so the limit doesnt exist
#help-26
cedar wagon
Have you study the monotonic behaviour of un somewhere ?
vital moth
yes
u_n isnt monotonic
but the even and odd subsequences are
here
this is proof that the even subsequence is increasing
i didnt prove that the odd subsequence is decreasing but a way similar to this proves that it is decreasing
cedar wagon
It doesn't converges actually
vital moth
basically it is literally the same process , you change the base case from u_0<u_2 to u_1<u_0 and then you flip every inequality here and change the subscipt of each u to the correct odd one
vital moth
It doesn't converges actually
i know that the whole sequence will not converge from the solution
but i am trying to reach that
but how did this happen
the limit of the even subsequence must exist
and it cant be this number
since u_n<1
cedar wagon
This works iff the sequences is converging iirc
vital moth
This works iff the sequences is converging iirc
but it is
the even subsequence is convergent
since it is increasing and bounded above
cedar wagon
It doesn't imply that the sequence is no ? Or im tired
vital moth
it doesnt
but the limit that we were looking at is for the even subsequence no ?
so what does the original sequence have to do with that
cedar wagon
How do you get the fixed point of the even sequence ?
vital moth
i get it by using the fact that is convergent
and then naming the limit l_1 and using the theorem here
cedar wagon
But a_2n will be a different sequence from a_n
You can't take the expression of a_n
Or u_n
vital moth
ah wait
i get what you mean
the function that i need here is fof not just f
since in the subsequence {u_{2n}} the subscripts of the terms differ by 2
so using the given recurrence relation, $u_{2n+2}=(1-(1-u_{2n})^2)^2=u_{2n}^2(2-u_{2n})^2$ right?
thorny flameBOT
cedar wagon
Thats too much 2 
thorny flameBOT
cedar wagon
Ive never used even and odd subsequence, i usually stopped at sequence being not monotonic so can't tell about converging caracteristic
But yeah for sure
You can't use u_n expression as u_2n
vital moth
Ive never used even and odd subsequence, i usually stopped at sequence being not...
if i write this the professor will say that he couldnt decide the mark and in the end chose to put 0
cedar wagon
To me it just seemed like turning around the pot and after 10 lap, go in it
But you're surely taught a method that is working
Thats just looking bad long
vital moth
so from here $l_1=l_1^2(l_1^2-4l_1+4)\implies l_1^4-4l_1^3+4l^2-l_1=0$ and the question had a hint that gave the roots of this which are $0,1,\frac{3\pm\sqrt{5}}2$
thorny flameBOT
vital moth
But you're surely taught a method that is working
at first i didnt know why it worked
but i asked here and it turns out that it works since these 2 subsequences cover the whole sequence
cedar wagon
,w (3-sqrt(5))/2
thorny flameBOT
vital moth
(3+sqrt(5))/2 is excluded
cedar wagon
Pog
vital moth
The minus is actually fitting in the interval
yes it fits
now u_{2n}>u_0 for all n in N since {u_{2n}} is increasing
so u_{2n}>1/2>(3-sqrt(5))/2>0
cedar wagon
Indeed
cedar wagon
now u_{2n}>u_0 for all n in N since {u_{2n}} is increasing
You don't include 0 in N ?
vital moth
i include it
vital moth
So its >=
ah yes indeed
now for {u_{2n+1}}, u_{2n+1}<1/2 and u_{2n+1} is decreasing
so l_2 is either 0 or (3-sqrt(5))/2
hmmm
cedar wagon
Better bet on 0 tho
vital moth
but why
cedar wagon
You don't know much between (3-sqrt(5))/2 and u_2n+1
vital moth
yes so what should i do
vital moth
cedar wagon
Can you prove that there exist no n odd in N such that Un < (3-sqrt(5))/2 ?
In that case you could say that this is a fine main character for the limit
Else its 0 the best one
vital moth
Can you prove that there exist no n odd in N such that Un < (3-sqrt(5))/2 ?
do you mean >?
rigid ivy
@vital moth sorry I am busy with something buy I'll catch up when I can
vital moth
<@990146348517060619> sorry I am busy with something buy I'll catch up when I ca...
np you dont have to be sorry
cedar wagon
do you mean >?
I mean that for all odd n, Un is > than the ugly term yeah
vital moth
anyway i am near the end thanks to the help of msg you can check the solution later if you want
rigid ivy
Have you proven that every bounded sequence has a converging subsequence?
vital moth
Have you proven that every bounded sequence has a converging subsequence?
not sure but probably yes
cedar wagon
He found the even one
vital moth
I mean that for all odd n, Un is > than the ugly term yeah
let me give it a thought
cedar wagon
||I cheated and there is a n||
Think of what you've done with the even sequence
Im heading out but using the limit of even sub seq gives u a term of odd sub seq that will be interesting for proving what we need, gl
vital moth
Im heading out but using the limit of even sub seq gives u a term of odd sub seq...
i will keep this in mind tysm
if i reach the solution can i ping you to check it ?
ah maybe i got it
$l_2=l_2(\lim_{n\to\infty}u_{2n}-1)(l_2^2-3l_2+1)=0$?
thorny flameBOT
neon iron
im scared of calculus now
vital moth
im scared of calculus now
btw thats analaysis not calculus
neon iron
oh
vital moth
calculus is only computational
vital moth
you can say that analysis is the proof version of calculus and more
no need to be scared of it
neon iron
are you looking at this?
vital moth
all of advanced mathematics is like this and more
neon iron
i think im going to be scared
vital moth
are you looking at this?
that is considered on the normal to easy side ngl
neon iron
and im still in algebra II 😭
vital moth
and im still in algebra II 😭
no need to be in a rush
you can accelerate your learning if you self study ahead
but you dont need to do that
do that if you like to do it
so i suppose you are in high school ?
vital moth
ohhh
this server is like a gold mine
for people who like math/want to discuss math
you can probably always find people knowledgable in the area of math you are looking for here
neon iron
in which year are you
10th; sophomore
neon iron
lolloolo
vital moth
what are you planning to do in university
vital moth
lolloolo
yea its not like i am in my 70s
i just finished high school not long ago and now i will be entering university
vital moth
<@&286206848099549185>
rigid ivy
<@&286206848099549185>
@vital moth catch me up
vital moth
<@990146348517060619> catch me up
alright
so i proved that both the odd and even subsequences converge
vital moth
then i used this and the theorem you mentioned at the beginning
vital moth
i reached this and oncluded that l_1=1
then i used the same method
for the odd subsequence
rigid ivy
so i proved that both the odd and even subsequences converge
what's the problem again?
vital moth
but this time i was left with 2 choices that cant be turned into 1 as easily as for l_1
given a sequence ${u_n}$ defined by the recurrence relation $u_{n+1}=(1-u_n)^2$ with $u_0=\frac 12$ and in a previous part i proved that $0<u_n<1\ \forall n\in\mathbb{N}$ \ Study the convergence of ${u_n}$
thorny flameBOT
vital moth
thats the main problem
rigid ivy
This may not converge
vital moth
so basically the goal i want to reach now is to find l_2 so that i can compare it with l_1 and check if {u_n} converges or no
vital moth
This may not converge
indeed this will not converge
i am in the last step now
which is to find l_2 which is the limit of the odd subsequence
thorny flameBOT
SWR
vital moth
so again using this and the theorem you mentioned , $l_2=l_2^2(l_2^2-4l_2+4)$ so that $l_2^4-4l_2^3+4l_2^2-l_2=0$
rigid ivy
,w plot (1-(1-x)^2)^2
vital moth
the roots are 0,1,(3+-sqrt(5))/2
rigid ivy
My theorem assumes your series converges, which it doesn't
vital moth
(3+sqrt(5))/2>1 so it cant be l_2
rigid ivy
unless you mean the even sequence
vital moth
unless you mean the even sequence
i mean the odd sequence
its the same way i used to find the limit of the even sequence too
so your theorem works here since both of these sequences converge
vital moth
**SWR**
thats why i am using this relation instead of u_{n+1}=(1-u_n)^2
so what happened so far is as follows :\${u_{2n}}$ and ${u_{2n+1}}$ are convergent. I want to find there limits. Since ${u_n}$ isnt necesarily convergent it is not possible to use your theorem for the original recurrence relation given by $u_{n+1}=(1-u_n)^2$ but since the odd and even subsequences converge then it is possible to use your theorem for the recurrence relations given by $u_{2n+2}=(1-(1-u_{2n})^2)^2=u_{2n}^2(2-u_{2n})^2$ and $u_{2n+3}=(1-(1-u_{2n+1})^2)^2=u_{2n+1}^2(2-u_{2n+1})^2$. Now let $l_1$ and $l_2$ be the limits of ${u_{2n}}$ and ${u_{2n+1}}$ respectively. the above new recurrence relations are continuous functions of $u_{2n}$ and $u_{2n+1}$ so it is possible to use your theorem for them. so now to find $l_1$, we have $f(l_1)=l_1\implies l_1^2(2-l_1)^2-l_1=0\implies l_1^4-4l_1^3+4l_1^2-l_1=0\implies l_1=0$ or $l_1=1$ or $l_1=\frac {3\pm\sqrt 5}2$, now $\frac{3+\sqrt 5}2>1$ so $l_1\neq \frac{3+\sqrt 5}2>1$ and $u_{2n}>u_0=\frac 12$ since ${u_{2n}}$ is increasing, then $l_1\neq 0$ and $l_1\neq\frac{3-\sqrt 5}2$. Hence , $l_1=1$
using the same way i want to find $l_2$, $l_2\neq\frac{3+\sqrt 5}2$ since $\frac{3+\sqrt 5}2>1$. and combining the 2 facts that $u_{2n+1}<u_0<1\ \forall n\in\mathbb{N}$ and that ${u_{2n+1}}$ is decreasing implies that $l_1\neq 1$ which leaves us with either $l_2=0$ or $l_2=\frac{3-\sqrt 5}2$
my problem is finding which one it is
why is this bad tex lol
thorny flameBOT
thorny flameBOT
vital moth
there was a small typo
for the last 2 i wrote l_1= instead of l_2=
so i corrected it
so thats pretty much everything (excluding the proof of monotoneness of both subsequences)
ah wait a second
isnt it sufficient to say that u_1=1/4<(3-sqrt(5))/2
and since {u_{2n+1}} is decreasing
then u_{2n+1}<u_1<(3-sqrt(5))/2 for all n in N
and also since it is decreasing
the limit cant be (3-sqrt(5))/2
because u_{2n+1}=<1/4 for all n in N
which means that l_2 must be 0
it cant be none of the above 4 values since it converges and if it converges it must be one of them
@rigid ivy what do you think
thorny flameBOT
SWR
vital moth
yes
rigid ivy
What do your even and odd subsequence converge to you said?
vital moth
because u_{2n+1}=<1/4 for all n in N
this has to be =< not just < i fixed it
vital moth
What do your even and odd subsequence converge to you said?
the even one converges to 1 and the odd one converges to 0
which means that {u_n} doesnt converge
rigid ivy
the even one converges to 1 and the odd one converges to 0
and you seek to prove this now, correct?
rigid ivy
you may be overthinking your problem very hard
vital moth
but i want you to check if i am correct or no
rigid ivy
I think you can just use the standard definition of convergenece
rigid ivy
but i want you to check if i am correct or no
So you want to see if your approach works?
vital moth
the even one converges to 1 and the odd one converges to 0
in principle it works if i reach this conclusion
but if you want you can double check if i did something wrong or missed something along the way
vital moth
I think you can just use the standard definition of convergenece
is that so
rigid ivy
sure okay
rigid ivy
is that so
yeah it's very easy
vital moth
is it a proof by contradiction ?
rigid ivy
My theorem helps you find convergence. Not so useful for proving nonconvergence
vital moth
ohh wait
$\exists\varepsilon >0\ \forall\delta >0\ \forall n_0,n\in\mathbb{N}$ such that $n\geq n_0\implies |u_{n+1}-u_n|\geq\varepsilon$
thorny flameBOT
rigid ivy
vital moth
if i can prove this then i will have proved that {u_n} is not cauchy
which means that it is not convergent
vital moth
so take $\varepsilon=2$ then $|u_{n+1}-u_n|=|u_n^2-3u_n+1|>|-3u_n+1|>|-3+1|=2=\varepsilon$
thorny flameBOT
vital moth
is this it ?
thorny flameBOT
SWR
vital moth
ah yes you are right
rigid ivy
This is wrong:
$$|u_{n+1}-u_n|=|u_n^2-3u_n+1|$$
thorny flameBOT
SWR
vital moth
isnt u_{n+1}=u_n^2-2u_n+1
vital moth
I thought it was n+2
ohh
rigid ivy
yeah that is correct
vital moth
then where did i go wrong
rigid ivy
How did you get this?
$$|u_n^2-3u_n+1|>|-3u_n+1|$$
thorny flameBOT
SWR
rigid ivy
by using u_n^2>0
this logic isn't sound
vital moth
why
rigid ivy
why do you think it is?
vital moth
i mean u_n^2-3u_n+1>-3u_n+1
ah wait
maybe the lhs here is >0 and the rhs is < 0?
is that why my claim isnt correct ?
rigid ivy
i mean u_n^2-3u_n+1>-3u_n+1
$u_n^2-3u_{n+1}>-3u_{n+1}$
thorny flameBOT
SWR
rigid ivy
Use triangle inequality
vital moth
so you mean $|u_n^2-3u_n+1|\leq |u_n^2|+|-3u_n+1|<|-3u_n+1|$?
vital moth
Use triangle inequality
alright this works
rigid ivy
so you mean $|u_n^2-3u_n+1|\leq |u_n^2|+|-3u_n+1|<|-3u_n+1|$?
this isn't trianlge inequality. You have it backwards
vital moth
ah wait i meant to write =<
thorny flameBOT
rigid ivy
ah wait i meant to write =<
And hence why it does not work
vital moth
ahaa i see
rigid ivy
It doesn't prove anything for you
rigid ivy
lower bound the even sequence and upper bound the odd sequence
try this 👆
vital moth
lower bound the even sequence and upper bound the odd sequence
even sequence is bounded below by 1/2
and the odd sequence is bounded above by 1/2
now it is correct
rigid ivy
and the odd sequence is bounded above by 1/2
Find a better upper bound
rigid ivy
there you go
that's basically your proof
every odd term is less than 1/4 but every even is greater than 1/2
vital moth
ah yes and that doesnt even require proving anything else
like the odd sequence is decreasing
and the even one is increasing
oh wait it does
right ?
or maybe i can just prove that these are bounds of both sequences using induction without mentioning increasing/decreasing ?
topaz sinewBOT
@vital moth Has your question been resolved?
ionic plaza
can anyone help me work these step by step and give me hints to solve these?
smoky sparrow
can anyone help me work these step by step and give me hints to solve these?
you need your identities
try factoring 9x^2 - 6x + 4 and 3x^2 + 13x + 4
for 3x^2 + 13x + 4 in fact you must have (3x .....)(x .....), because 3 is a prime number
and then you also need a^2 - b^2 for 9x^2 - 1
and a^3 + b^3 for 27x^3 + 8
smoky sparrow
what identities?
a^2 - b^2 = (a + b)(a - b), difference of two squares
a^3 + b^3 = (a + b)(a^2 - ab + b^2), sum of two cubes
ionic plaza
which method of factoring should i choose for the top left beginning with 9x^2
smoky sparrow
which method of factoring should i choose for the top left beginning with 9x^2
there's a million ways to factor
ionic plaza
multiply A and C and see what adds to B?
smoky sparrow
hold on 9x^2 - 6x + 4 can't be factored
ionic plaza
how do you figure?
smoky sparrow
the discriminant is less than zero, don't worry if you don't know what that means
ionic plaza
oh ok
smoky sparrow
the discriminant is the part under the square root
so b^2 - 4ac, and that is negative
ionic plaza
i see
smoky sparrow
so yeah just leave it
ionic plaza
for the quadratic eqn
ok so now what should my thinking be
approaching the rest?
can you hint me along?
smoky sparrow
okay now try factoring 3x^2 + 13x + 4
ionic plaza
3 times 4 equals 12
smoky sparrow
you must have (3x ......)(x ......)
smoky sparrow
3 times 4 equals 12
yep!
ionic plaza
12 nd 1 add to 13
smoky sparrow
cool, exactly
ionic plaza
3x+12 and x+1
smoky sparrow
3x+12 and x+1
okay this is not correct
it's (3x + 1)(x + 4)
by FOIL, the outside is 3x * 4
the inside is 1 * x
smoky sparrow
i dont see that automatically
you have to learn to put the terms in the right places
ionic plaza
that may be easy for you
but not for me
how do i approach beginning to see how it works
smoky sparrow
how do i approach beginning to see how it works
try learning the box method I guess
ionic plaza
what do you do?
smoky sparrow
read the page to find out
smoky sparrow
no
ionic plaza
well, how do you think about it to approach it?
smoky sparrow
but then I'm trying to look at it from your perspective
smoky sparrow
well, how do you think about it to approach it?
that's not going to help you
cause I explained how I would do it and you were confused
smoky sparrow
when should i look to use this method?
whenever you have a quadratic and you want to factor it
the box method is most useful for a not equal to 1 however
you had 3x^2 there, so when you don't have x^2 basically
ionic plaza
i am asking chatgpt to give me examples to work using the box method and im dominating them now.
thank u
smoky sparrow
no worries!
ionic plaza
ok now, diff of cubes on denom on left
let me think for a sec
(9x^2+4)(3x+2)
maybe?
oh wait...there is a sum of cubes formua
smoky sparrow
a^3 + b^3 = (a + b)(a^2 - ab + b^2), sum of two cubes
nope
ionic plaza
formula
smoky sparrow
yeah use the formula
ionic plaza
ok one moment
ok this is where i am at
what is my thinking now?
how do you turn a quadratic expression into vertex form from 3x^2+1
smoky sparrow
,w (3x + 2)(9x^2 - 6x + 4) simplify
thorny flameBOT
smoky sparrow
aside from that small mistake where you didn't square properly, (3x)^2
cancel cancel, you're nearly there
ionic plaza
oh yes i solved it, its x+4 over 3x-1)(3x+2)
that was a long problem lol!
ok now want to look at this one?
smoky sparrow
okay cool let's move onto q4
smoky sparrow
try multiplying the numerator and denominator by xy
ionic plaza
is that because it is a GCF of both?
smoky sparrow
yeah, so you have x and y as denominators
you need to multiply by xy to clear both of those fractions
smoky sparrow
yeah, keep going
smoky sparrow
ah, xy * y = xy^2
ionic plaza
im not seeing wher ei would multiply xy by y
smoky sparrow
im not seeing wher ei would multiply xy by y
you need to find xy * (y + 1/x)
by the distributive law, that equals xy * y + xy * 1/x
topaz sinewBOT
@ionic plaza Has your question been resolved?
fleet roost
topaz sinewBOT
fleet roost
This was one of the question for my admission test.
misty mirage
This was one of the question for my admission test.
You think you got in?
misty mirage
well what I would do is intergrate the function and add + c at tje end and then put that equal to the 3 over five and solve for C
hmm
fleet roost
well what I would do is intergrate the function and add + c at tje end and then ...
Ik that much. but integrate it if you can.
I tired everything that comes in my mind.
misty mirage
man I just finished differentiation Im not the right person for this
misty mirage
its a little different where Im from, its not really my grade
fleet roost
its a little different where Im from, its not really my grade
Ahh ok sorry.
I would still love if someone could solve this question let alone do the integration.
versed cairn
I would still love if someone could solve this question let alone do the integra...
maybe u=tan^-1 x?
fleet roost
maybe u=tan^-1 x?
I tried but doesn't seem to go too far tbh.
versed cairn
I tried but doesn't seem to go too far tbh.
it doesnt? it seems to work
or maybe im just wrong
fleet roost
it doesnt? it seems to work
or maybe im just wrong
Can you show me ur work please. I might be wrong too.
crystal goblet
it does work.. you have $\int e^{4u}cos^2(u) du = \int e^{4u}\left(\frac{1 + cos(2u)}{2}\right) du$ which is certainly doable by ibp
versed cairn
yeah that
thorny flameBOT
4573r01d|)d357r0y3r 45²
crystal goblet
good god that was bad
fleet roost
it does work.. you have $\int e^{4u}cos^2(u) du = \int e^{4u}\left(\frac{1 + cos...
How do you get that?
crystal goblet
substituting $u = \arctan x$ in $\int e^{4\arctan x}\frac{1}{(1+x^2)^2} dx$
thorny flameBOT
4573r01d|)d357r0y3r 45²
fleet roost
yeah
I get it now
THanks
dumb me
leaving a easy 6 marks question
behind
.close
potent night
For part d I found the transformation matrix in the standard basis
potent night
but how do i convert it into the basis B
ik the formula is transformation matrix = A^(-1)MA where M is is the transformation matrix and A is the change of matrix
now the problemis
for B
our M is 3 by 2
and A is 2 by 2
so i cannot mulitply A inverse with M so wtf do I do
topaz sinewBOT
@potent night Has your question been resolved?
thorny mango
so i cannot mulitply A inverse with M so wtf do I do
tame smelt
topaz sinewBOT
tame smelt
I do not understand this subject at all, and I need help to learn it
better picture
worthy storm
what is the goal? simplify?
worthy storm
,rotate
thorny flameBOT
tame smelt
yeah
worthy storm
so then you have 5xy/25
tame smelt
yes
worthy storm
do you see any more you can do?
worthy storm
how about the 5 and 25?
worthy storm
yea so you can cancel a factor of 5 on top and bottom
worthy storm
$$\frac{5xy}{25} = \frac{xy(5)}{5(5)}$$
thorny flameBOT
Bungo
worthy storm
yep and now you can get rid of the 5 above and below
worthy storm
ok
tame smelt
i lost it
found it
done
alright so
reducing is not that bad
thanks
ill probably return with other things soon
.close
topaz sinewBOT
topaz sinewBOT
Available help channel!
eternal spoke
if f prime prime prime (a) > 0, does it mean that f changes curvature from right to left at that point?
topaz sinewBOT
if f prime prime prime (a) > 0, does it mean that f changes curvature from right...
eternal spoke
or is it now always the case
vast lotus
"curvature from right to left" what does that mean ?
can you write it in terms of increasing/positive/nonincreasing terms
eternal spoke
"curvature from right to left" what does that mean ?
f curves
for example x^3
for x<0 its a right curve
and for x>0 its a left curve
vast lotus
I don't get it
In your example for f(x) = x³
f'''(x) = 6 > 0 for any x
topaz sinewBOT
topaz sinewBOT
Available help channel!
golden lark
I don't understand this question
I somehow got x²y²/y⁴
topaz sinewBOT
I don't understand this question
I somehow got x²y²/y⁴
keen raptor
cancelling out a y^2 on each side
golden lark
Wouldn't it be y^-2
keen raptor
no
golden lark
OH
keen raptor
x^2y^-2 maybe
golden lark
Oh I see
thorny mango
I don't understand this question
I somehow got x²y²/y⁴
x²y²/y⁴
= x²y²/y²y²
= x²/y²
golden lark
x²y²/y⁴
= x²~~y²~~/y²~~y²~~
= x²/y²
O cos U split y⁴ into y²y²
thorny mango
O cos U split y⁴ into y²y²
Exactly.
golden lark
Alright tysm
feral bluff
simplifiy
neat cedar
topaz sinewBOT
topaz sinewBOT
Available help channel!
stark crater
topaz sinewBOT
stark crater
is this Gram shimdt formula correct ?
stark crater
yeah it is correct
doesn't v2 denominator term should be | |v||^2 as well?
topaz sinewBOT
@stark crater Has your question been resolved?
topaz sinewBOT
@stark crater Has your question been resolved?
stark crater
.close
topaz sinewBOT
stark crater
.reopen
topaz sinewBOT
✅
stark crater
Any idea how to do this ?
I think it starts from the identity A*A^-1= I but I don't know how can I actually implement it here
<@&286206848099549185>
tough nest
thorny flameBOT
Katharine
lusty cedar
Theres info missing from this question right?
topaz sinewBOT
Theres info missing from this question right?
lusty cedar
im not going insane right?
cunning kayak
spring ravine
hi wtf..
sinful flint
!status
topaz sinewBOT
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
civic badge
let's say aiza's average speed is a
and manu's is m
we know that aiza's average speed is 10km/hr greater than manu's
could you write an equation to represent that?
civic badge
yes
civic badge
what other info do we have
civic badge
can you represent that with an equation
spring ravine
uhm] ok ill try
wispy pier
All pfps are b/w
civic badge
that's fine
aiza finished the race a half an hour faster than manu
so whatever time manu got
it's 30 minutes slower than aiza
does that help?
spring ravine
do i have to use different letters..s.s.s...
civic badge
not necessarily
civic badge
a = m + 10???
you just have to be clear that this is in km/hr
and the equation where manu is 30 minutes slower than aiza is in minutes
or you could have it in hours if you convert
spring ravine
huh
im gonnacry
ok
does that mean theres supposed to be another equation with a
spring ravine
m = a - 30??
spring ravine
can i convert the minutes into km/hr
civic badge
you can convert them to hours
civic badge
yeah
ok
so here
the race is 60 km
and manu took 0.5 hours more than aiza to complete the 60km
and aiza's pace was 10km/hr greater than manu's
civic badge
and aiza's pace was 10km/hr greater than manu's
how can we rewrite this
civic badge
the race is 60 km
knowing this
spring ravine
im sorr
civic badge
dw it's fine
i just realized my explanation was bad
we want to find the time manu took to complete the race
and we know that time = distance/speed
distance being what here?
civic badge
so T(manu) = 60/Speed(manu)
civic badge
so m = a + 0.5????
come bacck to this now
spring ravine
so.... time = 60/ a + 0.5?
civic badge
yes
spring ravine
yaey
civic badge
aiza finished the race in an hour less than Manu
so the time manu took - the time aiza took = 1 hour
so T(manu) - T(aiza) = 1 hour
spring ravine
so.... 60/a + 0.5 - 60/m + 10 = 1 hour ..???????
civic badge
m + 10 is in minutes remember?
spring ravine
wat
civic badge
oh nvm that mb
spring ravine
isnt it in km/hr
civic badge
no you're right
spring ravine
pls dont give me heart attacks im gonna crye
spring ravine
oh
civic badge
because it's already in km/hr
so 60/(a + 0.5) - m + 10 = 1 hour
we want to solve for m
so we don't want the a
how else could we write T(manu)?
spring ravine
oh
spring ravine
ohh h h hhh h hh.. . . .. . .
civic badge
so we want the same for manu's
spring ravine
but thats still aizas time rihgjjt/t/t/t//tt/t.t.t.t.t
spring ravine
im going to beat someone up
civic badge
and T(manu) = a + 0.5
civic badge
right but
T(manu)
can be rewriten as
T(manu) = 60/m
b/c it's the average speed 😭
i complicated it wayyy too much
so basically
we have
(60/m) - (60/m+10) = 1 hour
spring ravine
im five seconds away from banging my head against a wall dont do this to me pelase
civic badge
i genuinely feel so bad
sorry for wasting your time 😭
we can still finish the problem though
civic badge
yes
spring ravine
how thehell
civic badge
so multiply M(M+10) on both sides
lcm of the denominators
is that not what you were saying 😭
civic badge
so multiply M(M+10) on both sides
.
so we'd hav
M(M+10)((60/M) - (60/(M+10))) = 1(M(M+10))
which simplifies
(M+10)(60) - (60)(M) = M(M+10)
does that make sense?
spring ravine
wtf
can we not just
first simplify to m^2 + 10m and then do whatever the hell you're saying
civic badge
yes
spring ravine
IM GOING TO CRASH OUT
civic badge
we could
spring ravine
its still
gonna be
m^2 + 10m - 600
either way
DONT DO TS TO ME
ohmygodddddd
its
ok
its
ok
civic badge
😭
spring ravine
its a quadratic equation EITHER WAY
mortal thunder
m² + ||30||m - ||20||m - 600
civic badge
yes
civic badge
m² + ||30||m - ||20||m - 600
thank you 😭
mortal thunder
MY BLOOD PRESSURE
Is that "Blood Thrust"/"Area" = Blood "Pressure"
spring ravine
what
civic badge
huh
spring ravine
im
do you know
do you know
how
how
UGH
OH MY GOD
dont pmo
sighs whatever
thanks for the help either way
mortal thunder
civic badge
moral of the story
mortal thunder
Super Saiyan "Yiren" with Blood Manipulation mwahaha
spring ravine
whats your cirrculum @civic badge
civic badge
wdym
spring ravine
???... your cirriculum..??
spring ravine
sighs what country
spring ravine
AMERICA.????/??//
civic badge
yes
spring ravine
oh
thats cool
thanks for the help btw
im sorry for getting so pressed dhdhdhdjdjdkj
civic badge
thanks for being patient with me 😭
civic badge
im sorry for getting so pressed dhdhdhdjdjdkj
respectable crashout
spring ravine
i have a math exam tmr & im so cooked
civic badge
just believe
spring ravine
thats
called delusion
i just want to pass.
50% above please pelase please please please
ok.
.close
vocal island
why is this wrong?
topaz sinewBOT
why is this wrong?
vocal island
3 img
it's a bit of a long read
but i don't see where i went wrong
solution is way simpler but i don't see why this doesn't work
coarse tusk
by substituting x=sin(theta) you're assuming that x lies in [-1, 1], which isn't necessary
wispy pier
(1+x)/x²(1-x²)
= 1/x²(1-x)
= (1-x+x)/x²(1-x)
= 1/x² - 1/x(1-x)
wispy pier
my solution
Yeah there are mistakes !
vocal island
Yeah there are mistakes !
where
coarse tusk
(1+x)/x²(1-x²)
= 1/x²(1-x)
= (1-x+x)/x²(1-x)
= 1/x² - 1/x(1-x)
you mean 1/x^2(1-x)
the first term should just be -1/sin(theta), not -1/sin(theta)cos(theta)
the rest looks good, discounting the domain issues
but again this is not the right way to solve this at all
vocal island
the first term should just be -1/sin(theta), not -1/sin(theta)cos(theta)
OHH
ur right
it's sec * cot not sec * csc
grizzled sandal
why is this wrong?
ur tryna solve the first integral right?
this
go for partial fractions decomposition i suggest instead of trig sub
because the denominator is factored
topaz sinewBOT
topaz sinewBOT
Available help channel!
wispy pier
Is there some way to find the length of latus rectum of a parabola, from its general equation directly.
topaz sinewBOT
Is there some way to find the length of latus rectum of a parabola, from its gen...
wispy pier
Ping Kindly !
wispy pier
Huh it really is called latus rectum
I edited the question
sweet shard
Oh no directrix is outside the parabola
mortal thunder
Is there some way to find the length of latus rectum of a parabola, from its gen...
Yes
What is the "general equation" of parabola?
wispy pier
Yes
To clarify,
By general eqn I mean,
ax²+2hxy+by²+2gx+2fy+c = 0
Where, ab-h² = 0, and assume it's a central conic.
mortal thunder
Hmmmmm
wispy pier
Help me ! 😭
The method ik involves rotation of axes by 1/2arctan(2h/(a-b)) to eliminate the xy term, and then express it as a standard parabola.
wispy pier
Gimme a sec
Surely!
sweet shard
<@&268886789983436800> spam or bot
meager dawn
don't spam in the help channels.
wispy pier
Thanks !
meager dawn
see #❓how-to-get-help and find an open help channel if you have a question
wispy pier
Hey if anyone replies, kindly ping me !
glad arch
mortal thunder
Hey if anyone replies, kindly ping me !
wispy pier
<:sad:532803514892812288>
Hey hi !
mortal thunder
Hmm, it's a lengthy method. But here it is:
wispy pier
You can just say in short, in case I came across it earlier!
mortal thunder
Suppose you have ax² + 2hxy + by² + 2gx + 2fy + c = 0
The only way I could find was to rewrite it as (ax + by)² + 2gx + 2fy + c = 0
Plug X' = ax + by, Y' = bx - ay
Eliminate xy term, get "A"
That's the neatest way to get the size of latus rectum ._.
Am sorry, I probably did learn a more convenient way, but I just don't remember. It's been 3 years ;-;
wispy pier
All these methods are really lengthy !
mortal thunder
;-; sry!
wispy pier
Am sorry, I probably did learn a more convenient way, but I just don't remember....
No problem!
wispy pier
;-; sry!
I hate coordinate geometry for a reason!
And I'm sure most of the ppl do !
Anyways Thank you very much for your time and efforts !
.close
topaz sinewBOT
topaz sinewBOT
Available help channel!
stiff egret
How do you find the derivatives of exponentials like 2^x or 3^(x+3). Not these in particular but all
topaz sinewBOT
How do you find the derivatives of exponentials like 2^x or 3^(x+3). Not these i...
golden blade
How do you find the derivatives of exponentials like 2^x or 3^(x+3). Not these i...
the trick is to rewrite a^x as e^[xln(a)]
stiff egret
How does that help. Im mostly new to calculus
golden blade
because the derivative of e^x is again e^x
golden wigeon
How does that help. Im mostly new to calculus
$2^x = e^{x \ln(2)}$
thorny flameBOT
King Leo
golden blade
you take advantage of this property
stiff egret
So once you have it in that state how do u get the derivative. Or is that the derivative
golden wigeon
So once you have it in that state how do u get the derivative. Or is that the de...
Assume $y = e^{x \ln(2)}$
thorny flameBOT
King Leo
golden wigeon
And youre trying to find $\dv{y}{x}$
thorny flameBOT
King Leo
golden wigeon
The derivative of e^x is e^x, our function has an "inner" function (from teh chain rule)
Lets make a new function $u = x \ln(2)$
thorny flameBOT
King Leo
golden wigeon
$\dv{y}{x} = \dv{y}{u} \cdot \dv{u}{x}$
thorny flameBOT
King Leo
golden wigeon
@stiff egret do you know what to do from here
golden wigeon
Ok
So first, lets work with dy/du
Lets express y in terms of u
$$y = e^{x \ln(2)}$$
$$u = x\ln(2)$$
$$y = e^u$$
thorny flameBOT
King Leo
golden wigeon
@stiff egret now what is dy/du?
stiff egret
Um
golden wigeon
because the derivative of e^x is again e^x
Its very similar to this, all that changed is a single variable
golden wigeon
So e^u= the deravative of u right?
?
wind whale
e^u is the derivative of e^u
stiff egret
Yea
stiff egret
I have no clue
golden wigeon
**King Leo**
Now, you need to find du/dx
golden wigeon
I have no clue
You dont know what dy/du is?
stiff egret
I do kinda but not really
golden wigeon
I do kinda but not really
In general, $\dv{x} e^x = e^x$
thorny flameBOT
King Leo
stiff egret
Yea i understand that
golden wigeon
Can you try to fix this to match up with our e^u?
stiff egret
Umm one sec
Oh so the deravative of e^u is e^u so e^x ln(2) is e^x ln(2) so thats the answer?
stiff egret
No
golden wigeon
Oh so the deravative of e^u is e^u so e^x ln(2) is e^x ln(2) so thats the answer...
No
golden wigeon
**King Leo**
@stiff egret you also need du/dx
stiff egret
Im not good at understanding du/dx or when theres two variables like that
golden wigeon
Im not good at understanding du/dx or when theres two variables like that
Start with:
$$\dv{u}{x}$$
But:
$$u = x \ln(2)$$
so you end up with:
$$\dv{x} \qty(x \ln(2))$$
thorny flameBOT
King Leo
golden wigeon
(and remember: ln(2) is a constant)
golden wigeon
So is that 1
No, ln(2) does not equal 1
stiff egret
What i thought if u have ax the deravative would be a
golden wigeon
What i thought if u have ax the deravative would be a
Exactly
(thats when a is a constant)
wind whale
you missed an equal sign i think @golden wigeon missed that
golden wigeon
But in this case, assume a = ln(2)
stiff egret
Yea wait
golden wigeon
you missed an equal sign i think <@1214057610857160835> missed that
?
stiff egret
I think i wrote it wrong
golden wigeon
Is the deravative of (x ln(2)) ln(2)
Oh i see
stiff egret
D/dx of (x ln(2) =ln(2)
golden wigeon
@stiff egret were you saying that d/dx (x ln(2)) = ln(2)
stiff egret
So now we need
golden wigeon
So now we need
Do you know what dy/du is
stiff egret
dy/du ?
So dy/d (x ln(2)) or can u not have d on the bottom alone? Or something else
wispy gulch
why dont u just take log on both sides of the original equation and then differentiate wrt x
golden wigeon
why dont u just take log on both sides of the original equation and then differe...
Wont you have to use implicit differentiation then?
wispy gulch
for eg like this
golden wigeon
for eg like this
That is unnecessarily complicated
stiff egret
Do you know what `dy/du` is
Im sorta confused on this for dy/du if you simplified out the u wouldent we be solving for deravative of y
golden wigeon
Im sorta confused on this for dy/du if you simplified out the u wouldent we be s...
We certainly are solving for the derivative of y... with respect to u (instead of x)
wispy gulch
That is unnecessarily complicated
oh i learnt it like this and looked simple, ig its the way a person learns first..
stiff egret
We certainly are solving for the derivative of y... with respect to u (instead o...
Wouldent both sides of the equation simplify to like ln(2)=ln(2) or am i thinking of it wrong
golden wigeon
Wouldent both sides of the equation simplify to like ln(2)=ln(2) or am i thinkin...
Youre thinking of it wrong
Remember:
$\dv{x} e^x = e^x$
thorny flameBOT
King Leo
golden wigeon
We simply have a slightly different expression that uses e^u rather than e^x