It’s the black one
#help-28
light saddle
But that’s a period although you can’t calculate it
Since you don’t know the grey dot you circled
sharp fable
maybe it isn't drawn to scale?
light saddle
Find two black points that make a curve like the ones you circled
sharp fable
wait
light saddle
The dots
sharp fable
light saddle
You got it
light saddle
Exactly
sharp fable
1.358 + 0.642
light saddle
Go on
sharp fable
2.000
light saddle
That’s your period well done
light saddle
You’ll probably be able to do it yourself
light saddle
For the first one?
sharp fable
for all of them.
light saddle
Yup
light saddle
You as well
sharp fable
.close
distant halo
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distant halo
Should i solve using internal division or externaal division
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@distant halo Has your question been resolved?
wanton cedar
I think unless it's mentioned specifically to do it using external division formula , it usually means use the internal division formula
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@distant halo Has your question been resolved?
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glossy lintel
Prove that if a>b, then an ellipsis (x^2/a^2) + (y^2/b^2) = 1 is contained into the disc X^2+y^2 <= 1 and in the rectangle |x| <= a, |y| <= b, of which the vertices are the 4 points (+-a, +-b)
light sonnet
Prove that if a>b, then an ellipsis (x^2/a^2) + (y^2/b^2) = 1 is contained into ...
Please claim a new channel and do NOT delete the first message of any future channel you claim.```.close
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rare haven
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rare haven
Need help on this question
What’s the equation of the graph
Or am I just suppose to draw transformations on it
light sonnet
It just wants transformations of that graph
rare haven
Like do I have to find the equation and put transformations
Or I can just draw them
out
little jungle
I’ve tried many methods but I cannot for once get it right. I followed what the book did, what the professor did, what the internet showed me, I cannot figure it out! Can someone at least point me in the right direction
fathom saddle
I like it. Are you sure they don't want + C instead of + K?
Not that it matters mathematically, but the thing might be programmed to look for a + C
Ooh, wait maybe they want y = (all that)?
little jungle
Oh
I put the C before this in my previous 4 tries
I’ll try that y=
They do wtf
Thank you I was just gonna take a zero for this one or request an extension lol
It’s due in a few hours
Life saver
viral wyvern
molten tundra
hi
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@molten tundra Has your question been resolved?
molten tundra
anyone... <@&286206848099549185>
austere breach
I’m looking
austere breach
with this
Ok so this is solely conceptual
Do you think the number of quarters you can have is continuous?
@molten tundra
austere breach
Yeah
molten tundra
Do you think the number of quarters you can have is continuous?
wouldnt it be a straight line at one y axis
then another straight line at another
austere breach
Yes
molten tundra
no
austere breach
Think about the values measured by each function and ask yourself whether they can realistically be measured with decimal points
Like, 2.5 degrees Celsius versus having 2.5 quarters
austere breach
They can’t
molten tundra
so they should kinda be something like thsi
austere breach
Yeah
molten tundra
ok thanks
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tepid valve
hi may i know how to do this
full forumBOT
hi may i know how to do this
astral sinew
Just solve for θ:
θ = 2arcsin(t)
Then cosec(θ) = cosec(2arcsin(t)), then use reciprocal and double angle formula to simplify. Same for tangent
tepid valve
cosec(2arcsin(t)) = 2arcsin(t)?
short bridge
No.
Don't go that way. Just use the triangle method.
That's simpler.
Draw a right triangle where one of the acute angle is $\frac{\theta}{2}$.
glossy valveBOT
Enemagneto
compact crag
hi for this would the answer be 58713.56
full forumBOT
hi for this would the answer be 58713.56
torn jolt
hi for this would the answer be 58713.56
na, use compound interest formula
a = p(1 + r/n)^nt
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Available help channel!
simple ridge
hello everyone isnt S here a vector space because $\varphi_0$ is a linear map from S to F
glossy valveBOT
calculus is fun
simple ridge
and why is $\varphi\ \emph{uniquely}$ determined by $\varphi_0$
glossy valveBOT
calculus is fun
stiff summit
hello everyone isnt S here a vector space because $\varphi_0$ is a linear map fr...
S is the generating set of E, it may not be a vector space
stiff summit
and why is $\varphi\ \emph{uniquely}$ determined by $\varphi_0$
because of linearity, since all elements of E are sums(or products, not enough context to know) of those in S, if you know the image of S, then you can get the image of E
stiff summit
oh wait nvm
why nvm
simple ridge
$\varphi_0$ isnt linear
stiff summit
oh
glossy valveBOT
calculus is fun
simple ridge
$\varphi_0$ cant even be linear because we want to extend it to a linear map $\varphi$
glossy valveBOT
calculus is fun
simple ridge
because of linearity, since all elements of E are sums(or products, not enough c...
but elements in E may be written in more than one linear combination of vectors of S
onyx glen
but elements in E may be written in more than one linear combination of vectors ...
our consistency condition, namely that sum λ^i x_i = 0 => sum λ^i φ_0 x_i = 0, takes care of that
glossy valveBOT
calculus is fun
onyx glen
what are $\varphi_0 x_i$ if not vectors in $E$ tho
glossy valveBOT
Ann
simple ridge
they are vectors in F
simple ridge
hello everyone isnt S here a vector space because $\varphi_0$ is a linear map fr...
because $\varphi_0 : S \to F$
glossy valveBOT
calculus is fun
onyx glen
ah wait hold on yes
but then again S ⊆ E so then like
the x_i themselves live in E
i think it would be best to actually write out the proof of why this condition ensures phi is well-defined tbh
simple ridge
but isnt it not a necessity to have the span S of a vector space E where $S \subset E$?
glossy valveBOT
calculus is fun
onyx glen
can you say this again but less broken
glossy valveBOT
calculus is fun
simple ridge
because S can be a span of E and satisfy $S \not\subset E$
glossy valveBOT
calculus is fun
simple ridge
oh wait nvm it is a necessity
simple ridge
i didnt get what you mean
onyx glen
"S is a span of E" is incorrect wording
simple ridge
its correct to say S spans E instead is that what you mean
onyx glen
you can say "S is a spanning set of E", or "S spans E", or "E is the span of S"
simple ridge
ohhh i see now
simple ridge
the x_i themselves live in E
ok what does this imply
simple ridge
maybe i should write out the proof that phi is well-defined
here is the proof
onyx glen
not quite how i would have written it but yes same idea
simple ridge
hello everyone isnt S here a vector space because $\varphi_0$ is a linear map fr...
condition 1.12 is this
onyx glen
there is a slight sleight of hand
the consistency condition can and should be restated as "linear combintions which get sent to 0 must get mapped to 0 by phi, and phi_0 must guarantee this"
simple ridge
it can be restated like that why should it
onyx glen
because the linear combination in question is actually not written as sum λ^i x_i
but rather sum λ^i x_i + sum (-μ^j) y_j
you understand that this is still a linear combination of members of S, yes?
simple ridge
yea because its a sum of scalar multiples of vectors of S
ok now how is $\varphi$ uniquely determined by $\varphi_0$ if vectors of E can be expressed as linear combination of vectors of S in more than one way
glossy valveBOT
calculus is fun
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@simple ridge Has your question been resolved?
onyx glen
ok now how is $\varphi$ uniquely determined by $\varphi_0$ if vectors of E can ...
we've just proved that if we express the same vector in two different ways (one way as sum lambda^i x_i and the other as sum mu^j y_j), the value of phi at that point will be the same no matter which expression we go with
simple ridge
we proved this starting from how we defined phi in terms of phi_0 when we were proving the converse
simple ridge
we've just proved that if we express the same vector in two different ways (one ...
but how will we prove that it is uniquely determined by phi_0 at the start
simple ridge
np i will try to rewrite it
now what you are talking about is essentially used to prove uniqueness and phi being well defined in the second half of the proof
when we where trying to prove the other way
simple ridge
the author said conversly so he proved it in the 2 directions to have a complete proof
onyx glen
can you perhaps screenshot the entire proof in one piece
onyx glen
ok alright so then
he's saying phi_0 is extendable iff (1.12)
extendability => (1.12) is obvious
(1.12) => extendability is what weve been trying to prove thus far
simple ridge
ok but how does extendability implying (1.12) prove that phi is uniquely determined by phi_0
simple ridge
its fine answer this question in any way you want
of course if this doesnt bother you
onyx glen
anything i say now will make you confuse yourself even further.
simple ridge
dw ill manage
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@simple ridge Has your question been resolved?
simple ridge
anything i say now will make you confuse yourself even further.
so can you explain please
onyx glen
no
simple ridge
if you explain i may get it if you dont i wont (assuming that what you will say will just confuse me more)
if you dont want then isnt there any other way that you can use which you think doesnt confuse me more
onyx glen
i mean ok like fine let me lay this bullshit out in a way that i think is the clearest i can possibly manage.
this will naturally lead to 4 day's worth of bot messages left with ❌'s but whatever.
simple ridge
because i just wont leave this question unsolved in my head
onyx glen
@simple ridge i need your consent to use different notations than your book.
this is hard requirement, i will not post any proof without you giving consent
simple ridge
i am in
simple ridge
ok tysm ill be grateful
onyx glen
ok let's see if this will work at all
LMAO OK SHIT BREAKS THE LIMIT LMAO
ok let me try again
\begin{enumerate}
\item we start with a vector space $E$, a generating system $S$ therein, and a set map $\varphi_0: S \to F$.
\item we extend $\varphi_0$ to a linear map $\varphi : E \to F$ by linearity'', i.e. thusly: $$\mbox{if } x = \sum_i \lambda_i x_i, \mbox{where }x_i \in S, \quad \mbox{then} \varphi(x) = \sum_i \lambda_i \varphi_0 (x_i).$$ \item because $S$ is a generating system for $E$, this covers all possible vectors $x \in E$. \item the above extension is a direct consequence of the requirement that $\varphi$ be linear. assuming $\varphi$ exists at all, there cannot be any other linear map which agrees with $\varphi_0$ and is linear. (this proves the can be extended in at most one way'' bit)
\item however the definition as written has a problem. sure it gives $\varphi(x)$ a value for every $x$, but how do we know it is internally consistent, a.k.a. well-defined? i.e. if some vector $x$ can be expressed as two different linear combinations of $S$, how can we be sure that evaluating $\varphi(x)$ through those two different linear combinations won't result in two different values, which would of course be disastrous and ruin everything?
\item the answer is the condition numbered (1.12) in the book, namely:
$$\sum_i \lambda_i x_i = 0 \implies \sum_i \lambda_i \varphi_0(x_i) = 0.$$
\item {[}sidenote: intuitively, this condition implies that $\varphi_0$ must respect whatever linear relations may be present between vectors in $S$; for example, if $x_1, x_2 \in S$ and $x_2 = 7x_1$, the condition says $\varphi_0 x_2 = 7 \varphi_0 x_1$. the same can be said of linear combinations involving 3 or more vectors, of course.{]}
\item if $\varphi$ is a nice and consistent/well-defined linear map, then condition (1.12) follows immediately --- simply apply $\varphi$ to both sides of point (5), and you'll get exactly the right-hand side of the implication.
\item thus our interest will lie in proving that if (1.12) holds, then $\varphi$ is well-defined.
\end{enumerate}
(this is only the first half)
(stand by as i fix the tex)
glossy valveBOT
Ann
onyx glen
\begin{enumerate}
\setcounter{enumi}{9}
\item to do this, let us consider a vector $x \in E$ expressible as a linear combination of $S$ in two different ways:
$$x = \sum_i \alpha_i x_i = \sum_i \beta_i x_i.$$
\item {[}sidenote: unlike the book, i'm having both sums run over the same $x_i$! this can be done by padding either sum with zero-coefficient terms as necessary. for a finite example, $10x_1 + 2x_2 = 4x_2 + 9x_3$ would become $10x_1 + 2x_2 + 0x_3 = 0x_1 + 4x_2 + 9x_3$. this will make it convenient.{]}
\item now apply the definition of $\varphi$ given in point (2) to both linear combinations. we should now get
$$\sum_i \alpha_i \varphi_0(x_i) \overset?= \sum_i \beta_i \varphi_0(x_i),$$ or equivalently
$$\sum_i (\alpha_i - \beta_i) \varphi_0(x_i) \overset?= 0.$$
\item of course, point (10) implies that
$$\sum_i (\alpha_i - \beta_i) x_i = 0.$$
\item but now we can apply (6) to (13) and get (12), exactly as we wanted!
\end{enumerate}
augh hold on
ok gaming
glossy valveBOT
Ann
simple ridge
everything is clear except in point 4 why there cant be any linear map that agrees with $\varphi_0$ other than $\varphi$\
glossy valveBOT
calculus is fun
onyx glen
because any such map could only differ from phi on points which couldn't be reached by linear combinations of S
on points within the span of S, your hand is forced
oh i just noticed a very minor typesetting fuckup in point 2. the word "then" should have a space after it. oops.
simple ridge
ok but these maps will also be extensions to phi_0 which contradicts the uniqueness of this map
onyx glen
ok but these maps will also be extensions to phi_0 which contradicts the uniquen...
but are there any such points on which our impostor map could differ from phi?
no, there are not!
simple ridge
why
onyx glen
S is a generating set for E
by definition that means all of E can be reached by linear combinations of S
so there is nowhere for the impostor (a hypothetical second map that agrees with phi_0, is linear but differs from phi) to be sus (differ from phi)
simple ridge
oh so any point in the domain of the extension will be the same for all different extensions if they exist
and they will all map to the same elements because they all cover the same elements in there domain
onyx glen
could say that sure
simple ridge
thus they are all the same
simple ridge
oh i just noticed a very minor typesetting fuckup in point 2. the word "then" sh...
you are concerned with this while you wrote alot in a short period of time if i write all of these then it will take me forever and it will be full of mistakes
you will probably see the word sum appearing instead of $\sum$
glossy valveBOT
calculus is fun
onyx glen
took me 20 minutes in all to write it, then another 10-15 to fix the tex
and chainsaw it into two halves to get around discord's 2k character limit per msg
and frantic googling for how to make an enumerate start with something other than 1
cloud shore
Bro what's going on 💀
onyx glen
i shat out the equivalent of a 2-page proof written in my own words for why a set map defined on a generating system can be extended to a linear map on the entire space iff it respects any and all linear relations within said generating system
simple ridge
**Ann**
i have a question why do we know for sure that 6 ensures preserving linear relations
onyx glen
i mean, $\sum_i \lambda_i x_i = 0$ \textbf{is} a linear relation.
glossy valveBOT
Ann
simple ridge
i mean couldnt it just be 0 without preserving linear relation
onyx glen
"without not"?
i mean ok like
really i should have said "means" and not "implies"
bc $\sum_i \lambda_i x_i = 0 \implies \sum_i \lambda_i \varphi_0 (x_i) = 0$ is what it \textbf{means} for $\varphi_0$ to preserve that linear relation.
glossy valveBOT
Ann
onyx glen
like thats what we mean when we say said linear relation is preserved.
simple ridge
but doesnt linear relation mean the relations that linear map preserves
onyx glen
no
i think you are confused about the meaning of the word "linear relation"
a linear relation is simply an equation which says "<some linear combination of some vectors> = 0"
or <some linear combination> = <some other linear combination>, which is functionally the same thing
simple ridge
ohh ok
onyx glen
and we say that a map preserves a linear relation if it remains true after applying the map to every vector involved in it.
really, linear maps are by definition those maps which preserve all linear relations.
simple ridge
**Ann**
here we are assuming that $\varphi_0(\sum_{i}{(\alpha_i-\beta_i)x_i})=\sum_{i}{(\alpha_i-\beta_i)\varphi_0(x_i)}$ is that right
glossy valveBOT
calculus is fun
onyx glen
which point
onyx glen
then no. not only did i not assume that, phi_0(that sum) doesn't even make any sense
we applied formula (2) to $x = \sum_i \alpha_i x_i$ and to $x = \sum_i \beta_i x_i$
glossy valveBOT
Ann
onyx glen
we now have two things we want to be equal but don't know yet that they are
hence the $\overset?=$
glossy valveBOT
Ann
simple ridge
thats why i said linearity of phi o noticed how stupid my question was
tysm for your work and explanation sorry for wasting your time and have a nice day/night
you helped me alot
.close
dire quest
Im just wondering about limits
full forumBOT
Im just wondering about limits
dire quest
what is meant by "We say that L is the left limit of the function fat a point a if we can get f(x) as close as we want to L by taking x to the left of and close to a, but not equal to a
.
What is meant by "as close as we want to L"
proper hawk
it keeps getting closer to a
dire quest
correc
proper hawk
while being less than a
dire quest
yes, it could also be larger than a or?
proper hawk
for left limit it need to be less than a
dire quest
agred
proper hawk
for right limit it needs to be larger
dire quest
agreed, but if we had a right side limit
yes
Ok so we have something aproaching a
oh and another thing, does this go for both infinite and "regular" limits?
I retract that statement
nvm
but yeah we have something aproaching a
x
proper hawk
for inifinite there will not be left/right
dire quest
yea, we can talk about that later but x aproaches a
proper hawk
the function we are taking limit on
proper hawk
what is meant by "We say that L is the left limit of the function fat a point a ...
the function you said
dire quest
I guess my confusion is about how a function that visualy passes through a point can aproach it
if that makes sense
proper hawk
passes through?
atomic venture
What is meant by "as close as we want to L"
as close as we want means we can make, in the number x-a, a as small as we want
dire quest
atomic venture
so small that it looks like it equals x, but it doesnt
dire quest
I get that, thank you @atomic venture
i wondering if we have this graph
and it aproaches 1
our limit aproaches 1
dire quest
how do you know?
atomic venture
the curve approaches 3 from both sides
atomic venture
mhm
dire quest
Im confused about how it "aproaches 3", what happens at the point 3
at the exact point
L = 3
atomic venture
limits dont consider f(1), they consider f(0.9999999...) and f(1.00000...1)
dire quest
This might be a weird question but why?
proper hawk
the function might not be defined on x=1
atomic venture
because sometimes f(1) is undefined
dire quest
Ok, lets say it is then
proper hawk
so we use 0.999999 and 1.000000001 instead as they are close to 1
atomic venture
so that we dont have to consider f(1)
proper hawk
yes
we choose points close to it instead
It's like an approximation but infinitely accurate
dire quest
Is there any irl examples where the point is not defined
im just having a hard time wrapping my head around the concept
not defined
dire quest
y = x/x would just be a straight line right
proper hawk
yep
proper hawk
oh the epsilon delta definition
proper hawk
umm no
dire quest
for every delta there exists an epsilon right
dire quest
We need a master of limits here i believe <@&286206848099549185>
ive tried to wrap my head around the concept this week, aint that easy
But lim(fx) moves toward a point a, our x coordinate which has a corresponding L value. right?
dire quest
yes
proper hawk
the limit might not exist at some cases though
dire quest
well, why do we need to find the a value which then = L instead of going the other way around and finding the Y value first 
which then would correspond to our a
c in this example
proper hawk
umm so using the definition
for all x with distance less than delta
f(x) must have distance less than epsilon
proper hawk
yes
dire quest
"for all x with distance less than delta"
proper hawk
delta is a distance where x is allowed to be in
say c = 1 and delta = 0.1
then x can be any number from 0.9 to 1.1
proper hawk
no green
dire quest
yeah yeah
x = a right? or is a = L
yeah
a is the point we are aproaching
scratch my question
ok i understand now, delta is the area where x can exist
well
what happens if we go beyond this area? What defines the area?
proper hawk
ok so
basically we are given the purple line
and we find the green line that is inside the purple line
oh wait
oh that's how it works
dire quest
yes because f(x)= L
well isen't L just the point we are apraoching?
How do we find the accepted values L?
and if (a) values = L values why cant we do it both ways? Back to my previous question i guess
proper hawk
so if we have x=a and we claim the limit = L
we use the epsilon delta definition to check
if the definition is satisfied then L is the limit
dire quest
we use the epsilon delta definition to check
Check what
proper hawk
check if L is in fact the limit
we don't find L using the definition
we check if L is the limit using the definition
that's why it's not used practically
dire quest
ait bet
but how do we find the purple
how do we know where we cant go any further right or left
i dont get this at all
we find L
lets say its 3
L = 3
what would the purple and green mean
proper hawk
purple is given by epsilon, the distance from L
green is given by delta, the distance from a
proper hawk
it's not about a single distance
the definition means that for any epsilon distance
you can choose a delta distance
such that green line doesn't exceed purple line
for large epsilon distance it's trivial to choose delta
so the important part is for a infinitely small epsilon
dire quest
I understand that but im having a hard time understanding what those fields really mean
epsilon and delta
what is the difference between being right inside and right outside of the field with regards to our limit
proper hawk
as I said focus on when epsilon and delta are infinitely small
dire quest
We are infinetly close to L right
proper hawk
yes
In other words, the definition means that for points x infinitely close to a, f(x) would be infinitely close to L
dire quest
that makes sense
but why cant we also say that for point y infinetly close to L, f(x) would be infinetly close to a
dire quest
hmmm
dire quest
im gonna ask my prof tomorrow
do you want me to dm you the answer or?
thx for your help though
proper hawk
oh that'll be great
wise seal
I’ve been given a differentiate function which is f(lnx)=ln^2x+1 and I have to find f’(0). I found the derivative of the f(lnx) which is 2lnx/x but I don’t know what I should do next
devout harbor
yeh. sus
wise seal
is this the original problem you were given?
Yes yes
austere breach
First you need to find f(x)
torn jolt
send a picture of your textbook question
frosty geyser
what is
wise seal
It’s in Greek
frosty geyser
ln^2x+1
torn jolt
send it anyways
devout harbor
it is undefined at 0 bro 🙂
wise seal
Okay okay
narrow helm
also question is f(lnx)
frosty geyser
is it $\ln{x+1}^2$?
glossy valveBOT
Jigglyproff
narrow helm
and you need to find f'(0)
narrow helm
so x = 1 is what you need
wise seal
narrow helm
and differentiate f(lnx) using chain rule too
wise seal
is it $\ln{x+1}^2$?
No no
austere breach
You literally just take out the ln(x) because it’s f(ln(x))
frosty geyser
so its $(\ln{x+1})^2$?
glossy valveBOT
Jigglyproff
narrow helm
gpt
narrow helm
<@&268886789983436800>
frosty geyser
I'm in office
you arent meant to give solutions you know
devout harbor
I confirmed on that
austere breach
What is gpt?
devout harbor
ah alright.
narrow helm
chatgpt
austere breach
Oh
wise seal
so what am I supposed to do
austere breach
Like I was saying, you want to find f(x)
narrow helm
differentiate LHS also
narrow helm
Like I was saying, you want to find f(x)
no need
austere breach
It’s easy though
narrow helm
why extra effort
austere breach
You just take out ln(x) and it becomes a power function
wise seal
wait I don’t get it
austere breach
Also you are looking for f’(x), not f’(ln(x))
narrow helm
Also you are looking for f’(x), not f’(ln(x))
you are looking for f'(0)
so either works
@wise sealdifferentiate both LHS and RHS using chain rule
wise seal
ohhh
austere breach
you are looking for f'(0)
It’s not defined at 0 if you leave it in its current form
narrow helm
you dont need to sub 0
wise seal
I still don’t understand how to find the f(x) tho
austere breach
I still don’t understand how to find the f(x) tho
It’s a simple composition of functions. Basically, you are taking ln(x) and applying some f(x) to it. In this case, it’s just f(x) = x^2 + 1
wise seal
So I can use this to find the f’(0)
austere breach
Once you know what to look for, you end up with a function that is dirt easy to differentiate
narrow helm
ok, both methods are correct, in case you need another method, I'll leave it here
wise seal
ok, both methods are correct, in case you need another method, I'll leave it her...
I don’t think I’ve ever done your method before so I’m not sure I understand it completely
glossy valveBOT
Dyssrupt
wise seal
oh I get it but why did you multiply f’(lnx) with 1/x
narrow helm
i didnt multiply
i differentiated using chain rule
it was a composition of two functions, one is f(x) and other is lnx
wise seal
ohhhhh now I get it
narrow helm
yeah, direct answer
you can also find f(x) first, but in some questions finding f(x) becomes difficult.
wise seal
Yes I understand
I have another one similar exercise
it says that the function is twice differentiable which is f(x•lnx)=e^x+1 and I have to find f”(0)
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molten wadi
[ \lim_{x \to c} f(x) = A ] A is a constant %Does someone know how the Limit of this function is A.
glossy valveBOT
Figure
proper hawk
which function?
molten wadi
F(x)=A
molten wadi
its a line that goes through A
placid oar
exactly. In other words, for every input x, its output is A
molten wadi
but I thought limit had to approach A
placid oar
well, it does. x approaches c, therefore f(x) approaches A
molten wadi
but F(x) equals A
molten wadi
how can you approach something if you are it
placid oar
how can you approach something if you are it
I don't understand what you mean
molten wadi
like if a x approaches c =1 and the function is f(x)= 5. I get x approaches c but how is f(x) approaching's f(c) if f(x) is F(c).
placid oar
i am very confused
just imagine it in your head. You said f(x) = A is a horizontal line that passes through A. Now, move along the x axis. What happens? Nothing, you stay at y = A. And that's the case for any x value. Now when x goes to C, it doesn't matter what C is, because the output is always the same, A
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dull tinsel
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torn jolt
very lost
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ripe shale
Which part? What do you understand?
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torn jolt
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torn jolt
<@&286206848099549185> is this correct?
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Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
iron needle
pale dome
you should try to simplify the fractions before adding/subtracting them
worthy pasture
If you need help with the sum of fractions here's how you can do it: on that case 5/5 and 3/9 can be simplified
5/5 = 1 and 3/9 = 1/3
So rewriting it we have: 9/7 + 1 - 1/3
pale dome
!nosols
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pale dome
dw, now you know 👍
worthy pasture
So rewriting it we have: 9/7 + 1 - 1/3
And then you transform the fractions so that you have a common denominator on each of them and then just add
Let me know if you need any help with that👍
torn jolt
If you need help with the sum of fractions here's how you can do it: on that cas...
I think i get it but how do i simplify the fractions?
worthy pasture
Well, to begin with simplification we can start with one thing: a number divided by itself is always one (with the exception of 0)
For example: 3/3 =1
Or 4/4, 20/20 and so on
So 5/5 is also equal to one
And with the 3/9, here's how you can do it:
You can simplify it by decomposing 3 and 9
worthy pasture
You can simplify it by decomposing 3 and 9
With prime factors
When you do that you will find out that 9 is nothing but 3 × 3
So that fraction is the same as 3/(3×3)
And then canceling out 3 on each side: we get that this fraction is just 1/3
Let me know if I made it clear👍
torn jolt
You can simplify it by decomposing 3 and 9
are you sure you can decompose 3 with prime factors though? because 1 is a neutral factor
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torn jolt
3 is already a prime
so i assume this would be it? or nah
im not sure which one is right or if any of them are
11/21 or -11/21
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torn jolt
isn't any of these values
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spiral vigil
it starts underneath the √ sign so unless you can justify pulling it out it needs to stay there
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worn matrix
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worn matrix
should i continue?
or is there any short method
in starting i divided numerator and deno. with $\cos{4\alpha}$
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yajatk07
worn matrix
yeds
onyx glen
so if we abbreviate $\sin(2n\alpha)$ and $\cos(2n\alpha)$ as $c_n, s_n$ respectively, we get... $$\frac{c_2 \frac{s}{c} - s_2}{c_2 \frac{c}{s} + s_2} \overset?= -\frac{s^2}{c^2}$$
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Ann
onyx glen
does that help us?
$\frac{c_2 \cdot \frac{s}{c} - s_2}{c_2 \cdot \frac{c}{s} + s_2} = \frac{(c^2 - s^2) \cdot \frac{s}{c} - 2sc}{(c^2 - s^2) \cdot \frac{c}{s} + 2sc} \ = \frac{(c^2 - s^2)s^2 - 2s^2c^2}{(c^2-s^2)c^2 + 2s^2c^2}$
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Ann
worn matrix
why is there 'n' and i think youre respectively is wrong?, you mean sin(2a) as s_n?
onyx glen
yes sorry i switched them up on accident
i am just introducing multiple abbreviations at once
i like to abbreviate cos(x), cos(2x), cos(3x), cos(4x), ... as c, c_2, c_3, c_4 etc.
ofc lone c can be written as c_1 if need be but also if it helps de-clutter i drop the 1
except here i've decided to do it while also absorbing the twos bc both sides are obviously functions of 2α and not just of α alone
does that make sense to you
onyx glen
$c$ and $s$ stand for $\cos(2\alpha)$ and $\sin(2\alpha)$, $c_2$ and $s_2$ stand for $\cos(2 \cdot 2\alpha)$ and $\sin(2 \cdot 2\alpha)$
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Ann
onyx glen
what do you mean s/c?
s/c means s divided by c
in this case, s/c = sin(2α)/cos(2α) = tan(2α) ...
you know that tan is sin divided by cos, yes? you're not under any prohibition to use that knowledge or anything, are you?
worn matrix
you know that tan is sin divided by cos, yes? you're not under any prohibition t...
no
onyx glen
as in "no i don't know" or "no i'm not under a prohibition"?
onyx glen
i applied double-angle identities and then multiplied num and denom by sc
in that order
worn matrix
**Ann**
why have you used the 2 at the bottom, shouldnt it be 4?, according to your, "i like to abbreviate msg"
worn matrix
oh yeah got it
but wait, im done wth the first one, let me understand the second one too
onyx glen
had i used 2 and 4 as subscripts, it would've created a temptation to use more double-angle identities that would have made our life a lot messier
onyx glen
well now you have $\frac{(c^2 - s^2)s^2 - 2s^2c^2}{(c^2-s^2)c^2 + 2s^2c^2}$
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Ann
onyx glen
why not expand on the num and denom and see what happens
worn matrix
yeah, let me do it
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worn matrix
$\frac{-s^{4}-c^{2}s^{2}}{c^{4}-3s^{2}c^{2}}$
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yajatk07
onyx glen
that three looks sus
are you sure you did it correctly on the bottom
$-s^2c^2 + 2s^2c^2 \neq -3s^2c^2$
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Ann
worn matrix
oh yeah, i did mistake
onyx glen
you should now have $\frac{-s^4-c^2s^2}{c^4 + s^2c^2}$
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Ann
onyx glen
try factoring on the top and bottom
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granite hollow
What does 3^x mean?
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What does 3^x mean?
granite hollow
I'm not finding anything online
One of my questions is to find where f(x) = x^3 + 3^x is discontinous
nimble fable
$3^x$
glossy valveBOT
Gamma
dense vector
3 multiplied x times
nimble fable
Three to the power x
grave elm
3 multiplied x times
This definition works for positive integers only
granite hollow
also is tanx 0/1?
worn matrix
also is tanx 0/1?
?
grave elm
also is tanx 0/1?
Were you trying to calculate limit to 0 or sth
grave elm
One of my questions is to find where f(x) = x^3 + 3^x is discontinous
3^x itself is continuous as well as x^3
granite hollow
grave elm
3^x itself is continuous as well as x^3
x^3 is just x*x*x and 3^x has a bit more complex definition, but it's continuous everywhere
granite hollow
@grave elm ok thanks that makes sense. I guess it also makes sense just to look and see that since I cant cancel out anything or set the denominator = 0, there is not anything that can be found as discontinuous.
also... would you mind helping me with f.? unit circle makes no sense to me... it was never taught. I watched a Khan Academy video but that didn't help much.
grave elm
start by rewriting tanx as a fraction of 2 continuous functions
grave elm
Do you know how to rewrite tan(x) as a fraction of 2 other functions?
granite hollow
no
grave elm
tan = sin/cos, do you know this?
granite hollow
yeah I learned that a couple mins ago 😂
grave elm
?? I dont know what exactly do you mean, but anyway fraction of 2 continous functions is discontinuous if the denominator is 0, yeah.
granite hollow
well I just said that because thats what my unit circle shows
granite hollow
grave elm
oh, anyway where do you think is sin(x) / cos(x) discontinous
granite hollow
maybe when cos(x) = 0
grave elm
Yes, because when cos(x) = 0, sin(x) / cos(x) = sin(x) / 0 which is not even defined
okay, now when does cos(x) = 0
cos is the x-coord on unit circle and sin is the y-coord
granite hollow
+/- pi(2n+1)/2
grave elm
yes, exactly
granite hollow
okay, now when does cos(x) = 0
when x is not moving only y ??
grave elm
watch the first few seconds
can you answer where does cos(x)=0?
what does theta have to equal?
grave elm
the video makes no sense to me
The video shows how sin and cos are defined on unit circle
first of all, in the center of circle, do you see that angle theta?
can you see how the angle increases?
granite hollow
oh yes
grave elm
for example at this point, the theta could be something around 45° or pi/4 radians
grave elm
for example at this point, the theta could be something around 45° or pi/4 radia...
and the cos(theta) is aproximately equal to sin(theta) and it's somewhere around 0.7
granite hollow
for example at this point, the theta could be something around 45° or pi/4 radia...
how do you know pi/4
granite hollow
the entire circle or just the corner?
grave elm
hence half circle is pi, quarter circle is pi/2 and 1/8 of circle is pi/4
grave elm
the entire circle or just the corner?
I mean this point, here theta approaches 2pi
hence half circle is pi
granite hollow
i see
grave elm
etc
granite hollow
what does theta have to equal?
so theta has to equal 90
granite hollow
what is a radian...
im actually so screwed once i graduate high school i feel failed
grave elm
That's the another unit of angle, like 360° corresponds to 2pi radians
grave elm
nice, what about other solutions?
granite hollow
wym
grave elm
solutions to cos(x) = 0, you found one of the solutions which is pi/2
but there are more solutions
granite hollow
just pi?
grave elm
hence half circle is pi
pi isnt a solution of cos(x) = 0, see this picture, here theta = pi (which is 180°) but cos(theta) seems to be -1
granite hollow
oh
grave elm
What about this point? The cos(x) is so tiny that it's merely visible
granite hollow
what are you asking
grave elm
I am asking when does cos(x) = 0, you already found one solution (that is pi/2)
grave elm
What about this point? The cos(x) is so tiny that it's merely visible
here the cos(x) is quite small and very close to 0, so that could be a solution
grave elm
okay lets get back to the video https://www.youtube.com/watch?v=Dsf6ADwJ66E
grave elm
for example at this point, theta seems to be around 60°, cos(theta), which is cos(60°) seems to be around 0.5 and sin(theta) is slightly larger, maybe like 0.8
do you understand how is sin and cos defined on unit circle
granite hollow
yes ok that makes snese
grave elm
okay, now describe what do you see on this picture
I care about value of theta, value of sin(theta) and value of cos(theta)
just aproximately
eyeball it
grave elm
theta probably like 170
nope
grave elm
sin 0.9
okay sin and cos are fine, except for one detail, when the green line goes in left direction the cos is negative and when red line goes in downward direction, the sine is negative
It's quite weird but that's just how it works
granite hollow
oh so it would be -0.9 and -0.3
grave elm
yeah exactly, it sort of describes the coordinates
like sin is y-coord and cos is x-coord
grave elm
would it be like -100
yep this is fine
There is also a way to describe that angle by a positive number, try it
granite hollow
260?
granite hollow
let me think
grave elm
you got x=90 for now
granite hollow
and that was pi/2?
grave elm
yeah, 90° corresponds to pi/2 radians
granite hollow
1.5pi?
grave elm
Yep
grave elm
would it be like -100
good job, now lets return back to this idea, you found other way to describe that angle
it's the same angle but described by negative number, instead of going 260 degrees counterclockwise, you can go 100 degrees clockwise and get to the same position
granite hollow
yep
grave elm
now what if you go all the way around (that's 360° or 2pi radians) and then go 270 degrees counterclockwise, will that work?
granite hollow
no
grave elm
why not?
granite hollow
cos will still be 0
grave elm
so 360 + 270 is still a solution
granite hollow
yep
grave elm
as well as 360 + 90
granite hollow
so is that where the n comes from?
grave elm
as well as n*360 + 90
granite hollow
because you cant write out every possible solution
grave elm
as well as n*360 + 270
grave elm
so is that where the n comes from?
Exactly
you can go twice around the circle and reach the same point, you can go billion times around it and reach the same point
granite hollow
yup
grave elm
so now you have 360n + 90 and 360n + 270, or 2pi*n + pi/2 and 2pi*n + 3pi/2
when you inspect those 2 things, you can actually see that you get every solution every new pi you go in any direction, like you start at pi/2, then pi/2 - pi = -pi/2 is a solution
pi/2 + pi = 3pi/2 is a solution
pi/2 + n*pi is a solution
granite hollow
ok
grave elm
pi/2 + n*pi is a solution
now when you factor out pi out of that, you get
pi(n+1/2) is a solution
now n usually just means the positive integers, so to cover cases like n=-1 or n=-2, you just add +- in front of it and you get your final solution, that is +/- pi(n+1/2)
granite hollow
ayee thats it???
grave elm
thats it
grave elm
+/- pi(n+1/2) is equivalent to +/- pi(2n + 1)/2
grave elm
what is college math gonna look like
Once you get solid base, college topics wont be much harder
you just need to practise unit circle and trig functions a bit
granite hollow
ok thank you so much
grave elm
youre welcome
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grave elm
u too
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torn jolt
no idea where to even start with this one
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spiral vigil
hmm
spiral vigil
no idea where to even start with this one
what distances can you label in that diagram? consider where all the right angles are
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spiral vigil
well... no not really, that's the maximum bend but we're just looking at an arbitrary bend
spiral vigil
yep good start
torn jolt
meaning OC = r - CD
spiral vigil
yes true
torn jolt
CD = r - OC
spiral vigil
consider where all the right angles are
spiral vigil
well... those are right triangles
torn jolt
ohh i think i see what i can do
nvm actually
cos theta = 1 - theta^2 / 2 so that gives me a clue
sin theta = O/r
ohh ok lol
i got i)
wicked zodiac
how would i move the sqr of 5 up there
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how would i move the sqr of 5 up there
rough tundra
how would i move the sqr of 5 up there
$\tfrac{2}{3\sqrt{5}}=\tfrac{2}{3\sqrt{5}}\cdot\tfrac{\sqrt{5}}{\sqrt{5}}$ :)
worn matrix
$\pi$
glossy valveBOT
yajatk07
wicked zodiac
how wouuld u do 2 * sqr of 5
glossy valveBOT
yajatk07
worn matrix
?
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minor needle
How do i approach this problem?
spiral vigil
when would a system of equations have no solutions? can you give an example of one?