tidal storm
If I let N = 5q + r, where 0 <= r <= 4
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neon rivet
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hot herald
for 20b)
why did you replace x with 1 as well in you calculation for a_1?
also be more careful with signs and operations when subbing in your r
and for the second question, you're dealing with infinite series with |r| < 1
so you could've use the infinite series formula directly
BUT you used the wrong value for r in the denominator as well when plugging values in
@neon rivet
hot herald
why did you replace x with 1 as well in you calculation for a_1?
hot herald
didn't answer my first question
and yes, r is x-2
but you didn't substitute properly
by writing it as you have, you ignored the order of operations
neon rivet
oh the brackets
is there a video on this? I think I need to touch on this
I watched few but they werent a help
hot herald
this again is a basic algebra concept
you already know the formula
the rest is just substituting in the right values
and not screwing up manipulation
and for the third time
why did you replace x with 1 as well in you calculation for a_1?
neon rivet
and for the second question, you're dealing with infinite series with |r| < 1
so...
I understand this except the last line
BUT you used the wrong value for r in the denominator as well when plugging values in
hot herald
what is the value of r in this question?
neon rivet
1-r
hot herald
yes, and what value did you use for r in the denominator
neon rivet
ohhh
neon rivet
and for the third time
> why did you replace x with 1 as well in you calculation...
Im trying to think
hot herald
consider
$$g(x) = kx^2$$
what is $g(3)$?
glossy valveBOT
ραμOmeganato5
neon rivet
k3^2
hot herald
yes, same principle applies in your question
$a_{\red{n}} = T_{\red{n}} = (\blue{x}-2)^{\red{n}}$
glossy valveBOT
ραμOmeganato5
hot herald
for the first term, you want when the red n is 1
the blue x doesn't get changed
similar to how k stays as k
only the relevant variable is affected
hot herald
not really
restive geyser
It's just recognising algebra
And not trying to do everything by rote memorisation
hot herald
apply basic definitions and properties, and don't overthink
neon rivet
I wish it was color coded 😢
restive geyser
It doesn't really need to be, tbh
hot herald
if the index starts at n=1
to get the first term
sub n=1
not much more to it
restive geyser
I suppose the only problem would occur if your x's and n's look very similar (which is a common thing with some people's handwritings; if it is with yours I highly recommend fixing it fast)
neon rivet
yup okay
hot herald
() and make sure you're using the right values
if that's due to rushing, slow down
watching a vid won't really solve that
and as mentioned before
most of these issues are algebra related,
so you still need more practice with that
neon rivet
so a_1 = x-2
void nova
Yes
neon rivet
I thought r = x-2
void nova
Why?
neon rivet
because its in the bracket
void nova
It is correct, but the justification isn't too much
neon rivet
and raised to n power
void nova
and raised to n power
Alright
hot herald
a_1 and r can be the same
void nova
It is correct, but the justification isn't too much
The best answer would have been:
because a GP is in the form a•r^n
(or, because the ratio of consecutive terms is always x-2)
hot herald
apply the basic definition
and if they end up being the same value/expression
cool
0 + 0 + 0 + ...
series that could be considered arithmetic and geometric
with a_1 = 0
d = 0 and r = 0
hot herald
don't need those () on the numerator
neon rivet
wait why not?
void nova
Because of, again, basic algebra
hot herald
nothing else acting on it
neon rivet
we only use it when theres mulitplication of the terms
void nova
It's just one term, alone
hot herald
horizontal line already indicates division of the whole chunk above
void nova
or division
But not when you write out the fraction bar, only in plain text when you have the / or : symbols
hot herald
same reason why () weren't needed on the denominator
and you had no issue not putting those
but if you were comminicating in text
in a single line
then you would put parentheses as that doesn't have the advantage of the horizontal fraction line
frigid carbon
which q we on?
glossy valveBOT
neon rivet
and for the second question, you're dealing with infinite series with |r| < 1
so...
also with this, the question is asking for Tn
not sum to infinity
hot herald
misread that part, in which case you applied the wrong thing
you attempted to apply the formula for S_n, S_inf
hot herald
you had the formula for S_n, and used the wrong values and expressions
you're again trying to do stuff resembling a sum
neon rivet
so my r and a is wrong
hot herald
no, the a and r are fine
what you're doing with them is the issue
in general if you were given a and r, ignore the values in this question
in a geometric sequence, in terms of a and r, what would be:
the first term?
second term?
third term?
the general term?
deduce from above the general term from the above
it would be in definitions that you should have on hand,
if you can't remember
neon rivet
Tn=a_1(1-r^n)
neon rivet
from the top of my head but yes its incorrect
hot herald
if you don't know it, don't guess
apply basic definitions / properties
and/or look up the official resources
in a geometric sequence, in terms of a and r, what would be:
the first term?
don't overthink, these are not intended to be trick questions
neon rivet
evaluate T1
neon rivet
to get a
hot herald
forget about the values
neon rivet
hot herald
the first term is a, use that
and again the things you just posted are for the sum / series
the question is a little poorly worded but
you DO NOT CARE about those for what the question is asking
the general term T_n
in general if you were given a and r, ignore the values in this question
in a geometric sequence, in terms of a and r, what would be:
the first term? (that's just a)
and applying the properties of geometric sequences / GP
how would you get and what would be the second term?
neon rivet
the question is a little poorly worded but
you DO NOT CARE about those for what ...
wdym
you DO NOT CARE about those for what the question is asking
the general term T_n
hot herald
S is for the Sum
T is for the individual terms
neon rivet
Yep
hot herald
so you're applying all this stuff with S and sums, which isn't what the question is asking for
applying all those sum formulae is useless here, you care nothing about the infinite sum
you are given the series
and have identified the a and r
and then you should apply the formula for the general term of a GP
using those a and r because that's all is needed
neon rivet
so its just the formula issue
hot herald
yes
you're being asked for
T_n,
apply the formula for T_n
do exactly what the question asks
hot herald
they want the nth term in the series
if they wanted the sum, they would've used S instead of T
neon rivet
whenver T comes up
neon rivet
I should immeditely know its a sequence
hot herald
read the context of the question
hot herald
some people have horrid choices for variables
don't blindly assume
neon rivet
and told us to find Tn
do you have advice for remembering the formulas
theres like 4 formulas I think
2 for GP and AP
hot herald
apply basic definition/properties
learn how they're obtained
neon rivet
apply basic definition/properties
learn how they're obtained
what is that?
hot herald
like knowing the definitions
and doing a bit of algebra, will give you those nice simplified formulas
delicate torrent
Can anyone locate for me where is the original question?
hot herald
in a geometric sequence, in terms of a and r, what would be:
the first term?
second term?
third term?
the general or nth term?
deduce the general term from the above
glass frigate
Can anyone locate for me where is the original question?
its pinned
frigid carbon
2 for GP and AP
What are they
delicate torrent
I'll reply to this message so anyone can see the questions. Which question is he on now?
spice knot
for reference, usually the first is shown in the pinned comment so that you dont have to do this
hot herald
question is pretty much done
just seeing if there is anything needs clarifiying
delicate torrent
How about the other question?
delicate torrent
also need help with this q
This one specifically
hot herald
clarified that the question wanted T_n
i'm assuming h knows that now since there wasn't anymore questions on it
delicate torrent
clarified that the question wanted T_n
i'm assuming h knows that now since there...
Alright
neon rivet
ill name all of them:
For AP:
Tn=a+(n-1)d
Sn= n/2 (2a+(n-1)d) or n/2 (a+l)
For GP:
Tn= a * r ^ n-1
Sn= a_1(1-r^n)/1-r im not sure about this one
Sum to infinity
S_inf= a/1-r for -1<r<1
is that all the formulas I need to remember?
or am I missing something
delicate torrent
ill name all of them:
**For AP:**
Tn=a+(n-1)d
Sn= n/2 (2a+(n-1)d) or n/2 (a+l...
If you are not sure then you can just try to prove it again
frigid carbon
ill name all of them:
**For AP:**
Tn=a+(n-1)d
Sn= n/2 (2a+(n-1)d) or n/2 (a+l...
those are basic ones so yh, better if you remember
neon rivet
If you are not sure then you can just try to prove it again
prove iit again?
delicate torrent
prove iit again?
prove
delicate torrent
prove iit again?
Yeah
frigid carbon
prove iit again?
what doubt you got?
delicate torrent
I can sure to help you prove that
neon rivet
sure
delicate torrent
prove iit again?
Can you rewrite the geometric series?
In a form of something like 1 + 2 + 3 + ... + n
Not the sigma notation
neon rivet
wait why do we prove it?
delicate torrent
wait why do we prove it?
To confirm your assumptions are correct
frigid carbon
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neon rivet
its ok dw Ill move on
pastel badge
I don't know what I am doing wrong here: Consider the arithmetic series: $u_n = 7n - 5$. What the term of (less order? Welp, it's the one that appears first) that is bigger then 55. And here's my reasoning for the exercise
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$55 < u_n$
glossy valveBOT
ℕ∈ℝD ALERT: Gonçalo Gonçalves
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then I just simplified that and got $8.57 < n$
glossy valveBOT
ℕ∈ℝD ALERT: Gonçalo Gonçalves
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and then I rounded it to 9
elder sandal
I don't know what I am doing wrong here: Consider the arithmetic series: $u_n = ...
u looking the first term that exceeds 55?
elder sandal
and then I rounded it to 9
sounds correct
elder sandal
and then I rounded it to 9
okay so
since you rounded it up to 9
what you're looking for is $u_9$
glossy valveBOT
4-aminopyrimidin-2(1H)-one
elder sandal
,, u_9=7(9)-5
scarlet pike
pastel badge
yeahh
glossy valveBOT
4-aminopyrimidin-2(1H)-one
elder sandal
the rest works out
gentle hollow
hi can i get hints for this question, ive only established that they could be all even, all odd, or 2 even 2 odd, but this hasnt been too useful so far
queen gull
Use the fact that everywhere the integers are positive
Otherwise there are options other then everything being the same
Like the n'th row having value n
onyx glen
very obtuse hint: well ordering principle
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snow mural
Looking for help
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snow mural
I don’t understand B
Like I mean the working are copied I don’t understand why it’s OR=2OP+2OT
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knotty grail
Like I mean the working are copied I don’t understand why it’s OR=2OP+2OT
can you determine the direction of p + t using the parallelogram law
snow mural
can you determine the direction of p + t using the parallelogram law
Not really
No
Parallel = vector same?
snow mural
Yes
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upper scaffold
do i have the correct solution to this problem about the limit of indeterminate forms? this is the problem:
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upper scaffold
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gritty rose
mistake here
you can split the limit of the product to a product of limits only if both are finite
1/(1-x) diverges as x goes to 1
a lot of this can be simplified if you set t = 1 - x in the beginning
upper scaffold
but taking the natural logarithm of the exponential was the correct step tho right?
gritty rose
for your approach yea
upper scaffold
okay i set t = 1 - x. but i still cant split the limit of the product to a product of limits because 1/t diverges. what should i do?
gritty rose
this is indeterminate form 0/0 with denominator = t
upper scaffold
ah so i just apply L'Hopital's rule okay
gritty rose
or is it
gritty rose
yea i take it back it's easier if you just leave the 1/t in the exponent
or MAYBE
split the outer log as log(a/b) = log(a) - log(b)
you'd have to put the - sign with the numerator
only log[log(1/(1-t))] is defined for t > 0
torn jolt
And here is my solution:
may i ask what program you're using
upper scaffold
may i ask what program you're using
its just Obsidian MD with the latex suite extension for easier latex writing
torn jolt
its just Obsidian MD with the latex suite extension for easier latex writing
YOU CAN USE LATEX WITH OBSIDIAN? thanks
upper scaffold
YOU CAN USE LATEX WITH OBSIDIAN? thanks
yep. get latex suite community extension so you can use keyboard shortcuts to easily type the different arithmetic operations
upper scaffold
this is indeterminate form 0/0 with denominator = t
wait how is it an indeterminate form of 0/0? if i split the outer log i get ln(ln(1 - t)) - ln(-t), and if t approaches 0 from the right, wouldn't you get an indeterminate form INFINITY - INFINITY for the simplified outer log expression?
gritty rose
that's just the numerator
you should try to see if you can do lim [log(a) - log(b)] / t = lim log(a)/t - lim log(b)/t
upper scaffold
so i end up with this:
upper scaffold
you should try to see if you can do lim [log(a) - log(b)] / t = lim log(a)/t - ...
can i do this even if one of the limites diverge?
or is it just like in the limit of products, i cant split it into a product of limits if one of the limits diverge?
gritty rose
which one diverges
upper scaffold
dont they both diverge? they have the form infinity / 0
gritty rose
i guess as a separate l'hopital problem you have to prove
,w lim t to 0+ log(log(1-t)/(-t))
glossy valveBOT
gritty rose
this is indeterminate form 0/0 with denominator = t
to be able to show the numerator goes to 0 and prove what i claimed here
slightly more straightforward once you swap lim log = log lim
upper scaffold
okay i now see where the 0/0 comes from and then i just apply the L'Hopital's rule. thank you. i'll solve it on my own now. 🙇🏼♂️
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upper scaffold
yea i take it back it's easier if you just leave the 1/t in the exponent
out of curiosity, what idea did you get that made you think it was easier to leave 1/t in the exponent?
gritty rose
i was wrong since it wasn't 
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distant moat
is this correct
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is this correct
distant moat
(ignore the middle step)
why can we apply the summation to just i in the second term and take the n out
but in case of just n we have to apply the summation to it and make it n^2
queen crater
but in case of just n we have to apply the summation to it and make it n^2
That's not the case
n is a constant in both contexts
umbral dome
you can take the n out of the first one as well, but it would need to have something in so you would be summing 1
distant moat
n is a constant in both contexts
then the answer would be n + n^2(n+1)/2
queen crater
No, n^2 + n^2(n+1)/2
distant moat
but apparently it should be n^2 instead of n in the first term
distant moat
No, n^2 + n^2(n+1)/2
yeah how
why do i have to apply the summation to n and make it n^2
queen crater
,, \sum_{i=1}^n n = n \sum_{i=1}^n 1 = n \cdot n
glossy valveBOT
Nel
distant moat
what about n*i
umbral dome
,, \sum_{i = 1}^n n = n \sum_{i = 1}^n 1 = n (\underbrace{1 + 1 + \cdots}_{n\ \text{times}}) = n \cdot n
distant moat
when there is n*i we can take out the n out of the summation and just apply it to i
queen crater
Same thing except the 1 is i, so the last n is the n'th triangular number
queen crater
when there is n*i we can take out the n out of the summation and just apply it ...
That's exactly what I did here though...
I took the n out
distant moat
oh i see
gritty rose
some index funny business
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marsh summit
how can i prove this
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how can i prove this
velvet sedge
Do you know the definition of surjective?
marsh summit
yes
velvet sedge
What is the definition of surjective?
marsh summit
that means tha fx have at leaste on solution
void nova
Mmhh
next surge
Prends la définition avec les quantificateurs
velvet sedge
if f:A->B then f being surjective means that for all b in B (the codomain) there exists an a in A (the domain) s.t. f(a)=b
velvet sedge
how can i prove this
Prove the definitions are equivalent
i.e. you can get one definition from the other
next surge
how can i prove this
- Tu écris proprement les définitions (donc ici, la définition d'une fonction surjective et celle de l'image directe d'une fonction)
- Tu résonnes par double implication, c'est-à-dire que tu supposes f surjective et tu montres que ça implique que f(E)=F, puis tu supposes que f(E)=F et tu montres que f est surjecctive
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next surge
@marsh summit Tu t'en sors ?
marsh summit
<@1182657367851147285> Tu t'en sors ?
la j ai supposé que f surjective et j essaye de demontrer que f de E est egale a F
j ai montré que f de E esi inclut dans F la j essaye de mq F inclut dans f de E
la j ai fait les doubles implications dans la premiere j ai supposé que f surjective dans la desieme j ai supposé que f de e est egale a f
marsh summit
<@1182657367851147285> Tu t'en sors ?
mrc bcp t le boss
next surge
Bah c'est bon alors t'as fini
next surge
mrc bcp t le boss
Quand t'as une question comme ça, pense vraiment à réécrire les définitions et de bien les manipuler
Pareil, quand tu vois un $\forall x$, tu penses à fixer un $x$
glossy valveBOT
Tristan
next surge
(Pourquoi ? Parce que ton assertion est vraie (normalement) pour n'importe quel x choisi, donc en prenant un x quelconque tu dois réussir à montrer ce que tu veux)
marsh summit
(Pourquoi ? Parce que ton assertion est vraie (normalement) pour n'importe quel ...
daccort mrc bcp pour ton explication et tes conseilles t un frere mrc mille fois
next surge
daccort mrc bcp pour ton explication et tes conseilles t un frere mrc mille fois
T'es en quelle classe ?
marsh summit
T'es en quelle classe ?
je suis au maroc premiere chez nous c desieme anné lycé jsp c quoi en france
next surge
Par rapport au bac t'es où ? Tu le passes l'année prochaine ?
marsh summit
oui
marsh summit
mrc une derniere chose deso de te deranger tu peut m expliquer pq la premiere c une egalité et la desieme seulement une inclusion je l ai est deja demontré juste l exos c juste par curiosité
next surge
Attends j'ai fait des fautes de frappe je remodifie
Pour deux ensembles A et B, une égalité ça veut dire que A est inclu dans B et que B est inclu dans A. Donc la première te dit que $f(A\cup B)\subset f(A)\cup f(B)$ et $f(A)\cup f(B)\subset f(A\cup B)$.
La deuxième te dit que $f(A\cap B)\subset f(A)\cap f(B)$ mais que $f(A)\cap f(B) \not\subset f(A\cap B)$.
glossy valveBOT
Tristan
marsh summit
c est bon j ai compris merci infiniment il y a pas egalité car l autre inclusion n est pas tjr vraie
next surge
Exactement !
Et très souvent, tu peux trouver un contre exemple pour montrer que quelque chose est faux (ça peut être sympa d'en trouver un).
marsh summit
Et très souvent, tu peux trouver un contre exemple pour montrer que quelque chos...
ah oui je vais assayer d en trouver
next surge
Le mieux c'est de prendre des fonctions et des ensembles simples, cherche pas compliqué
marsh summit
ok
vrm mille merci pour tous j adore ta facon de simplifier et d expliquer et une derniere faveure si j amais je ne comprent pas un truc je peux suciter ton aide si cela ne te derange pas
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neon rivet
How do we do 4
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How do we do 4
neon rivet
We find the first derivative and second derivative
dense smelt
How would you imagine to find a minimum using the derivatives
neon rivet
Set the first derivative equal to 0 for st. Points
neon rivet
And plug in the st points into the second derivative to check the nature of the st. Points
neon rivet
what's the derivative
Gradient slope at a point
dense smelt
I mean, calculate it
neon rivet
-sqrt(3)sinx + cosx
neon rivet
Wait
keen lily
-sqrt(3)sinx + cosx
If you rearrange this it will work
neon rivet
I just want to know what’s my steps because I already tried this problem but it was a ramble
Is the question asking for the minimum value of the function?
dense smelt
Is the question asking for the minimum value of the function?
Yes
Find the minimum value of 1 + sqrt(3)cos(x)+sin(x)
for x between 0 and 2pi
dense smelt
Right, what are those points
neon rivet
To find the nature
dense smelt
and what is that second derivative
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@neon rivet Has your question been resolved?
dense smelt
work away
dense smelt
mightve had a sign error in the -sqrt(3)sin(x)+cos(x) = 0 telling you -sqrt(3)sin(x) = cos(x)
I think that should say just sqrt(3)sin(x) = cos(x)
but you do still get tan(x) = 1/sqrt(3)
neon rivet
wait tell me the steps
after finding the st points
so one st point
will give me
a maximum and other will be a minimum
because of my errors?
and we just use the one that is the maximum
and plug it into the original function
what value should it give?
the answer key said -1
but I also have -1 for the second derivative test
dense smelt
why did you a negative assigned as both a maximum and a minimum for the second derivative test
and your arithmetic for the pi/6 input is just wrong
neon rivet
why did you a negative assigned as both a maximum and a minimum for the second d...
wdym?
dense smelt
wdym?
Why did one -1 get a max and the other a min
why these answers
the left is wrong too
It should be positive, even
neon rivet
I meant maximum for both
after getting the maximum point, I need to sub it into the original equation right
to get a value
and that would be my final answer
dense smelt
I meant maximum for both
Well, the number isnt correct on the first
dense smelt
after getting the maximum point, I need to sub it into the original equation rig...
But yes
neon rivet
rn I am just messing up with arithmetic
dense smelt
plug in the critical values and th endpoints
and see what is the maximum output
You have to check the endpoints separately since the derivative tests cant help there
neon rivet
yes
dense smelt
So, check and see what they are
neon rivet
So, check and see what they are
did i do something wrong here?
for b
im getting 0
neon rivet
cos-sin?
dense smelt
I see your sign is messed up at the end here
neon rivet
the last line is the only mistake?
dense smelt
and this is wrong
dense smelt
the last line is the only mistake?
and I think it should be sin-cos
You have some sign issues going on in the middle
Since this cos-sin integral gives a negative area
dense smelt
I got the answer by fixing the last answer
the last line was wrong but you had sign errors above too
since the integral of cos-sin should give -2sqrt(2)
neon rivet
why did I get to the right annswer?
dense smelt
Because you flipped the integral sign and flipped signs in the middle
those errors cancelled out
dense smelt
sinx + cos x
I think yes
Since derivative of sin is cos, and cos is -sin
But your integral should have been sin-cos to start with, so this sign change cancelled out
dense smelt
yep
neon rivet
now i got the negative
dense smelt
but your integral should be sin-cos
neon rivet
it supposed to be positve
dense smelt
since sin is above cos
dense smelt
use a little critical thought
dense smelt
yep correct
and you get 2 sqrt(2) at the end, like you already saw the negative of with cos-sin
So basically complete
neon rivet
alright ty
Im just gonna try and finish the previous question we discussed
quesion 4
I have a small question
should i say set f'(x)=0 for st point or st points or st point/s
because when i write that we dont know how many st points there will be
dense smelt
should i say set f'(x)=0 for **st point** or **st points** or **st point/s**
Just find when f’(x)=0
And not worry so much about it it’s plural
You can change it later when you know
full forumBOT
@neon rivet Has your question been resolved?
frozen blade
can i get a little help on some of these, i’m a little ocnfused
ebon fulcrum
For which ones do you need help on?
frozen blade
i mostly just need a little explanation, but one and two are good! i’m unsure on how to start it and i kust learned it today
ebon fulcrum
Right, so you could either use chain rule, or you could expand, then apply power rule
Although I don’t know if you have to apply power rule or if you’re free to use what you want?
ebon fulcrum
Got it
frozen blade
we’ve done power rule, constant rule, constant multiple rule, and thats it
ebon fulcrum
Ahh I see
We could start with the first one
You would want to start with expanding so you can apply power rule
frozen blade
so it would turn into (4x^2 +1)(4x^2+1)?
ebon fulcrum
Yup
frozen blade
then you would apply the rule?
frozen blade
i’m not dxacrly sure, is it with the powers?
like if its 1 its linear
soemthing like that i think😭😭
ebon fulcrum
$\frac{d}{dx} x^n=nx^{n-1}$
glossy valveBOT
FriedRiceKing
ebon fulcrum
The reason why you have to expand first before using power rule is that we have (4x^2+1)^2
So we can’t directly apply it yet
frozen blade
ohhh alright alright
ebon fulcrum
(x itself has to be raised to the power)
ebon fulcrum
What do you get after expanding?
frozen blade
to get rid of the 2 exponent outside the parenthesis i expanded it into (4x^2+1)(4x^2+1)
ebon fulcrum
Yes
You want to get the expression to a form that you can differentiate
Can you differentiate now, with the rules you have learned?
ebon fulcrum
You can’t, since you have a product
$\frac{d}{dx}[f(x)+g(x)]= \frac{d}{dx}f(x) + \frac{d}{dx} g(x)$ for all differentiable f and g functions, but $\frac{d}{dx}[f(x) \cdot g(x)]\ne \frac{d}{dx}f(x) \cdot \frac{d}{dx}g(x)$
glossy valveBOT
FriedRiceKing
ebon fulcrum
(In fact, you will be learning product rule, that is, how to derive the product of two functions, soon)
frozen blade
**FriedRiceKing**
yea i have something lik this in my notes
ebon fulcrum
Ahh I see
frozen blade
or it mentions differentiating
ebon fulcrum
Right
Anyhow, we essentially cannot differentiate yet with only the derivative rules you have seen
What can we do to be able to differentiate the expression?
(Hint: take advantage of the sum rule and difference rule)
frozen blade
do we use one of the expressions? or no
ebon fulcrum
I mean, we have the expression (4x^2+1)(4x^2+1) right now
We can’t differentiate it right now with the rules you have seen
So, what can we do to differentiate?
ebon fulcrum
Yeah! Exactly
frozen blade
i dont know why i missed that omg
ebon fulcrum
This way we can take advantage of all the rules we know
ebon fulcrum
i dont know why i missed that omg
No problem with that!
frozen blade
so it would be 16x^4+8x^2+2?
ebon fulcrum
+2?
glossy valveBOT
FriedRiceKing
ebon fulcrum
Let’s do this step by step. We can use one rule immediately, can you see which one?
frozen blade
the sum rule?
ebon fulcrum
Haha yeah
frozen blade
the three is suppoed to be an exponent
ebon fulcrum
16*4(x^3) I suppose you mean
frozen blade
ohh okok so i don’t have to only one rule?
there will be certain rules for each problem?
that would make it easier to complete compared to others
ebon fulcrum
Yeah, you use whatever rule is necessary
As you go on, you’ll see that sometimes applying some rules are easier than others, though
But yeah, for these kind of derivative problems you just try to use your rules to derive, and often you have to do some algebra to be able to do that
frozen blade
ohhh alright
that makes a lot of sense, i thought that each problem had to only be done by one rule
ebon fulcrum
That would likely be impossible lol
frozen blade
yeah thats probably why i wasnt getting it😭😭 thank you so much though
ebon fulcrum
No prob!
*if you have other questions please ask, but if not remember to close the channel
twin wolf
need help with starting with part b)
full forumBOT
need help with starting with part b)
twin wolf
guessing i need to use strong induction
spice ivy
what kind of maths is this?
dense smelt
guessing i need to use strong induction
Suppose it’s true for n
Can you prove it for n+1
dense smelt
what kind of maths is this?
Don’t just jump in asking what everything is
dense smelt
Can you prove it for n+1
Namely, suppose $F_n \geq \phi^{n-2}$, then can you show $F_{n+1}\geq \phi^{n-1}$
glossy valveBOT
Sharp, Archon of Severing
twin wolf
Suppose it’s true for n
im not supposed to use the F_n = bla bla / sqrt(5) right im only just using the information from the question given?
dense smelt
im not supposed to use the F_n = bla bla / sqrt(5) right im only just using the ...
Yeah
The recurrence relation
twin wolf
okay hold on let me try
dense smelt
Yep yep do try
twin wolf
Yep yep do try
so F_n+1 = F_n + F^n-1
F_k >= phi^(k-2)
F_k-1 >= phi^(k-3)
so can we say something like
F_k+1 = F_k + F_k-1 >= phi^(k-2) + phi^k-3)
dense smelt
so F_n+1 = F_n + F^n-1
F_k >= phi^(k-2)
F_k-1 >= phi^(k-3)
so can we say somet...
Yes
How do you get phi^(k-1) now
twin wolf
How do you get phi^(k-1) now
we can write
phi^(k-2) + phi^(k-3) = phi^(k-1) right
and we know phi^2 = phi + 1 so im assuming u do some factoring with that to try and substitute it in
dense smelt
we can write
phi^(k-2) + phi^(k-3) = phi^(k-1) right
Yes, why
dense smelt
well phi^k-2 + phi^k-3 is phi^k-1 from the terms in the sequence
I mean, why is this true
twin wolf
oh well
you can factor phi^k-3 from phi^k-2 + phi^k-3 to get phi^(k-3)*(phi + 1)
and phi + 1 is phi^2 from part a
so phi^k-3 * phi^2 = phi^k-1
twin wolf
So there you go, that’s b
thanks sharp 
and for part c i can just use normal induction?
dense smelt
thanks sharp <:catking:812386361297862747> and for part c i can just use normal ...
Try it :3
full forumBOT
@twin wolf Has your question been resolved?
sonic stratus
If i might give some input, you have to use your own proof from a)
pearl nebula
I know that then range of the inverse is the domani of the function but still why is the logic behind why I got my range wrong? Is the range of the function not the range of the equation?
spice ivy
are u trying to find the domain of g-1(x)
pearl nebula
The range of it
spice ivy
yea
lime ether
a range is a set of values
lime ether
you need to give a subset of real numbers
pearl nebula
I switched the x and y values from the original function and isolated the y
pearl nebula
Yes
spice ivy
ok in that cause the range isnt much harder to find
u said
g(x) has the domain x E [0, 4]
yes?
pearl nebula
I kinda see it cus it doesn’t pass the vertical line test but I thought algebraically that was right idk why
Ah yes
spice ivy
hmm
whether or not its a function
well
it does pass the vertical line test
u are neglecting the original domain
pearl nebula
But doesn’t it cut two lines…
lime ether
But doesn’t it cut two lines…
🤔🤔🤔
lime ether
the domain is restricted to [0, 4]
pearl nebula
Ah no I tried sketching the inverse
pearl nebula
But assuming idk the domain of the original function how would u find that the inverse’s range is that?
lime ether
it doesn’t have an inverse on R
pearl nebula
Ah
glossy valveBOT
_rayn123_
spice ivy
do u know how to find that
pearl nebula
I thought you could also just solve it
pearl nebula
Isn’t it just all real numbers
spice ivy
so jumping into inverses is probs hard
pearl nebula
For the domain
pearl nebula
Isn’t it like f(x) is less than or equal to 16
lime ether
🤔
lime ether
16 - x^2 <= 16
spice ivy
im not good at english
pearl nebula
Ah it’s ok thanks for the help my brain is fried
spice ivy
y E (-∞, 16]
pearl nebula
Ah okok I think it’s the same thing
pearl nebula
But so for finding the range of the inverse I cannot just solve for y? Must I always need to find the domain of the original function?
spice ivy
i mean with something like g(x) u can just find the domain and range of g^-1(x) in your head
i dont think they will ask u to show working out for something that simple
pearl nebula
Ah ok thanks you’re overestimating how good my head is
spice ivy
u are reflecting g(x) over the line of y=x
pearl nebula
Also like if I need to find the range of fg^-1
spice ivy
yes...
pearl nebula
I also just find the domain of fg right
spice ivy
allgs
pearl nebula
.close
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Available help channel!
full birch
What are some of the most important and useful ideas in math that are helpful for physics.
Especially forming equations in physics; particle and quantum physics, radiation decay, and subjects related to these.
and if there's more important math topics related to other subjects in physics other than the ones I've mentioned yeah.
I wanna know on major things to master, as I plan on pursuing physics.
full birch
I'd prefer if you say specific ideas such as integration instead of whole concepts such as calculus, but if it's really not say-able otherwise and every component of a concept is highly useful, then sure
<@&286206848099549185> 
digital hemlock
dimensional analysis
full birch
hmm dimensional analysis
digital hemlock
wiki
slender onyx
What are some of the most important and useful ideas in math that are helpful fo...
calculus of variations, specifically the euler lagrange equation, powers lagrangian and hamiltonian mechanics, slick alternatives to newtonian mechanics
full birch
calculus of variations, specifically the euler lagrange equation, powers lagrang...
hmm I see
will check it out yeah, what pre requisites do they have in order to learning them though?
slender onyx
also the list of connections between physics and PDEs is a mile long. any decent PDE book should cover them
slender onyx
will check it out yeah, what pre requisites do they have in order to learning th...
as far as computing, just how to solve ODEs and work in typical coordinate systems
full birch
I see
would you recommend me a book that you think is great for someone who tries to go through intuitively?
perhaps even an online resource if not
slender onyx
from feynmans lectures https://www.feynmanlectures.caltech.edu/II_19.html
it should be a nice intro for you if youre learning from a physics pov and wanna skip a bunch of math formalism
full birch
it should be a nice intro for you if youre learning from a physics pov and wanna...
Yeah thank you
Feel like the question has been resolved
.close
full forumBOT
slender onyx
np
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Available help channel!
lean rock
An arrows-only metacategory satisfies the objects-and-arrows axioms when the identity arrows are taken as the objects.
full forumBOT
An arrows-only metacategory satisfies the objects-and-arrows axioms when the ide...
@lean rock Has your question been resolved?
lean rock
I can see the objects are sort of redundant
cross dom f and cod f as objects of f, and you still have the arrows dom f and cod f
associativity is missing maybe
id
composition
associativity
cross objects ⟶ arrows still commute
full forumBOT
@lean rock Has your question been resolved?
lean rock
The matrix A = \begin{pmatrix}
1 & 1 & 3 \
3 & 3 & 2 \
1 & 2 & 3
\end{pmatrix} is invertible.
glossy valveBOT
kytsu
The matrix A = \begin{pmatrix}
1 & 1 & 3 \\
3 & 3 & 2 \\
1 & 2 & 3
\end{pmatrix} is invertible.
```Compilation error:```! Missing $ inserted.
<inserted text>
$
l.50 The matrix A = \begin{pmatrix}
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.
LaTeX Font Info: Trying to load font information for OT1+lmr on input line 5
0.
(/usr/local/texlive/2023/texmf-dist/tex/latex/lm/ot1lmr.fd```
slate violet
,w inverse of {{1,1,3},{3,3,2},{1,2,3}}
glossy valveBOT
slate violet
ok sure... haha
lean rock
逆行列あり⇔行列式≠0
slate violet
you'd try to eliminate the rows to make 0s
slate violet
Gaussian elimination with row operations?
yes exactly
lean rock
I'll try, because I need to practice/improve the workflow of that computation
full forumBOT
@lean rock Has your question been resolved?
lean rock
you'd try to eliminate the rows to make 0s
because there is two zeros and one term ≠0 in the matrix at 4. I just crossed the corresponding row and column, because the 0 determinants of the 0 entries don't contribute much if at all to the determinant
It's a recursive formula somehow
dull grove
It's a recursive formula somehow
Is that uni maths ?
slate violet
you can also learn it in HS
lean rock
Is that uni maths ?
Or online, pick up a textbook
dull grove
if you're Indian you will
Yea
dull grove
Or online, pick up a textbook
Ohh
lean rock
Work through the exercises here, people are helpful, maybe help
At uni you are recommended textbook, and then you are tested of the knowledge of the exercises of the textbook/workbook, and the theorems proofs, theory which is in the books
so if you can solve the exercises, and understand the theorems, proofs, then you completed the subject at uni,
also you can learn Lean4, or some other proof assitant programming language and test your own proofs
Easy
growth mindset + mindfulness ⟶ you can learn anything
wise hornet
It's a recursive formula somehow
There is one thing to note about Gaussian elimination, some operations don’t preserve the determinant. I don’t remember the exact name, but the way you can represent applying a row operation onto a matrix is by performing the operation on identity and then doing matrix multiplication (one side you get the operation applied to a row, the other side to a column). This will allow you to see how applying a row operation changes the determinant because for any square matrices $\det AB = \det A \det B$ (also as an aside, do you see how there is a homomorphism here?)
The name is elementary matrix
glossy valveBOT
Beth
lean rock
Thanks @wise hornet
lean rock
There is one thing to note about Gaussian elimination, some operations don’t pre...
So when I scaled R1 down by 3, determinant was scaled down by 3 also
lean rock
The name is elementary matrix
I found this, but I don't yet understand it, so I am deciphering it, looking also for the homomorphism which is a morphism is the category of groups I believe
wise hornet
I found this, but I don't yet understand it, so I am deciphering it, looking als...
So, it’s not a group homomorphism unless you do some restrictions. The general linear group (of n x n matrices) to the non zero real numbers makes the determinant a group homomorphism
lean rock
So, it’s not a group homomorphism unless you do some restrictions. The general l...
f(A ∘ B) = f(A) ∘ f(B)?
wise hornet
I found this, but I don't yet understand it, so I am deciphering it, looking als...
Theorem 2.21 will give you what you want (it clarifies things I think)
wise hornet
f(A ∘ B) = f(A) ∘ f(B)?
Yes. But try to not write the compose symbol here. It’s better to omit it as we don’t write $A \circ B$ for matrix multiplication
glossy valveBOT
Beth
wise hornet
Or if you want the commutative diagram corresponding to a homomorphism between objects in the same category
But that’s of course a bit long lol
And not as insightful since sometimes you need additional things besides $f(ab) = f(a)f(b)$
glossy valveBOT
Beth
lean rock
Or if you want the commutative diagram corresponding to a homomorphism between o...
Do you think this is related?
lean rock
It preserves composite like det(AB)=det(A)det(B) maybe
wise hornet
Monoid homomorphism are like group homomorphisms but require identity to be preserved
wise hornet
It preserves composite like det(AB)=det(A)det(B) maybe
Well now we aren’t composing we are multiplying. So it preserves multiplication of matrices
wise hornet
Do you think this is related?
So the second diagram here is f(ab) = f(a)f(b). The third one is identity being preserved
With $\eta$ being the embedding of 1 into $M$ and similar for $\eta’$ but for $M’$
glossy valveBOT
Beth
wise hornet
(In fact you can identity 1 in the monoid with this map which I think is what is being done here)
lean rock
(In fact you can identity 1 in the monoid with this map which I think is what is...
I am grateful for your help @wise hornet! I am fitting these concepts together. So there is a relationship of these patterns and the row operations / determinant? Which is nice, because I found the calculation quite boring but now I see it has interesting patterns. One excerpt is from Linear Algebra by Meckes, and the other is Categories by Mac Lane
full forumBOT
@lean rock Has your question been resolved?
full forumBOT
@lean rock Has your question been resolved?
lean rock
So the second diagram here is f(ab) = f(a)f(b). The third one is identity being ...
I learn here M,M' are monoids, and f could be det : A ⟼ det(A)
det : GL(n) ⟶ 𝔽^×
μ : A × B ⟼ AB
f ∘ μ : A × B ⟼ AB ⟼ det(AB)
I don't yet know what's 1⟶M via η
wise hornet
With $\eta$ being the embedding of 1 into $M$ and similar for $\eta’$ but for $M...
This
lean rock
multiplying the elementary matrices do a transformation each
1. swap ⟂ rows
2. scale (≠0)
3. shear/add row
and something cool is that maybe these elementary matricesare like the primes of GL(n)? as if having 2,3,5 to build every number?
but I don't see the elementary matrix nor the row operation yet in the diagrams
wise hornet
So all groups are monoids. But group homomorphisms dont need f(1) = 1 because of inverses which make f(ab) = f(a)f(b) sufficient
lean rock
inclusion of 1 in M is eta?
wise hornet
and something cool is that maybe these elementary matricesare like the primes of...
This is a good idea. Primes are actually a general thing and appear often in certain types of rings (in particular primes can be defined in commutative rings with identity). but we can also take about generating the general linear group from elementary matrices which is more akin to what you are saying (I don’t know if this actually works)
wise hornet
inclusion of 1 in M is eta?
Yes it’s eta(1) = 1
lean rock
This is a good idea. Primes are actually a general thing and appear often in cer...
Yes this
wise hornet
I learn here M,M' are monoids, and f could be `det : A ⟼ det(A)`
det : GL(n) ⟶ �...
In what you have written you don’t even need a field as the F suggests. You can take a ring of matrices from any ring
lean rock
In what you have written you don’t even need a field as the F suggests. You can ...
Any monoid to any monoid
wise hornet
So take the units of a ring (there might not be any which is why it’s easier to work in fields)
lean rock
With the required struture, which I don't yet know
lean rock
determinant more special
wise hornet
determinant more special
Yes unfortunately context gets in the way of generality
lean rock
So take the units of a ring (there might not be any which is why it’s easier to ...
det : GLₙ(R) → Rˣ
det(1) = 1ˣ ∈ Rˣ
wise hornet
det(1) = 1ˣ ∈ Rˣ
Well, this you don’t have to write. Because you have a group!
It might be beneficial to show that a function $f: G \to H$ is a group hom iff. $f(1) = 1, f(a^{-1}) = f(a)^{-1}$ and $f(ab) = f(a)f(b)$
glossy valveBOT
Beth
wise hornet
(Where G and H are groups of course)
lean rock
I started constructing the proof
∀a,b,c: a*(bc) = (ab)*c — associativity
∃1: ∀a: 1a = a1 = a — identity
∀a: ∃a⁻¹: aa⁻¹ = a⁻¹a = 1 — inverse
maybe these axioms are like sort of elementary matrices, or prime basic building blocks also
Thank you very much @Beth
wise hornet
I started constructing the proof
∀a,b,c: a*(bc) = (ab)*c — associativity
∃1: ∀a...
This is the definition of a group, provided you have a closed binary operation
full forumBOT
@lean rock Has your question been resolved?
silver pecan
full forumBOT
https://www.desmos.com/calculator/fbjfalyulx
silver pecan
This is my attempt to make a roller coaster in desmos
couple issues:
- i think I am missing some force?
- the step seems to be too large, but it seems desmos doesnt support decimal values for recursive sequences. any other idea to make it smoother?
and really just like a review as to what i can improve
this has taken me forever lol
and sorry if this seems more like a physics problem
for anyone reading this lmk if you need help understanding some function, i tried my best to name/organize them
steel solar
2. the step seems to be too large, but it seems desmos doesnt support decimal va...
Is there anyway to find the function normally?
You are also definetely missing some force lmao
What values of t is this meant to be defined for?
Like the rollercoaster starts off to the right and then goes left, was that intentional?
silver pecan
Like the rollercoaster starts off to the right and then goes left, was that inte...
maybe
the idea is there is some initial velocity
i guess ideally
i want it to go somewhat up the big hill
then fall back down
i can't tell if its already doing that
if it is, it goes by way too quickly
i'm thinking maybe i should increase the step on t to be 10
and then compensate for that during calculations
but idk how to do that
desmos doesn't seem to allow decimals for the recursive functions, so i couldn't do like t/10
also, maybe i should use summation instead of recursive formula? honestly i dont really know the difference
silver pecan
Is there anyway to find the function normally?
and what do you mean by "normally"?
steel solar
and what do you mean by "normally"?
w/o recursion
steel solar
also, maybe i should use summation instead of recursive formula? honestly i dont...
Yeah I think desmos takes decimal inputs for sums...don't know if its accurate though
silver pecan
w/o recursion
well the problem is
velocity = sum of the accelerations for all time that has passed
position = sum of the velocities for all time that has passed
so maybe it is possible to skip the recursion/summation, but idk how
silver pecan
Yeah I think desmos takes decimal inputs for sums...don't know if its accurate t...
i guess im just kinda confused as to what the "step" is in a summation
like, it calculates every 1? then how does it work with a decimal
steel solar
like, it calculates every 1? then how does it work with a decimal
That's a long story
There are several yt videos on it though
atp i feel this is out of my league, sorry
silver pecan
haha no worries
this is crazy complex lol, i started this project thinking i'd only have to worry about derivatives
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@silver pecan Has your question been resolved?
silver pecan
<@&286206848099549185>
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@silver pecan Has your question been resolved?
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@silver pecan Has your question been resolved?
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silver pecan
.reopen
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✅ Original question: #help-28 message
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@silver pecan Has your question been resolved?
unreal wren
Can someone help me solve this? (No calculator)
full forumBOT
Can someone help me solve this? (No calculator)
unreal wren
,rccw
glossy valveBOT
unreal wren
The only idea I have is to use arcsin, but this is a no calculator question
grizzled nest
you're almost there
you could figure out the angle of the two triangles you drew
at the origin
wise tartan
The only idea I have is to use arcsin, but this is a no calculator question
you could use the arctan a - arctan b formula
unreal wren
you could use the arctan a - arctan b formula
It exists?
Damn..
But is there a way to solve it without trig?
Cause this is an 8th grade question, we are actually not supposed to use trig
grizzled nest
actually, those two are the same triangles
unreal wren
Oh wait
onyx glen
hold up
unreal wren
Lol that's genius
onyx glen
do you know complex numbers
unreal wren
do you know complex numbers
Yes
onyx glen
if you do then theres a comparatively straightforward way of doing this problem
oh good
arg( (6+8i)/(8+6i) )
that in fact is it
onyx glen
i mean you gotta simplify the fraction but like, still
slate violet
you'd have to convert it into $re^{i \theta}$ form using Euler's identity
glossy valveBOT
south
unreal wren
Can someone tell me the steps to find the arg
slate violet
based on that the angle is just arctan(y/x)
onyx glen
arctan(im/re) is the prototypical move
onyx glen
you're in quadrant 1 so theres no bullshit
grand hatch
you still need to know arctan(3/4) it looks like
unreal wren
you still need to know arctan(3/4) it looks like
Really?
slate violet
arctan(7/24) but like
unreal wren
arctan(7/24) but like
It's a no calculator question..
grand hatch
you can get arctan(7/24) if you know arctan(3/4)
unreal wren
I don't know how you can do this without a calculator
That's why I'm so confused
grand hatch
well i suppose it helps to have the value of arctan(3/4) memorised. then you can use the fact that (8,6) is the reflection of (6,8) about y=x, so the angle is ||2*(45-36.8) ||
grand hatch
yeah the answer is an integer as well
onyx glen
arctan(7/24) ≈ 7/24 rad = 7/24 * 180/pi degrees...?
unreal wren
Wait let me process this
wise tartan
Cause this is an 8th grade question, we are actually not supposed to use trig
how is this 8th grade
unreal wren
how is this 8th grade
That's why..
onyx glen
nah im just yapping
grand hatch
8th grader knows arctan(x) = x for small x?
unreal wren
Avg Asian things
But I rlly wonder if there's any way to do this in an 8th grade way
No complex numbers, no arc tan
slate violet
it's easy if you have more advanced knowledge yes
wise tartan
plot it probably
unreal wren
plot it probably
Please explain further
grand hatch
yeah bruh at this point just use a ruler and protractor
slate violet
yeah I think they want you to use your protractor
draw to scale and do it that way
well, it's the same angle as (3, 4) and (4, 3) by scaling
unreal wren
yeah I think they want you to use your protractor
We weren't told to bring protractors
Only pencil, eraser, sharpener
No ruler too
💀
grand hatch
but are you allowed to?
slate violet
Well still, at least I already know the solution
ask them how they did the problem then