mebarka
#help-27
woven radishBOT
cold bough
$(x+J)(y+J)=xy+J$ since $J$ is an ideal
woven radishBOT
everg
sand dove
exactly, $(r+J)^n = r^n + J(....) = r^n + J$
cold bough
so (x+J)^n=....
woven radishBOT
rafilou2003
cursive steppe
so you mean that for any element $x \in R$ (even if x is not in J) then $xJ=J$ ?
woven radishBOT
mebarka
sand dove
yes, by definition of an ideal
sand dove
**mebarka**
(of course this equality is only true if you view xJ and J as elements in R/J)
cursive steppe
(of course this equality is only true if you view xJ and J as elements in R/J)
that I don't understand, because in the quotient these elts are zeros... ?
cursive steppe
so I am supposed to consider the equality $xJ=J$ only in the quotient since there are non-examples (take $R=\mathbb{Z}$ the ideal $J=2\mathbb{Z}$ and 3*J is strictly included in J if I'm not mistaken) ?
woven radishBOT
mebarka
analog crypt
log𝑎(3𝑥 + 4) − log𝑎 𝑥 = 3log𝑎2
devout snowBOT
log𝑎(3𝑥 + 4) − log𝑎 𝑥 = 3log𝑎2
analog crypt
How to solve?
neat solstice
for x?
analog crypt
Yes.
rugged sparrow
log𝑎(3𝑥 + 4) − log𝑎 𝑥 = 3log𝑎2
Have you been given any range for alpha?
or any other information
neat solstice
log(a)-log(b)=log(a/b)
rugged sparrow
log𝑎(3𝑥 + 4) − log𝑎 𝑥 = 3log𝑎2
well, just turn the coefficients into powers and then use what martin wrote
neat solstice
yep
analog crypt
ok
neat solstice
in case you don't know:
a * log_b(c)=log_b(c^a)
analog crypt
.close
next nymph
devout snowBOT
next nymph
@tacit brook
Can you give me an example of why they should not be negative (question N.1)?
If σ = -5, then I(x0) = ]x0+5; x0-5[
Isn't that correct?
tacit brook
that correct, but this interval is empty because x0+5 is greater than x0-5
next nymph
Isn't an interval 'reversable'?
tacit brook
no, the first number always has to be lower than the second, otherwise its empty
tacit brook
its basically in this context just a convention and would also work in the definition of a limit with the exteriors included
next nymph
Ok, but why does that convention exists?
tacit brook
for example in the limit definition you could choose the interval [x0-0,x0+0] = [x0,x0]. if you take closed intervals. this would not make sense because then you cant choose any numbers from the interval as x != x0
An accumulation point is a point for which each neighborhood also includes another point. so you could say that there is not empty space around that point in the domain. x0 has to be a accumulation point because otherwise you cant choose small enough neighborhoods of x0 for which l is close enough to f(x)
next nymph
An accumulation point is a point for which each neighborhood also includes anoth...
So it means that if x0 was not an accumulation point, then there would not be an existsing I(x0)?
tacit brook
I(x0) would just consist of x0 for small interval sizes
next nymph
I don't understand.
What happens if x0 is not an accumulation point in the function's domain?
tacit brook
for example consider the domain $[-\infty,-1] \cup {0} \cup [1, \infty]$. here 0 is not an accumulation point because the interval ]-1,1[ in the domain just consists of 0
woven radishBOT
Robin123
tacit brook
so a function defined for this domain cannot have a limit at 0 because its not defined for values near 0
next nymph
So, x0 cannot be 0?
next nymph
So, saying "lim x->0 f(x)" has no sense because 0 is part of the domain but is not an accumulation point?
OOOH, so it means that i cannot study the function for values that tend to 0 because there are no values near enough to 0 to study the behaviour of the function?
tacit brook
yeah
next nymph
DAMNNNN
but wait, wouldn't the limit just be 0?
lim x->0 f(x) = 0
So the function actually exists for x0...
tacit brook
thats not how a limit is defined. the limit says what the function is approaching regardless what the actual value at x0 is
next nymph
So you mean that now we are not calculating the limit but just verifying it?
And to verify a limit, x0 must be an accumulation point?
tacit brook
yes
next nymph
Ok, so if x0 is not an accumulation point and the exercise asks me to verify the limit of x that tends to x0, than i can say that it cannot be verified because x0 is not an accumulation point?
tacit brook
yeah, you can say the limit does not exist in this case
tacit brook
again, it does not matter what the value of f(x0) is, so x0 is excluded.
next nymph
My book showed an example where x can be equal to x0 and an example where x must not be equal to x0, so it says that in general we do not consider x = x0.
Is it right or what?
tacit brook
what do you by an example where x can be equal to x0?
tacit brook
as i just said because the value of f(x0) does not matter. so youre right to say that we dont consider x = x0
next nymph
Why doesn't it matter?
tacit brook
the limit only tells you where the function is approaching to near x0
next nymph
OHHH
SO, x=x0 would be considered as 'calculating the limit', while 'x!=x0' as 'verifying the limit'?
DAMNN
tacit brook
no, f(x0) is not the limit. the limit can also have some other value than f(x0) at x0
next nymph
My book shows :
k = f(x0)
lim f(x)=k: { k != f(x0)
x → x0 f(x0) doesn't exists
Is that why x!=x0 than?
Because we cannot determinate generally what happens when x=x0?
tacit brook
yeah, basically
next nymph
Ok, and N.5?
My books says that this definition doesn't allow us to calculate the value of the limit but it only verifies if that value is the limit of the function or not.
Correct?
tacit brook
yeah thats correct.
tacit brook
no problem
next nymph
.close
devout snowBOT
@odd abyss Has your question been resolved?
cedar yarrow
Yes
devout snowBOT
@odd abyss Has your question been resolved?
vestal dirge
I am given this integral and they want me to solve it using complex integration
vestal dirge
my idea would be to write sin as a complex exponential and then try solving these 2 integrals and trying to get something in the form e^(-x^2) because I know how to integrate that
and then close the conture with cauchys integral theorem Ill try this for now
calm wren
hi how to find the reflection matrix for y = (-1/sqrt(3))x ?
devout snowBOT
hi how to find the reflection matrix for y = (-1/sqrt(3))x ?
calm wren
I tried 2 different methods
but they both are giving different answers and I'm not sure where I messed up
the first method I tried to use slopes/intersections/midpoints
and the second way:
I tried rotating it first and then reflecting over the x-axis
devout snowBOT
@calm wren Has your question been resolved?
devout snowBOT
devout snowBOT
Available help channel!
bleak needle
devout snowBOT
bleak needle
this is the solution for a question my teacher posted
how did she do this part:
i dont get how this part makes sense
supple knot
yea she messed up
the bottom is a continuation from above
the bottom left side should read $[\sin(\alpha) + \cos(\alpha)]^2$
woven radishBOT
riemann
supple knot
she pulled the ol' freshman's dream
queen zephyr
damn that ligatures 🥵
devout snowBOT
Available help channel!
cunning burrow
hey all, having some trouble with a problem. I'm unsure why pi/3 is incorrect here.
devout snowBOT
hey all, having some trouble with a problem. I'm unsure why pi/3 is incorrect he...
@cunning burrow Has your question been resolved?
smoky nimbus
hey all, having some trouble with a problem. I'm unsure why pi/3 is incorrect he...
You should use the unit circle for that part
smoky nimbus
hey all, having some trouble with a problem. I'm unsure why pi/3 is incorrect he...
This part here, when you were subtracting pi from 4pi/3 doesn't help because all you did was shifted 4pi/3, yes pi/3 is a solution but it's a solution based on 4pi/3
dry rover
hello, why is $\mathbb{Z}{/}0\mathbb{Z}=\mathbb{Z}$?
devout snowBOT
hello, why is $\mathbb{Z}{/}0\mathbb{Z}=\mathbb{Z}$?
woven radishBOT
lilisworld
devout snowBOT
sullen island
.reopen
devout snowBOT
✅
sullen island
hein ? @dry rover
sullen island
is it because O divides every integer even if the value would be infinity?
c'est quoi le rapport avec Z/0Z ?
dry rover
c'est quoi le rapport avec Z/0Z ?
la classe de 0 c'est Z?
sullen island
nan
la relation d'équivalence qui définit Z/0Z, c'est a ~ b <=> 0 | a-b
ça arrive seulement quand a=b ça
donc tes classes d'équivalence c'est les entiers, isolés les un des autres
dry rover
ah oui je vois merci
sullen island
d'où Z/0Z = Z
dry rover
en effet
stark sinew
c'est aussi utile de noter que 0Z c'est juste l'element 0, alors c'est comme faire le quotient par rien, alors on n'a pas fait le quotient du tout
modest spire
could anyone help me with 4? im not sure i did it right
devout snowBOT
could anyone help me with 4? im not sure i did it right
@modest spire Has your question been resolved?
devout snowBOT
@modest spire Has your question been resolved?
modest spire
.close
pseudo palm
devout snowBOT
pseudo palm
this is what i’ve done so far but it’s wrong
oh wait
i figured that shit out
.close
flat marlin
What happens with concavity and inflection when my second derivative is a whole number
devout snowBOT
What happens with concavity and inflection when my second derivative is a whole ...
wooden wraith
flat marlin
Ohh that makes sense
devout snowBOT
@flat marlin Has your question been resolved?
halcyon mango
Use the limit definition to find the expression for the area under the graph of f(x)=6-x on the interval from 1 to 5.
devout snowBOT
Use the limit definition to find the expression for the area under the graph of ...
urban harbor
yes, looks right for a right-hand riemann sum
restive river
Stuck on this question, any help?
devout snowBOT
Stuck on this question, any help?
restive river
Don't know how to convert those secs and min into 1
restive river
2250 secs for sebastian
dim knot
2250 secs for sebastian
no, express the entire time in minute, not in seconds (cuz the question asks for running rate at kilometres per 'minutes' )
restive river
37:50 mins
pseudo basin
no, express the entire time in minute, not in seconds (cuz the question asks for...
minutes per km, not km per minute.
restive river
37.5 mins.. 37.5/10=3.57mins/km
pseudo basin
anyway that's correct for sebastian, now do alexander.
possibly convert them both into min&sec later.
restive river
Alexander= 3195 seconds in total
3195/60=53.25 mins
53.25/15=3.55
3.55mins/km for alexander
pseudo basin
,calc 53.25/15
woven radishBOT
Result:
3.55
ok yeah
devout snowBOT
@restive river Has your question been resolved?
restive river
If I want to make this ratio into simple form I will have to first convert it in to a whole number?
summer harbor
Question about piecewise functions: What combinations of circle (open, shaded) connects the functions and disconnects the functions? (graph)
devout snowBOT
Question about piecewise functions: What combinations of circle (open, shaded) c...
summer harbor
Does open and open circle of 2 unique graphing shapes/lines connect and act as one function?
devout snowBOT
@summer harbor Has your question been resolved?
modest dagger
@summer harbor is there supposed to be a picture or diagram that goes with this question?
Or do you happen to have a picture of the full question?
summer harbor
none, it came from my curious brain because its important for my exam
modest dagger
Ah it is unclear to me what you are asking.
summer harbor
so imagine this is the graph, as you can see there are 2 shaded circles in each starting point
my question is, do they connect, such that they become one and acting function? because in technical this is a piecewise
modest dagger
As you have drawn them they do not connect. Are you asking how many ways we can connect them up into one function?
summer harbor
yes they do not connect when looking at it visually, but my classmates said that the type of circle (open and shaded) has something to do with their connection
modest dagger
Sometimes people draw an unshaded circle to show when points are being excluded from a graph
I can draw an example of this.
summer harbor
mhm
modest dagger
For example here
There is a curve f
The curve f has that little open circle at some point (x,y)
summer harbor
mhm
modest dagger
That notation means this function/graph does not include that specific point.
summer harbor
ohhh
modest dagger
Filling it in/shading it would just mean the curve includes the point.
modest dagger
Yeah exactly
There's a tiny infinitely small point at (x,y) missing from the graph of f
Sometimes this is important at endpoints of curves like you mentioned.
Let me draw this too
summer harbor
but without having to literally connect them, is it possible to use shaded and unshaded circle to indicate that graph 1 is part of graph 2?
like connecting disconnected parts
without having to like literally connect
modest dagger
but without having to literally connect them, is it possible to use shaded and u...
Typically people would draw something like this and have some wording telling you whether or not the whole diagram is the graph of f or whether the two pieces are graphs of two different functions.
Well let me rephrase this
modest dagger
For this kind of graph where just the one dot is missing people usually always mean that this is one single graph for one single function f.
Whether or not the dot is filled or not.
modest dagger
With this one people will usually explain whether this all one graph of one function f or if it is meant to be two graphs of two different functions
modest dagger
graph 1 is part of graph 2
Idk what graph 1 and graph 2 mean to you
summer harbor
Idk what graph 1 and graph 2 mean to you
the drawing
modest dagger
In the vast majority of cases if there is a doubt you can just ask whoever made the graph to clarify whether it's all a graph of a single function or if it is meant to be something else.
modest dagger
the drawing
Which drawing and in that drawing what specific things are you referring to when you say graph one and graph two?
summer harbor
modest dagger
Okay, yeah for that diagram what I'm saying is that it's ambiguous enough that if you are unsure you should ask the person who drew it what they mean.
Here I meant the graph to be of one curve f
But perhaps another author might draw something similar and mean two different curves.
summer harbor
this particular drawing, the curve with the shaded circle and the curve with the unshaded, they act as one curve
modest dagger
In a book there is usually surrounding paragraphs of text that clarify this.
modest dagger
this particular drawing, the curve with the shaded circle and the curve with the...
This is what I meant but in another context somebody might want the left curve to be a curve g and the right one to be a curve f
Hence the ambiguity.
summer harbor
In a book there is usually surrounding paragraphs of text that clarify this.
so even in piecewise functions, a part-curve with a shaded, and a part-curve with open circle can mean different things?
modest dagger
So the wording piecewise here isn't exactly referring to graphs that are chopped up or not like in these pictures.
The piecewise wording refers to a function you DEFINE in pieces/cases.
summer harbor
The piecewise wording refers to a function you DEFINE in pieces/cases.
a function containing subfunctions
that seems right
modest dagger
The formula f here is an example of a piecewise function
We defined it in two cases.
The first case is when x is between 0 and 1 (excluding 1 itself.
The second case is when x is larger than 2 (including 2 itself).
summer harbor
mhm
modest dagger
This function has the graph shown below.
summer harbor
The first case is when x is between 0 and 1 (excluding 1 itself.
these are also known as boundaries right
modest dagger
You see how there's no ambiguity about which pieces of the figure correspond to f when I tell you what f is in this way?
summer harbor
yes because it was clarified
modest dagger
these are also known as boundaries right
Yeah you can think of the inequalities involving x as boundaries in a sense.
Yeah
Lemme give another
Now there are two functions but still no ambiguity.
You see?
summer harbor
yea
modest dagger
But if I just gave you the figure here and told you it represented two functions p and q you'd have no idea which one was p and which one was q so you'd have to ask me.
modest dagger
One thing I was gonna say here is that if this graph is all for one function f
The function is undefined between the two endpoints.
We can pick different ways of shading or not shading the endpoints to come up with new technically different functions.
modest dagger
For example we can swap the shaded vs unshaded endpt or shade both or neither etc
Doing this gives us functions that are technically different because they differ at a handful of points from our original f.
You also don't always have gaps with piecewise functions
summer harbor
but what abt in cases like this
modest dagger
For example something like this can occur where the endpoints connect up real nicely
modest dagger
but what abt in cases like this
I'd say for something like this they mean this graph represents a single function
summer harbor
I'd say for something like this they mean this graph represents a single functio...
i see
modest dagger
Because they use the wording "the function" which means a single function
summer harbor
but in the case of piecewise function, what combinations of circles do we like have to identify to check if
its a single function, or 2 unique functions
modest dagger
The circles don't tell you that
modest dagger
See here
summer harbor
wat
modest dagger
If I erase the formulas I wrote defining f and g
summer harbor
it wont mean anything
modest dagger
And gave you just the graph and told you it represented two functions f and g but nothing else.
You wouldn't know which one was f and g unless you asked me.
The circles don't help you figure this out.
They just tell you when a point is included in part of a graph and when it is specifically being excluded from a part of a graph.
summer harbor
They just tell you when a point is included in part of a graph and when it is sp...
okay, time to change the question, and im really sorry
so using circles, what combinations are used to include that the point is part of a graph? aside from close and open circle
modest dagger
Idk what you mean?
summer harbor
Idk what you mean?
like for example
summer harbor
if we fill the circle on f(x) = 1; is it still part of the graph?
idk how to word im sorry
modest dagger
For this picture?
summer harbor
yea
modest dagger
If we fill the circle on that figure the formula for f hasn't changed, so we'd be drawing the figure of some new function that was very similar to f but is not the same as f.
The new function would have all the infinitely many points of f, but it would have one infinitely small tiny point that f does not have.
At the filled in endpoint
modest dagger
summer harbor
no... not that
modest dagger
If we draw the top formula, f we have to draw something like in the figure I drew earlier with the endpt at x=1 filled.
summer harbor
i meant this
modest dagger
If we draw the f squiggle function, we have to draw almost the exact same figure but we have to fill in the x=1 endpt so it's a technically different function.
summer harbor
hypothetically
summer harbor
ohhh
modest dagger
But now the figure shows my f squiggle here instead of f
summer harbor
ohh
modest dagger
i meant this
If you fill in the dot the picture isn't of f anymore it is of f squiggle
(For the left half of the picture at least. The right half is still of g.)
summer harbor
im confused about the boundaries..
summer harbor
i meant this
in the boundary; 0 <= x < 1; what does x symbolize?
because i thought, x is our starting point, turns out it's 0 and 1.
modest dagger
x is the variable
For the functions
Here 0<= x < 1 just tells us x is being restricted to be between 0 and 1 (including 0 but excluding 1)
You can sort of think of this as the values of x can start at 0 and proceed up t0 (but excluding) the value 1.
summer harbor
ah
modest dagger
So for example x can be 0, it can be 1/3, it can be 1/2 it can be 0.7777777776626378484736
It can be a lot of things
Not just 0
Also never 1 because our inequality excludes 1
modest dagger
It just takes practice
summer harbor
in piecewise right, its always vertical testing
vertical line testing
to check if function
modest dagger
Yeah, the vertical line test works to check if any graph of a curve is the graph of a function
So piecewise curves are just a special case of this.
So it works for piecewise stuff too
summer harbor
yes
modest dagger
The main idea is that we define functions to be rules that allow us to assign each element of the domain to exactly one element of the codomain/range.
The vertical line test is basically just trying to visually check if you can find a place where you can take one element of the domain and find two elements of the codomain/range that correspond to it I suppose.
Mmm have you talked about functions as input output machines?
That is kind of a good way to think of it.
summer harbor
yeah
modest dagger
Functions like f are supposed to take inputs like x and always spit out only one output, we call that output f(x).
When the vertical line test fails you end up finding a value x where you can find two different output f(x) values.
summer harbor
theres a reason why a common x cannot spit out 2, it'd be considered ar elation
modest dagger
So the thing you're testing can't be a function.
summer harbor
however, 2 unique x values can have a common y
modest dagger
theres a reason why a common x cannot spit out 2, it'd be considered ar elation
Yes definitely!
Relations can have 2 or more in fact.
summer harbor
yeye
modest dagger
however, 2 unique x values can have a common y
This is also true.
summer harbor
relations are like analogies
modest dagger
Yeah I suppose there is a way to think of them like that.
Oh shoot I gotta go. If you have more questions you can post them here and somebody will probably see them.
summer harbor
alr
modest dagger
You can also close and reopen a channel if you wanna start from scratch too ofc.
summer harbor
yea
paper radish
hey
twin wedge
i guess
paper radish
so it just determines the vertical parts of the graph right
twin wedge
yes
short shore
power series representation of 1/(9+(4x)^2)
devout snowBOT
power series representation of 1/(9+(4x)^2)
short shore
is it ((-1)^n)((4x^2n)/(9^n+1))
devout snowBOT
@short shore Has your question been resolved?
devout snowBOT
@short shore Has your question been resolved?
thorny pawn
How do I prove/disprove if f from Z to Z f(n) = n^3 -2n^2 +2n -1 is injecting/surjective
devout snowBOT
How do I prove/disprove if f from Z to Z f(n) = n^3 -2n^2 +2n -1 is injecting/su...
thorny pawn
We’re not allowed to show that it’s an increasing function
eager lodge
why
thorny pawn
That’s just what the professor says
n^3 -2n^2 +2n -1 = m^3 - 2m^2 + 2m -1
How do I show n=m
bitter snow
here is an idea but i dont know ....
n^3-2n^2+2n-1 = (n-1)(n^2-n+1)
but we can prove that n^2-n+1 ≥|n| its always bigger than 0
so if m is not n then one of them is bigger and you can see that cant be true
devout snowBOT
@thorny pawn Has your question been resolved?
neat solstice
I dont think we can prove this by using n=m
Or well
It would work for the real numbers
But at that point we are effectively showing that it always increases
wanton canyon
devout snowBOT
wheat pawn
the probability of an event is 1 minus the probability of the opposite event
wanton canyon
oh sorry
i meant like
the p(x<=2)/p(x<=5)
like how do they simplify the fraction to thta
wheat pawn
it is in the first line
wanton canyon
wdym
wheat pawn
it says where both probabilities come from
devout snowBOT
@wanton canyon Has your question been resolved?
devout snowBOT
@wanton canyon Has your question been resolved?
compact tree
hi, i have a chance and statistics question. i have a table with weights of sugar cubes, and the question is "i want to have atleast 9 grams of sugar what is the chance a have 9 grams with only two sugar cubes?
devout snowBOT
hi, i have a chance and statistics question. i have a table with weights of suga...
@compact tree Has your question been resolved?
devout snowBOT
@compact tree Has your question been resolved?
devout snowBOT
@compact tree Has your question been resolved?
devout snowBOT
@compact tree Has your question been resolved?
supple knot
hi, i have a chance and statistics question. i have a table with weights of suga...
!original
devout snowBOT
hi, i have a chance and statistics question. i have a table with weights of suga...
Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
compact tree
You need more then 9 grams of sugar, what is the chance u grab 2 sugarcubes and they weigh 9 grams or more? [I have been given a table of 500 weights of sugar cubes that vary around the 4.5 grams]
near stone
how big is 220X1802MM?
!occupied
devout snowBOT
how big is 220X1802MM?
Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).
devout snowBOT
@compact tree Has your question been resolved?
summer harbor
hi guys, im still confused about circles in piecewise functions (shaded and unshaded)
devout snowBOT
hi guys, im still confused about circles in piecewise functions (shaded and unsh...
summer harbor
using the types of circles, which combinations connect the lines/slopes/shapes? (ie. close and shaded), and how do u write the domain and range?
how does the former have a union in the domain while the latter doesnt?
visual hazel
how does the former have a union in the domain while the latter doesnt?
the domain of the first is not continuous but the second one is
summer harbor
the domain of the first is not continuous but the second one is
what does that mean?
twin stump
I think a more appropriate term is connected
For the domain, imagine projecting each point on the graph to the x-axis
For the first one you get two disconnected segments
visual hazel
I think a more appropriate term is connected
yeah thats better
summer harbor
I think a more appropriate term is connected
yeah
are closed and open circles connected?
twin stump
it depends
summer harbor
like shown here
visual hazel
so the first one, the first function ends at 4 but the second function doesnt start at 4
visual hazel
it just means if its a closed or open interval
summer harbor
it just means if its a closed or open interval
and that depends on the inequality right
visual hazel
filled in circle means closed, hollowed means open
summer harbor
less than vs less than or equal to
twin stump
For jump discontinuities:
If an open circle is directly over an open circle, it is disconnected.
If an open circle is directly over or below a closed circle, it is connected.
If a closed circle is directly over a closed circle, it is not a function. (It fails the vertical line test)
visual hazel
less than vs less than or **equal to**
closed is less or equal, open is strictly less
summer harbor
For jump discontinuities:
If an open circle is directly over an open circle, it...
ohh thank god
in piecewise functions, if one of 3 formulas isnt a function, does that mean the main function is not a function anymroe?
twin stump
Yes but I've never seen a question that asked if a piecewise function was a function
also the domain here would be $[-4, 2)\cup (2, 4]$
woven radishBOT
WHAT?
summer harbor
also the domain here would be $[-4, 2)\cup (2, 4]$
huh?
twin stump
I slightly changed the example
summer harbor
ohhh
summer harbor
Yes but I've never seen a question that asked if a piecewise function was a func...
ive seen worksheets
that ask if
is it a function?
twin stump
Well, yes, if a piecewise "function" is defined using a "function" that isn't a function, that piecewise "function" isn't a function.
summer harbor
so hitting 2 birds with one stone
if one formula out of 2 or 3 formulae in the function isnt a function
boom its not a piecewise anymore
twin stump
I'm not sure if they would do this, but if we're discussing the possibilities of something defined piecewise not being a function, it is possible that they define the piecewise function in such a way that the conditions overlap
like maybe they say
f(x) =
2 if x < 1
3 if x > -1
then f(0) = 2 and f(0) = 3
summer harbor
like maybe they say
f(x) =
2 if x < 1
3 if x > -1
also
those are
open circles right
twin stump
Yes, those are open circles
summer harbor
albeit its infinite therefore we cant place rhem
so it doesnt really affect when verticla tested
summer harbor
For jump discontinuities:
If an open circle is directly over an open circle, it...
here
twin stump
We can consider this function f as a relation rather than a function and then discuss the domain and range of this function.
f(x) =
2 if x < 1
3 if x > -1
Is just like the relation R where xRy if and only if y = 2 and x < 1 or y = 3 and x > -1.
Then, Dom(R) is the set of all real numbers and Ran(R) is {2, 3}
summer harbor
We can consider this function f as a relation rather than a function and then di...
wait wait
how are they not piecewise
twin stump
It doesn't make sense to discuss a jump discontinuity since it is not a function
summer harbor
It doesn't make sense to discuss a jump discontinuity since it is not a function
yeah but; theyre closed circles and theyre disconnected
twin stump
I don't think you will be asked to find the domain and range of things that are not functions unless you are in a much more advanced class
summer harbor
yeah
summer harbor
We can consider this function f as a relation rather than a function and then di...
wait hold up, where is function f
twin stump
yeah but; theyre closed circles and theyre disconnected
what do you mean they are closed circles and disconnected
summer harbor
For jump discontinuities:
If an open circle is directly over an open circle, it...
based from here right
twin stump
wait hold up, where is function f
I replaced f with R so that I was writing xRy rather than y=f(x)
by the way, a function $f$, in set theory, is just a relation such that for every $x$ in the domain of $f$ and $y_1$ and $y_2$ in the range of f, $xfy_1$ and $xfy_2$ implies $y_1=y_2$. Then, for any $x$ in the domain of $f$, $f(x)$ is said to be the unique number $y$ in the range of $f$ such that $xfy$
woven radishBOT
WHAT?
summer harbor
**WHAT?**
yeah,
for every x input is a unique y input
2 x inputs can share a common y input but 1 input cannot have 2 y inputs
twin stump
exactly
summer harbor
We can consider this function f as a relation rather than a function and then di...
im just confused here
twin stump
how about we graph it
twin stump
lol
summer harbor
jfc im so sorry
but for jump discontinuity, the close and open circles must be parallel to each other
twin stump
They must be on top of one another
summer harbor
yes
twin stump
They must share the same x-coordinate
summer harbor
yesyes
so if one circle is one x coordinate away from another, its not connected therefore union
twin stump
Yes, in that case you would have to describe the domain using $\cup$
woven radishBOT
WHAT?
twin stump
Just as the domain is the collection of all real numbers x such that f(x) exists, the range is the collection of all real numbers y such that x exists for which y=f(x). To visualize the domain, you can imagine projecting all points onto the x-axis. To visualize the range, you can imagine projecting all points onto the y-axis.
summer harbor
yeah
also for the type of circle when writing the format for domain, which is which?
[] for what, () for what
devout snowBOT
@summer harbor Has your question been resolved?
summer harbor
why does the 2nd curve pass the asymptote yet is still considered a function
delicate fox
?
summer harbor
?
the one at point b and
point a
the red broken lines represent the asymptote
how is it that it still passes the asymptote yet consideres a function
wtf
delicate fox
how is it that it still passes the asymptote yet consideres a function
why would the asymptote matter?
its not a real line
it just describes what the function looks like at large enoguh values of x
instead of saying "the function gets close to x" you say the asymptote is x=0
summer harbor
mhm
summer harbor
instead of saying "the function gets close to x" you say the asymptote is x=0
wait why
delicate fox
because its an easy way to describe it
summer harbor
its not a *real* line
yeah, but even if it is an imaginary, if represented, arent they supposed to not touch
delicate fox
no?
why wouild it matter
again it represents large values of x
around the origin (0,0) x isnt large and as you can see the function isnt similar at all to x=0
summer harbor
yeah
quaint citrus
<@&268886789983436800>
summer harbor
.close
stark zenith
x^5-5x^4-7
devout snowBOT
x^5-5x^4-7
stark zenith
Find local mins and maxs
So far
I have f’’
Found the crit points at
4 and 0
4 is the minimum
But idk what 0 would be
An inflection point?
violet wind
f'' is zero there?
the way I would solve is keep taking derivatives until one isn't zero but idk the popular calc 1 technique
stark zenith
I figured it out actually
violet wind
f''(0)=0 right
stark zenith
Yea
Sorry i meant that
Yea
Wait no
Wait yea
Yea
And then i plugged in -1 and 1 for f’
And it told me if it was the local or max or an inflection point
violet wind
that's a little sketchy but
stark zenith
Is it?
violet wind
it will probably give the right answer
stark zenith
Idk any other way lol
violet wind
?
Oh oops
f''''(0) = -120
3rd derivative being the first non-zero derivative would make it be neither min nor max
4th derivative being the first non-zero just acts the same as 2nd
violet wind
x^5-5x^4-7
Not unless this is already f'
violet wind
4th derivative being the first non-zero means it acts the same as second derivative
stark zenith
Ive never went past the 3rd derv tbh
violet wind
So negative meaning there's a local max
violet wind
Yeah you have to keep taking them to avoid guessing
stark zenith
So if f’’’’ would’ve been postive
violet wind
Hmm
stark zenith
It would be local min?
violet wind
Or maybe you could make a sign chart for f' or such
violet wind
It would be local min?
Yeah exactly
violet wind
then u gotta take more derivatives
violet wind
hmm
stark zenith
Inflection?
violet wind
if all the derivatives are zero then you probably have to make a sign chart or something
That's possible but rare
Sign chart for f' would show increasing/decreasing sections
devout snowBOT
@stark zenith Has your question been resolved?
stark zenith
.close
devout snowBOT
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Available help channel!
tribal kernel
Hi guys! I'm trying to find the domain of the following function, and to do that I have to take the union of the interval of each subfunction. but I have no idea on how I am suppose to find the interval of x = 0 (and I have to do it without graphing btw)
any help please?
devout snowBOT
Hi guys! I'm trying to find the domain of the following function, and to do that...
restive river
Domain of a function is just the set on which the function is defined
In the second case, that part of the domain will just be {0}
It doesn't necessarily have to be an interval
restive river
Not more than
For second case, you can only take x=0
Because it is defined that f(0) = -1/2
tribal kernel
since it's closed interval with one element it's only 0
yeah! now I understand!!!
thanks mate!!!
restive river
No problem
tribal kernel
.close
tardy magnet
Hi
tardy magnet
i need help with areas
oops i didnt realize i solved question d so i just need help with c
devout snowBOT
@tardy magnet Has your question been resolved?
tardy magnet
<@&286206848099549185>
empty glacier
<@&286206848099549185>
There are 4 triangle in the picture, you can calculate the area of the the not shared triangles
wet hatch
Can someone help me fill out the nmr data? Or better yet help me learn how to do it on my own
devout snowBOT
Can someone help me fill out the nmr data? Or better yet help me learn how to do...
@wet hatch Has your question been resolved?
devout snowBOT
@wet hatch Has your question been resolved?
devout snowBOT
@wet hatch Has your question been resolved?
solar yoke
Can someone help me with #5 please
devout snowBOT
Can someone help me with #5 please
solar yoke
If XZ and ML are congruent by the Line Perp to Transversal Theorem then does that make all of them = to 33
Making measure Z = 33
H e l p
devout snowBOT
@solar yoke Has your question been resolved?
distant sorrel
devout snowBOT
distant sorrel
this is the question
can someone please explain the explanation
cause i am not following at all
unique prawn
the relation between radius and h is given
unique prawn
since you require the rate of change of height so convert the whole eq in terms of height
unique prawn
why did 1/3 disappear
by differentiating the volume
distant sorrel
but in the next step 1/3 disappeared and h^3 turned into h^2
distant sorrel
by differentiating the volume
differentiating ?
unique prawn
you have lerned differentiation right?
distant sorrel
oui
unique prawn
yeah so whats there that you dont get
distant sorrel
what is the volume
unique prawn
this is basic differenciation rule bro
distant sorrel
like
distant sorrel
ok
I get that
but in the next step
he replaces h with 3
why?
where did 3 come from?
is it the 3m?
ahhh
ok
yea
I get it now
thank you
.close
restive river
D
restive river
All fo em are hcf problems right?
stuck field
yeah it would appear so.
tranquil lagoon
devout snowBOT
Available help channel!
thorny parcel
how can we prove a statement for specific set
devout snowBOT
how can we prove a statement for specific set
thorny parcel
natural numbers set
like proving a statement hold for all k such that k is in the set {1,2,3,...,n-1}
i wanted to use induction but idk if induction can be used in such way
swift knoll
what's your problem specifically?
tribal sierra
like proving a statement hold for all k such that k is in the set {1,2,3,...,n-1...
Why can't you use induction here?
thorny parcel
how can we do so if induction prove it for all naturals n
but in my case i need it for all naturals less than some n
so it's not true for all
it's just for a specific range that depends on the chosen n
tribal sierra
Whats the actual problem
you can always assume it holds true for some arbitrary k in ur set then show it holds for some k+1 where k+1 <n
thorny parcel
you can always assume it holds true for some arbitrary k in ur set then show it ...
oh right
thorny parcel
Whats the actual problem
ill show it to you but it's written in arabic so maybe it wouldn't be clear
tribal sierra
can you translate it
thorny parcel
this is the part i was dealing with
devout snowBOT
@thorny parcel Has your question been resolved?
sage socket
devout snowBOT
sage socket
Hello,
I was following this video for finding the power series by differentiating and I couldn't quite follow the step I noted in blue
I understand what's going on for the lhs. We're just taking the derivative ( a few times but I'm not sure why) and on the rhs I'm not following.
lofty monolith
Bit confused
devout snowBOT
Bit confused
storm gyro
I believe what they want you to state is common to both is that they have the same coefficients
so the same coefficient matrix and inverse can be used to solve them
Or if you're not thinking in terms of matrices yet, the same operations applied to both systems
devout snowBOT
@lofty monolith Has your question been resolved?
hollow ivy
devout snowBOT
hollow ivy
i dont quite understand how t is being solved for here
am I diving the entire equation by log(2)?
sullen island
well you're dividing by -log(2)/30
or multiplying by -30/log(2) if you prefer
@hollow ivy
hollow ivy
so i divide everything by -30/log(2)
sullen island
yea
hollow ivy
poh misreadf
why am I doing this its to isolate 2 but does doing this get rid of the 30 under the t?
sullen island
yeah that's exactly what they got
high chasm
for the function Y = -X^2
devout snowBOT
for the function Y = -X^2
high chasm
with domain (0,inf), would the range be (-inf,0)?
frozen aurora
yes
high chasm
wouldnt it be (0, -inf)?
rustic zealot
not sure how to do a T~T
devout snowBOT
not sure how to do a T~T
jade oak
not sure how to do a T~T
To be able to add fractions you need a common denominator. To do this you find a common multiple between the two denominators, then transform each fraction so each has the same denominator
jade oak
rustic zealot
oooh
so that's why they multiplied the top and bottom by (x+4)?
of each fraction I mean
jade oak
yeah the left fraction got multipled by (x+4)/(x+4) and the right one was multiplied by (x-14)/(x-14)
jade oak
If you dont have any more questions and dont think you will have anymore yeah
rustic zealot
oh I definitely will, but not sure how soon
jade oak
hmm I dont think there is really any rules about it but since there are not any other available channels atm if it will be 10+ min I would close
rustic zealot
ok so I do apparently have a question alrdy xD
wait do I-
I am confusing myself dsajdksad
ok nvm I read something wrong, oops
I thINK I should be good for now, so I'll close
ty for your time :D
.close
topaz sequoia
I need help with factoring in exponential equations. I have no clue where to go or how to start. For example if i have 8 * 4^2x - 2 * 4^x = 0. I dont know when to factor, I kind of know how to by logic but It doesnt get me anywhere.
devout snowBOT
I need help with factoring in exponential equations. I have no clue where to go ...
topaz sequoia
I did bring 2 * 4^2x infront but then im left with
2 * 4^2x(3 * 1) - 2 * 4^2x
prime egret
You can denote 4^x as y that should make it easier to see
topaz sequoia
2y ( 4y - 1 ) = 0?
prime egret
Yes
topaz sequoia
But what do I do from here
prime egret
the product ab = 0 iff a=0 or b=0
topaz sequoia
Oh yeah,
2y = 0 --> y = 0 but because y = 4^x. 4^x =/= 0 so this cant be?
4y - 1 = 0 --> y = 1/4
y = 4^x 1/4 = 4^x
If that is right then what now
Nvm im still very confused
topaz sequoia
log 4^x = log 1/4
x * log4 = log 1 - log 4
x = (log 1 - log 4)/ log 4
x = -1?
I dont know how the log10 works but I was told that i dont have to write log thats base is 10
But that should be correct I hope
But what if all of the left side doesnt equal 0. For example its
2 * 5^3x = 10 * 5^2x
<@&286206848099549185>
empty glacier
log 4^x = log 1/4
x * log4 = log 1 - log 4
x = (log 1 - log 4)/ log 4
x = -1?
Yea
That's correct
topaz sequoia
But what if all of the left side doesnt equal 0. For example its
2 * 5^3x = 10 ...
but what about this? what can i do here?
.close
devout snowBOT
topaz sequoia
.reopen
devout snowBOT
✅
prime egret
Using that $a^{-1} = \frac{1}{a}$ gives u the answer
woven radishBOT
$Pure$
topaz sequoia
oh yeah
damn
I dont get this factoring at all. I cant understand when to factor and I cant see how to factor
prime egret
It comes with practice
topaz sequoia
I had to learn it in 2 weeks
prime egret
Do you mean factoring like polynomials
topaz sequoia
sorry i dont speak much english idk what thatis
I mean factoring like I just did, and a little more complicated with other stuff.
prime egret
Yeah the more practice questions you do the better you’ll get at it
topaz sequoia
Not really
2 * 5^3x = 10 * 5^2x how can i factor this if there are no similiar things
and they are on different sides
topaz sequoia
what
topaz sequoia
🙂
prime egret
Simplify 2 nd 10
topaz sequoia
confused
prime egret
You’d get $$5^{3x} = 5 \cdot 5^{2x} = 5^{2x+1}$$
woven radishBOT
$Pure$
topaz sequoia
I need to factor
prime egret
Why
topaz sequoia
Thats the excercise. I need to learn factor for the test. I can do without factoring that part i have learned
but factoring is a major part that i dont know how to do. And it will be in the test
prime egret
Well write 5^x as y and try to factor it
topaz sequoia
Can you always write something as y?
prime egret
You‘re just making it easier to manipulate it’s only symbolic
topaz sequoia
y = 5^x 2 * y^3 = 10* y^2
prime egret
Yeah
topaz sequoia
(2 * y^3 = 10* y^2) / 2
y^3 = 5 * y^2
y^3 -5 * y^2 = 0
y^2 (y -5) = 0
y^2 = 0 --> 5^x2 =/0
y - 5 = 0 --> 5^x - 5 = 0
5^x = 5
x = 1
?
Correct?
prime egret
Yeah
devout snowBOT
@topaz sequoia Has your question been resolved?
topaz sequoia
2 * 20^x - 2 * 4^x + 16 * 5^x - 16 = 0
what can i factor here or change to y
2*4^x?
.close
devout snowBOT
devout snowBOT
Available help channel!
pale loom
Question:
If you write on Sage Math:
mod(7^euler_phi(100),100) == 7
Result is True or False?
I know the result is false cause I tried it but what’s the logical thinking behind it?
devout snowBOT
Question:
If you write on Sage Math:
mod(7^euler_phi(100),100) == 7
Result is Tr...
devout snowBOT
@pale loom Has your question been resolved?
brisk totem
euler’s theorem
devout snowBOT
devout snowBOT
Available help channel!
crystal token
devout snowBOT
crystal token
I have a question is tan^2v = tan (v)^2
gaunt matrix
yes
prime egret
It means (tan(v))^2 yeah
gaunt matrix
maybe cross multiply both top and bottom
crystal token
Is there a difference between (sin v / cos v )^2 and ( sin (v)^2 / cos (v)^2 )
gaunt matrix
no
prime egret
You should’ve applied difference of two squares at the beginning
crystal token
You should’ve applied difference of two squares at the beginning
is it necessary?
prime egret
No but it’s easier
devout snowBOT
@crystal token Has your question been resolved?