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#help-36
tranquil pine
how it works '-
final saddleBOT
how it works '-
mellow axle
why are the two equal?
foggy minnow
think of u as u/1
tranquil pine
why are the two equal?
it is the question 
, actually the question is to show that a>0, a!=1, by the lim h->0 a^h-1/h=ln(a)
foggy minnow
Divide u/1 by the bottom fraction
faint locust
it is the question <:sadcat:547124024585486359> , actually the question is to sh...
Just simplify
mellow axle
LHS equals $u \div\frac{\ln (u + 1)}{\ln a}$
tranquil pine
faint locust
$\frac{u}{\frac{\ln(u + 1)}{\ln(a)}} = \frac{u \cdot \ln(a)}{\ln(u + 1)}$
Now, simplify the RHS too
tranquil pine
Now, simplify the RHS too
what is RHS
soft zealotBOT
M8 of 48
tranquil pine
The right hand side of the original expression
u* h?
faint locust
What is h?
tranquil pine
@faint locust
faint locust
how it works '-
Now why does that have to do with this
tranquil pine
Now why does that have to do with this
well, i found that equation from a forum answer for that question
"Being a>0, a!=1, show that
tranquil pine
yes
faint locust
Ok good, we can probably work with that
faint locust
$\lim_{x \to 0} \frac{e^x - 1}{x} = 1$. Now substitute $x = \ln(a^h)$ and "substitute" $h \to 0$ for $x \to 0$. We can do this because as $h \to 0$, we have $a^h \to 1$ and wso $\ln(a^h) \to 0$
faint locust
but how a^h - ln(e)?
What do you mean?
tranquil pine
What do you mean?
never mind
tranquil pine
$\lim_{x \to 0} \frac{e^x - 1}{x} = 1$. Now substitute $x = \ln(a^h)$ and "subst...
ok, but how it would be done as a answer to verify the equation
faint locust
Well, do what I described, plug in ln(a^h) into x and h -> 0 into x -> 0
tranquil pine
Well, do what I described, plug in ln(a^h) into x and h -> 0 into x -> 0
right 🤔
tranquil pine
Well, do what I described, plug in ln(a^h) into x and h -> 0 into x -> 0
can you help me with other question?
faint locust
We get $\lim_{h \to 0} \frac{e^{\ln(a^h)} - 1}{\ln(a^h)} = 1$.
Now what is $e^{\ln(a^h)}$?
tranquil pine
e^h*ln(a)?
tranquil pine
sorry, i dont know what was supposed to be
faint locust
It's x
tranquil pine
oh
faint locust
for ln(x) you basically ask "e to what value is x?"
And if you do e^("e to what value is x") you get x
tranquil pine
a^h
faint locust
Exactly
So we have $\lim_{h \to 0} \frac{a^h - 1}{\ln(a^h)} = 1$ now.
Now you said something about simplifying ln(a^h) earlier
faint locust
e^h*ln(a)?
Here
We do exactly that for the denominator
ln(a^h) = h * ln(a)
$\lim_{h \to 0} \frac{a^h - 1}{h \cdot \ln(a)} = 1$
tranquil pine
🤔, right
faint locust
Now, ln(a) is not dependant on h, so we might aswell get it out of the limit
Now can you see the next, last step?
faint locust
Look at this again and compare to what we have
tranquil pine
just get the 1/ln(a), and put in the RHS
final saddleBOT
final saddleBOT
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tranquil pine
$\lim_{x \to 0} \frac{5x - 1}{x} = \frac{e^{\ln(5x)}-1}{\ln(5)\cdot\ln(x)}=\ln(5)$
final saddleBOT
$\lim_{x \to 0} \frac{5x - 1}{x} = \frac{e^{\ln(5x)}-1}{\ln(5)\cdot\ln(x)}=\ln(5...
soft zealotBOT
Cosm
viral phoenix
x^4 + x^2 - 1
Wolfram says this is factorable, is there any way I could reasonably factor it myself?
final saddleBOT
x^4 + x^2 - 1
Wolfram says this is factorable, is there any way I could reasonab...
fathom walrus
try making a substitution
k = x^2
and see if u can make anything of it from there
@viral phoenix
viral phoenix
oh yeah
how did i not think of that
thanks
so we have $x^2 = \frac{-1 \pm \sqrt{5}}{2}$ so $x = \pm\sqrt{\frac{-1 + \sqrt{5}}{2}}$
soft zealotBOT
Captn
vagrant wraith
x not in row space of A implies x is in null space of A, could i just check whether this statement is true?
final saddleBOT
x not in row space of A implies x is in null space of A, could i just check whet...
desert mantle
what are your thoughts
vagrant wraith
I don't think so
I had a friend use this as part of a proof but I think they meant something else
It was related to proving something about linear regression that the vector $x$ that minimises the mean square error of $||Ax-b||$ is found by $A^TAx=Ab$, they proved it by saying that if $Ax$ minimises the mean square error, then for $w=Ax-b$, $w$ is not in the column space of $A$ and thus not in the row space of $A^T$ and thus in the null space of $A^T$
soft zealotBOT
Iusgnol
vagrant wraith
x not in row space of A implies x is in null space of A, could i just check whet...
So im wondering whether their proof was incorrect or whether there was some other additional detail that makes this statemenet true
desert mantle
well that proof doesnt work anyway cause w could be 0
vagrant wraith
oh its considering the case where
our vector b doesn't lie on the column space of A so w will be non-zero
desert mantle
I don't think so
try going through a few examples. e.g. 2x2 matrices with lots of 0 or 1 entries
vagrant wraith
$A^T = \begin{pmatrix}
1 & 2 \
2 & 4
\end{pmatrix} \$ and $w = \begin{bmatrix}
1 \ -1
\end{bmatrix}$, should suffice right as a counterexample
soft zealotBOT
Iusgnol
vagrant wraith
Clearly not in the rowspace and a simple multiplication gives us (-1,-2)
desert mantle
yes also works
final saddleBOT
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sacred geyser
can someone help me prove that triangle KIE is equal to triangle BIE (sas criterion)
final saddleBOT
can someone help me prove that triangle KIE is equal to triangle BIE (sas criter...
@sacred geyser Has your question been resolved?
lunar skiff
can someone help me prove that triangle KIE is equal to triangle BIE (sas criter...
what is given in the question?
sacred geyser
what is given in the question?
AB, AC are tangents. BK // ED, I is midpoint of ED, and BC is perpendicular to OA at H
final saddleBOT
@sacred geyser Has your question been resolved?
solar shard
solar shard
can someone please help with this question, im so lost
magic sparrow
.close
final saddleBOT
final saddleBOT
Available help channel!
oak inlet
final saddleBOT
pliant shore
If you're adding the terms from j = 1 to j = n
That's the same as adding the terms from j = 1 to j = n - 1
And then you still need to add on the last term, so the term when j = n
oak inlet
so I was thinkingof using the sum formula thing'
pliant shore
so I was thinkingof using the sum formula thing'
What's inside the actual sum doesn't matter
This question is a conceptual question
oak inlet
oh
pliant shore
It's testing your understanding of what summation means
oak inlet
I get a bit confused w the recursion part
pliant shore
I get a bit confused w the recursion part
Yeah so basically it's asking you what is ? in sum(n) = sum(n - 1) + ?
oak inlet
uhh
pliant shore
If you're adding the terms from j = 1 to j = n
Go back to this if you're stuck
final saddleBOT
@oak inlet Has your question been resolved?
oak inlet
Go back to this if you're stuck
could you explain the sum part doesnt matter
pliant shore
could you explain the sum part doesnt matter
You don't need to calculate the sum from j = 1 to j = n
Cause all of the answer options depend on y, which is the sum from j = 1 to j = n -1
oak inlet
so do i replace it with just y
oak inlet
You don't need to calculate the sum from j = 1 to j = n
would it end up being the third option?
pliant shore
would it end up being the third option?
Nope, that's a misunderstanding of the question
pliant shore
If you're adding the terms from j = 1 to j = n
If you go back to this
You have y + something else
The third option is not in that form
oak inlet
I see you're leaning towards y + n can you expand on the reasoning for it?
final saddleBOT
@oak inlet Has your question been resolved?
fathom walrus
callate monkey
pliant shore
<@&268886789983436800>
fathom walrus
@daring saddle
pliant shore
callate monkey
Wrong language
fathom walrus
is that portugese
pliant shore
It's Portuguese
fathom walrus
oh
pliant shore
Yes
fathom walrus
same thing basically
sturdy ocean
.close
winter skiff
How did the numerator 2 become -i1.09?
final saddleBOT
How did the numerator 2 become -i1.09?
thick breach
How did the numerator 2 become -i1.09?
after multiplying both numerator and denominator by the conjugate of the denominator
final saddleBOT
Available help channel!
dense coral
final saddleBOT
dense coral
!status
final saddleBOT
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
@dense coral Has your question been resolved?
magic sparrow
dense coral
https://media.discordapp.net/attachments/1064643750493691967/1226332777612312677...
thanks Austin
magic sparrow
by positive, does it mean positive semidefinite
wintry talon
Or positive definite?
dense coral
it means the operator is self adjoint and has no nonegative eigenvalues
dense coral
hm okay
magic sparrow
the best kind of matrix 😄
dense coral
we didn't call it that
but that makes sense
...I still don't know how to do this though 
magic sparrow
just spitballing
but maybe you could use that since A is positive semidefinite it has a square root
and if you could decompose A_k to have a square root aswell
then it too is psd
since PSD <=> has a square root
wintry talon
Also a thought:
You could simply think about what the adjoint means computationally for complex matrices
dense coral
conjugate transpose?
wintry talon
Then removing a row and column of the same order will maintain this property why?
It's a counting argument I believe
Yes
You just need to verify that removing the same row ri and column ci will maintain the hermitian property of the matrix
dense coral
why is that sufficient
wintry talon
A hermitian matrix is semi definite, you just need to show it's positive
dense coral
right... but me thinks the positive part is harder
I don't think I know how deleting a row and column affects the eigenvalues
dense coral
why is that?
wintry talon
Let's walk through this
What does removing a row from a matrix do
Essentially, it reduces the image of the map to a subspace (not necessarily proper) of the output space
dense coral
okay
wintry talon
Similarly removing a column simply restricts the map to a subspace (again not proper) of the input space
In some sense, we're considering the same exact map, just over a restricted input and output
dense coral
yes, I agree
magic sparrow
and a matrices eigenvalues being positive tells you that the output space is essentially in the same direction as the input space
make it smaller and the direction isn't impeded
wintry talon
Therefore, the sign of the determinant cannot change. Try to make this more formal
dense coral
and a matrices eigenvalues being positive tells you that the output space is ess...
what do you mean by direction?
wintry talon
Yes it is about the flippedness of the map
magic sparrow
it's a heuristic
wintry talon
If a man standing up becomes flipped upon transformation (and not rotation), the determinant is negative
dense coral
hm
we never really talked much about reorienting space via linear transformations
I only know a bit about it
hmm
magic sparrow
a visual example could be something like this aswell
positive numbers are numbers that have square roots
(real numbers)
visually
this is saying you can take the number
multiply by itself
and still be positive
facing the same (positive) direction
and in terms of matrices
PSD matrices are the only matrices with square roots
maybe it makes sense that we have the same intuition here
wintry talon
Yes I think it is similar
magic sparrow
but also because the eigenbasis determines your map, and for Av=lambda v if lambda > 0 then all you're doing is positively scaling your vector, this is basically in the same direction
dense coral
hm, it's a lot harder for me to think about complex matrices 😵💫
magic sparrow
and it determines it for your entire map
dense coral
but also because the eigenbasis determines your map, and for Av=lambda v if lamb...
okay, that makes some sense
wintry talon
In fact, it is true that the eigenvalues of a hermitian matrix are real
If I remember correctly
dense coral
yes, we talked about this property of self-adjoint ops
wintry talon
Yes
dense coral
so I have a positive matrix A
I want to show that removing the kth row/col still preserves it's positivity
so I need to show that the new matrix is self-adjoint and has positive eigenvalues
wintry talon
Yes
dense coral
self adjoint means I need to show that A = A*, where A* is the conjugate transpose
wintry talon
Yes
dense coral
and positive means showing... what exactly?
wintry talon
All it's eigenvalues are positive
dense coral
yes, but how do I do that?
magic sparrow
self adjoint means I need to show that A = A*, where A\* is the conjugate transp...
sorry, doesn't this guarantee positivity?
dense coral
why would it
final saddleBOT
@dense coral Has your question been resolved?
dense coral
.close
final saddleBOT
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distant ridge
Hi I want to know if this reasoning is correct
final saddleBOT
Hi I want to know if this reasoning is correct
trail mango
can't hurt to include what you're trying to prove
distant ridge
my bad
Prove that for any real x there is a unique integer n such that n <= x < n + 1
i meant to start with n>n'
so 0<n-n'
final saddleBOT
@distant ridge Has your question been resolved?
distant ridge
aight i give up
final saddleBOT
final saddleBOT
Available help channel!
fading fossil
Can't simplify rational function
If it looks like a 5 it's actually an S
Assessment questions, don't solve just help please 🥲
final saddleBOT
Can't simplify rational function
If it looks like a 5 it's actually an S
Assessm...
fading fossil
<@&286206848099549185>
sinful folio
is there a specific form it needs to be in?
fading fossil
Question says rational functions to partial fractions
sinful folio
do you know what partial fractions are?
fading fossil
do you know what partial fractions are?
Kinda, maybe, no, been following tutorials, just trying to answer the questions
I just think it's supposed to look simpler
sinful folio
so partial fractions are a specific form of writing rational functions
and there's a specific way of doing them
and tricks and what not
but ultimately, you want to get something of the form $$\frac{A}{x - a} + \frac{B}{x - b} + \frac{D}{x - d}$$
soft zealotBOT
Shell
sinful folio
the top may be functions of some kind
and the bottom (denominator) is just factors of x
it could be even just 2 or more terms
it depends on how many solutions the denominator has
i think in this case you might instead get something like $$\frac{A}{ax^2 + bx + c} + \frac{D}{x - d}$$
soft zealotBOT
Shell
sinful folio
because (x^3 + 10x^2 + 27x) has two complex roots
fading fossil
I think I got that with the x out the front
I'm pretty sure from what I got that I need to factor but can't
sinful folio
because (x^3 + 10x^2 + 27x) has two complex roots
yeah thats what i meant by this
you cant actually factor x(x^2 + 10x + 27) further
because x^2 + 10x + 27 doesn't have real solutions
I dont know if you're expected to know what complex numbers are yet, but if you dont this basically means there's no solutions
fading fossil
I mean it's University level so probably but I'm pretty sure that we're not supposed to for these questions
Going to assume that it's finished then, thanks
.close
final saddleBOT
final tangle
@fading fossilcomplex numbers aren't needed, since you can't factor the quadratic further, the decomposition will be in the form
$$\frac{As + B}{s^2+10s+27} + \frac{C}{s}$$
soft zealotBOT
ℝαμΩℕωⅤ
fading fossil
<@1202960668056354869>complex numbers aren't needed, since you can't factor the ...
Ok, can do
Thanks
Do I have to close it again?
.close
final saddleBOT
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fading fossil
Limit question, x to infinity
Am I on the right track, if yes what happens when dividing by x the dom goes to 0 before the num
If no, I have no idea, use the x^10?
final saddleBOT
Limit question, x to infinity
Am I on the right track, if yes what happens when ...
@fading fossil Has your question been resolved?
fading fossil
<@&286206848099549185>
Sorry, last one for a while
digital steeple
!original
final saddleBOT
Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
left wolf
hmm?
fading fossil
My working so far
digital steeple
My working so far
fading fossil
Original problem, 4.b.
left wolf
My working so far
how did the x^7 in the denominator inside the root come in the second step
fading fossil
That'd be cause I only half know what I'm doing and couldn't find anything similar online, nothing useful from learning material provided
left wolf
ill tell sth
take out x^ 12 in the numerator
simplify the thing inside the root in the denominator
the denominator inside the root will be x^2
take that out
you get x^6 in the denominator
fading fossil
Ok, I'll do the x^12 for numerator
Little not sure about the denominator but ill give it a shot, looking at it as an individual bit I think I have a tutorial for that
Thanks
left wolf
is answer 20/7?
fading fossil
I'll keep that in mind
I'm a pity points kinda guy, not expecting the best results
final saddleBOT
final saddleBOT
Available help channel!
mortal bison
Hi, I was wondering why can I not evaluate e^2x-1 using the wonderful limit and get the second fraction as 1/2x^2.
final saddleBOT
Hi, I was wondering why can I not evaluate e^2x-1 using the wonderful limit and ...
mortal bison
I get the wrong result when I do so
final saddleBOT
@mortal bison Has your question been resolved?
mortal bison
<@&286206848099549185>
dusky trellis
i guess you can, that's the right way...
final tangle
what wonderful limit
final saddleBOT
@mortal bison Has your question been resolved?
final tangle
product law doesn't apply as one of the limits doesn't exist
final saddleBOT
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Available help channel!
clear vessel
is the mark scheme wrong?
final saddleBOT
is the mark scheme wrong?
soft zealotBOT
proud plover
what is the relation between d and h?
final saddleBOT
pliant shore
The MS must be wrong
final saddleBOT
Available help channel!
untold hare
Is this the right way to prove this ?
final saddleBOT
Is this the right way to prove this ?
strange fractal
It is not a proof, but it is the correct answer to this question.
final saddleBOT
final saddleBOT
Available help channel!
true prairie
People
proud plover
I'd like your opinion
what is the purpose of the equation?
true prairie
Integration
trail mango
true prairie
Ik
trail mango
what is this question
true prairie
what is this question <:hmmge:1153028294552342579>
I myself not very sure
proud plover
I was thinking by parts as the first step
true prairie
proud plover
then I gave up
trail mango
what is “an opinion on where the power belongs”
vast summit
Where fo you think the power belong ?
i really hate this way of writing it, but most often I see that it usually means ln((1+x)^5)
true prairie
I mean that 5
true prairie
Is it for the parenthesis inside the ln
proud plover
😭
vast summit
which seems stupid since the 5 can just be taken out
true prairie
Or is it the whole ln that is raised to power 5
trail mango
whichever one makes it feasible to antidifferentiate
vast summit
yeah its stupid notation but its really common for some reason
true prairie
which seems stupid since the 5 can just be taken out
True
But yeah if it's inside then this will be very ez
proud plover
once u remove the 5 from the integral, follow up with by parts and the solution follows if i thought of it correctly
true prairie
It's probably outside
true prairie
But then the integral will become super easy
This sheet contains some hard integrals
solar glade
Do both, that way will be more entertaining
final saddleBOT
warm python
Is the second form even integrable though?
true prairie
Is the second form even integrable though?
Won't know till i try
solar glade
Yes
stoic pawn
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sonic crystal
a) Show that for all real numbers $x,\ e^{x}\ge x+1$.
b) Sow that for all $x>0,\ \ln x\ \le x-1$
final saddleBOT
a) Show that for all real numbers $x,\ e^{x}\ge x+1$.
b) Sow that for all $x>0,...
soft zealotBOT
water beam
sonic crystal
need help starting with a)
feral ginkgo
what is taylor expansion/definition of exponential in the form of a series
sonic crystal
i havent learnt taylor stuff yet
ocean lintel
just look at the derivative
sonic crystal
of e^x?
ocean lintel
they're equal at 0
Then if e^x grows faster it's greater on R+
If it increases slower on R- then it's greater there too
simpler option:
Did you study convexity ?
sonic crystal
simpler option:
Did you study convexity ?
i dont think im familiar with that term
sonic crystal
yeah i know concavity
ocean lintel
do you know how it relates to the tangent ?
sonic crystal
if a function f is concave up then the slope f' is increasing so the slope of the tangent is increasing
and then the opposite for concave down
ocean lintel
*to a single given tangent
sonic crystal
so how do i use this ?
ocean lintel
*to a single given tangent
so another statement
final saddleBOT
@sonic crystal Has your question been resolved?
modest roost
i create the table, just check if the mean, v and sd are correct and also if they are how should i plot it on a binomial graph please
final saddleBOT
i create the table, just check if the mean, v and sd are correct and also if the...
@modest roost Has your question been resolved?
modest roost
please can somebody help me
modest roost
is there someone on this discord ffs?
final saddleBOT
@modest roost Has your question been resolved?
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hidden prairie
im looking to find the derivative of this equation...I found the answer to be 2/(2+2x^2) but the website says the answer is 0..what is correct?
final saddleBOT
im looking to find the derivative of this equation...I found the answer to be 2/...
final saddleBOT
@hidden prairie Has your question been resolved?
hidden prairie
arctan gets 1/(1+x^2) arccot gets -1/(1+x^2)
I think arcctg is a typo and it means arctg or am I wrong?
warm ether
if the thing youre using says the answer is 0, then no
if it was a typo your answer would still be wrong
plus it would be weird to write it like that if so, therefore i highly doubt it
dense coral
I think arcctg is a typo and it means arctg or am I wrong?
arccotangent, probably
still dragon
How do I intergrate 1/144 ln|sec(12x)|tan(12x)?
final saddleBOT
How do I intergrate 1/144 ln|sec(12x)|tan(12x)?
still dragon
How do I intergrate 1/144 ln|sec(12x)|tan(12x)?
<@&286206848099549185>
lyric isle
<@&286206848099549185>
first you can take the 1/144 out the integration
still dragon
first you can take the 1/144 out the integration
$\frac{1}{144}\int ln|sec(12x)|tan(12x)$
soft zealotBOT
morphine_addiction
still dragon
Now what?
lyric isle
wait
simple kelp
What are the vertical lines? Module? Cos is symmetric
still dragon
What are the vertical lines? Module? Cos is symmetric
Modulus bc that's how it's written in the formula booklet
lyric isle
**morphine\_addiction**
you can integrate this by parts i guess
still dragon
you can integrate this by parts i guess
A 3rd time?
simple kelp
You can have Cos^2 in denominator if you break up the Tan and Sec if it helps
still dragon
You can have Cos^2 in denominator if you break up the Tan and Sec if it helps
But isn't the sec inside a ln so it would actually do anything
still dragon
Are they not?
Only sec
simple kelp
Why don't you just use the integral function of secant?
still dragon
Why don't you just use the integral function of secant?
?
simple kelp
Choose any
Tho your expression doesn't have the dx part, I'm not sure it's important
still dragon
Hiw would I use it tho bc I'm trying to intergrate ln(sec(12x))
simple kelp
Oh
lyric isle
Choose any
the sec is inside the ln
simple kelp
Forgot about the ln
still dragon
So how would I do the q?
lyric isle
any way
lyric isle
**morphine\_addiction**
use integrating by parts
still dragon
use integrating by parts
A 3rd time tho? The q only says to use it twice
simple kelp
It's beyond me but I tested your integral in the mathdf calculator and it found a solution so if you are really interested you may try understanding what it does. It shows steps
still dragon
It's beyond me but I tested your integral in the mathdf calculator and it found ...
Apparently this was the answer
still dragon
Learn from it!
Ngl I got no clue whats going on
simple kelp
I don't know almost anything about integrals but we can try to understand it together!
still dragon
.close
final saddleBOT
lyric isle
I don't know almost anything about integrals but we can try to understand it tog...
it needs practice
still dragon
I'm pretty sure u have to do u substitution along with double integration by parts
candid zephyr
I need help can somebody help me though 😭
Give me a radical expression using 3 different operations. Must consist of numbers and variables inside and outside the radicals. Its final answer should be negative.
lyric isle
Apparently this was the answer
i thinks if you derive the (sec(12x)^2tan(12x)) instead of integrating it would be easier
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tiny gorge
that $f^{n+1}(z)$ should have abs vals too, who wrote this
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Bungo
tiny gorge
a is where the series is centered (it would be a=0 in this case)
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grizzled token
a is where the series is centered (it would be a=0 in this case)
Could you explain why
I’m guessing because it’s near the point
0.1
tiny gorge
because the taylor series for sine (centered at x=0) is x - x^3/3! + ...
and they're using the approximation x - x^3/3!
tiny gorge
yea
grizzled token
Its just a cut off Taylor series/ Taylor polynomial
tiny gorge
i mean they don't explicitly say it but that's what you should deduce here
tiny gorge
Its just a cut off Taylor series/ Taylor polynomial
yep, so the taylor remainder theorem applies
grizzled token
And z is a number beteeen a and x
grizzled token
So I can use any number?
tiny gorge
no
you want to choose a worst case number for your bound
i.e.
choose the z between a and x that maximizes |f^(n+1)(z)| and use that maximum for your bound
that way it'll be true no matter which z it is
grizzled token
So between 0 and 0.1
grizzled token
N is 3
tiny gorge
they're approximating sine using x - x^3/3!
grizzled token
Right ?
tiny gorge
so you could use n = 3
however
the n=4 term for the sine taylor series is zero
so you could view it as them approximating using x - x^3/3! + 0
and use n=4
and you'll get a tighter bound
grizzled token
But does 3 also work
tiny gorge
they'll both be true
n=4 will give you a better (smaller) bound
who knows what the question intends
i suggest doing it with n = 3 first
grizzled token
So n = 3 a = 0 x = 0.1 and z is
tiny gorge
then you can try using n=4 and compare
tiny gorge
ok so you want to consider f^(4)(z)
so the first thing to ask is, what is f^(4)?
it's the 4th derivative of sine
which is what?
grizzled token
Sinx
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simple wave
Hi
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sacred lynx
In ABC right triangle AC=2+sqrt3 BC=3+2sqrt3. There is a cirlcle thats on the AC side it crosses in point M and toches the hypotnus in point D need to find the black parts area
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In ABC right triangle AC=2+sqrt3 BC=3+2sqrt3. There is a cirlcle thats on the A...
warm python
Try finding the area of the triangle first
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tall epoch
please help me out with this question! i was taught it earlier but forgot again :((
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please help me out with this question! i was taught it earlier but forgot again ...
ancient basalt
maybe use sin(63)?
tall epoch
maybe use sin(63)?
so sin(63) x 17/18?
ancient basalt
ig
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tranquil pine
welp
slow warren
So look at the pattern. How many more matchsticks are being added each pattern?
tranquil pine
+4
sleek jungle
Basically, look at the base number of sticks and how much are added each time
kind gorge
So, she has 6 matchsticks at the start, every next pattern requires adding 4
tranquil pine
ye
slow warren
Yes
So first subtract the matchsticks from the first pattern (6) from the total amount of matches
tranquil pine
oh k
slow warren
What that means is that .75 is remainder
tranquil pine
ye
slow warren
.75 of a matchstick isn’t 1 matchstick so that means that there are a few matchsticks left over
kind gorge
First pattern is the 6 you subtracted, so that already starts you at 1
You have 59 whole patterns that you can complete on top of that
You're 1 matchstick short of finishing the 60th pattern (excluding the 1st). That's why you don't round up
slow warren
So now you know that you have 59 full patterns excluding the first pattern of 6 matchsticks
So
sleek jungle
My head is going straight to summations and series lol
tranquil pine
- that pattren gives 60 with one remaining
tranquil pine
My head is going straight to summations and series lol
∑ n = 1 5 n = 1 + 2 + 3 + 4 + 5 = 15
slow warren
+ that pattren gives 60 with one remaining
Well 60 full patterns with some left over
kind gorge
Well, you have .75 pattern remaining, with each pattern (after the 1st) being 4 matchsticks, .75 of 4 is 3 matchsticks remaining
slow warren
But that is the most you can do with the amount of matchsticks you can have
tranquil pine
k
slow warren
It’s addition-
tranquil pine
no
sleek jungle
Assuming this is a circle, isn't the radius just 9+12?
No
tranquil pine
@sleek jungle how do
sleek jungle
Idk tbh
tranquil pine
damm
kind gorge
Alright, since I don't know how this works then, I'll try to learn something as well.
Why is this not just a quarter circle with 21cm being the radius?
Do we not know whether the corner in the bottom left is the center?
Am I misunderstanding the question?
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pliant birch
help (simplify)
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help (simplify)
brave shoal
help (simplify)
Have you tried anything yet
pliant birch
no
pliant birch
division
tranquil pine
what does : mean?
Ratio it means division we do
pliant birch
its just division aka /
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tranquil pine
A jacobson radical for a non-commutative ring can be defined as the intersection of the annihilators of all simple right R-modules
Since an R-module consists of not only a ring R but also a set (let's call it M), can the set M be absolutely any set whatsoever in the definition of a jacobson radical?
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A jacobson radical for a non-commutative ring can be defined as the intersection...
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weary junco
,, \lim_{x\to\infty} (2)^x
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,, \lim_{x\to\infty} (2)^x
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Oshiri
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Oshiri
weary junco
is this infinity, negative infinity or DNE?
pliant shore
this equals infinity right?
Yes
pliant shore
**Oshiri**
Well, when x is even here, then this goes to +infinity
But when x is odd, this goes to -infinity
So what can you conclude?
weary junco
DNE?
pliant shore
Yep
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bitter nexus
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bitter nexus
i don’t know where to start on this
wide grove
What do you know about reflections and compressions/dilations/stretches?
...and translations...
It's a puzzle. Start with the graph of f(x).
What can you do to it to get g(x)?
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hoary oxide
What is the difference between a formal inference proof and an english proof?
zenith pollen
formal inference probably doesn't have any words, just symbols and applying theorems to logic expressions
hoary oxide
gotchya
onyx wadi
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onyx wadi
I need help with this integral
vital surge
does that say this
vital surge
is this an estimation problem or is it legitimately asking u to evaluate that integral with a closed form
tranquil pine
what’s a closed form
onyx wadi
The question is to "calculate this integral"
vital surge
weird
vital surge
that makes more sense
since unless you use a calculator this just does not have a nice solution
hearty shore
Who is good at combinatorics pls i posted problem no one helped me to solve
hearty shore
I must find bijection beetween these two stets
Pk is subset with cardinal k
E={1,…,n}
zenith pollen
I must find bijection beetween these two stets
I helped with this problem a few days ago (A and B are the left and right sides of the equation) #help-8 message
unless you just want a bijection formula to go from x,S to z,P
hearty shore
This help to solve the problem yh but my teacher want us to solve with bijective proof
And damn thats so hard
hearty shore
unless you just want a bijection formula to go from x,S to z,P
How tf im supposed to find one?
This is similar problem but this one is kC(n,k)=nC(n-1,k-1)
zenith pollen
I think it matches because P_k-1 is just subsets of [[m-1]] so 2^(m-1)
if you take for granted both sets count subsets with a "leader", like {1,4,5} and {1,3}, then A is (leader,group) and B is (leader,subset of P_k-1), and you can adjust those a bit to get an actual bijection formula
you'll have to shift the P in (z,P) over a bit so that it goes from 1 to m and includes the leader z
but then it should just match up perfectly
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restive copper
not understanding what B is asking for
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not understanding what B is asking for
restive copper
primal sierra
it should cut once at the bottom and once at the top
(of the line)
what is the length of AB?
restive copper
what is the length of AB?
9.12
primal sierra
interesting
primal sierra
im like 90 percent sure that AB should be less than 9.5
restive copper
im like 90 percent sure that AB should be less than 9.5
i mean it is
primal sierra
but we need two intersections right
primal sierra
yea could be a scaling error
they shoulda given lengths
so that you could double check that your size is correct
restive copper
yea could be a scaling error
do u know which would be ideal
restive copper
lol
primal sierra
also what windows version is this my dude
restive copper
it's from pdf
restive copper
also what windows version is this my dude
re-printed and got AB at 10cm
restive copper
that should work ithink
did I do c right
primal sierra
seems like it works
I cant completely check your values, but you seem to know what you are doing
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autumn geode
$cos\left(\left(2n+1\right)\theta \right)=\sum _{r=0}^n\binom{2n+1}{2r}cos^{2n+1-2r}\left(\theta \right)\left(cos^2\left(\theta \right)-1\right)^r$
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$cos\left(\left(2n+1\right)\theta \right)=\sum _{r=0}^n\binom{2n+1}{2r}cos^{2n+1...
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nosqldb
autumn geode
need a hint to prove this
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autumn geode
<@&286206848099549185>
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dusky trellis
**nosqldb**
this becomes of the form of binomial expansion (RHS)... if you take (2n+1) = power (let m), and 2r as k basically so you see mCk . (cos theta)^m-k . (-sin theta)^k
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lyric oxide
Suppose f is a function from A to B were A and B are finite.
Prove:
-
If f is 1-1 and |A| = |B|, then f is onto.
-
If f is onto and |A| = |B|, then f is 1-1.
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Suppose f is a function from A to B were A and B are finite.
Prove:
1. If f is 1...
lyric oxide
I don't know how to start
iron knoll
Help me out with the english terminology here,
1-1 means injective and onto means surjective?
lyric oxide
Yes
iron knoll
- f is injective, |A| = |B|
Assume f is not surjective.
Then, there exists k>0 elements bi ∈ B, i =1,2,...,k such that there is no a ∈ A satisfying f(a) = bi;
If that is the case, for the other |B| - k = |A| - k elements of B, we have |A| elements of A.
You can't map them one-to-one since |A| > |A| - k (pigeonhole principle), which contradicts our hypothesis. Hence, f must be surjective.
You can do something similar with the item 2. Give it a try, assume it is not injective and get to a contradiction
is that clear?
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tranquil pine
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tranquil pine
,rotate
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tranquil pine
I'm so confused here what is going on is the answer wrong
I ended up getting negative one to the k over k factorial
static plinth
can you show your work
tranquil pine
can you show your work
static plinth
did you get the same derivatives as the answer showed above?
tranquil pine
Yes
static plinth
so by the taylor series formula we would have $\frac{(-1)^n n!(x-1)^n}{n!}$
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y0shi
static plinth
since our derivative is just that first bit there
and the factorial would cancel out
static plinth
yw!
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ivory fog
if x and x_0 are roughly similar, why do we draw the line after the first three terms
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if x and x_0 are roughly similar, why do we draw the line after the first three ...
ivory fog
and make a claim that everything after the third term is roughly 0
Is there a rule of thumb about how many terms are sufficient for an approximation that I am missing?
(For taylor expansions)
zenith pollen
basically just the first term that's interesting
usually you'd just go for the linear first term but it turns out to be 0
ivory fog
They used the first two non-zero terms (just summed up)
royal gust
Sounds like this is a heuristic argument anyways, so you could be correct that sometimes this argument doesn't work.
You might be looking for a proper proof.
ivory fog
I see, so it's just a hand-wavy explanation that says it's close enough
Was just making sure I wasn't missing anything thanks 🙂
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river thistle
I have a question regarding Euler's formula for geometry (if I'm correct, it's F + V = E + 2). Does my equation seem right? I can't help but feel like I'm missing something.
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I have a question regarding Euler's formula for geometry (if I'm correct, it's F...
river thistle
zenith pollen
looks good to me
weak tartan
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weak tartan
idk if im over complicating it but do i just multiply the orginial det by 2?
or mb im missing smth
proud plover
det(cA) = c^n det(A)
weak tartan
fire
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shy steeple
aight whats the technique for this one
final saddleBOT
aight whats the technique for this one
shy steeple
i don't really get what's happening in the solution, you can see it below
shy steeple
aight whats the technique for this one
my first thought is factor
,w factor x^2-3x-10
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shy steeple
,w factor x^2+6x+9
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shy steeple
aight so after factoring we get $\frac{(x+2)(x-5)}{x-5}(\ln(\frac{(x+3)^2}{x^2(x+3)}))$
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alshfik (ő/هو)
shy steeple
my idea is to see where the rational thingamabob might have an asymptote, but we don't need to worry about those because there aren't any (its x+2 with a hole at (5,7))
for ln i think we need to see where the inner rational function is either zero (so the ln has an asymptote) OR the inner function has an asymptote (which would correspond to the ln also having an asymptote)
is this idea correct?
that inner function doesn't equal zero anywhere because it has a hole somewhere on x=-3
so we can only look for its asymptotes, which is x=0
and then we'd like to double check with the other rational function here, make sure its defined, blah blah blah
you substitute, you get x=0 is the only asymptote
,w graph \frac{(x+2)(x-5)}{x-5}(\ln(\frac{(x+3)^2}{x^2(x+3)}))
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shy steeple
and it's wrong
oh I see
its fine if the ln's rational function is only approaching zero because the ln will still shoot off to negative infinity
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sand pumice
i got partial credit tho?
uh okay
i retried a different question and got it wrong still
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lament garnet
how is this a rectangular hyperbola? its been a while since ive done conics
fathom linden
it's in the form (x - a)^2 - c^2(y - b)^2 = k^2 which is a shifted hyperbola equation
you may want to review the conic equations
lament garnet
isnt it usally like this
pliant shore
isnt it usally like this
Yes
fathom linden
sure
fathom linden
isnt it usally like this
you could also have it in this form
there are several equivalent forms
fathom linden
the = 1 helps to dissect the aspect ratio of the hyperbola better
(directly from the a^2, b^2 terms)
pliant shore
Yeah so swapping x and y means that you reflect the hyperbola across y = x
fathom linden
!!!
lament garnet
so like this
lament garnet
Yeah so swapping x and y means that you reflect the hyperbola across y = x
(x) - (y) gives you east west and (y)-(x) gives you north south
then to graph that its 1/2 down, no x translation
asymptote is smth like
pliant shore
Yeah
lament garnet
right and for generalised form?
should be this? it works idk if any way to write neater
x-h = plusminus (by/a -k)
nvm that its not it
tranquil pine
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tranquil pine
So usually I'd try to find the equation of the curve and then equate that to the tangent but I don't know how
The MS subs in the x and y values but I don't know why that works
Ok I understand the bulk of it now
Still don't understand why you'd sub in the x and y though
What are they
exotic solar
I'm thinking about the Syracuse conjecture... There's what I found :
$U_0=p$
[ U_{n+1}=\frac{2+7U_n-(2-5U_n)\cdot(-1)^{U_n}}{4} ]
But how to find the explicit form, ie U_n according to n and p?
final saddleBOT
I'm thinking about the Syracuse conjecture... There's what I found :
$U_0=p$
\[ ...
worldly vale
Surround maths in $ like this $U_0 = p$
exotic solar
,tex [ U_{n+1}=\frac{2+7U_n-(2-5U_n)\cdot(-1)^{U_n}}{4} ]
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Roi H. Clem
Roi H. Clem
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quickdoom
quickdoom
signal vector
Even if you get a correct recurrence relation I don't think anyone here will be able to help you with solving it
exotic solar
OK thank you,
But I tried on an excel file, and it returned the same result for the 1000 firsts numbers :
Oh I found my mistake : I replaced the plus by a minus...
So the good expression is :
[ U_{n+1}=\frac{2+7U_n-(2+5U_n)\cdot(-1)^{U_n}}{4} ]
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Roi H. Clem
exotic solar
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