there's no need to introduce x
#help-36
final tangle
start with
$$\sqrt{33-18\sqrt{2}} = a\sqrt{3} = b\sqrt{6}$$
then square both sides
$$\underbrace{(\sqrt{33-18\sqrt{2}})^2}{33- 18\sqrt{2}} = \underbrace{( a\sqrt{3} = b\sqrt{6})^2}{\text{expand this}}$$
soft zealotBOT
ℝαμΩℕωⅤ
final tangle
no need to introduce x at all
craggy bane
no need to introduce x at all
i get 3a^2 + 6b^2 + 6ab sqrt 2
final tangle
whoops just realised i typoed a = instead of a +
craggy bane
whoops just realised i typoed a = instead of a +
yea i notice that
final tangle
then equate the rational and irrational components
final tangle
not quite
the irrational part on the left side is
**-**18sqrt(2)
not just 18sqrt(2)
due to how squaring works, the work on the left side isn't really affected
final tangle
yes
final tangle
that seems way too high
craggy bane
that seems way too high
final tangle
,rotate
soft zealotBOT
final tangle
that won't lead to a being 81 or 4
craggy bane
ouh
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delicate fiber
\
delicate fiber
how they get 2root3
delicate fiber
blud its a cubic
agile coral
No, you find speed with derivative
agile coral
Then you plug in 2 sqrt3
tiny gorge
he's asking how they determined t = 2sqrt(3) to begin with
agile coral
i think that's the solution in the first screenshot
O
delicate fiber
b
agile coral
Yeah, then quadratic
shadow tinsel
You dont equate the displacement to 20
proud plover
You dont equate the displacement to 20
u literally do
solution to 2x^3-24x = 0
is 2√3
one of the
solutions
delicate fiber
the +20 already encodes the displacement
No it doesnt
shadow tinsel
No it doesnt
it does
tiny gorge
plug in t = 0, you get x = 20, which is the initial displacement
the solution is wrong though:
if you plug in t = 2sqrt(3) you get x = 20, not x = 0
delicate fiber
if you plug in t = 2sqrt(3) you get x = 20, not x = 0
Yea thats because the initial position is x = 20, not x = 0
tiny gorge
Yea thats because the initial position is x = 20, not x = 0
oh wait "initial position"
yea they should have set that equal to 20, not 0
it's wrong either way haha
that at least makes the problem tractable
delicate fiber
so 2root 3 is correct
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faint locust
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silk ibex
hey there's an integration which am trying to do
final saddleBOT
hey there's an integration which am trying to do
fathom walrus
send
silk ibex
hey
can anyone help me solve this integration
solar glade
tranquil pine
@silk ibex Hi Hayao Miyazaki
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rich forum
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craggy bane
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craggy bane
Can someone tell what to do next after found a and b ?
I kinda confused it ain't work for √a - √b to become a√3 + b√6
final tangle
continue from
p=9 or p=2
as mentioned yesterday the two lines you did after are incorrect
craggy bane
what do you recommed then ?
final tangle
how did you get a= 81 or 4 from
p=9 or p=2
craggy bane
how did you get a= 81 or 4 from
> p=9 or p=2
b = -3/a
craggy bane
solving higher degree eq ?
final tangle
tell me the exact calculations you did and the reason you did them that got you from
p = 9 or p= 2
to
a = 81 or a=4
where did 81 come from
where did 4 come from
craggy bane
where did 81 come from
first a^2 then 9^2 = 81 same as 2^2 = 4
craggy bane
yea i squared
final tangle
but why
craggy bane
to solve for a ?
final tangle
note that in the first like
p = a^2
and if p = 9
that gets you the equation
a^2 = 9
craggy bane
oh wait i get wym
final tangle
to get the possible values of a
that's what you should be solving
craggy bane
to get the possible values of `a`
that's what you should be solving
should be 3 and -1 instead
final tangle
where -1 coming from
craggy bane
a = 3 , b = -3/a then b = -3/3 = -1
final tangle
oh, ok
you should be clear in what you numbers represent
since you replied to a message involving a only
that implied that you got a=-1
,w (3sqrt(3)-sqrt(6))^2
final tangle
looks good
craggy bane
owhhh
final tangle
to be more rigorous, you should also ensure that your result
3sqrt(3)-sqrt(6)
is actually positive
you may need to consider the other case as well if its negative
as the solutions to a^2 = 9 are a=3 or a=-3
craggy bane
to be more rigorous, you should also ensure that your result
3sqrt(3)-sqrt(6)
is...
is that acceptable when sqrt(-3) ??
craggy bane
where's that coming from
nah im just curious that how come sqrt (-3) too, since you have mention that other case where a = -3 or 3
final tangle
wdym how come sqrt(-3) too
there is no sqrt(-3) anywhere here
where did taht come from
is this part of a different question?
craggy bane
is this part of a different question?
nvm prb i still need to do more practice
let me more familiar with this topics
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violet mountain
can someone help me understand the part i circled and arrowed?
final saddleBOT
can someone help me understand the part i circled and arrowed?
autumn oracle
Meth
tulip coyote
Because |a| = a when a >= 0, and -a when a <= 0, they're splitting it up so they can make use of those
autumn oracle
can someone help me understand the part i circled and arrowed?
Yk the definition of |x|?
violet mountain
Yk the definition of |x|?
yes but i dont understand why its a neccessary step
autumn oracle
yes but i dont understand why its a neccessary step
So this is integration
Which is the area under the curve
But |x| is a piecewise function
So you integrate one part of the function in its domain, and then the other in its domain
|x| = x when x >= 0
So for the case where the inside is greater than 0, you integrate the inside
|x| = -x for x < 0
so you integrate -ve of the inside when it is less than 0
Then you add the areas
violet mountain
ok so its less about something specific with absolute value and more about adding both areas together?
autumn oracle
Yes
violet mountain
gotcha
autumn oracle
Since its a piecewise function, you have to break up the integral
violet mountain
i get where 1 and 4 for the integral come from but where is the 3 from or why do we have it
autumn oracle
i get where 1 and 4 for the integral come from but where is the 3 from or why do...
We have to split up the function based on the values of x
violet mountain
so if it was |t -5| the number would just be 5 instead?
autumn oracle
The function $f(t) = t-3$ is less than 0 when t < 3, and greater when t >= 0
soft zealotBOT
math X meth ✓
autumn oracle
so if it was |t -5| the number would just be 5 instead?
Kinda
It would depend on the limits of integration
But yes
violet mountain
violet mountain
i should probably know this but i forgot, do we use b or a in the integral to check if the result of |x| x, x>= 0 or |x| = -x, x < 0
old hornet
How can we show 2^(n-1) + 2^(n-2) + ... + 4 + 2 + 1 <= 2^n + n by induction?
Base n=1 is good, and we assume the above, but now what? Inequality is less familiar to me.
tulip coyote
You'd then have ${\color{green} 1 + 2 + 4 + \ldots + 2^{n - 1} } + 2^n$ and want to show somehow that this is at most $2^{n + 1} + (n + 1)$
soft zealotBOT
@tulip coyote
old hornet
You'd then have ${\color{green} 1 + 2 + 4 + \ldots + 2^{n - 1} } + 2^n$ and want...
How can we replace it when we have an inequality? I only know how to replace it when we have an equality, not an inequality.
tulip coyote
How can we replace it when we have an inequality? I only know how to replace it ...
Similar to equalities: you know e.g. that if you had c <= d, that c + p <= d + p (similar for reversed or strict inequalities)
old hornet
Similar to equalities: you know e.g. that if you had c <= d, that c + p <= d + p...
Thoughts on my proof?
<= 2^k + 2^k + k
<= 2^(k+1) + k
<= 2^(k+1) + (k+1) ∎
tulip coyote
Thoughts on my proof?
<= 2^k + 2^k + k
<= 2^(k+1) + k
<= 2^(k+1) + (k+1) ∎
that's it, yep! the last line basically using that 0 < 1 after all haha
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real ridge
1+1
vivid walrus
<@&268886789983436800>
sturdy ocean
.close
tranquil pine
AD= BC
BD =CD
Angle DBC = ANGLE DCB = 20°
find x
final saddleBOT
https://media.discordapp.net/attachments/1230158901555757149/1230177725772206214...
tranquil pine
I have solved it a bit
but im now confused
i need to find angle x
but i havent found angle dac
loud sundial
My first instinct is to place triangle BDC over triangle ADC such that BC coincides with AD and BD coincides with DC
alternatively you can trig bash
🤷♂️
tranquil pine
can you teach me both ?
loud sundial
My first instinct is to place triangle BDC over triangle ADC such that BC coinci...
well there isn't rlly any specific concept to teach here
it's just an idea
loud sundial
alternatively you can trig bash
let BD=DC=1 without loss of generality, then you can get BC using basic trig
then just bash the living daylights out of the law of sines
tranquil pine
if we place the triangle bdc such that ad is common line
How do we find the angles in red ?
fallen jasper
you can use external angle theorem for triangle adc
tranquil pine
you can use external angle theorem for triangle adc
that would give me angle dac + x = 40
fallen jasper
that would give me angle dac + x = 40
yea and you can find angle dac using sum of angles in triangle adc
you will find angle adc in terms of x and then equate it in that equation
tranquil pine
oh ok
tranquil pine
we can apply alternative angle propert y
since
now since bb' || bc
@loud sundial @fallen jasper
someone did the question. can you confirm if its correct
https://cdn.discordapp.com/attachments/1230158901555757149/1230551105461817446/rn_image_picker_lib_temp_ec2aa4d1-49df-447b-b51c-17c6ed0bf877-2.jpg?ex=6633bb06&is=66214606&hm=18619de4d804d16d74cc4388578e1d6feeaa5d27c4ef0a8c821c9d73684d3d71&
i dont understand how they got 60 and 80 degrees
loud sundial
<@560122149126275093> <@917727061890727936>
someone did the question. can you ...
yeah that's fine
the 60 degree angle in the left triangle comes from the fact you have two equal sides (B'C and B'D')
and a 60 deg angle in between
angle B'D'C' = angle BB'C = 140 deg
angle B'D'C = 60
so CDC' = 80
tranquil pine
equilaterial triangle property ?
loud sundial
equilaterial triangle property ?
yeah that triangle is equilateral
tranquil pine
yeah that triangle is equilateral
but how are we so sure that D'C =DC =DD'
we just constructed two a triangle and joined the points together
maybe D'C is line bisector , thats why ?
loud sundial
<@560122149126275093> <@917727061890727936>
someone did the question. can you ...
These two sides are clearly equal
The third side wouldn’t necessarily be equal to those two
But the fact that there’s a 60 degree angle in between makes that the case
B/c the other two angles:
- are equal
- add to 120 deg
loud sundial
B/c the other two angles:
- are equal
- add to 120 deg
Meaning they’re both 60 deg -> triangle is equilateral
tranquil pine
what about 80 ?
loud sundial
angle B'D'C' = angle BB'C = 140 deg
.
loud sundial
<@560122149126275093> <@917727061890727936>
someone did the question. can you ...
These two are both 140
tranquil pine
which property tells us that ?
loud sundial
which property tells us that ?
You copied the same triangle over…
So the angles will be the same
tranquil pine
thanks @loud sundial
understood it
i have one more question
should i send the ss
loud sundial
Yeah
tranquil pine
^translation: O centered half circle, AOBC is a square. What is the angle DEA
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tranquil pine
.close
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arctic forge
hello. How come that exponential functions can be inverted into logarithmic functions? their range is not equal to all real numbers, so it sdhouldnt be inversible. right?
final saddleBOT
hello. How come that exponential functions can be inverted into logarithmic func...
sleek quartz
did you mean domain? the domain of $\sqrt x$ is also not that of $x^2$
soft zealotBOT
b
arctic forge
did you mean domain? the domain of $\sqrt x$ is also not that of $x^2$
no, I mean the range
the range of an exponential functions is restricted
sleek quartz
oh mb lol
sleek quartz
the range is (0, infinity)
warm ether
you mean if y=2^x and then the inverse is y=log_2 (x)
the range of 2^x is all the positives
thats also the domain of logs, no issue
sleek quartz
and the domain of ln is (0, infinity)
arctic forge
in order for a function to be resversible
one of the conditions is that the range must be all real numbers
in spanish the name is
sobreyectiva
warm ether
thats not a condition for something to be reversible, it just has to be a one to one function
arctic forge
what ive been taught is this
"To make a function reversible, it must be injective and surjective. In other words, it must be bijective."
arctic forge
exponential functions arent surjectibe
trail hearth
The exponential function is not invertible as a function from R to R, but it is invertible as a function from R to (0,infty)
Under the definition you gave
arctic forge
The exponential function is not invertible as a function from R to R, but it is ...
okay getting somewhere
how can I restrict a function to reverse it
it seems arbitrary
what are the steps
does it also mean that I must limit the range of a logarithmic function?
versed crater
You want to restrict it so that it becomes surjective on the domain
arctic forge
wow so many helpers
versed crater
And monotonic as well
midnight sparrow
Exponential is bijective under certain conditions
trail hearth
does it also mean that I must limit the range of a logarithmic function?
You don't need to since the logarithmic function has range equal to R
versed crater
You could if you wanted to
sleek quartz
it seems arbitrary
because it is
like sqrt(x) earlier, i think it makes sense to say that we always pick the positive solution
but ultimately it is arbitrary
arctic forge
what are the steps
how do I restrict the range
lets say that I am asked to find the inverse of the following function:\
f(x)=(5 to the power of x+3) - 2
how would you do it
sleek quartz
that's just $x=\log_{5}\left(y+2\right)-3$
mathematicians have already decided that the range on ln should be restricted to (0, infinity)
soft zealotBOT
b
sleek quartz
it's not really something you decide yourself
midnight sparrow
It's bijective on ]-2, +inf[ that's your domain when you Can have an inverse
sleek quartz
^
i forgot to say that because you use ln(y+2), you should note that the range is now (-2, infinity)
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raven obsidian
brain fart, i cant find numbers for 245
final saddleBOT
brain fart, i cant find numbers for 245
raven obsidian
i know 9 works as 2 log base7 3
but what terms include 3 and 5 to get 245 ;-;
this is what i tried the first time and realized i couldnt use 7
tulip coyote
this is what i tried the first time and realized i couldnt use 7
I think they want you to evaluate log7(7)...
abstract bramble
is it not like this previous problem
the base is the same as its argument
which means...?
raven obsidian
oh.
raven obsidian
which means...?
would just equal the 2 if im interpreting what you're saying
how tf did that work
raven obsidian
is it not like this previous problem
guessing it wouldve worked for this as well with 2 log base 2
welp now i know. thanks
.close
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raven obsidian
.reopen
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✅
raven obsidian
what
ln 1 -> ln e 1 = 0, e^0 = 1
sqrt e = blah blah
division of 0 by any number = 0 no?
blissful meadow
$\frac{1}{\sqrt{e}} = e^{\frac{-1}{2}}$
soft zealotBOT
Azyrashacorki
raven obsidian
oh hold on, let me think that through
would = -1/2 because ln and e cancel out
i see
raven obsidian
**Azyrashacorki**
wait where did the -1 come from? qoutient property?
im trying to expand it on paper
blissful meadow
1/x = x^{-1}
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arctic forge
do all log functions have their range equal to R
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shadow tinsel
f is continuous on the domain [0, 2]. f(0) = 0 and f(2) = 4. does there exist some value c (in the domain of course), such that f(c) = c^3?
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f is continuous on the domain [0, 2]. f(0) = 0 and f(2) = 4. does there exist so...
dusk loom
!status
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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
1
grim nebula
shadow tinsel
yeah nonzero
<@&286206848099549185> anyone reckon they could come up with a solution and reasoning ?
mental dragon
what's the question again?
shadow tinsel
f is continuous on the domain [0, 2]. f(0) = 0 and f(2) = 4. does there exist some value c (in the domain of course), such that f(c) = c^3?
mental dragon
f is continuous on the domain [0, 2]. f(0) = 0 and f(2) = 4. does there exist so...
c=1
shadow tinsel
how do you know?
mental dragon
The domain is inclusive
foggy minnow
also like
It’s a random function right
What if my function was literally f(x) = x^3
Then for all values in the domain wouldn’t c be any value or something
shadow tinsel
What if my function was literally f(x) = x^3
it is random but f(2) = 4 so it cant be x cubed
mental dragon
Every value from 0 to 2, inclusive. Now, just think of a number that when cubed, is equal to itself. There are only two numbers like this: 0 and 1
mental dragon
Every value from 0 to 2, inclusive. Now, just think of a number that when cubed,...
@shadow tinsel this help?
shadow tinsel
the condition f(c) = c^3 just means that a point (c, c^3) exists on the function
c doesnt have to equal c^3
shadow tinsel
read the question
mental dragon
my bad I'm not wearing my glasses
mental dragon
f is continuous on the domain [0, 2]. f(0) = 0 and f(2) = 4. does there exist so...
Intermediate value theorem
So basically
Let's get a function, g(x) = x^3
you need a value in the interval [0, 2] where f(x) = g(x)
aka an intersection point
shadow tinsel
nonzero
mental dragon
according to the intermediate value theorem, if g(x) takes on all the values between g(0)=0^3 and g(2)=2^3=8.
shadow tinsel
yeah nonzero
stated up here mb
mental dragon
there must be one value, x, in [0,2], such that g(x) will have a value between 0 and 8, which is the range of this function
Am i misunderstanding something?
shadow tinsel
it is a criteria, is there a non-zero value of c i.e (0, 2]
mental dragon
do you need a specific number?
or just reasoning as to whether or not that number exists
what did I say that was incorrect?
shadow tinsel
how can f(x) be x^3/2 when f(2) = 4?
shadow tinsel
he probably means (x^3)/2
ohhh yeah ur right
so the answer is there does not exist a number c such that f(c) = c^3?
due to a counterexample being sufficient to prove its false
yeah the function isnt specified f is just every function that goes thru (0,0) and (2,4), so sometimes f(c)=c^3 and sometimes its not
but the answer is still a general no?
.close
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peak kernel
how would i simplify this?
final saddleBOT
how would i simplify this?
peak kernel
ive made arctanh into arcsinh/arccosh but where to from there?
or is it a different way entirely?
final saddleBOT
@peak kernel Has your question been resolved?
shadow minnow
ive made arctanh into arcsinh/arccosh but where to from there?
You could try to use the formula for arctanh
With logarithms
And then use the formula for sinh with exponents to cancel them
Also I don’t believe that arctanh=arcsinh/arccosh
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whole pecan
If I have log base 2 of x = 3
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If I have log base 2 of x = 3
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earnest yew
is there a way to solve this limit without fully facotring the denominator cuz it would be too long
final saddleBOT
is there a way to solve this limit without fully facotring the denominator cuz i...
agile coral
Why are you factoring?
earnest yew
Why are you factoring?
how do u do it then
agile coral
If the denominator is going to 0
...
mystic night
This limit is just a limit of a fraction so you look at the top part : what is its limit ?
and same for the bottom part
You only have to do fancy stuff if it's a special form like 0/0 or infinity/infinity
But here it is not the case
earnest yew
This limit is just a limit of a fraction so you look at the top part : what is i...
ohh okay
earnest yew
This limit is just a limit of a fraction so you look at the top part : what is i...
also, is the vertical asymptote -3 and 3 in this question since it's a square number?
mystic night
yes both -3 and 3 cancel the denominator so they will be vertical asymptotes
earnest yew
yes both -3 and 3 cancel the denominator so they will be vertical asymptotes
alright thank you
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tranquil pine
what type of function is a floor / ceil
final saddleBOT
what type of function is a floor / ceil
@tranquil pine Has your question been resolved?
wind needle
i dont understand why if A is invertible matrix then its row equivalent to matrix I
final saddleBOT
i dont understand why if A is invertible matrix then its row equivalent to matri...
wind needle
and does it go both ways too ?
as if I is row equivalent to A that means A is invertible matrix ?
blissful meadow
If it's invertible, the equation Ax = b has a unique solution, so in particular when you row reduce it there has to be n pivots. The matrix is nxn, so n pivots means exactly that the result is the identity.
But you can think of it in terms of row operations as well. Row equivalence of A to X means that you have elementary row operation matrices such that En ... E2 E1 A = X. Then if A is invertible, we can RREF it to some matrix that is also invertible, but the resulting matrix is already row reduced, and since it's invertible it has to have n pivots, i.e. it's the identity.
wind needle
now wait i didnt rlly understand everything u said since im still learning all that but
i understood that if A is invertible matrix
that means it has only on solution
if its not
then it has either 0 solutions or infinite
so wait
A is our matrix
x is the thing
this
blissful meadow
If A is invertible, then the equation Ax = b has a unique solution. For any b
The point is, you can RREF it, and the only way it has a unique solution for any b is that the RREF has n pivots.
But if the RREF has n pivots, it has to be the identity
wind needle
ok so if the RREF has no row that is all 0 and A belongs to Mij(R) (i=j) that means it for sure have one solution
and thats why to find out the invertible A we do this A=I and then we change their places
like
[A|I]
then if we switch their places we get the invertible A
and thats why if A is row equivalent to I then its invertible
right ?
blissful meadow
Yeah exactly.
It's like that method of finding inverses you've probably seen where you rref [A | I] and the right side becomes the invese.
Then you can see that the A and Ainv are both row equiv. to I
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alpine ferry
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alpine ferry
hello, I need help with my homework I feel like Im doing something wrong
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alpine ferry
<@&286206848099549185>
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@alpine ferry Has your question been resolved?
alpine ferry
<@&286206848099549185>
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digital steeple
whats the qeustion?
stone rivet
ok so there are 5 calcul like this on but i understand nothing realy nothing so if someone can explain me one calcul to do the rest
final saddleBOT
ok so there are 5 calcul like this on but i understand nothing realy nothing so ...
carmine current
ok so there are 5 calcul like this on but i understand nothing realy nothing so ...
you know the formula?
$$
(a+b)^2
$$
soft zealotBOT
notwish
carmine current
?
stone rivet
i dont know this one
soft zealotBOT
notwish
carmine current
ok so there are 5 calcul like this on but i understand nothing realy nothing so ...
this follows same pattern
$$ (10 + _)^2 = _ + _ + 16a^2 $$
soft zealotBOT
notwish
Compile Error! Click the 
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stone rivet
ok but what im censed to put in the calcul
carmine current
ok but what im censed to put in the calcul
so now see we got 16a^2
we can convert it to (4a)^2
right?
stone rivet
mmmmhhh i thing yes
carmine current
**notwish**
so now compare this pattern
to get (4a)^2
we need 4a in the left hand side tooo
so the first box will have ?
stone rivet
(10+4a)^2 ?
carmine current
(10+4a)^2 ?
yesss
carmine current
**notwish**
now use this formula and find the rest
stone rivet
it is (10+4a)^2=4a^2+10+16a^2 ?
carmine current
it is (10+4a)^2=4a^2+10+16a^2 ?
no-
stone rivet
(10+4a)^2=10^2+4a^2=16a^2
sleek quartz
16a^2 is just (4a)^2
which are equal?
carmine current
so the middle one should be 2 x 10 x 4a
carmine current
ok so there are 5 calcul like this on but i understand nothing realy nothing so ...
see this
sleek quartz
using $a$ ambiguously probably isnt helping
soft zealotBOT
b
sleek quartz
$(x + y)^2 = x^2 + 2xy + y^2, x = 10, y = 4a$
soft zealotBOT
b
stone rivet
im lost
sleek quartz
here just watch this tbh https://www.youtube.com/watch?v=4OhmtEDCFAA
carmine current
using $a$ ambiguously probably isnt helping
right
@sleek quartz hey can you come to #help-7｜zen1thxyz
stone rivet
here just watch this tbh https://www.youtube.com/watch?v=4OhmtEDCFAA
so if i understand it good the result is
(10+4a)^2=10^2+40a+16a^2
soft zealotBOT
b
sleek quartz
yeah you can just do 100 lmao
soft zealotBOT
b
final saddleBOT
final saddleBOT
Available help channel!
tranquil pine
guys
static plinth
$\lim_{n\to \infty}\frac{e^n}{\left(1+\frac{1}{n}\right)^n}$?
soft zealotBOT
y0shi
static plinth
do you remember what the definition of e was?
tranquil pine
in the denominator
static plinth
oh
static plinth
$\lim_{n\to \infty}\frac{e^n}{\left(1+\frac{1}{n}\right)^{2n}}$
soft zealotBOT
y0shi
tranquil pine
do you remember what the definition of e was?
no (1+1/n)n^2
static plinth
is the entire thing raised to the second power or only n
tranquil pine
its rasied to n^2
static plinth
no parenthesis?
tranquil pine
nope
tranquil pine
do you remember what the definition of e was?
what definition
static plinth
$\lim_{n\to \infty}\frac{e^n}{\left(1+\frac{1}{n}\right)^{n^n}}$
tranquil pine
i know several definitions
scarlet sequoia
$\lim_{n\to \infty}\frac{e^n}{\left(1+\frac{1}{n}\right)^{n^2}}$
soft zealotBOT
rafilou2003
scarlet sequoia
not sure this would work well, depending on what binomial approximation you used
I think first take the ln
(natural log)
factor nicely
tranquil pine
not sure this would work well, depending on what binomial approximation you used
(1-n)e^n
i got this after using it
is it valid
scarlet sequoia
oh no it doesn't work at all
tranquil pine
as i can put 1/n = h where h tends to 0
tranquil pine
oh no it doesn't work at all
why
scarlet sequoia
as i can put 1/n = h where h tends to 0
if your h is 1/n then when h tends to 0, ALL of the n tend to +infinity at the same time
you can't select a few well chosen n to tend to infinity first
tranquil pine
wouldn't it mean the same thing h tends to 0 or n tends to infinity
in this case
as we are only talking about +ve infinity here
tranquil pine
Rewrite clearly what you tried to do
e^n ( 1 + 1/n)^-n^2
now i applied binomial approximation
as 1/n would tend to 0
scarlet sequoia
So you used $(1+x)^\alpha$ as $x\to 0$ right?
soft zealotBOT
rafilou2003
tranquil pine
yes
scarlet sequoia
That's when alpha is CONSTANT
scarlet sequoia
anyways
tranquil pine
so what were you suggesting
scarlet sequoia
take the natural log
scarlet sequoia
you wanted a suggestion? Or an explanation of the whole solution?
tranquil pine
do you mean i write it again in the exponential form
like
take ln
then
apply the limits
scarlet sequoia
well yeah you have lim (...) and you could write it as lim (e^ln(...))
scarlet sequoia
yep
now factor the exponent nicely
and you can use expansion of ln. Be careful to include ENOUGH terms
tranquil pine
hmm true
i can use it
what is it ?
lol
i dont remember
and i dont wanna work out all derivatives to compute it rn
scarlet sequoia
yep
so you should easily figure out how to get ln(1+1/n) from the expansion I gave you
tranquil pine
2 terms will do ?
scarlet sequoia
See what happens when you get two terms
scarlet sequoia
yes
tranquil pine
hey i wanted an advice
should i solve less problems and learn further topics
or solve tons of problems on the topics i learn then move on to the next ones
i dont really know how much to solve
scarlet sequoia
That really depends on how fast of a learner you are, and how you learn the quickest
Try everything and see whether you understand topics better with a lot of exercises or not
tranquil pine
whats better
tranquil pine
That really depends on how fast of a learner you are, and how you learn the quic...
also how do i even know if i learned it completely or not
i use book so i don't really get an idea about it
scarlet sequoia
also how do i even know if i learned it completely or not
That's up to your intuition, it builds up over time
you're free to go learn other topics and go back to that one at any time, and compare whether you know more about some topics than others
tranquil pine
you mean i can learn multivariable calculus without practicising tons of integrals
in single variable calculus
scarlet sequoia
If you feel like you're ready then attempt it
and if you don't or feel like you're lost while learning, practice more
tranquil pine
from where do you know a good source
scarlet sequoia
I have no good sources for you I'm afraid
tranquil pine
and if you don't or feel like you're lost while learning, practice more
hmm never tried this ive learned things linearly
okay
thanks
.close
waxen pine
final saddleBOT
waxen pine
is there a formula for this?
abstract bramble
rewrite as a geoseries
maybe
yeah
the denominator factors
oh but it's not a perf square
waxen pine
rewrite as a geoseries
yeah
how
tihs doesnt seem like common ratio
oh you said canno tnvm
static plinth
limit comparison test
abstract bramble
how do you find the sum
static plinth
no idea
abstract bramble
ok
static plinth
doesn’t seem like something you should find the sum for honestly
final tangle
factorise denominator
then pfd
final tangle
partial fraction decomposition
abstract bramble
it'll be telescoping
terse folio
Telescoping series
static plinth
oh then that’s even better
final tangle
🔭
terse folio
look at first 2 terms
final saddleBOT
@waxen pine Has your question been resolved?
waxen pine
okay i did psf
sorry was doing smth else
yeah
i see the telescoping
i think thats the name
thats crazy
its like magic
-1 +3 makes it connect
wait
it doesnt
becuase the psf the tops are differnet
did i do my pfd
wrong
found the mistake
ty for the help guys
.close
final saddleBOT
final saddleBOT
Available help channel!
tranquil pine
final saddleBOT
tranquil pine
I rearranged both for sint and cost
Addition formula for the y parametic
Then subbed in the x parametic
Then equated sin^2 t + cos^2t
Then switched it for 1 and cancelled down
It didn't get me the answer
final saddleBOT
@tranquil pine Has your question been resolved?
tranquil pine
<@&286206848099549185>
sharp goblet
@mahdi_gd hii
tranquil pine
?
tranquil pine
.close
final saddleBOT
final saddleBOT
Available help channel!
real reef
Hello! Does anyone how to use Mathematica 6.0 with Decarta2D? I have an exercise to solve for tomorrow and I don't know how.
final saddleBOT
Hello! Does anyone how to use Mathematica 6.0 with Decarta2D? I have an exercise...
candid hull
real reef
Thank you. Imma post it soon.
The converse of Ceva's theorem states the following: Let ABC be a triangle, and let points A' lie on (BC), B' lie on (CA), and C' lie on (AB). If: (A' B | A' C) * (B'C|B'A)*(C'A|C'B)=1, then the lines AA', BB', CC' are concurrent.
I have also the problem solved, i need it in Mathematica
final saddleBOT
@real reef Has your question been resolved?
final saddleBOT
@real reef Has your question been resolved?
raven obsidian
why did the base change to 2 on step 2? i thought the change base formula stated: log (base b) of x = log (base a) of x / log (base a) of b
final saddleBOT
why did the base change to 2 on step 2? i thought the change base formula stated...
raven obsidian
that was on google's calculator
fast arrow
Yes it does state that
They just chose a to be 2
Has nothing to do with the 2 as the parameter
raven obsidian
could i do the same even though the original base is 8?
fast arrow
Yeah, the original base doesn't matter
raven obsidian
because i see thjey used it to make the top of the fraction to equal 1 in the following step
raven obsidian
Yeah, the original base doesn't matter
oh
fast arrow
The new base can be any number
sharp goblet
?
raven obsidian
so it's my choice to make it into whatever number that works for me?
odd... i guess that makes sense?
sharp goblet
can you clearly say your question ?
raven obsidian
why did the base change to 2 on step 2? i thought the change base formula stated...
here
i was just wondering why the base went from 8 to 2
sharp goblet
i was just wondering why the base went from 8 to 2
you got the answer maybe
raven obsidian
without prior knowledge of that, it seems like the method isnt useful if i cant envision the rest of the steps
sharp goblet
But theres an easier way go compute your result
raven obsidian
i figured i could also try inverse property
raven obsidian
yeah that way i understand, thanks. just had to make sense of the other one but oh well
.close
final saddleBOT
final saddleBOT
Available help channel!
jolly vector
final saddleBOT
loud sundial
Is ABCD a square? If so, 3D Pythagorean theorem.
jolly vector
yea ik pythageoream
but
i duno
whixh ones to take
like which linea
bec bh
i duno
loud sundial
The height is kinda obvious
For the two other sides, assuming ABCD is a square, it doesn’t matter
Since all the sides are the same anyway
jolly vector
.close
potent lance
stuck on the graphing part
final saddleBOT
stuck on the graphing part
karmic kindle
remote mauve
What exactly do you need help with the graphing
remote mauve
If it’s not asking to label everything on the graph, then I think simply graph the ellipse
remote mauve
It is
(x+3)^2 means the x goes from 0 to -3, and (y-6)^2 means they goes from 0 to 6
Also, the distance from the vertex to vertices and co-vertices is the square root of the numbers below the squared x and y
remote mauve
Rotate it 90 degrees
remote mauve
Yes
remote mauve
Np
potent lance
how i close
remote mauve
.close
raven ember
Hello, Im trying to do the Quine McCluskey method to simplify an expression and I was wondering if it can happen that you are eliminating more than one variable for example with 2 and 3 here I would only get wy, no?
final saddleBOT
Hello, Im trying to do the Quine McCluskey method to simplify an expression and ...
@raven ember Has your question been resolved?
raven ember
<@&286206848099549185>
final saddleBOT
@raven ember Has your question been resolved?
final saddleBOT
@raven ember Has your question been resolved?
final saddleBOT
@raven ember Has your question been resolved?
raven ember
.close
final saddleBOT
wild geyser
final saddleBOT
Available help channel!
plucky bridge
hello how to do this question
final saddleBOT
hello how to do this question
plucky bridge
Also is the determinant of a matrix equal to the scalar triple product of its rows?
Like somebody has said that to me
tranquil pine
invertible is matrix that has an inverse ryt
plucky bridge
what is ryt
gritty drift
Also is the determinant of a matrix equal to the scalar triple product of its ro...
yes
formal trail
remote island
please explain both the "addition" and the multiplication by 12
final saddleBOT
please explain both the "addition" and the multiplication by 12
versed crater
They are properties of the determinant
Do you have notes on the properties of determinants?
remote island
They are properties of the determinant
versed crater
I’m asking you
versed crater
Do you know what the definition of the determinant is?
remote island
just that if it's != 0 then the matrxi is invertible
remote island
i was absent
remote island
please explain both the "addition" and the multiplication by 12
this is from the notes
versed crater
I actually don’t know any good videos that explains determinant nicely
remote island
I actually don’t know any good videos that explains determinant nicely
could you plz just can say the relevant things to solving exercises
i don't have to intuitvely understadn what determinants are
at least rn
versed crater
It’s a sort of thing you can do to a matrix
It has some interesting properties
Notably, it’s linear in every row and column
And if you swap 2 rows the sign changes
versed crater
Notably, it’s linear in every row and column
If you imagine the 2 straight lines is a function that takes a bunch of column vectors in and returns a number
versed crater
please explain both the "addition" and the multiplication by 12
The first line where they merged the 2 determinants together they use linearity
remote island
wdym by using linearity
what exactly do you mean that it's linear in ever yrow and column
versed crater
Just like how if A(x+y) = A(x) + A(y) this is linearity
It’s linear in its (first and only) entry
But say my function takes more than 1 input
remote island
Just like how if A(x+y) = A(x) + A(y) this is linearity
i don't udnerstand what you mean
versed crater
Have you not been studying linear algebra
remote island
for me linearity is y = mx + b
versed crater
Do you know what a vector space is?
remote island
don't remember
but could you plz
just note all the things i need to know for solving problems
versed crater
Do you know what a linear transformation is?
versed crater
You should learn the theory first or it won’t make much sense
remote island
i dont have time
versed crater
If you don’t have time to learn something you won’t be able to learn it
You don’t tell the IRS I didn’t file my taxes because I didn’t have time
remote island
just expalin this qwuickly pls
versed crater
I can’t because you lack the background knowledge to understand the explanation
You’re asking how a plane flies without knowing any physics
There really isn’t much to say besides “it just does”
remote island
dude just explain ther ule
without saying why it works
like you have two determinants
when can you add them
and how do you do so?
i know how to compute determinants
<@&286206848099549185>
also what does D mean here
remote island
ok so it's just a different notation
i'm familiarw ith it
@versed crater could you pls say how did they compute all these determinants so quickly?
versed crater
you just look at it and do it in your head
3x3's are not too hard when it's all 1's and 0's
remote island
also what does D mean here
ok, when do you use this method andw hen gaussian elimination?
remote island
also what does D mean here
also is there any name for the above?
how do you solve stuff like
final saddleBOT
@remote island Has your question been resolved?
nocturne vale
how do you solve stuff like
For the first question, let's try to simplify our systems
remote island
also, what does this ^-1 mean here
remote island
For the first question, let's try to simplify our systems
ok, how?
like wdym by simplify here?
midnight sparrow
also, what does this ^-1 mean here
It's thé inverse of the Matrix like if you multiply it by the original you get the identity
remote island
It's thé inverse of the Matrix like if you multiply it by the original you get t...
oh gotcha
midnight sparrow
Maybe for the transpose write the coefficients and try to solve the system
It's the same for the second
And I think for the third if you multiply both matrices you will get something maybe
remote island
Maybe for the transpose write the coefficients and try to solve the system
it ought to be somethign else
this was before systems of equations
remote island
And I think for the third if you multiply both matrices you will get something m...
3rd what?
midnight sparrow
? Wdym
remote island
? Wdym
we didn't do systems of equatiosn before this problem
so the teachers wants you to solve it another way
midnight sparrow
For the third multiply both matrices and you'll get a diagonal matrices maybe which IS very easy
midnight sparrow
So what's your lesson
remote island
Maybe for the transpose write the coefficients and try to solve the system
transpose is this, right?
but im not sure to what problems are you referring to with other questions
nocturne vale
sorry my phone dead
remote island
no worries
nocturne vale
but yh take transpost
midnight sparrow
Maybe I don't use thé right term I'm not english
remote island
Maybe I don't use thé right term I'm not english
just say what are you referring to
you just said "second" and "third"
second and third what
midnight sparrow
Of what you have send
remote island
but yh take transpost
how? idk what X is
remote island
And the other IS third
i sent 4 pics
nocturne vale
how? idk what X is
ok for the first, set the second term as A
remote island
3rd in order or thed one marked 3
remote island
ok for the first, set the second term as A
nocturne vale
XA*A^-1=XI=X
remote island
also is there any name for the above?
nothing below the above was solved
remote island
ok for the first, set the second term as A
what's the point
midnight sparrow
When multiply by identity it do nothing
remote island
XA*A^-1=XI=X
where did you get A^-1 from
nocturne vale
A^-1 is A transpost
midnight sparrow
Here it's the case