confusing, but I understand your overall approach and your approach makes sense. Let me see if I can help clean up your explanation a bit
fervent anchor
I was able to simplify the RHS to get $2 - \frac{1}{k+1}$ and can show you, if i...
please do
granite shore
I think you basically want to argue that $Y-\left{m\right}=X$ has $k$ elements, so $\mathcal{P}(X)$ has $2^{k}$ subsets by the induction hypothesis. Next, consider how many new subsets would be formed containing element $m$. Well, this is exactly $2^{k}$ additional subsets (each one of the previous ones with $m$ in it). Thus, $\mathcal{P}(Y)$ has $2^{k} + 2^{k} = 2 \times 2^{k} = 2^{k+1}$ subsets.
glossy valveBOT
granite shore
please do
My thought process was I needed to cancel $k(k+1)$ in the denominator, so I asked if I was able to remove the $+1$ in the numerator preventing me from factoring.
glossy valveBOT
granite shore
Since removing the 1 would make this fraction smaller (thus subtracting LESS than before), the overall inequality does not change. In other words, removing the 1 actually makes the RHS larger than the previous step.
So to finish up my thoughts about Thm 4. Your idea is absolutely correct and valid. It just needs to be "cleaned up" a bit to make it stronger.
granite shore
sure! ill also be online in about 3 hours (Twitch) so may have to wait until closer to then
i need to adult for a few minutes lol
@fervent anchor dont forget to .close if youre done!
fervent anchor
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sterile cloak
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@sterile cloak Has your question been resolved?
rapid cloak
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rapid cloak
It’s just to calculate the standard deviation
If that helps
Am I wrong to add up the sum of each column?
ebon glacier
so you do know the formula for standard deviation right?
$\sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2}$
glossy valveBOT
Davit
ebon glacier
you have 7 terms, so replace N with 7, Xi is the current terms you have, myu is the mean or the arithmetic average of the set
rapid cloak
Would xi-u be the sum of the 7 xi-u’s?
Ok I got it, so the standard deviation is 6.268
Would you recommend adding the columns 2 and 3 like I did 1 to get the totals?
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rapid cloak
<@&286206848099549185>
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torn jolt
can someone help me solve this
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can someone help me solve this
torn jolt
i dont think my class has ever covered this at all but im curious
i only know power and energy here
cursive dock
if you put N watts in, you would get 0.05N watts out
so if you get 50W out, then 0.05N = 50
cursive dock
where did you get 2.5?
torn jolt
5% of 50
cursive dock
ohh - nono
e.g. if you put 100W in, then you would get 0.05*100=5W out
so yes if you put 50W in you would get 2.5W out
but you want to do this backwards
torn jolt
oh
cursive dock
you're putting N watts in, and getting 0.05*N watts out - which is equal to 50
torn jolt
so its 50 divided by 0.05?
cursive dock
yeah
cursive dock
I think the efficiency question is completely separate?
torn jolt
=5%
cursive dock
it talks about a pulley
torn jolt
oh ur right
this question sheet im using is ancient its full of mistakes
ty
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fast marlin
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desert cypress
how would i pick my x and y
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how would i pick my x and y
desert cypress
to sub in
smoky wing
context
smoky wing
desert cypress
desert cypress
<a:thinkspin:586078649149554690>
it sent finally 😭
i got my mapping from the function
smoky wing
what does that even mean
smoky wing
,rcw
glossy valveBOT
smoky wing
[ x \mapsto \df{x}{2} - \df{3\pi}{8} ]
[y \mapsto 5y - 3 ]
glossy valveBOT
Pure
desert cypress
yes
smoky wing
? this doesn tmake much sense so far what are you trying to do?
desert cypress
its asking us to state 5 points on the graph
smoky wing
any 5 points?
smoky wing
,rccw
glossy valveBOT
desert cypress
how would i get their x and y
tho
yea bit i need both x and y
i duno where x is
when y is at a max or min
smoky wing
[\mr{5\cos (2x - \df{3\pi}{4}) -3= 2}]
[\mb{5\cos (2x - \df{3\pi}{4}) -3= -8}]
solve these two
glossy valveBOT
Pure
desert cypress
how would i even solve that
smoky wing
[\mr{\cos (2x - \df{3\pi}{4})=1}]
[\mb{\cos (2x - \df{3\pi}{4})= -1}]
glossy valveBOT
Pure
desert cypress
where did 5 cos go
smoky wing
[\mr{5\cos (2x - \df{3\pi}{4}) -3= 2} \mg \rightarrow \mr{5\cos (2x - \df{3\pi}{4}) = 5} \mg\rightarrow \mr{\cos (2x - \df{3\pi}{4})=1}]
same for the other one
glossy valveBOT
Pure
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torn jolt
need help with time series ML model
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need help with time series ML model
torn jolt
if anyone can look over my code
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torn jolt
<#406129911992942593> for CS server
thx bro
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tender zinc
What is this called and how do I do this? I'm guessing you use the formula for whichever is across the given number but I'm not sure. So like you start across the 9 and if its a 45 degree angle then you do y = x radical 2 and stuff
grim skiff
What is this called and how do I do this? I'm guessing you use the formula for w...
Those are special right triangles, more specifically, 45 - 45 - 90 triangles
tender zinc
Those are special right triangles, more specifically, 45 - 45 - 90 triangles
thanks I'll look up a video
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glad whale
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glad whale
what triple would work with this
I thought maybe 3,4,5 if i multiply by 4
But no ;/
limber pilot
I would start off by finding the hypotenuse using the pythagorean theorem
Then I would simplify the triangle down as much as possible
Then sin(x) = y/h
cos(x) = x/h
tan(x) = y/x
csc(x) = 1/sin(x)
sec(x) = 1/cos(x)
cot(x) = 1/tan(x)
glad whale
ohhhh
Ok
I try something
wait
so a^2+b^2=c^2
which is for the hyp , adj and opposite
c^2 for hyp?
uh
they want both
waiiit
.close
soft agate
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buoyant pewter
what are you precisely supposed to do with it ?
factor it ? or you need a vertex form ?
or you search zeros of this fucntion ?
tulip oriole
seems like simplifying
rough tundra
if you're factoring where a doesn't equal 1 I love the slide and divide method :)
https://valenciacollege.edu/locations/lake-nona/documents/Method3-SlideandDivide.pdf
buoyant pewter
$ax^{2}+bx+c=a\left( x-x_{1} \right)\left( x-x_{2} \right)\text{ }\text{ for }\text{ }\Delta =b^{2}-4ac>0\\\text{where: }\\x_{1}=\frac{-b-\sqrt{\Delta }}{2a}\\
x_{2}=\frac{-b+\sqrt{\Delta }}{2a}$
glossy valveBOT
Joanna Angel
buoyant pewter
and
$ax^{2}+bx+c=a\left( x-x_{0} \right)^{2}\text{ }\text{ for }\text{ }\Delta =b^{2}-4ac=0\\\text{where: }\\x_{0}=\frac{-b}{2a}$
glossy valveBOT
Joanna Angel
soft agate
what are you precisely supposed to do with it ?
the teacher's final answer was
(2x+5)(3x-1)
buoyant pewter
the teacher's final answer was
(2x+5)(3x-1)
ye si agree with it
so you can use first method
i showed you above
soft agate
but u cant divide
soft agate
i showed you above
im confused
buoyant pewter
i do not use division
soft agate
still
soft agate
**Joanna Angel**
there's so many numbers
digital seal
buoyant pewter
still
you can also do it like this:
$6x^{2}+13x-5=\left( 6x^{2}-2x \right)+\left( 15x-5 \right)=\\2x\left( 3x-1 \right)+5\left( 3x-1 \right)=\left( 3x-1 \right)\left( 2x+5 \right)$
glossy valveBOT
Joanna Angel
soft agate
how did u get -2x
buoyant pewter
because 13x = -2x + 15x
i selected such wya, because later, i get two parenthesies, where
i can take out something and , parenthsies will be identical (3x-1)
you just have to remember that this method does not rely on a pattern, but on observation
that's why earlier, I showed you general methods to deal with each case without using observation
and now, you have to decide, what you like
soft agate
oh
soft agate
because 13x = -2x + 15x
yes ik how to do that
what would happen to 15
thats what im confused on
buoyant pewter
in second parenethesis, was 15x - 5, so i took out 5, to get 3x - 1, inside
because
in first parenthesis i got later, 3x - 1 too
soft agate
in second parenethesis, was 15x - 5, so i took out 5, to get 3x - 1, inside
so then 18x and -6
buoyant pewter
so then 18x and -6
nom not 18, if you have 15x - 5 , you get: 5(3x - 1)
soft agate
oh
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soft agate
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kindred sail
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kindred sail
im completely confused
glacial pasture
whats confused you?
kindred sail
how do the roots and g(2)=168 become that polynomial? do I make use of the factor theorem?
glacial pasture
if by factor theorem you mean writing the below, then yeah
g(x)=ax(x+3)(x+4)(x+5)
you then use g(2)=168 to find a
kindred sail
so would g(2)=168 be a factor in this case?
glacial pasture
thats not a factor no
kindred sail
ohhh wait wait
glacial pasture
its just something to help find a, since that scaling factor cant be found from the roots alone
glacial pasture
yup
kindred sail
ohhh thank you thank you
glacial pasture
nw
kindred sail
okay wait so
I got a=0.4
how do I determine which polynomial is the right one
wait no
holon
nvm yea
its 0.4
so what do I do with a to get the polynomial
glacial pasture
if by factor theorem you mean writing the below, then yeah
g(x)=ax(x+3)(x+4)(x+5...
if youve said a=0.4
then you can just expand it
kindred sail
wdym
glacial pasture
expand the brackets, then you should have one of the answers
kindred sail
sorry im still a little confused what u mean by expand the brackets
glacial pasture
for example if you had (x+a)(x+b)
expanding those would give x^2+ax+bx+ab
that kind of thing
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torn jolt
I need help on further simplifying this problem
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I need help on further simplifying this problem
glossy valveBOT
proper hawk
first you should simplify -15/6
proper hawk
no I mean -15/6 is not in simplest form
torn jolt
I know
glossy valveBOT
WhereWolf(ping if needed)
torn jolt
so just move it to the bottom?
This was the answer that was provided on the question
???
I do not know how my math teacher got that.
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torn jolt
NO
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@torn jolt Has your question been resolved?
torn jolt
no
queen crater
Just press the ❌, no need to reply
You got to x^(-15/6) y^(5/6)
-15/6 = -5/2, right?
@torn jolt
rocky spear
was i correct here
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was i correct here
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@rocky spear Has your question been resolved?
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tight heath
if the black dot is down then i look down and if its up then i look up right
tight heath
(ping when u reply pls ty)
sick karma
if the black dot is down then i look down and if its up then i look up right
what do you mean by "look down/up"
tight heath
what do you mean by "look down/up"
sick karma
I mean yeah, but it's because the black dot shows the actual value of the function at that x
tight heath
I mean yeah, but it's because the black dot shows the actual value of the functi...
so if that black dot was up for example
sick karma
yes
sick karma
finding them graphically is what you're doing right now; analyzing a graph and checking what the limits are for all shown x values
finding numerically is just taking a given function f(x) and computing its limits
like it just means exactly what you think it means
tight heath
like it just means exactly what you think it means
ooooh
so finding limits numerically
is like
when it says
f(x) = x^2+4 / x+2
x -> 2+
right?
tight heath
ohh ok great, but now the part that confuses me the most is
does this mean less than -4?
(which is -4.1)
sick karma
limit of f(x) as x approaches -4 from the left
tight heath
where is the left and right
sick karma
uh
I mean just look at the x value youre finding the limit of
and determine which side is left and right
which is probably the easiest step
tight heath
doesnt matter how long the distance is
ahh so distance doesn't matter
sick karma
you don't really view limits in terms of distance because you're finding the value f(x) approaches as x goes infinitesimally close to some value
tight heath
and this 5, it doesn't say if its 5+ or 5-, so do i just treat it as 5
sick karma
basically yeah
I mean you also treat 5- and 5+ as 5's but the - and + just indicate which direction you're approaching the value from
tight heath
I mean you also treat 5- and 5+ as 5's but the - and + just indicate which direc...
so can is ay that 5+ is 5.1
sick karma
thats not what a limit means
tight heath
that's what the teacher told us
sick karma
thats a way of computing what it approaches
tight heath
like act as if its 5.1
sick karma
by increasing 5 by a small amount
but thats not explicitly what a limit is
but sure
it could also be 5.01
or 5.000001
sick karma
only thing that matters is that the value is extremely close to, but not equal to, 5, and is logically consistent with the direction you're going
eg if you were approaching from the left you would have something like 4.9999
sick karma
for a piecewise to be continuous like that for all real #s then 3x^2 has to equal ax-4 at the point where the piecewise changes
plot this piecewise in desmos and you'll get an idea
tight heath
ahh i get the idea now
sick karma
otherwise you would have a jump discontinuity which is not continuous
tight heath
otherwise you would have a jump discontinuity which is not continuous
yuppp
the teacher failed to explain this for a whole week
and u explained it in 1 discord message
great
sick karma
it be like that sometimes
sick karma
I'm not sure I understand but lets go through each one
a) you approach 0 from the left, anything to the left of zero is negative
resulting in a negative value
b) as you approach 1+ you'll have some value that is slightly larger than 1
for example 1.00001
tight heath
yes
sick karma
so when you plug this into the function, the denominator x-1 is positive because you have 1.00001 - 1
tight heath
then the answer will be positive
sick karma
yes
tight heath
so its 0+
sick karma
well no for that one the limit is 1+ as it shows
tight heath
which lesson is that
1.2 Finding limits graphically and numerically.
1.3 Finding limits analytically + (L'hopital rule)
1.4 Continuity and one sided limits
1.5 Infinite limits
2.2 Basic differentiation rules and rates of changes
2.3 Product and quotient rules
2.4 The chain rule
2.5 Implicit differentiation.
Finding limits numerically
sick karma
uh
tight heath
ok i think i get that part
sick karma
I think it's numerically? I would check on what is covered in "infinite" limits
tight heath
i think its infinite limits
sick karma
it could fall under that one cause the result is infinity unless it just talks about x -> inf
tight heath
yup
ok good i understand infinite limits fully
1.2 Finding limits graphically and numerically.
1.3 Finding limits analytically + (L'hopital rule)
1.4 Continuity and one sided limits
2.2 Basic differentiation rules and rates of changes
2.3 Product and quotient rules
2.4 The chain rule
2.5 Implicit differentiation.
sick karma
one more thing
tight heath
one more thing
ya?
sick karma
numerically means testing small increases in values like we did with 1.0001 and all that, analytically means you solve for the limit using algebra/calc manipulation
anyway for this part
tight heath
numerically means testing small increases in values like we did with 1.0001 and ...
oh so theres a difference, ok
tight heath
anyway for this part
i just take the denominator and do =0
sick karma
you find where f(x) goes to infinity and usually that is found through the denominator
yes
sick karma
because theres a vertical asymptote there
it just goes to infinity which is associated with undefined
tight heath
so what does it mean when the answer is equal undefined
oh
so DNE and undefined are the same?
tight heath
i dont think so
because in our exam paper
i answered a question and put it as DNE
it was correct
but my friend had it as undefined and it was wrong
let me show u
sick karma
ok yeah
so when you're dealing with limits
you use DNE if lim(x->a-) is not equal to lim(x->a+)
but if you're dealing with JUST evaluating the function at that point, f(a) = undefined
hold up
tight heath
🤔
sick karma
is this your answer
sick karma
you use DNE if lim(x->a-) is not equal to lim(x->a+)
yeah so for the DNE answer it's right because as I said here
limits are not equal
sick karma
no
sick karma
yes
tight heath
i dont get how i did C
tight heath
ye i know
i was confused on whether to choose DNE or undefined
now i get it
but i dont get why is it DNE and not 1 or -1
sick karma
because limits show what value you're approaching
left limit shows you're approaching 1
right limit shows you're approaching -1
they're not the same so the limit cannot exist
when you're evaluating the function, then you use the black dot
because that's the actual point
sick karma
because as I said
limits show what value you approach
not the value at that point itself
the two limits left and right very clearly meet at the same point
so the limit exists, and f(x) approaches 4
but f(x) is not 4 as shown by the discontinuity
sick karma
yes
tight heath
abit more and abit less
tight heath
true
tight heath
we just showed this by the discontinuity at x = 1
do u think u can still answer my questions? i feel like i asked too much
sick karma
yeah
tight heath
yeah
okay thanks a lot
how is it DNE for lim x-> -1
oh wait i see it
why is f(-1)=0?
oh the black dot
so let me make sure i understand this correctly
if it says lim f(x) as approaches -1 for example
i find -1**+** and -1**-** then check if they're equal, if they are then its that number else DNE
tight heath
also applies if they're different infinites, such as -inf and inf
did i draw it correctly
for the blue one
sick karma
yes
tight heath
yes
(sorry for the amount of questions im asking)
i have my final exam tomorrow and im trying to wrap everything up
now this, i dont get anything in it
sick karma
the interval is [-3,3] and you want to check if any points or range of x values are discontinuous
first thing to do is analyze the function and look at what possibly could make it discontinuous
what do you see?
tight heath
what do you see?
well, from what i understand that i should just plug in the number and forget the pluses and minuses at the top of the number
since that's what they're doing
sick karma
look at the function
theres a constant 3 which doesn't change anything
then you have a square root
what do you know about the domain of square roots?
tight heath
what do you know about the domain of square roots?
0 to inf?
sick karma
yes, what does that mean for the inside of the radical?
tight heath
yes, what does that mean for the inside of the radical?
no idea
sick karma
the radical cannot be <0
because doing so yields an imaginary value
so for your problem, 9 - t^2 cannot be negative
from (-3,3), is the square root imaginary?
tight heath
from (-3,3), is the square root imaginary?
nope
sick karma
good
so you have part of the interval down
now you need to test the endpoints
to see if they can be included in the range of continuity
if they yield a non-real value, then they can't
tight heath
oooooooooooooooooh
ok i think i get it now
so i have to make sure
that the range
between them
is a real value
all of them
sick karma
yes
tight heath
yup got it
implicit differentiation is when u do derivatives and include y prime when u do the derivative of y right
sick karma
yes, aka dy/dx
sick karma
I'd remember them
tight heath
when solving problems
sick karma
removable - remove just one point and put it somewhere else
jump - entire function jumps suddenly to somewhere else
tight heath
what do u think it can ask me about them
sick karma
basically anything we've just gone over
tight heath
maybe explain the 2 types of discontinuities?
sick karma
maybe
sick karma
jump - entire function jumps suddenly to somewhere else
but this list should be all you need for that
tight heath
ok i'll keep them in mind
removeable is when u remove just one point and put it somewhere else
but are u sure about the jump one
does the entire function jump
or some of it
tight heath
ah ok
tight heath
got it
we replace x with 4 and not 4+ because it didnt specifiy, if its from the left or from the right, correct?
and if we get 0/0 when doing limits we use the L'hopital rule
sick karma
we replace x with 4 and not 4+ because it didnt specifiy, if its from the left o...
yes
sick karma
yes
tight heath
is this a rule i'm supposed to know
sick karma
1st one is easy af to derive using lhopitals so I wouldnt
2nd one too
but if you want to then sure
tight heath
ok no more board pics
oh wait there is still 3 more
let me check if theres anything i dont get
ahh this
isn't this called a piecewise function
sick karma
yes
sick karma
yes
I would use closer values to the limit
but if thats what your teacher said then i guess it's fine
tight heath
and for these i just treat the sign of the upper number and the down number
and multiply them
so + divide by - is -
so i just put -∞
sick karma
yes
but make make make make make sure you don't confuse the sign on the limit value
with the actual sign of the function
tight heath
ahh like -1-
sick karma
like in general
for example x -> 1-
for that function it would not be negative despite the sign there (which only indicates which direction youre approaching from)
tight heath
yup yup i get what u mean
i think thats it
now lets just make sure i know the topics
1.2 Finding limits graphically and numerically.
1.3 Finding limits analytically + (L'hopital rule)
1.4 Continuity and one sided limits
1.5 Infinite limits
2.2 Basic differentiation rules and rates of changes
2.3 Product and quotient rules
2.4 The chain rule
2.5 Implicit differentiation.
1.2 is where it gives us the graph and asks us to find the limit, and numerically is where it gives us the function without a graph
what was 1.3?
sick karma
numerically - using actual values and plugging in to estimate value of limit
eg. 1.001
analytically - using algebra/calc rules to manipulate function and make the limit able to be evaluated at the point
eg. lhopitals, factoring out x terms, etc
tight heath
o so thats what it means
1.4 Continuity and one sided limits - when it says 1/0+ then it'll be = +∞
wait no
thats infinite limits
sick karma
again go ask your teacher if that's what infinite limits mean
tight heath
continuity is the removeable thing and the jump
sick karma
because I can't say for sure
tight heath
i remember it
1.5 Infinite limits - is where it has something /0
2.2 Basic differentiation rules and rates of changes - is basically the chain rule, product rule etc
sick karma
1.5 Infinite limits - is where it has something /0
again
idk if your teacher classifies 'limits that go to infinity' as infinite limits
so I would ask them that
tight heath
idk if your teacher classifies 'limits that go to infinity' as infinite limits
yea it is, i just made sure
tight heath
2.3 Product and quotient rules - easy af
2.4 The chain rule - easy af
wait the chain rule is like the power rule right
f(x) = (4x+1)^2
2(4x+1)(4)
- put the power down
- copy paste whats inside the bracket
- derivative for whats inside the bracket
sick karma
have you covered derivatives other than powers of x yet?
sick karma
then chain rule isn't just that
sick karma
- indentify functions that are contained within a function
- differentiate outside function while keeping the inside function the same
- multiply that with the derivative of the inside function
chain rule
sick karma
it's just $\frac{d}{dx}(f(g(x)))= f'(g(x))\cdot g'(x)$
glossy valveBOT
Triaxyz
sick karma
1. indentify functions that are contained within a function
2. differentiate out...
that's what I said in this
sick karma
I said it in a general way because you seem to only know the case for powers of x
tight heath
I said it in a general way because you seem to only know the case for powers of ...
probably
sick karma
thats the point
tight heath
2.5 Implicit differentiation - is when u use y'
sick karma
how will you know what to do if they ask d/dx (sin(4x))
tight heath
how will you know what to do if they ask d/dx (sin(4x))
they wont
because my teacher is the one who puts the exam (hopefully thats true)
and we haven't taken it as d/dx
sick karma
that's besides the point and you're still not sure if that's the case
so I would remember the general procedure
tight heath
true u got a point
well man i really hope tomorrow's exam goes well
im very worried
@sick karma appreciate you for staying w/me and answering all of my questions, it means a lot
sick karma
np
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i need help with three questions
hey guys?
i need help
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guys i need help
i aint gonna make it
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i need help
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lol?
your telling me
we got 20k
and not one can help??
its been like 5 hours
like i need help
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obsidian agate
Womp womp
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hearty prairie
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hearty prairie
in question ii how do they go from 1 to two
how does the -x become x^2 on the numerator
torn jolt
the multiplaied by x to have common denominator
10/x - x
10/x - x/1
multiply x/1 by x
10/x - x²/x
now you have common denomimator
now you can write
10-x²/x
do you understand?
torn jolt
you can only subtract and add fractions only if you have common denominator
hearty prairie
third step * sorry
they move the ten but how does it become 3x and what happened to the fraction
hearty prairie
yeah sure
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ancient wyvern
WHAT TO DO
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dreamy valve
I do not know what real roots is. What is the real root?
small spruce
Im assuming since you have the discriminant and knowning that it's above 0, you're already given 2 solutions
And since it also equals 0, that's another solution
so you have 3
im just spouting some random things, don't take me that seriously
small spruce
I do not know what real roots is. What is the real root?
a solution that is real
which is just a real number
dreamy valve
a solution that is real
okay, thanks
ancient wyvern
And since it also equals 0, that's another solution
how?
small spruce
from the original equation
as it equals 0
yea maybe thats not it
well actually
without proper reasoning
it is 3
since the discriminant gives u 2 already
and its a cubic polynomial
so there can't be two
so choose 3
queen crater
It's a quintic, not a cubic
ancient wyvern
you same guy as just now
small spruce
yuh
queen crater
The discriminant exists for any polynomial, of any degree
ancient wyvern
oh
queen crater
Whether it "works" or not depends on what you mean
ancient wyvern
i dont know
like can i get the number of real roots
i thought discrimant only give 1 2 or 0
queen crater
Not really
A zero discriminant tells you there is a multiple root, but other than that, I don't know how I would use it
I'm not sure there is a method other than taking the derivative and checking values on increasing or decreasing intervals
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ancient wyvern
I'm not sure there is a method other than taking the derivative and checking val...
how does that give x intercept though?
queen crater
A quintic is only going to have at most 4 local extremas, plus it will tend to +-infinity at +-infinity
A zero will only occur at a local extremum or between two consecutive extrema that have opposite sign
ancient wyvern
A quintic is only going to have at most 4 local extremas, plus it will tend to +...
is this memorization?
that a quintic has 4 extremas
queen crater
A polynomial of degree n can only have n-1 local extrema
It's easy to show: its derivative is a polynomial of degree n-1 and as such can only have n-1 roots
(at most)
ancient wyvern
It's easy to show: its derivative is a polynomial of degree n-1 and as such can ...
i see
if its like x^5+3x^3 +2
lets say
it onyl factors in the highest degree?
queen crater
?
gentle totem
congruent chords subtend congruent angles i believe
spiral vigil
that is correct
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valid path
if anyone understand lodev raycasting tutorial(https://lodev.org/cgtutor/raycasting.html#Textured_Raycaster) i want to ask him about this
H: hitpoint of the ray on the wall. Its y-position is known to be mapY + (1 - stepY) / 2
yDist matches "(mapY + (1 - stepY) / 2 - posY)", this is the y coordinate of the Euclidean distance vector, in world coordinates. Here, (1 - stepY) / 2) is a correction term that is 0 or 1 based on positive or negative y direction, which is also used in the initialization of sideDistY.
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noble pivot
why is sin(180-v)/cos(180-v) the same as -sin(v)/cos(v), but sin(90-v)/cos(90-v) not the same as -sin(v)/cos(v)
(in degrees)
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noble pivot
still confused
small pebble
I’ll reformat it with LaTeX
noble pivot
thx
tawny star
sin(90-v) = cos(v)
cos(90-v) = sin(v)
sacred sparrow
small pebble
$\frac{\sin(180-v)}{\cos(180-v)}=-\frac{\sin(v)}{\cos(v)}$
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Chixen
weary socket
#help
weary socket
Ok so I need to stick with one?
weary socket
.close
plush tiger
yo whats x and y and pleasee explain so a kindergartner can understabd
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slate meteor
Could someone tell me if i am doing this question correctly
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primal condor
Seems written ad linear algebra - Are they looking for a matrix answer?
slate meteor
Seems written ad linear algebra - Are they looking for a matrix answer?
nope just a function
slate meteor
ye
slate meteor
i thought u had to split it like 2x + 15, 5y + 20
primal condor
Best check, I think.
slate meteor
i think you might be right
primal condor
It reads like a matrix multiplication.
Hence my earlier question.
But always an option to just do the algebra.
slate meteor
x' = 2x+5y + 15?
ig these two functions here would be the correct answer ?
since it doesnt provide you with letters to encipher
slate meteor
i guess
primal condor
They want the deciphering function
the inverse, right?
BTW: Wolfram tells me that inverse [2 5;1 4] is 1/3*[4 -5;-1 2]
slate meteor
oh right but i have only ever inversed a one letter function
slate meteor
BTW: Wolfram tells me that inverse [2 5;1 4] is 1/3*[4 -5;-1 2]
uh not sure where all those numbers are coming from
ax + b = y and inverse would be x = a^-1(y-b)
primal condor
slate meteor
not sure
primal condor
$x'=2x+5y+15$
glossy valveBOT
G. Spark
slate meteor
oh and x + 4y + 20
primal condor
$y'=1x+4y+20$
glossy valveBOT
G. Spark
primal condor
Solve for x & y in terms of x' and y'?
primal condor
slate meteor
i assume i can let them both = 0 ?
primal condor
You don't know what they are.
You need to solve the functions
slate meteor
not sure i know how to do that
primal condor
\begin{bmatrix}x\y \end{bmatrix} = \begin{bmatrix}2 & 5\1 & 4\end{bmatrix}
\begin{bmatrix}x\y \end{bmatrix} + \begin{bmatrix}15\20 \end{bmatrix}
glossy valveBOT
G. Spark
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
primal condor
Inverse matrix function?
$\begin{bmatrix}x'\y' \end{bmatrix} = \begin{bmatrix}2 & 5\1 & 4\end{bmatrix}
\begin{bmatrix}x\y \end{bmatrix} + \begin{bmatrix}15\20 \end{bmatrix}$
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G. Spark
slate meteor
(4, -5 \ -1, 2)
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primal condor
we could subtract the [15;20] from both sides
$\begin{bmatrix}x'\y' \end{bmatrix} - \begin{bmatrix}15\20 \end{bmatrix}= \begin{bmatrix}2 & 5\1 & 4\end{bmatrix}
\begin{bmatrix}x\y \end{bmatrix} $
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G. Spark
primal condor
Then take an inverse
slate meteor
🤔
primal condor
$\begin{bmatrix}2 & 5\1 & 4\end{bmatrix}^{-1}\left(\begin{bmatrix}x'\y' \end{bmatrix} - \begin{bmatrix}15\20 \end{bmatrix}\right)=
\begin{bmatrix}x\y \end{bmatrix} $
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G. Spark
primal condor
Just one approach
slate meteor
this is alot of work for just 4 marks lmao
primal condor
Ok. Longer by algebra without matrices, I suspect.
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$1/3\begin{bmatrix}4 & -5\-1 & 2\end{bmatrix}\left(\begin{bmatrix}x'\y' \end{bmatrix} - \begin{bmatrix}15\20 \end{bmatrix}\right)=
\begin{bmatrix}x\y \end{bmatrix} $
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G. Spark
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weary socket
Hey can u help me
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twin wolf
How did they get the /2 from when taking out the 1/4
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How did they get the /2 from when taking out the 1/4
umbral dome
,,\frac{(x+1)^2}{4}=\frac{(x+1)^2}{2^2}=\left(\frac{x+1}{2}\right)^2
glossy valveBOT
cloud
twin wolf
Does this only work because it’s a square
If it was a number like 1/17 that was pulled out the integrand what would happen
umbral dome
you could pull in $\sqrt{17}$
glossy valveBOT
cloud
umbral dome
,,\frac{(x+1)^2}{17}=\frac{(x+1)^2}{(\sqrt{17})^2}=\left(\frac{x+1}{\sqrt{17}}\right)^2
glossy valveBOT
cloud
