sin^-1(x) is not a function because
#precalculus
quasi elbow
let's say we wanted to know sin^-1(1/2)
it would not only be pi/6
but infinite other values
hence we change the domain which changes the graph
and now if we calculate the value of sin^-1(1/2)
between -pi/2 to pi/2
it gives us only one value
hence satisfying the definition of f(x)
is that correct?
quasi elbow
sin^-1(x) is not a function because
with previous domain
daring tapir
Sorry i missed your message
daring tapir
sin^-1(x) is not a function because
It is a function cuz WE defined its range to be [-pi/2,pi/2]
daring tapir
hence satisfying the definition of f(x)
Yepss
quasi elbow
It is a function cuz WE defined its range to be [-pi/2,pi/2]
yes
i meant
for all real numbers
but yeah it makes now
daring tapir
for all real numbers
Sure then
quasi elbow
yes
quasi elbow
do not get (iv)
daring tapir
do not get (iv)
Uh it isnt even defined for a huge part of the interval
Doesnt make sense
daring tapir
what do we conclude
Ill let someone else answer that cuz im not sure
granite echo
looking at the graph of cos its not that hard to see the image set for the given range
which is [0,1]
granite echo
Uh it isnt even defined for a huge part of the interval
what do you mean?
quasi elbow
which is [0,1]
oh wait
so
the domain of cos^-1(x) is from -1 to 1
and -pi/2 is smaller than -1 and is therefore not between -1 and 1
so it is undefined
hence we ignore that part
between -pi/2 and -1
is that correct?
craggy obsidian
granite echo
between -pi/2 and -1
bro x is defined on R, y is defined on [0,1]
y =cos(x)
x=cos^-1(y)
the issue here is that x for each of y values can take infinitely many values
but if we define our question on $x \in [-\pi/2,0]$ then we can get definite answer
obsidian monolithBOT
Rootsyl
quasi elbow
how are we getting #3
quasi elbow
i get the procedure but why are we adding the angles? got it
daring tapir
what do you mean?
Arccos(-pi/2) and aint defined
granite echo
Arccos(-pi/2) and aint defined
yea
duh -pi/2 < -1
which is impossible
yep
sry
daring tapir
do not get (iv)
@winter comet can you comment on this
viscid thistle
for cos^-1?
daring tapir
Undefined
daring tapir
for cos^-1?
Yea
viscid thistle
cos^-1 is defined for 0 to pi
viscid thistle
0 to pi
daring tapir
Again
viscid thistle
0 to pi
daring tapir
What
obsidian monolithBOT
daring tapir
Just see in here
viscid thistle
im sorry
viscid thistle
yeah so cos^-1x is defined for [-1, 1] -> [0, pi]
daring tapir
yeah so cos^-1x is defined for [-1, 1] -> [0, pi]
Yea
jagged patrol
,w x
daring tapir
yeah so cos^-1x is defined for [-1, 1] -> [0, pi]
But the dumb ass que said -pi/2 to 0
viscid thistle
[-1.57079632679, 0]
daring tapir
I just wanna know what is the word i am supposed to use
daring tapir
Exactly
daring tapir
But i meam itis defined for more than half the interval
viscid thistle
oh yeah right
daring tapir
XD
daring tapir
Yeps
viscid thistle
they may not exist
viscid thistle
yes
daring tapir
But can we do so in here?
daring tapir
atleast in the problem statement
Op asked it cuz they didnt know lmaoo
Whats the point of asking them
daring tapir
phi
Its def not phi
Its either pi/2, pi
Or undefined
Def not phi
Ik you were being sarcastic
But still
viscid thistle
but we can't put -pi/2
daring tapir
-pi/2 doesnt make sense
viscid thistle
like the domain
daring tapir
like the domain
Yea it doesnt
daring tapir
Bbut ig you have to exclude
viscid thistle
not in domain means no range i think we can do exclude that
daring tapir
But on the other saying undefined is tempting af
viscid thistle
the x doesn't necessarily needs to be domain number
daring tapir
Does it not?
daring tapir
,w define domain
obsidian monolithBOT
viscid thistle
domain is where function is defined
daring tapir
Yea
tame pine
Wbat
daring tapir
domain is where function is defined
Ik i thoufht maybe i missed smthn in the definition
viscid thistle
they didn't say x is domain but its just that x is smth in a interval and see if it maps
to smth
tame pine
Yo
daring tapir
do not get (iv)
This
tame pine
Oh
viscid thistle
they didn't say x is domain but its just that x is smth in a interval and see if...
@daring tapir do u agree
daring tapir
I have the board
Its not board
viscid thistle
yes
tame pine
Ye
viscid thistle
[-pi/2, 0] is given
tame pine
Ye
viscid thistle
we can exclude the
tame pine
What
tame pine
,w arccos(-1.57)
obsidian monolithBOT
daring tapir
Or even undergrad
daring tapir
Lol
viscid thistle
range = {f(x) | x in B}
its only gonna take those which are defined @daring tapir
tame pine
Ye
viscid thistle
so we can do exclude
tame pine
Lol
viscid thistle
we went downhill
daring tapir
Are needed
XD
quasi elbow
no lmao
tame pine
Ye
quasi elbow
i am busy
with some stuff
for the next few days
rip exam
connect four?
good game
get rekt
skill issue
no
i go study now
bye
winter comet
<@764342735732867072> can you comment on this
when x is in [-pi/2, 0], uh.... even though arccos(x) isn't defined for some of the domain, the range still exists? i'd put what the range is even so :l
daring tapir
when x is in [-pi/2, 0], uh.... even though arccos(x) isn't defined for some of ...
What
"Id put what the range is even so" explain
winter comet
"Id put what the range is even so" explain
i would put the range for the interval [-1,0] is what im saying
the part of the interval that is defined
winter comet
if the entire interval was undefined id say undefined
but if only part of the interval was undefined...
quasi elbow
need help with this question
quasi elbow
but if only part of the interval was undefined...
but can we just do that?
say
if it were a normal trigonometric function
it'd still be the same?
winter comet
if it were a normal trig function it wouldn't be undefined in the domain 
quasi elbow
well tan(x) would be undefined at x=pi/2 and such but uh...
odd multiples of pi/2? or do i remember wrong
and
but what
what would be the answer
winter comet
odd multiples of pi/2? or do i remember wrong
uh its just x = pi/2 +- kpi for all integers k
winter comet
what would be the answer
[pi/2,pi] sorry i had to scroll up to see the problem LOL
winter comet
this feels wrong
i feel like you'd only consider the defined part of the domain is what im sayin tho
lol
quasi elbow
yeah i get that part
like why aren't we saying anything about the undefined part?
can we write it separately?
say it were a theoretical exam
winter comet
can we write it separately?
if you want to sure lol
I just feel like we would do this but we need an expert who actually knows LOL
it makes sense to me
thorny marlin
quasi elbow
need help with this question
.
exotic sun
viscid thistle
need help with this question
@quasi elbow
viscid thistle
need help with this question
construct a triangle and left an angle be alpha
for cosec(tan^-1 x)
like tan alpha = x
and now they are asking cosec(alpha)
viscid thistle
like tan alpha = x
since you know tan alpha you can write cosex alpha
similar way for the other part also
simplify it
you will be left with 2tan^- (smth) which can be rewritten to one of the options provided
'smth' is really sus though
leaden cave
could someone explain why this is true it may seem like a stupid question but I just got into math super late so it may be easy just I donβt get it lol
never mind Iβm just so stupid and figured out why lmao
winter comet
that wolphram alpha sucks at limits
why?
shadow ether
no clue why
shadow ether
i mean why did you say that π
i dont remember i put some limit going to infinity and it said infinity
when the answer was 3/4 or sm
winter comet
i dont remember i put some limit going to infinity and it said infinity
i've literally only seen wolfram alpha make a mistake once lol
proven comet
Could someone teach me limits
You still want to learn stuff about limits ?
I can teach you a thing or two. The basics, mostly
fading notch
anything specific you need help on?
proven comet
Can someone help me with this?
What part don't you understand ? Unless you're looking for the exact answer to your homework, in which case, I don't think you'll find much help here
zealous patio
I need help with the box part of the questions
proven comet
You mean the square part where you graph the function ?
proven comet
Well, domain, for all real functions, is R
As in, every real number
1/x or sqrt(x) functions do not apply since x is confined by a limit
Does that help ?
zealous patio
Ye
proven comet
So, in your case, number 2. : 4x^-2 doesn't have a domain of R since division by x=0 is impossible
The domain in this case would be {R/x=0}
every real number excluding x=0
Range is the equivalent for the Y-axis
So, what is the maximum value and the minimum value y reaches
In number 2., the function can never reach a negative, since x is squared, therefore always give a positive denominator
Looks a little something like this :
zealous patio
Ye
proven comet
And you may see quite clearly that is spikes rapidly toward positive infinity
Therefore, the range of said function 4x^-2 would be the interval of ]0, +infinity[
Excluding 0 since it gets near but never reaches, same goes for infinity
Last part : x-int(s) and y-int
Do you understand those ?
proven comet
ok ok
Well, for x-int(s), you're trying to find the exact values for which y=0, that is to say, at which x value does the function touch the X-axis
To find this one, it's quite simple. Simply take you function and equal it to 0.
4x^-2=0
divide both sides by 4
x^-2=0
so : 1/x^2=0
grand shard
i am like goku i have no limits
proven comet
But this case is impossible, because if you try to isolate x, you'll realize you have to divide by 0, with isn't possible, meaning there are no solutions for this y=0
1/0 = x^2
Do you get it or do you want another example ?
@zealous patio
proven comet
sure
I'll take a function that does have a solution
Let's say : f(x) = 1/2x^2 - 3
this
You want to find every value of x for which y=0
So you plug in "=0"
Then, you isolate x. Do you know how to isolate ?
zealous patio
Then times 2 each side
zealous patio
Sqr root it?
zinc compass
Heh I don't have any other resources. π¦
zealous patio
Now, if you square root, what's the solution ?
Um sqrt of 6
zinc compass
I bought a few books but they seem to difficult
zealous patio
Oh
proven comet
Because both of these answers can be squared to give 6
That's why the function is symetrical and hits the x-axis at 2 separate point :
The first point is -sqrt(6) and the second one sqrt(6)
Is that all for x-int(s) or do you need yet another example ?
if so we can move on to y-int
zealous patio
Its good
proven comet
aight
y-int is similar, just a little different
For this one, you're trying to find what the value for y is when x=0
Since the function, by definition, never has more than one y value for a single x value, this question can never have 2 answers
proven comet
If we take the same function again, now replace x with 0
Now, simply solve the equation
(it's quite simple)
zealous patio
0?
proven comet
no
zealous patio
-3
proven comet
yes
proven comet
That's why the function is symetrical and hits the x-axis at 2 separate point :
Looking at this graph, you can see the function traverses the y=axis at y=-3
If we go back all the way to the problem in your document
number 2 : 4x^-2
Replacing x with 0 would give yet again an impossible case, in with we would have to divide by zero
Therefore, the function 4x^-2 doesn't have a solution for y-int. Specifically, this particular function (4x^-2) doesn't cross any of the axis
zealous patio
Do i write none for the y int?
proven comet
For x-int(s) as well, yes
zealous patio
Ok
proven comet
As you can see, the function just doesn't touch those lines :
zealous patio
Yea
proven comet
Anyway, if you need a way to visualize what you're trying to calculate, you can try Desmos, it's what I used for those screenshots
bleak steeple
what is the equivalent of the us precalc curriculum in uk?
zealous patio
Ok
zealous patio
Anyway, if you need a way to visualize what you're trying to calculate, you can ...
Would the domain be (- inf, 0) U (0,inf) ?
proven comet
Would the domain be (- inf, 0) U (0,inf) ?
In this case, yes
I formulated it a different way ( {x E R / x = 0} )
But that's basically it
hasty jackal
4/0 has no solution
proven comet
Btw, for inf and 0, you need to exclude those
proven comet
4/0 has no solution
ik...
proven comet
4/0 has no solution
Did you read what I wrote or just looked at the picture ?
zealous patio
Btw, for inf and 0, you need to exclude those
?
hasty lily
I formulated it a different way ( {x E R / x = 0} )
What the fuck is that notation
proven comet
What the fuck is that notation
gimmie a moment
hasty lily
Just say $\mathbb{R} - {0}$ like a normal person
proven comet
?
Since x can only approach 0 or approach inf and never actually be equal to those, you have to exclude those out of the interval
obsidian monolithBOT
Catgod
proven comet
Like this :
]-inf, 0[ U ]0, +inf[
zealous patio
?
proven comet
Just say $\mathbb{R} - \{0\}$ like a normal person
That doesn't refer it to x
zealous patio
Outside brackets
hasty lily
That doesn't refer it to x
Itβs implied
hasty lily
$x \in \mathbb{R} - {0}$ like a normal person
obsidian monolithBOT
Catgod
hasty lily
You're complicating things
How come?
proven comet
@zealous patio ?
zealous patio
Im trying to find it
proven comet
oh ok
zealous patio
Like this
proven comet
Ok, so this notation means :
for every value ( { } ) of x ( x ) part of ( | ) the interval (x smaller or equal 2)
I described the meaning of every symbol
proven comet
I don't exactly know how your teacher expects you to notate it, but I would write the domain for 4x^-2 like this :
{ x | x β 0 }
So every value possible, except 0
zealous patio
Ok thats what i wrote before but
zealous patio
Ye i thought it was wrong
proven comet
No. It's just the specific way you wrote it in Discord isn't right
If you really wrote (-inf, 0) U (0, inf) in your paper, you'd imply that x can be 0
Unless you use outside brackets
zealous patio
Oh ok
proven comet
For the range, I'd write it like this :
{ y | y > 0 }
Since it's always positive and never reaches 0
But you can also say :
] 0, inf [
like that, specifically
zealous patio
Is the others like increasing and decreasing all right?
proven comet
I mean, as I was saying, since you used parentheses, it would be incorrect, but yeah, the interval in itself is correct
-inf to 0
zealous patio
I feel like increasing is worong
proven comet
0 to inf
proven comet
I feel like increasing is worong
No, it's fine
zealous patio
Even the symmetry and discont and end behaviors?
zealous patio
Ok
proven comet
Look man, I'd really like to go on to support you if you have more questions, but it's really late and I have College super early tomorrow
I kinda pushed it far by staying awake 'till now, but I really need to sleep
zealous patio
Oh ok im so sorry to make you stay late ty for the help
proven comet
Nah, I did this without being forced, it's my problem
Anyway, don't hesite if you have more question and see me online to tag me
Have a good one
π
proven comet
Ciao
hasty lily
,rotate
obsidian monolithBOT
hasty lily
Is number 3 correct?
Looks correct
zealous patio
Idk how to do symmetry
hasty lily
Thereβs no axis of symmetry so dw
zealous patio
So what do i write
hasty lily
So what do i write
βNo axis of symmetryβ
zealous patio
So none
hasty lily
Yeah
hasty lily
Is number 1 correct
Looks correct
zealous patio
Ty
hasty lily
Idk how to do symmetry
But No.3 is also an odd function
hasty lily
What symmetry would 4 be
I donβt think thereβs any symmetry
hasty lily
Yeah, no symmetry
zealous patio
Discontinuities would be x=0
quasi elbow
construct a triangle and left an angle be alpha
do not get it
what after tan(alpha)=x?
what do we do with the triangle
since
we do not know any of the values
do you mean
we write cosec(alpha) in terms of cos?
dry rune
Yo, happy halloween guys!
So, I just wanna know the very basics of calculus (I am really interested in maths XD)
Can anyone help? (I read in class 8 BTW. I know kinda younge but I wanna know so bad)
quasi elbow
ah, well, fair
quasi elbow
and now they are asking cosec(alpha)
if we do this
can we apply tan^-1(a)-tan^-1(b) somehow
viscid thistle
Yep!
quasi elbow
i did not get the 1 part earlier but yeah it makes now
alright sp
so*
a minute
i am getting
2tan^-1[(root(1+x^2)-1)/x]
is that correct?
viscid thistle
Yeah
quasi elbow
how are we supposed to think of all of this? like i feel there's certain methods to solve certain problems but what is the thought process
viscid thistle
how are we supposed to think of all of this? like i feel there's certain methods...
Since we have trig(inv trig(x)) we are using this process(thinking process)
viscid thistle
2tan^-1[(root(1+x^2)-1)/x]
Now assume this to be alpha and apply tan on both sides
Simplify tan(2theta)
Im getting tan inverse x
You should try it
I feel like there's a shorter method
daring tapir
,w e
obsidian monolithBOT
quasi elbow
Now assume this to be alpha and apply tan on both sides
the whole thing?
viscid thistle
Yeah
quasi elbow
don't you think that
viscid thistle
alpha=2taninv(...)
quasi elbow
this is too complicated
quasi elbow
Im getting tan inverse x
this is correct
viscid thistle
are we trying to apply the concept of tan[tan^-1(x)]?
Yeah
viscid thistle
I will ping you if I figure out a simpler method
viscid thistle
How to get the idea?
quasi elbow
if it were some other time, i would not be able to think of the substitution thing by myself
quasi elbow
How to get the idea?
yeah
viscid thistle
Since we have trig(inv trig(x)) we are using this process(thinking process)
.
You seea trig(invtrig(x))
quasi elbow
i mean yes but
viscid thistle
Yeah?
quasi elbow
do we generally use tan(x) in such kind of questions?
viscid thistle
No we could also use cot sin cos
But the thing is
In most cases we have tanx to be smth like x or 1/x
Where are when we use sin
quasi elbow
But the thing is
In most cases we have tanx to be smth like x or 1/x
why?
viscid thistle
We get x/ sqrt(1+x^2) smth like that
viscid thistle
I didnt get your question completely
quasi elbow
never mind
zealous patio
Can someone help me find the increasing and decreasing intervals
<@&286206848099549185>
slim oak
zealous patio
Ok
late carbon
You still want to learn stuff about limits ?
Yes please
proven comet
Ok, so for the basics, limits are a way to approximate a value for a function when x approaches an impossible value
Example, when you take a 1/x function, you can never reach x=0, 'cause that would divide by zero
But, using a limiting process, we can determine that lim (x -> 0), y approaches infinity
For that, you substitue a very small value for x, therefore closing into 0, like 1*10^-99
Does that help ?
Or is that a little too vague ?
proven comet
Yes please
?
late carbon
Is there a way you could dumb it down a bit more
proven comet
I'll reformulate what I just said
Ok. Let's imagine you need to find the result for the function : y = 1/x
when x is equal to 0
I suppose you know there is no way to actually put x = 0 in your calculator, because you would need to write 1/0, which is impossible
Right ?
So, the limiting process would be used to approximate a value for y
because you can't actually calculate it
Since you can't use 0 as a value to approximate it, you need to use a value that is close to it, such as
0.000000000000000000000000000000001
Do you understand that ?
late carbon
Yeah
proven comet
So, the limit as x approaches a value of 0 would be written like this :
lim (x -> 0)
Using a very small value for x would allow us to determine that y grows infinitely as x gets smaller
Therefore, we can write it like this :
lim (x -> 0) for (y = 1/x) is "inf"
"inf" meaning infinity
Is this more clear ?
late carbon
Yeah
proven comet
Aight, if you need more info about limits, I can help you, just a little later, I'm actually in class right now
If you don't mind waiting... maybe an hour ?
Is that okay with you ?
proven comet
@late carbon I'm back
quasi elbow
hi back
proven comet
:|
I knew that's what you were gonna say
And I was thinking : nah, he wouldn't.... But I knew you would
quasi elbow
no worries!
proven comet
Altho, I suggest you actually do so, 'cause that would make it easier for people to not get it wrong ^-^
quasi elbow
fair enough
proven comet
@late carbon
viscid thistle
I don't want to be that person but
Hey fudge! Do you care about pronouns?
late carbon
<@1045887118926418061>
Back
proven comet
Back
Nice
So, for limits, the other way they are most likely to be used is for series, usually infinite ones
Do you know anything about summations or products ?
Ξ£ & Ξ
late carbon
Uh I donβt think Iβm not even in hs yet
proven comet
Yeah, so that part might get a lot harder to understand for you
I can condense it for simpler terms
So, let's say you want to add multiple terms in a series
Like, add all the numbers from 1 to 10 (only integers)
You can write it like : 1 + 2 + 3 + ... + 10. But that's boring and annoying to write down
So, for simplification, you can notate it using the Capital Sigma letter from greek
Ξ£
This
You got it ?
late carbon
Yeah
proven comet
Alright, for starters, I'll do a summation of the integers from 1 to 3, since it's easy to calculate by hand
1 + 2 + 3 = 6
simple enough
Now, to write it as a summation, you formulate it like this
Sigma (summation of intergers) starting at n = 1 up to 3 of the series "n"
Mbd, kinda difficult to understand written like this
So this part, is the repetitive value. It's similar to a function in which you plot multiple values of n :
This one would simply be y = x if it were a function
Then, you have the starting value of n, at the bottom :
It's usually one, but it can be anything you want it to
Now, for the last part of the notation, you have the end value of n, sitting at the top :
So, to read this sum, you would first look at the function of n, in this case " n "
Then, look at the starting value, in this case 1
And the ending value up to which you add the integers starting from 1, in this case 3
Got it ?
late carbon
Iβm reading over and writing notes down 1 sec
proven comet
Iβm reading over and writing notes down 1 sec
You done ?
late carbon
Yeah just finished
proven comet
Aight. So just to make sure you got all that, just a quick exercise : calculate the value of this sum :
It's only three terms, you should be able to write it down
proven comet
Close
late carbon
31
proven comet
It's actually 24
late carbon
Oh
proven comet
So, the way to write it down is :
2(n=3) + 2(n=4) + 2(n=5)
so 2(3) + 2(4) + 2(5)
6 + 8 + 10, 24
You get it ?
late carbon
Yeah
proven comet
Aight
So, now, Products
It's pretty simple once you've understoof sums :
It's just a different symbol
With a product, instead of adding each term, you multiply them
So
(n=1) * (n=2) * (n=3)
1 * 2 * 3
6
Got it ?
late carbon
Yea
proven comet
Aight, so if I ask you to calculate this one :
As a fraction
I don't want the ugly decimal result
proven comet
So
(n=1) * (n=2) * (n=3)
Try to use the same pattern I used up there
Whats the function of n ?
late carbon
2 but you said it increases up to 4 right
proven comet
2 but you said it increases up to 4 right
That's the starting value
What's the function of n ? Where do you plug in your values of "n" ?
proven comet
So this part, is the repetitive value. It's similar to a function in which you p...
^this^
proven comet
Aight, so if I ask you to calculate this one :
So, can you tell me what the function of n is for this product ?
late carbon
1?
late carbon
If n is 1 than 1 divided by n is 1
proven comet
Yeah, but that's not the question when I ask : What's the function of n
Not saying you're wrong, but that's not the expected answer for that question
late carbon
What was the answer
proven comet
1/n
That's the function
The part to the side of the symbol is the function of n. The part where you plot the values of n from it's starting value to it's end value :
Is it easier to understand that way ?
@late carbon ?
quasi elbow
need help with the question
i tried simplifying sec(x) and tan(x)
but do not know how to proceed further
daring tapir
need help with the question
Wont you need to integrate that
But yea try the half angle identities
For sine and cosine
The result for this question in particular is really pretty
quasi elbow
okay
i am getting
f'(x)= tan^-1[(sinΒ²(x/2)+cosΒ²(x/2)+2sin(x/2)cos(x/2)]/[cosΒ²(x/2)-sinΒ²(x/2)]
daring tapir
f'(x)= tan^-1[(sinΒ²(x/2)+cosΒ²(x/2)+2sin(x/2)cos(x/2)]/[cosΒ²(x/2)-sinΒ²(x/2)]
Factorise
daring tapir
sorry, i am not good with the latex thing
This was readable so eh
daring tapir
but is it correct?
Yea
quasi elbow
[cos(x/2)+sin(x/2)]Β²?
daring tapir
sorry, i am not good with the latex thing
Also all you really need to know in latex is the frac func
So just like this
$$ \frac{a}{b}$$
obsidian monolithBOT
Ender Doesn't Mind
daring tapir
Youll figure out the rest along the way
quasi elbow
got you
daring tapir
[cos(x/2)+sin(x/2)]Β²?
For the numerator yes
quasi elbow
yeah
oh
wait
not sure if this is correct but
isn't there an identity for cosΒ²x-sinΒ²x
daring tapir
isn't there an identity for cosΒ²x-sinΒ²x
Uh there is sure..... but dont use that
Just use a^2-b^2
quasi elbow
but how do we proceed from there
quasi elbow
f'(x)= tan^-1{[1+tan(x/2)]/[1-tan(x/2)]}?
quasi elbow
isn't there an identity for cosΒ²x-sinΒ²x
what is it?
quasi elbow
Now think of some way to convert that whole thing into one tangent function
right
daring tapir
what is it?
Cos(2x)
quasi elbow
oh yes
do we
multiply by its conjugate or something
even if it's not in root
is that even valid
daring tapir
Nope.
quasi elbow
i think you're asking me to somehow apply the formula of either tan(a+b) or tan(a-b) by replacing 1 by tan(pi/4) but I do not see it
or I am totally wrong
daring tapir
i think you're asking me to somehow apply the formula of either tan(a+b) or tan(...
Yep
(Tan(a+b) and tan(a-b) are the same formula anyways lol)
quasi elbow
isn't that tan(a)+tan(b)/1-tan(a)tan(b)
daring tapir
isn't that tan(a)+tan(b)/1-tan(a)tan(b)
Ye
quasi elbow
bro
daring tapir
Sub a = pi/4
daring tapir
Hmmmm
quasi elbow
how do we solve inequalities in inverse trigonometric functions?
daring tapir
how do we solve inequalities in inverse trigonometric functions?
Example? But very loosely just graph them
leaden cave
so I have a random question but how do I like say this problem out
winter comet
so I have a random question but how do I like say this problem out
forgot the dx 
leaden cave
yeah ik I left it out on accident
winter comet
yeah ik I left it out on accident
oh lol
winter comet
so I have a random question but how do I like say this problem out
you mean like say it out loud?
leaden cave
ohhh okay thank you very much
near ravine
guys is stuff like logarithms and exponential functions (like solve for x where 53^(2x+1)-27^(x+3)-12=8) pre calc or algebra 2
it italicized part of that random example prob i gave cuz of the asterisks
also idk if the problem i sent has a solution i just said something random
viscid thistle
guys is stuff like logarithms and exponential functions (like solve for x where ...
It's mainly considered Pre-Calc but briefly introduced in Alg 2 to get a feeling for exponential growth and decay
jagged patrol
guys is stuff like logarithms and exponential functions (like solve for x where ...
alg 2
u dont do much with it until precalc tho
near ravine
alg 2
so lu
=ike =
whoops
i pressed enter instead of backspace
so like ln and solving stuff like this (image) is alg 2
(ive already solved these 2 problems)
vapid nova
so like ln and solving stuff like this (image) is alg 2
average alg 1 class
jagged patrol
so like ln and solving stuff like this (image) is alg 2
thats probably precalc since it isnt log10 or ln
but im sure a savvy alg 2 student could do it
not toooo difficult
uneven nexus
Is it true that a system of equations for a circle only contains the square terms of x and y, no terms like xy etc?
daring tapir
Is it true that a system of equations for a circle only contains the square term...
Yea
river drift
[ (x-3)^2 + (y+2)^2 = 4 ] is the equation for a circle, but so is: [ x^3 - 6x + y^2 + 4x + 9= 0 ] (because they're the same equation)
obsidian monolithBOT
cloud
uneven nexus
Ooh so not x^2 y^2, but square terms
uneven nexus
how to plot |x|+|y| β€ 2 quickly
river drift
split into cases by quadrant
viscid thistle
split into cases by quadrant
this^
uneven nexus
there's no other quick way?
viscid thistle
how quick do you consider βmanhattan distance from the origin is β€ 2, therefore itβs a diamond with corners at (0, 2) (2, 0) (0, -2) (-2, 0)β
uneven nexus
this sounds like some metric space stuff
echo sinew
Or you can do it by remembering the plot of y = |x|
viscid thistle
this sounds like some metric space stuff
i have absolutely zero idea what that is
echo sinew
Or you can do it by remembering the plot of y = |x|
So that you only have to graph (in the region -2β€xβ€2) the inequality:
-2 + |x| β€ y β€ 2 - |x|
dire wharf
Pls anyone to help me understand the inequality of cauchy schwarz ?
viscid thistle
what about it?
arctic meteor
$$\prod_{k=1}^{n} \frac{3k^2+5k+2}{3k^2+5k-2}$$
obsidian monolithBOT
wolly5114
arctic meteor
How do you solve a product series?
dire wharf
Factorise
daring tapir
,w 3k^2 + 5k-2
obsidian monolithBOT
soft cloak
GUYS
CAN SOMEONE EXPLAIN THIS
the red line is me and I don't know what I doing wrong
broken pecan
What is the original polynomial?
willow skiff
the red line is me and I don't know what I doing wrong
it seems the only thing that is wrong is the 5/36
try substituting the point in again
cause it's a problem with your arithmetic
uneven nexus
hmm any ideas
winter comet
hmm any ideas
whats k?
uneven nexus
integers>0
winter comet
then its infinity
lim x->kpi x/tan(x)
by direct substitution, this is like kpi / tan(kpi)
since tan(x) = sin(x)/cos(x),
tan(kpi) = sin(kpi)/cos(kpi)
sin(x) is 0 for pi, 2pi, 3pi, 4pi, ect.... basically, kpi
cos(kpi) is never 0 for kpi, only for pi/2 + kpi
so therefore tan(kpi) = 0 for any k, so ur gonna get lim x->kpi x/tan(x) = kpi/0 which blows up to infinity for every integer k>0
i know putting equals is a bit of a stretch but yk what i mean LOL
uneven nexus
aah yesyes ty
proven comet
Guys, I need help finding the zero for this function :
river drift
- -2k with k natural
- 1/2 + i*r, with r denoting the imaginary part of the nontrivial roots of the riemann zeta function
- special bonus root (secret)
winter comet
(does this secret happen to be on Wikipedia)
proven comet
1. -2k with k natural
2. 1/2 + i*r, with r denoting the imaginary part of the no...
I need a value of s, my fault for the wrong formulation of my question
A concrete one, that is
river drift
-2 then
fading monolith
Pls dont troll
proven comet
I mean, I did a science fair project with one of my buddies on the Zeta function, we got stuck near a value. Not a zero, obviously, else we'd be famous
But, yeah. I'm really interested in other's thoughts
Obviously, I'm trolling, this is not the right server for this, my questionning is way beyong #precalculus
river drift
proven comet
Close enough... But I was really interested in your personnal thoughts
@river drift
Anyway, gtg
vagrant nymph
what are the symbols?
silent prism
Guys, I need help finding the zero for this function :
Some pretty cool precalc work
viscid thistle
what are the symbols?
It represents factorial,
The exclamation mark(!) is an alternative symbol.
n! = 1.2.3.4 .... n
tall falcon
It represents factorial,
The exclamation mark(!) is an alternative symbol.
n! = ...
Woah cool
viscid thistle
Why do I feel like your reply is sarcastic π€
winter comet
It represents factorial,
The exclamation mark(!) is an alternative symbol.
n! = ...
same with gamma but you have to add 1 :P
viscid thistle
Nvm
tame pike
How do I graph the inverse?
tame pike
Iβm assuming I change its order
jagged patrol
Iβm assuming I change its order
f(x) is the inverse of x(f)
so yes
swap outputs and inputs for inverses
notation is f^-1
viscid thistle
x(f)?
tender questBOT
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
yes
soft cloak
π can u help plz
viscid thistle
you could start by finding its real root
jagged patrol
π can u help plz
use difference of cubes
a^3 - b^3 = (a-b)(a^2+ab+b^2)
do that then come back if you need more help
rough tide
Somebody correct me if I'm wrong because I cannot for the life of me find a direct answer to this online.
If I had a function, say f(x) = 2sin(2x), and plotted it, then I took another function r = 2sin(2Ξ) and plotted that on a polar graph, it seems to me that no matter what the function is so long as the only difference between the two is that x is swapped with Ξ and r is swapped with y, that for any point on f(x), (x,y), there exists a point on the polar variant by which the value of x and y are equal to theta and r respectively. For example, on f(x) there is a point (2pi,3), therefore on the polar graph there must be (3,2pi).
Am I wrong or is that like a no-brainer that I am heavily overanalyzing?!
summer ruin
there is a way to go from one to another, but it's not as simple as swapping the coordinates
also there is a problem in that in polar coordinates (3, 2pi) and (3, 4pi) are the same point
rough tide
I mean if it's limited from 0 to 2pi
unborn brook
This means rational functions are not always polynomial. So the intermediate value theorem is not necessarily true for rational functions, right?
viscid thistle
yes, for example, there is no value of x between -1 and 1 for which 1/x = 0
soft cloak
I have tried putting the roots. I don't know what you call it, but I put those in and whatever, but the equation is not lining up any tips
i have to make it like the blue line
viscid thistle
what do you mean by "putting the roots?"
obsidian monolithBOT
Nanoeo
fading monolith
There are some integrals with no antiderivative in terms of elementary functions
Like sin(x)/x
willow skiff
guyss, why this can't be integrated ??
you should ask yourself why the functions you've seen can be integrated in closed form
there are only so many names for infinitely many functions
with integration we only have the reverse chain rule (u-sub) and reverse product rule (by parts)
it's not like differentiation where we are pretty much guaranteed to come up with something in closed form
viscid thistle
(x-1)(x+1)(x-3)
check the y-intercept
you may need to multiply that by a coefficient
plucky fable
do any one of understand this
roots of unity are the BEST
and the roots form an n-gon!!
plucky fable
arctic meteor
$$\prod_{k=1}^{n} \frac{135*...(2k-1)}{246...*(2k+2)}$$
obsidian monolithBOT
wolly5114
arctic meteor
for the denominator i know what to do but what about the numerator?
is there a formula
?
winter comet
for the denominator i know what to do but what about the numerator?
what u do about the denominator?
pretty sure theres a formula for both
arctic meteor
@winter comet You write the denominator as $$2*(123*...*(k+1))$$
obsidian monolithBOT
wolly5114
arctic meteor
winter comet
<@764342735732867072> You write the denominator as $$2*(1*2*3*...*(k+1))$$
2^k * (1 * 2 * 3 * ... * (k+1)) u mean?
arctic meteor
This is for the numerator but I have no idea how these 2k terms come from!
arctic meteor
2^k * (1 * 2 * 3 * ... * (k+1)) u mean?
yes
winter comet
oh
winter comet
This is for the numerator but I have no idea how these 2k terms come from!
the product of (2k-1) is all the odd factors such as 1, 3, 5, 7, ect.
the product of (2k) is all the even factors such as 2, 4, 6, 8, ect.
so together, they make all the terms 1, 2, 3, 4, 5, 6, 7, 8, ect.
which can be represented the product from k=1 to 2n of k
so if you multiply and divide by these products, you get two products which you know their uh...partial sums but in product? LOL idk what its called
safe basin
so if you multiply and divide by these products, you get two products which you ...
This is gonna be crazy you ready
Drumroll please
π₯π₯π₯π₯π₯
Partial product
safe basin
Fr
winter comet
Why does x/(x^2) have an asymptote at x=0 while x^2/x has a hole at x=0?
think about what happens when x!=0
charred stirrup
Yo guys is 3x +1 actually possible to solve
river drift
solve for what? that's just an expression