why is that so ?
#help-10
pastel scaffold
timid silo
if you are standing on top of a hill, the height you are at is the same as the height if you were part way down the hill plus the height difference from part way to the top of the hill
edgy needle
help help
edgy needle
wut
proven zephyr
you probably should make the denominator positive
edgy needle
is it wrong tho
proven zephyr
no
wait
i was looking at the correct answer
wait.. which one is your answer im confused
edgy needle
oki ping me when ur done i need to select a movie to watch with mummy
top is mine
on computer is the real correct answer
proven zephyr
ohh
they're both the same
it's just that they multiplied the top and bottom with -1
proven zephyr
is this question from your school?
edgy needle
from my textbook
proven zephyr
idk if the denominator has to be a positive number
edgy needle
obtuse pebbleBOT
@edgy needle Has your question been resolved?
proven zephyr
it's still correct tho
obtuse pebbleBOT
floral garnet
Yeah i know just that taking small numbers will make it easier
proven zephyr
he was just doing a common trick
$$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$$
warm shaleBOT
MarveI
proven zephyr
but yeah it's easier
obtuse pebbleBOT
Available help channel!
pine fog
I now know how they solved this but i thought when you use 2 in the denomitor to cancel both terms i.e 2 and sqrt(2) but that just not even possible right? So they just factor out the sqrt(2) but if i do that then i would have to redistribute it over the 2 in the nominator so i am very confused rn? :/
obtuse pebbleBOT
I now know how they solved this but i thought when you use 2 in the denomitor to...
high lily
combining your terms and rationalising the denominator gets you
$$\frac{2+\sqrt{2}}{2}$$
this would be considered simplified
warm shaleBOT
ℝamonov
high lily
splitting the fraction again, would just lead back towards what you started with
pine fog
Ahhhh so they divide 2 with 2 which leads to 1 and factor out the sqrt(2) am i right?
pine fog
I dont really what to call it else but like the answer below taking out sqrt 2 which leads to 1 in the nominator
pine fog
that algebra calc you're using looks pretty trash
do you have any suggestion for better calculators XD
high lily
wolfram
pine fog
Ahh thanks
high lily
writing $\frac{\sqrt{2}}{2}$ is \textbf{objectively} better than writing $\frac12 \sqrt{2}$
warm shaleBOT
ℝamonov
pine fog
really I have heard mixed opions about this i.e that dont like radicals within fractions
high lily
imo (2+sqrt(2))/2 would be considered the most simplified form for this expression
pine fog
I think the text books just really like to see how far can a student go even though it looks ugly but that is my guess
thanks a lot again!
.close
night oyster
Its getting so lengthy!!
obtuse pebbleBOT
Its getting so lengthy!!
night oyster
No
kind hawk
do you know polynomial interpolation in general?
kind hawk
hmm well then it could be tricky
do you know that 3 points uniquely determine a quadratic function?
kind hawk
ok
so the LHS and RHS are each a quadratic
(the RHS is a quadratic with the x^2 term equal to 0)
night oyster
Yes
kind hawk
what is the value of each of these quadratics at the point a, b and c
night oyster
A= x² b= x c= constant , may be !!
night oyster
How ?
kind hawk
if we call the quadratic on the left f(x)
what is f(a), f(b) and f(c)
and if we call the quadratic on the right g(x)
what is g(a), g(b) and g(c)
night oyster
g(x) is the RHS ?
kind hawk
yes
night oyster
a , b , c
kind hawk
yes
so we have two quadratics f, g that are equal at three points a,b,c
what can we conclude about these quadratics?
night oyster
Identity
kind hawk
and we are done
kind hawk
we have seen that f(a)=g(a), f(b)=g(b) and f(c)=g(c)
night oyster
Hm
kind hawk
as these are both quadratics, this means that f(x)=g(x) for all x
but f(x)=LHS and g(x)=x, so LHS=x for all x
boreal patio
translation: first row: function ... is defined: f(x) = (1; x = 1/n for some n in Real numbers... 0; otherwise)... Find out if the function is Riemann integrable. if it is calculate ...
obtuse pebbleBOT
translation: first row: function ... is defined: f(x) = (1; x = 1/n for some n i...
boreal patio
i have the solution but i don't understand it... if its possible i would like to go voice chat to help... tnxx alot
obtuse pebbleBOT
@boreal patio Has your question been resolved?
fierce lagoon
You can't integrate that
At least, you can't integrate for all x
The problem is that at 1/3, 1/4, and so on, you have spasmodic outputs of 1
Integrating that gets 0
boreal patio
In the solutions it says that function is Riemann integrable and the sum is 0
fierce lagoon
Really
tardy epoch
1/n isn't dense in [0,1]
fierce lagoon
Idk the function seems rather spasmodic within the domain (0, 1/2]
Like it's dense sure
But it's spasmodic
But I guess it's dense enough to integrate
boreal patio
what means spasmodic? (im not familiar much with english definitions)
tardy epoch
In the solutions it says that function is Riemann integrable and the sum is 0
show your definition of the riemann integral for this f
tardy epoch
Oh you use partitions instead of upper and lower sums
tardy epoch
its in Slovene i will try to translate
Nah it's fine I understand what's going on
tardy epoch
The partition chosen just avoids all x=1/n points
boreal patio
how can i just avoid them... for me it makes sense to avoid all other points since their sum is obviously 0
because they lay on x axis
tardy epoch
The partition chosen just avoids all x=1/n points
Correction: finitely many points. They pick [0, delta] that has width going to zero but constant height
boreal patio
idk
this is as far as i understand
(green) epsilon > 0
<@&286206848099549185>
obtuse pebbleBOT
@boreal patio Has your question been resolved?
obtuse pebbleBOT
@boreal patio Has your question been resolved?
distant plank
Hello guys
obtuse pebbleBOT
Hello guys
distant plank
how can we find this limit :
,w limit ((product of 1 + k/n, k=1 to n))^(1/n) as n to infty
warm shaleBOT
distant plank
always taylor expansion i used
tardy epoch
,w limit ((product of 1 + k/n, k=1 to n))^(1/n) as n to infty
write unknown limit as y and find log(y)
log(y) = log( limit ( nth root (prod (...) ) ) )
distant plank
`log(y) = log( limit ( nth root (prod (...) ) ) )`
don't understand
well i understand
this is a nice strat
my plans was to use taylor expansion series (i have to use this) to appear : 1 + x + x^2/2... + o(x^n) which is equal to... e^x
tardy epoch
nah that's hella work
oh you have to?
uhhh i guess have fun expanding
$(1+1/n) (1+ 2/n) (1+3/n) + ... (1+n/n) = 1 + \frac{A}{n} + \frac{B}{n^2} + ...$
warm shaleBOT
riemann
distant plank
oh you have to?
yes i have to^^
obtuse pebbleBOT
@distant plank Has your question been resolved?
distant plank
**riemann**
how did you get this ?
tardy epoch
it's all the possible denominators. i just grouped them by powers of n
distant plank
ok
i first used expansion with (1+1/n)^(1/n)
but there is a problem with the order
the goal is to have 1 + x + x^2/2 + ... + x^n/n! but again, i don't find an usefull expansion
distant plank
i first used expansion with (1+1/n)^(1/n)
but i can only use this with 1/n --> 0
(1 + u)^(1/n) with u = k/n --> 0
so, this is the only way
for examples, the first three terms :
$(1+\dfrac{1}{n})^{\dfrac{1}{n}} (1+\dfrac{2}{n})^{\dfrac{1}{n}} (1+\dfrac{3}{n})^{\dfrac{1}{n}} ... (1+\dfrac{n}{n})^{\dfrac{1}{n}}$
😦
warm shaleBOT
Pako
distant plank
i will make an expansion in ordre 2
$=(1 + \dfrac{1}{n^2} + \dfrac{n^2 - n}{2} o(n^n))( 1 + \dfrac{2}{n^2} + o(n^n)) (1 + \dfrac{3}{n^2} + o(n^n))...(1 + \dfrac{n}{n^2} + o(n^n))$
i should make an expansion to ordre n
warm shaleBOT
Pako
obtuse pebbleBOT
@distant plank Has your question been resolved?
distant plank
ok, i have to make a DL to ordre n
chrome spade
Hello, I want to decode a collection of numbers into English words. This is a riddle from a game called Tibia and looks like this:
When the veils of shrouded truths are lifted, who can stand?
663 902073 7223 67538 467 80097
The answer is supposed to be that collection of numbers, but I don't know where to start or how to decode it
obtuse pebbleBOT
Hello, I want to decode a collection of numbers into English words. This is a ri...
robust idol
Just dropping in to say Tibia is the best game ever.
daring rock
I'm not familiar with Tibia. Are there any rules to the game that might be helpful?
Like restrictions on what kind of patterns/codes they might use?
chrome spade
Okey, there is a library where is a creature called "Bonelord"
It explain us that were an ancient races and they have their language called 469
a bonelord is this creature
There is a npc called A Wrinkled Bonelord
who tells us this:
Is not Tibia is 1.
I am not Blinky I'm 468468
0 is an obscene number
Also according to the Wikia (information from inside the game)
469 is the Bonelord language. In written form 469 appears as a coded sequence of numbers lacking any punctuation or spacing within the text. their native 'tongue' consists of a blinking code with each eye, where a blinking could mean some syllable, letter, or word.
His language consists of a blinking code with each eye, where a blinking could mean some syllable, letter, or word.
There are another kind of Bonelord that scream this:
Inferior creatures, bow before my power! | Let me take a look at you! | 659978 54764! | 653768764!
chrome spade
Hello, I want to decode a collection of numbers into English words. This is a ri...
this is just a short quote that the creator put in a forum poll, but there is a big collections of with just numbers
Like this:
561145727857261185764364672435345275601928895219735364672496847560 199684770908895219727816705121648561145191991180036468895219911800 65128
The npc also said that only a experienced person with mathematics knowledge can solve that or using the mathemagics
daring rock
Huh, this is interesting but I can't seem to find any recognizable patterns
The number 469 seems so familiar for some reason but I can't find any particularly interesting properties it has
All I learned from the internet is that it's an area code for Dallas, TX
chrome spade
Haha
Yes is something Interesting
But seen very difficult to understand
The npc said you need more eyes to understand hahaha
tranquil dust
what is this image of circle with trignometric function summarized?
obtuse pebbleBOT
what is this image of circle with trignometric function summarized?
nocturne minnow
You can probably look up trig functions on the unit circle or something similar
nocturne minnow
what is this image of circle with trignometric function summarized?
All I did was look up excsc, since I didn't know what that was, and it look me to the wiki page
wooden cipher
What the name of that one problem in math that asks to color the grid in the least amount of colors so that any two points with unit distance is colored different colors?
They found the upper bound to be 7 with some hex coloring and disproved cases with 2-5 colors
obtuse pebbleBOT
What the name of that one problem in math that asks to color the grid in the lea...
wooden cipher
Trying to figure out the name so i can look it again
willow ravine
It sounds similar to the calculation of graham’s number
wooden cipher
Oh hey poisoned!
willow ravine
It deals with colors and vertex of cubes in higher dimensions I believe
wooden cipher
Yeah i know that one
willow ravine
The number so big that there aren’t enough planch space in the universe to hold it if each digit was assigned one planch.
Something like that.
wooden cipher
Thats not what im looking fir tho
willow ravine
Ahh okay.
willow ravine
Hmm I’m not sure then.
wooden cipher
Anybody?
obtuse pebbleBOT
@wooden cipher Has your question been resolved?
fierce lagoon
All I did was look up excsc, since I didn't know what that was, and it look me t...
Wanna know who uses those stupid functions? Railroad engineers and cartographers.
mellow pulsar
obtuse pebbleBOT
mellow pulsar
what am i doing wrong?
slim cove
yeah, exactly
mellow pulsar
LMAO
i shouldve done that
maybe you can help me with this one too
its tiny ill try to make it bigger
A lamp with a parabolic reflector is shown in the figure. The bulb is placed at the focus and the focal diameter is 8 cm. (Assume the vertex O is at the origin. Let the x-axis be horizontal and the y-axis be vertical.)
robust yacht
obtuse pebbleBOT
distant moth
Just add them? 🤔
Oh lol i did not see the + and - signs inside💀
royal basin
there is a trick here that might help
write u := (1-x)^(1/3) and v := (1+x)^(1/3)
then u^3+v^3=2 and 2uv = u^2 - 15v^2
robust yacht
write u := (1-x)^(1/3) and v := (1+x)^(1/3)
Suddenly it's all possible thank you very much
obtuse pebbleBOT
Available help channel!
pliant drift
For 1/x = 1/y + 1/z I need to transpose for y, how would I go about doing this. I'm just stuck on how to take the 1 away really.
obtuse pebbleBOT
For `1/x = 1/y + 1/z` I need to transpose for y, how would I go about doing this...
lone oyster
well $\frac{1}{x}=x^{-1}$
warm shaleBOT
DJJ05
pliant drift
so it's essentially x^-1 = y^-1 + z^-1
which would be -y = -x + z?
kind hawk
no
you can't just take the ^-1 away like that
$y^{-1} = x^{-1}-z^{-1}$ or equivalently, $\frac1y = \frac1x-\frac1z$
warm shaleBOT
Denascite
kind hawk
and now you can do 1/(...) on both sides to get $y = \frac{1}{\frac1x-\frac1z}$
warm shaleBOT
Denascite
kind hawk
and then you can simplify the RHS to $y=\frac{xz}{z-x}$
warm shaleBOT
Denascite
obtuse pebbleBOT
@pliant drift Has your question been resolved?
pliant drift
I see, alright thanks for the help
Also, for q=bm+tp/a transpose for m would this be an answer?
m = (q)/(b) - tp
?
kind hawk
no
first we subtract tp/a from both sides to get $bm = q-\frac{tp}{a} = \frac{aq-tp}{a}$. Then we divide by $b$ to get $m=\frac{q}{b}-\frac{tp}{ab}=\frac{aq-tp}{ab}$
pliant drift
Oh sorry I forgot, q = (bm+tp)/a
warm shaleBOT
Denascite
kind hawk
first we multiply by $a$ to get $aq = bm+tp$, then we subtract $tp$ to get $bm=aq-tp$, then we divide by $b$ to get $m=\frac{aq-tp}{b}$
warm shaleBOT
Denascite
regal latch
Find the domain
obtuse pebbleBOT
Find the domain
regal latch
Wait actually the answer is infinity except x>4, right?
regal latch
And x CAN be > 4. So no, no exception to that.
So (4, inf)?
pine sail
Incomplete.
regal latch
x>4
pine sail
Yes, incomplete.
regal latch
Incomplete.
Oh idk what's the answer now
warm shaleBOT
What the hell am I doing here?
pine sail
But no, (4,inf) is correct
regal latch
But no, (4,inf) is correct
What's the answer?
regal latch
But no, (4,inf) is correct
Oh oh sry I read this message wrong as well
Thanks
.close
woven citrus
English translation of Q7: If the graph below demonstrates the function fx=cos(wx + pi/6) from -pi to pi, find the minimal positive period of fx.
I tempted to solve it with the use of phase shift, but got a wrong answer. May I ask what did I get wrong? Or is the minimal positive period different from the period in this case?
(As I am just secondary 3 and haven't learnt deeply about sinusoidal functions, I am not sure what "minimal positive period" actually means, so I am sorry if I made a silly mistake)
Thanks!
obtuse pebbleBOT
English translation of Q7: If the graph below demonstrates the function fx=cos(w...
woven citrus
4π/9-π/6=T/4
Oic, thanks!
compact shadow
Oic, thanks!
Sorry made a mistake, fixing
Should be like this
$\frac{4π}{9}-\frac{π}{6ω}=\frac{π}{2ω}$
warm shaleBOT
Cogwheels of the mind
brisk juniper
I'm having trouble with this question. I have to use l'hopital's rule to solve it.
obtuse pebbleBOT
I'm having trouble with this question. I have to use l'hopital's rule to solve i...
brisk juniper
lim𝑥→0(cos 𝑥) ^(7/𝑥^2)
I know I messed up somewhere and made it way more complicated than it had to be. I think I got up to the -7tanx/2x correctly on the right.
spice basalt
Your second application of l'hopital is erroneous, you took the derivative of the whole thing rather than derivative of top over derivative of bottom
calm pawn
i found the values for p and q
how do i factorize the expression completely?
p=-7 and q=16
obtuse pebbleBOT
i found the values for p and q
how do i factorize the expression completely?
p...
calm pawn
use polynomial long division
how do i do that
calm pawn
okk
compact shadow
i found the values for p and q
how do i factorize the expression completely?
p...
Told you , if you don’t want to factor it you can plug in x=1 and 2 respectively. That polynomial has roots 1 and 2, you will obtain two linear equations of p,q
obtuse pebbleBOT
@calm pawn Has your question been resolved?
sterile wing
obtuse pebbleBOT
@sterile wing Has your question been resolved?
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
light gull
can anyone explain why the mclaurin taylor series of e^x is
obtuse pebbleBOT
can anyone explain why the mclaurin taylor series of e^x is
light gull
instead of e^x+(e^x)x+(e^x)(x^2)/2!+...
timid silo
the definition of mclaurin series has $f^{(n)}(0)$, ie you evaluate each derivative at 0
warm shaleBOT
Toby
light gull
gotcha, thx
timid silo
the goal of the series is to convert a complicated function into a polynomial series
light gull
ope, I have another question
timid silo
go ahead
light gull
if the question was to evaluate the taylor series about say x=3, would e^x & same derivatives also be evaluated at e^3 w/ (x-3)^k instead of x^k?
timid silo
yup
light gull
ahhh, gotcha that makes more sense now. Thx
light gull
.close
obtuse pebbleBOT
harsh remnant
You mean like how they are applicable in real life?
blazing sentinel
google doesn't work?
harsh remnant
lol
obtuse pebbleBOT
@timid silo Has your question been resolved?
floral garnet
First is when u differentiate once and second is when u differentiate twice
That's it
latent wind
obtuse pebbleBOT
latent wind
Plz help
floral garnet
Which one
latent wind
61
floral garnet
Ok let me try
latent wind
Sure, btw i tried opening formula and then some weird stuff
latent wind
d(tan(x)=dx/(cos(x)^2)
Bro, not again,
latent wind
I don't understand ur concept sorry
visual wasp
(cos+sin)²
latent wind
(cos+sin)²
Yeah, done
compact shadow
Sure. Let him
latent wind
Done, *2sincos
floral garnet
2
visual wasp
2sincos
latent wind
Yeah
compact shadow
Strange that you can’t calculate integral of dt/(t+1)^2, that was all I was trying to say but you don’t get it
latent wind
theres a frac
Yeah, have solved as per 1/ forgot to show
visual wasp
try @compact shadow method
bc it is way more useful
mine works in this case with substitutions....
his method is more useful in many cases
compact shadow
He says he can’t understand the concept, maybe df=f’dx he wasn’t taught this
latent wind
u know substitution ?
Sort of...
floral garnet
Will sinx work tho
latent wind
Ok, let's get from starting, u are saying i need to let the term as dx/[cos(1+tanx)]^2
Hello? Am i alive?
floral garnet
Will integration by partial fractions work
latent wind
Idk, haven't studied it tho
floral garnet
Hmm
floral garnet
Ok, let's get from starting, u are saying i need to let the term as dx/[cos(1+ta...
Ok let's stick to this
latent wind
K, what next with this statement?
floral garnet
What
visual wasp
d(tan(x)=dx/(cos(x)^2)
this*
half plume
obtuse pebbleBOT
half plume
Hey is this correct ?
obtuse pebbleBOT
@half plume Has your question been resolved?
half plume
<@&286206848099549185>
shadow crater
u mean all is true or not?
hoary cargo
@half plume I don't think this is correct. You can't take induction step to be true beforehand.
half plume
<@761698471874002945> I don't think this is correct. You can't take induction s...
why cant i do that
?
hoary cargo
Do you know the difference between induction hypothesis and induction step?
half plume
I.A represents the basecase ; I.V is the Induction step ; Behauptung is the aasumption ; I.Beweis is the proof
hoary cargo
In I.Beweis you have already accepted the inequality, also you are not reaching at the right conclusion. You need to try different approach. Maybe use contradiction
obtuse pebbleBOT
@half plume Has your question been resolved?
slim cove
Wait what is the confusion here?
You can totally do this by induction
how did you get from here to here though
@half plume
cedar glade
2x^2 + 9x + 10 = 0
obtuse pebbleBOT
2x^2 + 9x + 10 = 0
cedar glade
I can’t find a way to factor this and then find the values of x
heavy beacon
quadratic formula
high lily
consider ac method if you want to factorise
cedar glade
Alr I’ll try those
high lily
if you're just after the solutions and nothing else, you can apply QF
cedar glade
Thanks for the quick and straight forward answers
cedar glade
if you're just after the solutions and nothing else, you can apply QF
Alright, thank y’all
.close
woven badge
obtuse pebbleBOT
mental socket
I have some difficulty 'seeing' the size difference when displayed in arcminutes/arcseconds. The moon is about 27'30". I have calculated a moon for my world which is 50'16.13".. Would that take up most of the sky or is it just almost 2x bigger than what our moon looks like... Can anyone help me picturing this?
obtuse pebbleBOT
I have some difficulty 'seeing' the size difference when displayed in arcminutes...
warm canopy
its roughly double yeah
if you want it to be easier to compare then just convert everything into arcseconds
mental socket
So in the night sky it would be a bit bigger, but not take up all of the night sky?
I've never worked with arc-anythings. 🙂 I didn't finish high school and have taught myself som emath, but didn't reach this subject yet 🙂
THanks for the help!
mental socket
Is there a way to get the size of an object based on arc seconds? Say an object has a distance of 474,982 km and a diameter of 6,945.6 km. The angular diameter is about 50'16.13". But looking up into the night sky.. how many cm would that look like? Does that make sense?
obtuse pebbleBOT
Is there a way to get the size of an object based on arc seconds? Say an object ...
@mental socket Has your question been resolved?
mental socket
.close
timid silo
(2 - h) ^ 2 + (1 - k) ^ 2 = (- 2 - h) ^ 2 + (3 - k) ^ 2. Answer should be h=-2 and k=-2
obtuse pebbleBOT
(2 - h) ^ 2 + (1 - k) ^ 2 = (- 2 - h) ^ 2 + (3 - k) ^ 2. Answer should be h=-2...
timid silo
.close
solemn ravine
if i have two sequences whose supremums are both 1
obtuse pebbleBOT
if i have two sequences whose supremums are both 1
solemn ravine
and i add them together componentwise
is the new sequence's supremum 2
(i.e is supremum linear)
novel knoll
yes
sterile wing
obtuse pebbleBOT
obtuse pebbleBOT
@sterile wing Has your question been resolved?
obtuse pebbleBOT
@sterile wing Has your question been resolved?
warm shaleBOT
forest sinew
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
formal egret
This is problem 21d of chapter 6.3 of Velleman's How to Prove It. Here are the assumptions.
$O(g) = {f : \mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+} | \exists a \in \mathbb{Z}^{+} \exists c \in \mathbb{Z}^{+} \forall x > a ( |f(x)| \le c|g(x)|) }$
$\forall n \in \mathbb{N} (f : \mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+} )$
$\forall x \in \mathbb{Z}^{+} (f0(x) = x )$
$\forall n \in \mathbb{N} \forall x \in \mathbb{Z}^{+} (f{n+1}(x) = 2^{fn(x)})$
I'm trying to prove this:
$\forall n \in \mathbb{N} (f{n+1} \notin O(fn))
My understanding is that the goal is equivalent to this:
$\forall x \in \mathbb{Z}^{+} \forall c \in \mathbb{N}^{+} \exists x > a (|f{n+1}(x)| > c|f_n(x)|)$
x must be larger a, so I'm assuming x's definition will need to involve a. I'm thinking c will need to be involved as well. I'm totally stumped and need a nudge in the right direction.
obtuse pebbleBOT
This is problem 21d of chapter 6.3 of Velleman's How to Prove It. Here are the a...
formal egret
I'm having trouble with the LaTeX, as you can see...
warm canopy
Start and end section with $stuff$
warm shaleBOT
ΣAC
tardy epoch
maybe send in separate messages
warm shaleBOT
dad
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
formal egret
This is problem 21d of chapter 6.3 of Velleman's How to Prove It. Here are the assumptions.
$$O(g) = ${ f : \mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+} | \exists a \in \mathbb{Z}^{+} \exists c \in \mathbb{Z}^{+} \forall x > a ( |f(x)| \le c|g(x)|) }$$
$$\forall n \in \mathbb{N} (f : \mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+} )$$
$$\forall x \in \mathbb{Z}^{+} (f_0(x) = x )$$
$$\forall n \in \mathbb{N} \forall x \in \mathbb{Z}^{+} (f_{n+1}(x) = 2^{f_n(x)})$$
I'm trying to prove this:
$$\forall n \in \mathbb{N} (f_{n+1} \notin O(f_n))$$
My understanding is that the goal is equivalent to this:
$$\forall x \in \mathbb{Z}^{+} \forall c \in \mathbb{N}^{+} \exists x > a (|f_{n+1}(x)| > c|f_n(x)|)$$
x must be larger a, so I'm assuming x's definition will need to involve a. I'm thinking c will need to be involved as well. I'm totally stumped and need a nudge in the right direction.
^Okay, that's what I was going for
warm shaleBOT
dad
formal egret
Okay, I think I'm done editing and it looks like what I was going for.
warm shaleBOT
riemann
formal egret
This is problem 21d of chapter 6.3 of Velleman's How to Prove It. Here are the assumptions.
$$O(g) = { f : \mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+} | \exists a \in \mathbb{Z}^{+} \exists c \in \mathbb{Z}^{+} \forall x > a ( |f(x)| \le c|g(x)|) }$$
$$\forall n \in \mathbb{N} (f : \mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+} )$$
$$\forall x \in \mathbb{Z}^{+} (f_0(x) = x )$$
$$\forall n \in \mathbb{N} \forall x \in \mathbb{Z}^{+} (f_{n+1}(x) = 2^{f_n(x)})$$
I'm trying to prove this:
$$\forall n \in \mathbb{N} (f_{n+1} \notin O(f_n))$$
My understanding is that the goal is equivalent to this:
$$\forall x \in \mathbb{Z}^{+} \forall c \in \mathbb{N}^{+} \exists x > a (|f_{n+1}(x)| > c|f_n(x)|)$$
x must be larger a, so I'm assuming x's definition will need to involve a. I'm thinking c will need to be involved as well. I'm totally stumped and need a nudge in the right direction.
warm shaleBOT
dad
obtuse pebbleBOT
@formal egret Has your question been resolved?
obtuse pebbleBOT
@formal egret Has your question been resolved?
timid silo
This is problem 21d of chapter 6.3 of Velleman's How to Prove It. Here are the a...
with my awake two braincells, here are some noncoherent thoughts:\begin{itemize}
\item absolute values are unnecessary
\item note that $2^x$ is increasing
\item it should be enough to prove it for $n=0$ and use induction
\item suppose you were given a and c. Then once you prove $n=0$, you can get an $x_0>a$.
\item Now for $n=1$, $x_1$ satisfies the inequality $$af_2(x_0)=a2^{f_1(x_1)}>2^{cf_0(x_1)}=2^cf_1(x_0)>cf_1(x_0)$$
\item continue via induction\end{itemize}
warm shaleBOT
Toby
formal egret
No problem lol
timid silo
but intuition says its probably induction
formal egret
Yeah, the chapter is about recursion and induction. I'm having trouble finding something that even has an acceptable base case.
timid silo
for the base case you want to find an x such that a2^x>cx
intuitively, 2^x grows much faster than cx/a
formal egret
Why $a2^x > cx$ and not $2^x > cx$
warm shaleBOT
dad
timid silo
because i misread the problem :)
formal egret
lol gotcha
timid silo
my example inductive step from before works then lol $$f_2(x_0)=2^{f_1(x_0)}>2^{cf_0(x_0)}=2^cf_1(x_0)>cf_1(x_0)$$
hold up
Z^+ is positive integers?
formal egret
Yeah
timid silo
ah ok
then we can use the fact 2^x is strictly increasing on Z^+
(or maybe you need to prove it)
warm shaleBOT
Toby
formal egret
Alright, thanks for some ideas I'll work with
cerulean pine
This is problem 21d of chapter 6.3 of Velleman's How to Prove It. Here are the a...
try induction, are you fine with the base case ?
formal egret
No, I don't know what value of x would satisfy the base case.
cerulean pine
No, I don't know what value of x would satisfy the base case.
you don't need an exact value, you can use the fact that 2^x grows way faster than cx
so that lim(x->+∞) 2ˣ/cx = +∞
this implies the existence of x
formal egret
But if c is a very large number, isn't $2^x > cx$ false for a large number of possible values of x?
warm shaleBOT
dad
cerulean pine
no, c can have any positive value, lim(x->+∞) 2ˣ/cx = +∞ will always be true
formal egret
So if x = 1 and c = 5, $2^x > cx$ is false but that's irrelevant to the problem because of the general trend?
warm shaleBOT
dad
formal egret
Well, x has to be larger than 1 to be larger than a, but you get what I mean.
cerulean pine
the result at x = 1 is irrelevant, you just want to assure that there is a certain x (it can be as large as you want) for which 2ˣ>cx is true
it is the case since lim(x->+∞) 2ˣ/cx = +∞
formal egret
Oh of course.
Okay, that makes sense.
I can try to do the induction step from here. Thanks for setting me on the right track.
cerulean pine
np
obtuse pebbleBOT
Available help channel!
feral sentinel
obtuse pebbleBOT
feral sentinel
hi, can someone tell me how to do this?
my attempt was 40c1 * 39c1 * 38c1 = 59280
@compact shadow
tardy epoch
What about 1,2,1
feral sentinel
one moment
wait
oh
they said "consecutive numbers cant be the same"
so the "1" in ur example is separated
ohh
didnt think abt that
compact shadow
6(40C3)+2((40C2) I think
feral sentinel
What about 1,2,1
since we can account for 1 as well
so every term other than the first we can choose 39 others
not sure if the logic is right
compact shadow
6(40C3)+2((40C2) I think
First term is for three numbers being different, second term is for first and third numbers being equal
feral sentinel
im not sure either 
feral sentinel
wait why is it 6
compact shadow
3!
feral sentinel
oh
compact shadow
I forgot !
upbeat island
im thinking 40C1 * 39C1 * 39C1
i think this is the easiest way to count here
tardy epoch
correct answer is 60840 from ans key
This is right
You pick the middle number first, then the other two
compact shadow
i think this is the easiest way to count here
Yeah
upbeat island
no need for C, just counting principle, 40 numbers, 39 to exclude the previous, then 39 again, including the first and excluding the previous
tardy epoch
no need for C, just counting principle, 40 numbers, 39 to exclude the previous, ...
Also this, but not terribly important
loud egret
obtuse pebbleBOT
wooden cipher
damn this problem is chain rule hell
noble kindle
This should be illegal
obtuse pebbleBOT
@loud egret Has your question been resolved?
obtuse pebbleBOT
royal basin
yes
alpine trench
I need help with these questions please!
obtuse pebbleBOT
I need help with these questions please!
obtuse pebbleBOT
@alpine trench Has your question been resolved?
timid silo
its not an answer, they are telling you what to do
and its t=0 because its height initially (ie height at time 0)
obtuse pebbleBOT
@alpine trench Has your question been resolved?
alpine trench
and its t=0 because its height initially (ie height at time 0)
Ok thank you
proven apex
Hey im at this work experience place for construction management and they just put this work infornt of me and i have no idea how to even start
obtuse pebbleBOT
Hey im at this work experience place for construction management and they just p...
@proven apex Has your question been resolved?
obtuse pebbleBOT
@proven apex Has your question been resolved?
distant ravine
Linear Algebra:
I'm looking at the Kernel of a transformation T: R^4 -> R^4, and it says "Ker(T) = Span(v1,v2,v3)". I take the 3 vectors and put them in a matrix to check if they're linearly independent, and I get this matrix. What does this mean if I've got more rows than columns?
obtuse pebbleBOT
Linear Algebra:
I'm looking at the Kernel of a transformation T: R^4 -> R^4, and...
distant ravine
I'm thinking that this means "There are two vectors here that are linearly dependent"
Also I noticed that v3 is made up of v1+v2, but I'm not sure how to prove that using the matrix.
obtuse pebbleBOT
@distant ravine Has your question been resolved?
obtuse pebbleBOT
@distant ravine Has your question been resolved?
distant ravine
.close
dusky helm
is it correct to write something like $$y \in (-\infty, 0) \cup [1, \infty)$$ as $ y \in R - [0, 1)$
obtuse pebbleBOT
is it correct to write something like $$y \in (-\infty, 0) \cup [1, \infty)$$ as...
warm shaleBOT
KtorGray
dusky helm
basically can you subtract an interval instead of a specific set (e.g. {0, 1}, {0} etc.) from a defined set (i.e. R in this case)?
kind hawk
yeah
you can subtract arbitrary sets from other sets
might just not always have a nice "alternative representation"
visual wasp
i think this works well, when you write A\B or A - B it means the same thing
dusky helm
i think this works well, when you write A\B or A - B it means the same thing
oh
thank you :D
.close
north cove
I have the Diophantine equation y^2 = 1 + x + x^2
How would you solve it if it has integral solutions only
obtuse pebbleBOT
I have the Diophantine equation y^2 = 1 + x + x^2
How would you solve it if it h...
north cove
I have some sloppy rough work that I can share
That gives us (2I - 2x - 1) (2I + 2x + 1) where I is an integer
timid silo
note that 3 is prime
north cove
note that 3 is prime
Right
timid silo
so each factor needs to be 1 or 3
timid silo
yeah
timid silo
and also for 1 and 3 the otherway round
timid silo
yup
flat rune
by the way is there a reason why you wrote I instead of y suddenly?
north cove
by the way is there a reason why you wrote I instead of y suddenly?
oh yeah
by I I mean any integer
timid silo
sure thing
fierce lagoon
Holy shit the real slim shady!1!1!1!1
obtuse pebbleBOT
@north cove Has your question been resolved?
timid silo
Could someone help me decipher what tf this means?
“What is the angle of the circular section that you obtain by developing the lateral surface of a cone that has an opening angle Alpha?”
???
(i have the solutions too but I dont even understand the question)
obtuse pebbleBOT
Could someone help me decipher what tf this means?
“What is the angle of the c...
north cove
Holy shit the real slim shady!1!1!1!1
😎
tired skiff
,rccw
warm shaleBOT
trail musk
forest yacht
The expansion diagram of a cone is a circular section
This is a cone, its lateral and diameter forms an isosceles triangle with open angle α
obtuse pebbleBOT
@timid silo Has your question been resolved?
rustic rock
trying to learn how to use substitution for indices and e
question: 2e^x = 7 * squareroot(e^x) -3
obtuse pebbleBOT
trying to learn how to use substitution for indices and e
question: 2e^x = 7 * s...
high lily
consider substituting something that looks ugly
proven zephyr
is e = euler number?
high lily
in an attempt to get something that looks nicer
rustic rock
is e = euler number?
yes
proven zephyr
oh ok
tired skiff
gotcha
rustic rock
wth theres a praying mantis on my sheet of paper
tired skiff
lol what
rustic rock
when looking at it, what did you consider as a substitution
e^x
high lily
doing that gets you
2u = 7sqrt(u) - 3
you'll still have a sqrt, so that doesn't really improve it that much
rustic rock
doing that gets you
2u = 7sqrt(u) - 3
you'll still have a sqrt, so that doesn't ...
yeah thats why i didnt think it would work
it wont be even
high lily
consider doing a substitution so that your resultant equation no longer has a square root
rustic rock
because its still square rooted
high lily
instead why not substitute the whole sqrt(e^x)
rustic rock
wdym?
proven zephyr
sqrt(e^x) = u
proven zephyr
yes
proven zephyr
yes
rustic rock
ohhh
proven zephyr
rustic rock
and then you can just solve as a quadratic with 2u^2 -7u + 3
high lily
missing = 0
rustic rock
my b
proven zephyr
idk why i thought = 0 was a surprised face
drifting ether
Where is the mistake in the top way of simplifying?
obtuse pebbleBOT
Where is the mistake in the top way of simplifying?
tardy epoch
sqrt(-1) = +- i
But i=sqrt(-1)
You can use e^(i 3pi)=-1 and then take the square root of both sides
drifting ether
Ah ok. This is for working with matrices so I definitely shouldn’t put +_1 as a value in a matrix
Thanks
.close
distant moth
obtuse pebbleBOT
forest yacht
have you try the condition of continuity?
distant moth
have you try the condition of continuity?
yes
i somewhat feel that the options are wrong
forest yacht
how about option c
distant moth
i feel like it should be a=2, b= R-{0}
forest yacht
why b can't be 0
distant moth
cuz for the continuity when we are checking it for x-->0+
if b=0
then it comes out to be -1
which doesnt equal the other side limit
distant moth
i think it should be ax+bx^2/2 or something
what should be?
forest yacht
f(x)=ax+bx^2/2 when x>0
forest yacht
it's definitely not continuous at x=0
distant moth
it's definitely not continuous at x=0
it is for b not 0
wait
ok so let me explain my process, i found the lhd first, its 2
then i wrote the expression for the rhd
and then i wrote the 2 sides continuities
the left continutiy is 0
the right side has an expression
so i equaled that expression to 0
this is the same expression in the numerator of the rhd
so now the rhd is in a 0/0 form, and i used l hopital
that gave me a
so lhd=rhd, i got a=2
if its differentiable, shudnt it be continuous?
forest yacht
if f(x) is continuous then e^2x-1=ax+bx^2/2-1 at x=0, which leads to 0=-1
the question itself is wrong
forest yacht
if f(x) is continuous then e^2x-1=ax+bx^2/2-1 at x=0, which leads to 0=-1
$f(0)=e^{2\cdot 0}-1=\lim_{x\to 0^+}ax+\frac{bx^2}{2}-1$
warm shaleBOT
JellyShark
distant moth
so i equaled that expression to 0
i input a=2 here, and solved the quadratic, it gives an expression in b
yea this question looks wrong lol
weird, lets see if anyone faces this in the class as well
thank you
.close
timid silo
obtuse pebbleBOT
timid silo
yeah uh
my broswer had gliched
so when i screenshotted my questions it posted an old pic
.close
tardy epoch
i call this state clopen
when the bot tells you it's closed, but you can still type so it's open
timid silo
it says help-10 lunar moon
warm canopy
It's a ticking time bomb
tardy epoch
@timid silo bot is slow
warm canopy
It's a ticking time bomb
tardy epoch
sometimes clopen states take 5 minutes
warm canopy
it says help-10 lunar moon
Open a new one
timid silo
oh i see so i should open a new one and im fine
tardy epoch
Open a new one
clopen a new one
obtuse pebbleBOT
Available help channel!
warm canopy
oh i see so i should open a new one and im fine
Yes
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
barren bough
A and E are both false correct? just making sure bc it doesnt say it can be two answers lol
obtuse pebbleBOT
A and E are both false correct? just making sure bc it doesnt say it can be two ...
royal shard
if you take the limit as x goes to a, that is not the same as saying x=a
if we think about the limit, my visual intuition is that we never reach a but get really close
and for any point before a, i think it is obvious that f' exists
timid silo
yeah brain fart, you're right
barren bough
well the corner on f(x) means that theres a jump discontinutiy on f'
so the left and right limits will be different
royal shard
correct
barren bough
gotcha, so I was right
royal shard
uhm
there are two limits for f'
that is correct
but they do exist
and are defined
im not into this topic though
so i dont know what your consensus is here
my assumption was that lim as x goes to a is defined as coming from the left
If the question implies that both limits are the same, then the statement is false
kind hawk
for a limit to exist, it has to not matter how we approach the point
in one dimension that means limit from the left = limit from the right
in 2 or more dimensions its more complicated because there are a lot of ways in which you can approach the point
mint tendon
^^
obtuse pebbleBOT
@barren bough Has your question been resolved?
barren bough
so when the two limits are not equal, then what?
yeah, the limit does not exist
versed coral
can someone help?
obtuse pebbleBOT
can someone help?
timid silo
can someone help?
what have you tried
sterile wing
can someone help?
That's just Sum (-1)^(k) k((n+1)Ck)/(k+1)
in the expansion of (1-x)^n first differentiate, the multiplying by x and integrating then putting x=1 will give you the answer
versed coral
in the expansion of (1-x)^n first differentiate, the multiplying by x and integr...
oh yeah im gonna try it
sterile wing
Cool
versed coral
.close
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
candid canopy
50, 3 and 4
obtuse pebbleBOT
50, 3 and 4
candid canopy
i should know this
feeling mad stupid
but ion know what to draw
hi citrus yass its me again i wouldve been able to do all this last year 😭😭
upbeat island
so for these i like to pick a point and count to the reflection line horizontally, and then when i get to the line i count that same amount vertically away from the way i came in