now focus on that blue component
again i must repeat, do not make any attempt at explicitly finding the inverse function f'(x)
#help-42
dull wagon
dull wagon
now is the time to do
Are you saying make 2x + cosx = 1
dull wagon
you are told the function is one-to-one
so there's only one solution
which can identifiable by inspection
next fossil
X = 0?
dull wagon
yes
next fossil
F inverse at 1= 0
potent lotusBOT
ℝαμOmeganato5
dull wagon
you should now have that
2 - sin(x)
that was f'(x) you took earlier
(i said no because that wasn't what i asked you to do at that stage)
use that to get f'(0) and you'll have your final result
next fossil
So it’s 1/2-sinx
next fossil
Ohhh
I see so I have to take the original make it equal to 1
Figure out what it equals to
Then take the derivative and plug in whatever the f-1 is equal to
Which is 0 and sin at 0 is 0
So I’m left with 1/2
dull wagon
given a formula, its usually a good idea to first plug in what you are given
and go from there
next fossil
Sounds good
calm coralBOT
@next fossil Has your question been resolved?
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indigo grove
just checking the resulting vector from cross multiplying
u<4,2,-1> and v<5,1,4> could be either
<9,-21,-6> for u x v and
<-9,21,6> for v x u
yea ?
velvet osprey
yes, order matters
calm coralBOT
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halcyon ore
Hi.
brazen elbow
heya
halcyon ore
.close 😭
calm coralBOT
brazen elbow
good question
unique jackal
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tough star
calm coralBOT
tough star
how do i find it algebraically?
hasty fiber
you can start by simplifying this
tough star
how
hasty fiber
recall your exponent laws, what is another way to write sqrt(3*x)?
tough star
(3x)^1/2
hasty fiber
sure but not quite what i'm looking for
you have an x^something in the numerator, it'd be nice to have an x^something in the denominator
tough star
(3^1/2)(x^1/2)
potent lotusBOT
Steakanator
hasty fiber
which leaves you with?
tough star
(x^3/2)/3^1/2
hasty fiber
indeed
hasty fiber
you do indeed
calm coralBOT
tough star
.reopen
calm coralBOT
✅
tough star
wait also
when you plug in x = 0 in the beginning
u get 0/0
which is indiscriminate
that leads to a hole at x = 0
but on desmos it shows as a defined point?
tall moon
desmos isnt really precise
it probably estimates it but it was so small it thought it was 0
i think
calm coralBOT
@tough star Has your question been resolved?
tacit moat
but on desmos it shows as a defined point?
if you slide a point to 0 though, it shows undefined
sudden yarrow
astral mural
Hello
astral mural
Why is it not possible to integrate $\int \frac{cos\left(x\right)}{x}dx$?
potent lotusBOT
Chaewon
astral mural
Are nonelementary integrals impossible to solve?
quaint sphinx
that highly depends on what you mean by “solve”
astral mural
Can we stick to knowledge in calc 3
quaint sphinx
then yes, treat them as impossible to solve
calm coralBOT
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tall moon
find all positive integers $(a,b)$ such that $9^a-3^a=b^4+2b^3+b^2+2b$
potent lotusBOT
skissue.in.a.teacup
tall moon
yea idrk
i tried mod 9 but that didnt really get anywhere
it can be factored into 3^a(3^a-1)=b(b+2)(b^2+1) but doesent seem helpful
mod 8 maybe?
ok yea no idk
gentle stream
find all positive integers $(a,b)$ such that $9^a-3^a=b^4+2b^3+b^2+2b$
wtf is this lol
i mean u can factor the RHS
tall moon
wtf is this lol
true
gentle stream
it can be factored into 3^a(3^a-1)=b(b+2)(b^2+1) but doesent seem helpful
i mean hopefully b, b+2 and b^2+1 are coprime or something close to coprime
then you can say i.e. b is divisible by 3^a or smth like that and looking at size should yield issues
tall moon
then you can say i.e. b is divisible by 3^a or smth like that and looking at siz...
err whar
gentle stream
as in hopefully b, b+2 and b^2+1 should be coprime
if they are coprime, then the fact that b(b+2)(b^2+1) is divisible by 3^a means one of them is divisible by 3^a
let's say b+2 is divisible by 3^a
then b ~ 3^a * stuff
so b(b+2)(b^2 + 1) ~ b^4 ~ 3^(4a) which is bigger than 9^a
tall moon
b and b+2 isnt coprime (gcd can be 2)
b and b^2+1 is coprime
b+2 and b^2+1 isnt coprime (gcd can be 5)
gentle stream
b and b+2 isnt coprime (gcd can be 2)
b and b^2+1 is coprime
b+2 and b^2+1 isnt...
oh perfect ty
lyric ravine
b and b^2 + 1 are coprime because 1(b^2 + 1) - b(b) = 1
gentle stream
well none of those gcds can be 3 so you can do stuff with that
gentle stream
if they are coprime, then the fact that b(b+2)(b^2+1) is divisible by 3^a means ...
anyway this would mean that the only possibility is b^2+1 is divisible by 3^a
do stuff with that and hopefully finish
tall moon
err this looks so wrong what
b^2+1=(k-1)3^a+3^a
b^2-(k-1)3^a=3^a-1=b(2+b)=b^2+2b
-(k-1)3^a=2b, since 3^a and 2b is positive, k-1<0 <=> k<1 which is not possible as k is a natural number, contradiction?
something aint right, 1,1 is a solution
which i would probably bet is the only solution
oh wait
so 3^a cant divide b^2+1, so 3^a must divide b(b+2), since only one of them can be divisible by a power of 3 (gcd is either 1 or 2), and b+2>b, so if 3^a|b then b>=3^a <=> 3^a(3^a+2)=3^a(3^a-1) which leads to a contradiction, so 3^a|b+2, b+2=k 3^a <=> b=k 3^a-2
×
k must be equal to 1 as b(b+2)=(k×3^a-2)(k×3^a)>=4×3^2a-2×3^a>3^2a-3^a
so b=3^a-2
b^4+2b^3+b^2+2b=b^2+4b+4-b-2=b^2+3b+2
b^4+2b^3-b-2=0
ok yeah thanks
.close
fluid cypress
Area under the curve
calm coralBOT
Area under the curve
fluid cypress
Ignoring part b
How would I do part a
the answer stated in the texbook is 0.2
i keep getting 0.72
velvet osprey
how are you getting 0.72
fluid cypress
i went ahead and calculated the area of a single cube
which was 0.2 x 0.2
0.04
then times that by 18 cubes
velvet osprey
uhhh lets see
the dimensions of these grid squares are 0.2 horizontally but 0.4 vertically by the looks of it?
which would mean 0.08 for each one
wait but are you sure your textbook is giving you 0.2 as the answer for this problem
cause that sounds ridiculously small
fluid cypress
waitttt hold up
velvet osprey
were you looking at the wrong answer key
fluid cypress
nvm
fluid cypress
were you looking at the wrong answer key
no
velvet osprey
oh then why the 0.2
fluid cypress
waittt
a sec
you might be right
Yea your right 🤦♂️
ive been looking at the wrong answer
thanks haha
velvet osprey
happens to the best of us
velvet osprey
ok so each square represents 0.08 units^2 yeah?
velvet osprey
this image is too crunchy for the paint bucket tool to work properly
but i count about 7+5+4+2+1=19 filled squares (more or less) which i've marked in green
and then some significant chunks of squares
which i think would probably add up to like 2 more
,calc 0.08*21
potent lotusBOT
Result:
1.68
so i guess that's close enough
calm coralBOT
Available help channel!
crystal breach
how do i know which point these pass through
calm coralBOT
how do i know which point these pass through
crystal breach
how many do i need for each
glad parrot
To be ok, i think 4 is enough
crystal breach
wait so i can just put any x value on this case and that gives me the y
4 for each?
glad parrot
wait so i can just put any x value on this case and that gives me the y
As long its not 0 yeah
glad parrot
Yeah
glad parrot
what does "on the same set of axes do"
On the same graph
You do the three function on the same graph
crystal breach
oh
also this topic is abt reciprocal and relation functions
are there any vids i can watch
glad parrot
You can check khan academy, but honestly this is not a really interesting exercice you have here
calm coralBOT
If you are done with this channel, please mark your problem as solved by typing .close
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velvet bronze
calm coralBOT
velvet bronze
Someone please help me with 2nd half of this question
glad parrot
Wait let me re-read it
dull wagon
can you shade in the region represented by the integral
velvet bronze
Wdym I just need to answer the 2nd half of the question that’s all the info we are given
Ima email my professor and ask him
I was wondering maybe I don’t know something
glad parrot
I think you should do what ramonov said before
dull wagon
to answer the question,
you need to know the link between integrals and (signed) area under a curve
first identifying the region of interest in the pic
will make the rest of the calculations trivial
fast marsh
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fast marsh
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cedar void
do you know the answer??
fast marsh
its (2)
velvet osprey
fast marsh
i found that x+y=16 and x^2+y^2 is smth like 225.xx or smth
which doesnt really make sense
cedar void
i found that x+y=16 and x^2+y^2 is smth like 225.xx or smth
that's exactly what I got.
fast marsh
fast marsh
that's exactly what I got.
yeah the question is not correct i think
cedar void
If option(2) is correct then the variance should be 8.16.
fast marsh
If option(2) is correct then the variance should be 8.16.
or 10.2 if sample varience
neither of which are even close to 27.xxx
cedar void
yeah
fast marsh
.close
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warm pewter
Need help with question b). Got the correct answer here but why is the displacement not 30.62499999999999 bc 30.625 would be the displacement from the top to the ground so using 30.625 means it already reached the ground.
plain bone
Need help with question b). Got the correct answer here but why is the displacem...
$30.624\overline{999} = 30.625$
potent lotusBOT
King Leo
velvet osprey
i mean if you wanna sign yourself up for using a dozen extra decimal places of precision that nobody else cares about
then nobody can really stop you
but part b just asks for the instantaneous velocity at the instant that the impact happens, but without accounting for the impact
warm pewter
Lmao right
Ohhh I GET IT NOW
Like I get they r technically the same number but i js didn’t get how u would calculate it
HAHAHAHAHAHAHAAHAHAHAA HELLO
Yes 😓 but in maths ughhhhhh why I need to learn physics
I do business further maths and maths 😓😓😓 do u also take a levels
If u took them now u gonna get A* everything based on what u helped me w
So what do you take?
Is it bad or good idk
WOOOOWWWWWWWWWWW
That’s so cool 😓 u doing like pure maths or engineering or?
I hate comp sci and physics so much 😓 I wanna study finance and accounting
Wowwwwww what kinds of things do u learn
AHAHAHAHAHAHAHAHAHA
Many people said that
But I just don’t like Econ too
🤣
This one?
LMAOAOAOAOA I’m just doing based on what I like tbh
Yep I guess so bc there will be calculus classes
Oh my how r u understanding all this 😃 I got so dizzy
I hope so 😃😃😃 right now I’m the worse in the class
Especially in physics bc I didn’t understand since gcses
Ooohhhh I’ve used it for the SAT but haven’t tried for other stuff
Did u take the SAT?
WHATTTTTTTTT
APs hard or nah
WHATTTTTTTTTT
NO WAY.
Howwww
Omgggg did it go well for you in the end?
Ok what this is so impressive. Graduating early is so impressive already and u also self studied
Aw man
WHAT THATS SO GOOD STILL?!!!
R u able to retake it any time
WHAAAAATTTT how much time in total?
I’m so sorry mannnn
but still the same uni that you would get in if u got a 5 right
I’m in year 12 rn
Ohhhhhh that’s goood stillllll
One more yrrrrr
But I’m going to uni early ahahaha
Wait so u went to uni early too
But like my smartness level is nothing compared to urs HAHAHAH
Asia ahahahahhaa
I would fail the APs
I’m not self teaching a levels but I’m failing
Yes u smart af
What really? A lot easier?
NAH AHAHAHAHAH
In my country
Ohhhhhhhhhhhh
How many classes did u choose?
That’s true
Whatttt
That’s cheating?
Oh you mean native to the language u took the test on
Ohhhhhhhhhhh I seeeeee
Then how did u manage to apply
Ahhhhhhhhhh
Was it easy for you
Nah bc I took first language of my native language but I got a worse score than English
And I’m not even that good at English
Oh yeah I forgot
AHAAHAHAHAHAHAHAAHAHA
Other than those subs what else did u do
calm coralBOT
@warm pewter Has your question been resolved?
wheat wyvern
calm coralBOT
wintry echo
Hi
wheat wyvern
Can I just u sub this?
ancient grotto
Can I just u sub this?
Yep.
wintry echo
Yes
wintry echo
Yes
wheat wyvern
got it thank you!!
wintry echo
You can also do u-sub initially
Indeed
ancient grotto
I will leave that as an exercise for you
wheat wyvern
You can also do u-sub initially
after factoring?
ancient grotto
You can try it if you want and I’ll post the answer
wheat wyvern
I'll try
ancient grotto
after factoring?
No, just let u=x^2-px
wheat wyvern
ancient grotto
what do I do with the p?
ax=ax-p+p
ancient grotto
so ax + p = ax + p?
$\int\frac{ax}{x^2-px}dx$
potent lotusBOT
;(
wheat wyvern
yes
ancient grotto
If $u=x^2-px$, then $du=(2x-p)dx$
wheat wyvern
du?
ancient grotto
aa
potent lotusBOT
;(
wheat wyvern
oh the dx is on the outside of everything
ancient grotto
So $\frac{a}{2}du=\qty(ax-\frac{ap}{2})dx$
potent lotusBOT
;(
ancient grotto
See where I am going now
wheat wyvern
ancient grotto
That works too
wheat wyvern
that is confusing me
Thank you for helping, but I think the partial fraction strategy is best for me in this case
ancient grotto
Thank you for helping, but I think the partial fraction strategy is best for me ...
Ok
I’m just saying, for things like $\int\frac{1}{x^3-1}dx$ you will need this technique
potent lotusBOT
;(
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solid horizon
Quick question, what does the ⊘ symbol mean between two vectors?
leaden thunder
show context
leaden thunder
this is from a paper?
not a math one it looks like
CS people make up their own symbols so i'd just read the linked paper Kingma & Ba
glass heart
unique jackal
smh when will they use 
as a symbol
shy estuary
X=[0,1] subset of R¹.
Ext(X)=(-inf,0)U(1,+inf)
Is this correct?
pallid halo
sure
remote mural
X=[0,1] subset of R¹.
Ext(X)=(-inf,0)U(1,+inf)
Is this correct?
yes
calm coralBOT
@shy estuary Has your question been resolved?
shy estuary
Thanks guys :)
drifting prawn
calm coralBOT
https://i.imgur.com/ECVDPUZ.png
drifting prawn
I thought that it would be increasing on (-inf,0)?
because the point 0 is the end of the increase
this is an error from cengage, right? lol
in my earlier answers, it said my answers were right for not including the last point in the increase/decrease
very odd...
edgy leaf
look at the definition of increasing/decreasing functions again
drifting prawn
https://i.imgur.com/a4fib05.png here's an example
drifting prawn
look at the definition of increasing/decreasing functions again
I will
the function f is said to be increasing when its graph rises and decreasing when its graph falls
edgy leaf
no
well it depends but in general increasing means non-decreasing and decreasing means non-increasing
constants satisfy both
drifting prawn
f is increasing when f(x)_1 < f(x)_2, whenever x_1 < x_2
edgy leaf
that is strictly increasing
drifting prawn
so f(0) = 5 for example so it would be false increase
like 5<5
uhh...
it's the same point it wouldn't make sense
x < x
"From the definition we see that a function increases or decreases on an interval. It does not make sense to apply these definitions at a single point."
untold drum
f is increasing when f(x)_1 < f(x)_2, whenever x_1 < x_2
there are different definitions for in-/decreasing. one of them is the definition you showed, which uses function values, another one would be via the first derivative.
in youre case you have to include 0 as for all x < 0 f(x) < f(0) whch meets the criteria in your definition.
edgy leaf
"From the definition we see that a function increases or decreases on an interva...
u are not applying it to a single point
drifting prawn
I don't understand because it's like doing 5 < 5 or 5 > 5 for decreasing definition
edgy leaf
https://i.imgur.com/ECVDPUZ.png
here take the domain [0, epsilon) for any epsilon>0
check if its decreasing
untold drum
I don't understand because it's like doing 5 < 5 or 5 > 5 for decreasing definit...
why 5< 5? first you need two different (!) x-values, x1 and x2, then compare f(x1) with f(x2).
drifting prawn
why 5< 5? first you need two different (!) x-values, x1 and x2, then compare f(x...
f(0) is the point at which the function is neither increasing nor decreasing
f(0) = 5
so
edgy leaf
u cant look at a point and say if its increasing or decreasing
drifting prawn
it would have to be x > 0 where it would start decreasing
edgy leaf
No
untold drum
the point is: your definition says f is increasing when f(x)_1 < f(x)2, whenever x_1 < x_2, so choose x1 and x_2 with x_1 < x_2
edgy leaf
is f(0)>f(x) for x>0?
drifting prawn
yes
edgy leaf
good, what does that mean
drifting prawn
it's decreasing
drifting prawn
same argument can be applied for the increasing side
https://i.imgur.com/BLyut5y.png why is this one different?
f(-inf)>f(-3) = inf > -1
-inf < -1
all true
for decreasing (-inf, -3)
but we use a ) for the -3 not a ]
<@&286206848099549185>
untold drum
https://i.imgur.com/BLyut5y.png why is this one different?
as said before there are different definitions of increasing. maybe the defintion for this example is another one.
drifting prawn
as said before there are different definitions of increasing. maybe the defintio...
im using the example from the book
and this video is of the book
untold drum
im using the example from the book
i dont know what you are saying. i dont know the book youre speaking of, i dont know the video you are speaking of. All i can say is there are different definitions, for example this one you posted earlier -> which results in including the endpoints in intervals. or a definition via first derivative which would result in excluding the endpoints.
drifting prawn
i dont know what you are saying. i dont know the book youre speaking of, i dont ...
I'm in pre calculus
the definition I gave is the only one for this section and should be for that video too
untold drum
then maybe the video is wrong.
drifting prawn
in my homework in the same section it says I'm doing it right excluding the transition point
even google gemini says the video is right
I'm going cray
loll
currently researching "continuity" and transition point etc
Why Include 0?
Continuity: The function is continuous at x = 0. There's no break or jump in the graph.
untold drum
even google gemini says the video is right
the video is right in the sense of the definiton with the first derivative.
drifting prawn
the video **is right** in the sense of the definiton with the first derivative.
unfortunately we haven't done derivatives yet so
Behavior Around 0: Even though the function's slope is momentarily zero at x = 0 (it's a critical point, specifically a maximum), we look at what's happening immediately before and after x = 0.
Just before 0 (e.g., x = -0.001): The function is slightly lower than at x=0, but it's still going uphill as you move towards 0. This means it's part of the increasing behavior.
untold drum
unfortunately we haven't done derivatives yet so
but gemini does.
drifting prawn
Just after 0 (e.g., x = 0.001): The function is slightly lower than at x=0, and it's going downhill as you move away from 0. This means it's part of the decreasing behavior.
this doesn't even make sense to me.
untold drum
this doesn't even make sense to me.
but its true.
drifting prawn
the only way I can kinda understand it is if 0 = 0.0000000000000000000000000001 for example
no idea
untold drum
i said it before, your definition requires two points to compare. so you need always a second value to compare with x = 0. if x < 0 then f(x) < f(0) -> increasing. if x > 0 then f(x) < f(0) -> decreasing.
and yes, with this definition i would say the answers in the other example are wrong.
drifting prawn
to make matters worse I think gemini is also giving me nonsense
devout sequoia
to make matters worse I think gemini is also giving me nonsense
!nogpt
calm coralBOT
to make matters worse I think gemini is also giving me nonsense
Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).
untold drum
to make matters worse I think gemini is also giving me nonsense
would never trust chatgpt or gemini oder such tools.
timid prawn
f(x)=(1-x) * e^(2x)
Find derivative of f at order 2024 and x=0
I somehow sometimes get the result 0 and 2^2024 by someways😇 😭
sweet stag
use leibniz rule
timid prawn
what have you done
Mac expansion of e^(2x) then I let (f^(2024)(0) (x^2024) (1-x) )/2024! = ...
sweet stag
huh?
velvet osprey
you mean maclaurin?
velvet osprey
Mac expansion of e^(2x) then I let (f^(2024)(0) (x^2024) (1-x) )/2024! = ...
but this notation is strange
leaden thunder
MacDonald expansion
velvet osprey
MacDonald expansion
not the place
timid prawn
If ya mean f^(2024)(0) ? I mean that the drivate order 2024 of f at 0
velvet osprey
no i mean the stuff that comes after it
it sounds like you have vaguely the right idea but can't put it into notation correctly
(also you swallowed about 1/3 of the letters in the word "derivative")
timid prawn
oh (1-x) e^2x ~ (1-x)+ (1-x)(2x) + .... (1-x)(2x)^2024)/2024!)+...
f^(2024)(0) x^2024 / 2024! = (1-x)(2x)^2024)/2024!)
velvet osprey
...
timid prawn
💀 I got D in calc 1 so I need to retake the programm for better score
velvet osprey
you should not be trying to write the whole thing as a single "calculation" with no words btw
timid prawn
calculus
timid prawn
nothing
velvet osprey
the idea is this: $$e^{2x} = \sum_{n=0}^{\infty} \frac{2^n}{n!} x^n$$
when you multiply this by $(1-x)$ and expand, there are two cases where an $x^{2024}$ term shows up:
\begin{itemize}
\item $1$ times the $x^{2024}$ term from $e^{2x}$
\item $-x$ times the $x^{2023}$ term from $e^{2x}$
\end{itemize}
potent lotusBOT
ann.in.a.teacup
velvet osprey
the coefficients of these are 2^2024/2024! and -2^2023/2023! respectively
add those, multiply by 2024!, and you have the answer to your problem.
calm coralBOT
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atomic python
hi
atomic python
could someone explain the concepts of stretching and adding in statistics
for example P(Y) = 1.5x + 5
how does it affect the mean, variance, and standard deviation of the original P(X)?
.close
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quaint sapphire
calm coralBOT
quaint sapphire
im asked to find the equation of the normal
ancient grotto
im asked to find the equation of the normal
!status
calm coralBOT
im asked to find the equation of the normal
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
so im just unsure if my differentiation is correct
ancient grotto
so im just unsure if my differentiation is correct
Send the work!
quaint sapphire
ancient grotto
This is not correct sadly
pliant coral
What is S lol
quaint sapphire
sorry i meant 5
ancient grotto
You either use product rule or use $\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}$
potent lotusBOT
;(
ancient grotto
What is S lol
That is a pretty readable 5
fickle musk
YOU can use quotient rule
pliant coral
I’d use the latter, don’t overcomplicate things
quaint sapphire
is x/x^1/2 the same as x(x)^-1/2?
ancient grotto
is x/x^1/2 the same as x(x)^-1/2?
Yes.
potent lotusBOT
;(
quaint sapphire
oh so x(x)^-1/2 is just x^-1/2
sorry i meant
x^1/2
okay thank you i got it
how do i find x
nvm
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quaint sapphire
calm coralBOT
quaint sapphire
is this correct
pallid halo
how did -x^-1 become +x^-2?
fickle musk
Did you differentiate 2
quaint sapphire
oh wait right
fickle musk
oh wait right
Also y ≠ dy/dx
barren burrow
ancient grotto
Again, you keep on either:
- Not using product rule
- Straight up miss the fact that you need to differentiate the other term as well (this is a result of 1.)
The first term is not correct. Try it again.
calm coralBOT
@quaint sapphire Has your question been resolved?
twilit lodge
haven't used polar functions since precalc so i need help with this problem bc i think im missing something fundamental
twilit lodge
the hardest part for me is figuring out the limits of integration
vocal summit
think about the values of theta where the radius becomes 0 to see the bounds
potent lotusBOT
tna
vocal summit
yes; also, there's 3pi/2.
vocal summit
well, you know the curve starts opening up at values of theta greater than pi/2
vocal summit
essentially, each point on the curve, you should be able to draw a straight line from the origin to it, where it's x coordinate should be $r\cos{\theta}$ and its y coordinate $r\sin{\theta}$
potent lotusBOT
fish
vocal summit
because of this, whenever a part of the curve has an x coordinate that's negative, that means theta should be greater than pi/2
personally, i like to think about it with lines from the origin that "generate" the curve, as seen in the picture.
manic vault
A
fossil forum
I dont understand how they know the discriminant is Less than 0 in the first line of the solution.
wide stirrup
what do u not get?
fossil forum
i just dont understand how they know the discriment is less than 0
with the first line of the solution
void umbra
Since it's positive for all values of x it will never intersect the x axis
wide stirrup
did u learn the rules
fossil forum
this part
wide stirrup
when its all real numbers
void umbra
Since it's positive for all values of x it will never intersect the x axis
So we know it has no solutions
void umbra
And as such the discriminant is less than 0
fossil forum
Since it's positive for all values of x it will never intersect the x axis
Wait how do we know its positive
void umbra
Stated in the question
wide stirrup
when u simplifyt the k over 8 is greater than 0
fossil forum
huh wait back to the basics whats a real number
void umbra
wide stirrup
its becomes k is greater than 0
devout sequoia
huh wait back to the basics whats a real number
not an imaginary one
void umbra
The quadratic is greater than 0 for all real values of x
fossil forum
not an imaginary one
so it can be negative too right?
void umbra
The quadratic is greater than 0 for all real values of x
So it's always positive
fossil forum
The quadratic is greater than 0 for all real values of x
ohh
devout sequoia
so it can be negative too right?
a real number can be negative yes
fossil forum
so if its greater than 0 we know that its always positive?
void umbra
Yes that's what positive means
fossil forum
ohhh i get it now
void umbra
Positive means greater than 0
calm coralBOT
fossil forum
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fossil forum
for the same question
the second part
should'nt it be k = 0 or k = 1
why is it just 1
prime skiff
if k = 0 its not a quadratic
void umbra
if k=0, the coefficient of x^2 is 0
fossil forum
k(k-1)=0
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wild marten
Most proofs I found of this online were through contradiction, would this still work?
calm coralBOT
Most proofs I found of this online were through contradiction, would this still ...
@wild marten Has your question been resolved?
tacit moat
Most proofs I found of this online were through contradiction, would this still ...
why's b a ratio of 2 integers?
it should be outside of Q, so there shouldn't be any p and q in the integers
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winged geyser
hello
winged geyser
id like to know if theres an easier way ti find this?
analog wasp
I would just plug in n=10
vocal pagoda
I would just plug in n=10
And that's the easiest way.
jaunty snow
that expression being the nth term means that you can replace n with any number to get the number in that place in the sdequence
so sub n=10 into it
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wild marten
most proofs i found online about this were through contradiction, I was wondering if this still worked
wild marten
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remote mural
How to solve these kind of questions ?
calm coralBOT
How to solve these kind of questions ?
bronze adder
Write each polynomial as a tuple in R^3, put them into the columns of a matrix, and reduce it to RREF
The LI columns of the RREF are also LI in the original matrix, so those form the basis
remote mural
tuple in R^3 means like this ?
a(1+x)+b(x+x^2)+c(2-x+x^2)+d(1+x^2)
(b+c+d)x^2 +(a+b-c)x +(a+c+d) ?
bronze adder
I mean the polynomial ax^2 + bx + c can be represented as (a, b, c)
do you know what an isomorphism between spaces is?
remote mural
nope
bronze adder
alr
potent lotusBOT
neon
remote mural
and its called as tuple ?
bronze adder
yeah the list (x1, x2, x3, ... , xn) is called an n-tuple
remote mural
i thought n-tuple means just n-dimensions
so the matrix would be
0 1 1 1
1 1 -1 0
1 0 2 1
remote mural
let's pretend that didn't happen
till where?
bronze adder
Nothing, it's gone now
bronze adder
I actually think you were meant to put the vectors in rows
bronze adder
,w REF of {{0,1,1},{1,1,0},{0,-1,2},{1,0,1}}
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tacit moat
,w RREF of {{0,1,1},{1,1,0},{0,-1,2},{1,0,1}}
bronze adder
if it is correct, then discount the vector corresponding to the (0, 0, 0) row
potent lotusBOT
bronze adder
so just RREF works huh
remote mural
its ref right?
remote mural
1 1 0
0 1 1
0 0 2
0 0 0
got it from this
after removing the 0 0 0 row
so now each row represents a vector,and all the 3 vectors are LI and span W ?
@bronze adder ?
neonnnnn
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@remote mural Has your question been resolved?
simple musk
did you found the basis? express vectors in S with respect to standard basis E = {1,X,X^2}
(1+X)_E = (1,1,0)
(X+X^2)_E = (0,1,1)
(2-X+X^2)_E = (2,-1,1)
(1+X^2)_E = (1,0,1)
then find a basis for the column space
remote mural
column space?
simple musk
just place the vectors as columns in a matrix and rref, the pivots give you which vectors are linearly independent
, w rref {{1,1,0},{0,1,1},{2,-1,1},{1,0,1}}^T
potent lotusBOT
remote mural
just place the vectors as columns in a matrix and rref, the pivots give you whic...
pivots?
leading 1's?
remote mural
so here the independent vectors would be 1+x , x+x^2 and 2-x+x^2
and hence they form the basis
simple musk
and hence they form the basis
yeah we found the basis
simple musk
so here the independent vectors would be 1+x , x+x^2 and 2-x+x^2
you can try to find the coefficients of the linear combination for which 1 + X^2 is in the span of this 3
do you want to find the coordinates?
remote mural
okay so
write down all the vectors in tuples and then
make a matrix with coloums as its vectors
Convert to RREF and then the coloums with leading ones would form the basis of Span of S
simple musk
more like, express S in standard basis of R2[X], which is E = {1,X,X^2}
place the coordinates wrt E as columns in a matrix, and rref, the columns with leading ones would indicate which vectors form a basis that spans S
remote mural
wouldnt it be R^3 ?
simple musk
did you found the basis? express vectors in S with respect to standard basis E =...
this step is crucial
E = {1,X,X^2}
(1+X)_E = (1,1,0)
(X+X^2)_E = (0,1,1)
(2-X+X^2)_E = (2,-1,1)
(1+X^2)_E = (1,0,1)
simple musk
wouldnt it be R^3 ?
yeah
remote mural
do you want to find the coordinates?
oh damn its 0,1/2,1/2
its the same as the one we got on RREF
remote mural
more like, express S in standard basis of R2[X], which is E = {1,X,X^2}
place t...
okay
thanks a lot!
one more thing
what does S \ {v} mean ?
simple musk
is set minus symbol
is like, v is linearly dependent of the other three
in this case v is 1 + X^2
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remote mural
calm coralBOT
remote mural
B AND C right ?
swift laurel
it's asking which is not correct
urban geyser
this is wrong, right?
because the last term in the top equation: (6/n) (n(n+1))/2
multiplied by the 2/n at the front of the equation
should not equal
(6n+1)/n ,
but should instead equal (6(n+1))/n , right?
urban geyser
velvet osprey
yeah looks like a missed pair of parentheses
velvet osprey
answer keys can contain errors
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tall moon
find all integers $a,b,c,d$ such that $a+b+c+d=ab=cd$
calm coralBOT
find all integers $a,b,c,d$ such that $a+b+c+d=ab=cd$
potent lotusBOT
skissue.in.a.teacup
tall moon
$\sqrt{abcd}=a+b+c+d\geq 4\sqrt[4]{abcd}\implies abcd\geq 256$ but this is only for pos
potent lotusBOT
skissue.in.a.teacup
tall moon
wlog a is 0, then either c and d is 0, wlog c is also the 0, them b=-d for one of them is 0
frozen rampart
Is it possible to sub smth for a
tall moon
call case 1 a,b,c,d>0
case 2 a,c>0, b,d<0
case 3 a,b>0 b,d<0
this is for case 3
bruteforced case 1, bottom counts for case 2 so i still dont count it
im not rly sure how to do case 2
wait hold on
ismt case 2 like just a mirrored case 1
oh wait no
slightly diffrent
lemme work it out
yes
because
i dont like coffee :(
fuck why doesent (1,-2,1,-2) show up
ab+cd-2a+2b-2c+2d=0 <=> (a+2)(b-2)+(c+2)(d-2)=-8 <=> (a+2)(2-b)+(c+2)(2-d)=8, sub -b=b' and -d=d' and you get (a+2)(b'+2)+(c+2)(d'+2)=8
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@tall moon Has your question been resolved?
calm coralBOT
@tall moon Has your question been resolved?
gentle stream
find all integers $a,b,c,d$ such that $a+b+c+d=ab=cd$
i guess you could rearrange to get
(a-1)(b-1) = c+d+1
(c-1)(d-1) = a+b+1
maybe size should get you a contradiction?
because we roughly have ab=c+d, cd=a+b
tall moon
err whar
tall moon
fuck why doesent (1,-2,1,-2) show up
honestly i just dont know why 1,-2,1,-2 doesemt show up here, and if i missed anymore
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tall moon
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gusty cliff
Can someone explain how shear works🤔
I saw the answer but I don’t understand why it’s like that
velvet osprey
shear*
"sheer" is a different word
anyway ok do you know your basics of linear algebra
gusty cliff
"sheer" is a different word
Yooo I’m sorry😭
velvet osprey
namely that the matrix of a transformation encodes the images of $\pmqty{1\0}$ and $\pmqty{0\1}$ under it
potent lotusBOT
ann.in.a.teacup
velvet osprey
more specifically
the 1st column of $M$ is the vector that $\pmqty{1\0}$ gets sent to, and the 2nd column is the same but for $\pmqty{0\1}$
potent lotusBOT
ann.in.a.teacup
velvet osprey
3blue1brown has a playlist called essence of linear algebra and he explains this basic fact about matrices pretty early on, i think chapter 3 or so
gusty cliff
**ann.in.a.teacup**
I do not understand what this means😔
gusty cliff
3blue1brown has a playlist called _essence of linear algebra_ and he explains th...
Lemme try
What do u mean by “ gets sent to”
Ohhh wait
velvet osprey
a transformation is a function from R^2 to R^2
it takes vectors as inputs and produces vectors as outputs
gusty cliff
So u mean that the first column affects (1,0) and the second column affects (0,1)
gusty cliff
it takes vectors as inputs and produces vectors as outputs
Mhm
velvet osprey
So u mean that the first column affects (1,0) and the second column affects (0,1...
i dont like that phrasing
but i guess sorta?
gusty cliff
It’s the way I understand it💀
So first column affects x direction and 2nd affects y direction?
velvet osprey
ggghhhh
yes kinda?
but like
i do think you should be sticking more closely to the wording that i gave you
because they explicitly tell you that (0,1) is sent to (5,1)
you gotta use that
gusty cliff
Okay it’s just I understand it better with caveman wording lol
Ohh I think I understand
Lemme draw
Is that what they meant?
It’s not to scale
But assume it is
velvet osprey
it's kinda so off the mark you cant even classify it as wrong
gusty cliff
Huhh😭
velvet osprey
the upshot is that since they say $(0,1)$ is mapped to $(5,1)$, the matrix for the shear is $\pmqty{* & 5 \ * & 1}$
where the asterisks are the other two entries which i am deliberately hiding for now
potent lotusBOT
ann.in.a.teacup
velvet osprey
no but seriously go watch essence of linear algebra
i dont think it will be a good use of my or your time to re-explain the stuff told in there
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tulip fossil
I'm trying to draw parallels between "a theorem which cannot be proven but is true" and a "language that is recognisable but not decidable"
tulip fossil
What's the intuitive way to draw such parallel
fringe reef
I'm trying to draw parallels between "a theorem which cannot be proven but is tr...
wdym by decidable language
tulip fossil
A Turing decidable language, recursive language
One for which we can construct such a Turing machine which not only tells if a strkng is in the language but also if a string is not in the language
tulip fossil
Recognisable langs are
Those which can only be recognised
If a strjng is in the lang it'll say sure it's in the lang
tulip fossil
But if a string outside the lang is given then it may or may not loo0
Loop
Loop forever
And not halt
Well tbh that's a bit of a jumbled way
Of saying Things
It's more appropriate to say that it's such a language for which we cannot construct such aturing machine which tells you if a string is outside or inside a lang
And can only construct that for telling if it's inside
tulip fossil
understood
Ok
Yes
Reiterating: I'm trying to draw parallels between "a theorem which cannot be proven but is true" and a "language that is recognisable but not decidable"
What's the intuitive way to draw such parallel
.close
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opaque yoke
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opaque yoke
could someone please help me with this problem
i canceled out the square root
so (x-2)-1
and then plug in f(x) into g(x)
so |x-4|-2-1=0
and i got 1 and 7 as the solutions
so the common difference is 6
and i got 13
but the answer is 9
pls explain
potent lotusBOT
Sepdron
opaque yoke
$\sqrt{(x-2)^2}$ what happens if x = 1?
it equals 1
oh
so it should be |x-2|
rigt
i forgot that rule
tacit moat
yup