k thx
#help-28
sacred sequoia
full forumBOT
@sacred sequoia Has your question been resolved?
rancid harbor
\section*{Exercise 53}
Suppose $f(x)$ is a polynomial of degree $n$ and that the values $f(0), f(1), \dots, f(n)$ are known. Describe a procedure for determining $f$, and justify that it works.
glossy valveBOT
alshfik
rancid harbor
Ok so my idea is use the consider the harder problem $f(a_0), \dots, f(a_n)$ instead and use the following lemma to split the case $n$ into $n-1$ cases (CORRECTION: SEE BELOW)
\textbf{Lemma 3.E53:} Let $f(x)$ be a polynomial of degree $n$. Let $a$ be a real number. Then, there is a polynomial $h(x)$ of degree $n-1$ such that $f(x)=(x-a)h(x)+f(a)$. \
Because $f(x)=f(x)-f(a)+f(a)$, we prove that $f(x)-f(a)=(x-a)h(x)$. Let $\langle c \rangle$ be a sequence such that $f(x)=\sum^{n}{i=0}c_ix^i$. Then
[f(x)-f(a)
=(\sum^{n}{i=0}c_ix^i)-(\sum^{n}{i=0}c_ia^i)
=\sum^{n}{i=0}c_i(x^i-a^i)
=\sum^{n}{i=1}c_i(x^i-a^i)]
For $i \geq 1$, $x^i-a^i=(x-a)h_i(x)$, where $h_i(x)=\sum^{i}{j=1}x^{i-j}a^{i-1}$. And so $f(x)-f(a)=\sum^{n}{i=1}c_i(x-a)h_i(x)=(x-a)\sum^{n}{i=1}(c)ih_i(x)$. And so $f(x)-f(a)=(x-a)h(x)$, where $h(x)=\sum^{n}{i=0}c_ih_i(x)$. Since $h_i(x)$ has degree $n-1$ for all $i$, $h(x)$ has degree $n-1$. \qedsymbol
glossy valveBOT
alshfik
rancid harbor
Ok so my idea is use the consider the harder problem $f(a_0), \dots, f(a_n)$ ins...
does this make sense? is the procedure fine? and most importantly: how the hell do I express the procedure?
rancid harbor
\textbf{Lemma 3.E53:} Let $f(x)$ be a polynomial of degree $n$. Let $a$ be a rea...
this lemma was given(sort of) by the textbook, so its used somewhere or there is another method
rancid harbor
Ok so my idea is use the consider the harder problem $f(a_0), \dots, f(a_n)$ ins...
i mean split the case of degree $n$ into $n$ cases of degree $n-1$ sorry
glossy valveBOT
alshfik
rancid harbor
so for example for $n=2$ you can use the lemma to obtain $f(x)=f(0)+xf_0(x), f(x)=f(1)+(x-1)f_1(x), f(x)=f(2)+(x-2)f_2(x)$, then you can keep subbing $x$ to obtain the values for $f_0(1),f_(2),f_1(0),f_1(2),f_2(0),f_2(1)$ and you've basically downgraded the problem
full forumBOT
@rancid harbor Has your question been resolved?
torn jolt
Trying to find the difference equation of
[
y_n = A3^n + B
]
torn jolt
Trying to find the difference equation of
\[
y_n = A3^n + B
\]
Took the determinant
[(y_n, 1, 1), (y_{n+1}, 3, 5), (y_{n+2}, 9, 25)] = 0
Which leads to the answer of
2y_{n+2} -16y_{n+1} + 30y_{n} = 0
I'm not sure if it is correct tho 
(Fourier analysis, just linear differences)
full forumBOT
@torn jolt Has your question been resolved?
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@torn jolt Has your question been resolved?
short siren
Why does bot not recognize the LaTeX? Is it down?
full forumBOT
@torn jolt Has your question been resolved?
torn jolt
Why does bot not recognize the LaTeX? Is it down?
probably
short siren
Trying to find the difference equation of
\[
y_n = A3^n + B
\]
Is this what you meant?
torn jolt
yea
topaz valley
full forumBOT
@torn jolt Has your question been resolved?
full forumBOT
@torn jolt Has your question been resolved?
dusk tangle
grim kestrel
nvm i got it
gray pagoda
I just would like to get a 2nd opinion in regards to this question:
I set up my integrals as ( 0 < z < 4) (0 < theta < 2pi) (0__<__r (radius) < 2)
i know how to do the problem but im confused if the integral with the z is correct. Cuz if not, would it then be (x^2 + y ^2 __<__z < 4) which is then converted to polar coords being (r^2 < z < 4) ?
gray pagoda
And i know the shape is a bowl like shape between the z axis of 0 and 4
<@&286206848099549185>
torn jolt
(0 < z < 4) (0 < theta < 2pi) (0 ≤ r (radius) < 2)
gray pagoda
oki then that moves me to here: how come this problem the interval is not (0 < z < 4) and instead is actually (0 < z < 4-r^2)
Its nearly the same shape but just flipped upsidedown (and disregarding the first octant part)
@torn jolt
gray pagoda
they both have r^2 haha
torn jolt
subtracting r^2 from 4? This is because the r^2 term in the equation can reduce the value of z, so the upper bound for z has to be adjusted
or am i missing the question
if yes, sorry xd, ill let someone else try
gray pagoda
i mean its oki but its just losing me why one question has one method while the other wont work the same way when i think it should
cuz after doing it, it actually makes more sense if it was r^2 < z < 4 instead of 0 <z <4
<@&286206848099549185>
<@&286206848099549185>
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@gray pagoda Has your question been resolved?
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@gray pagoda Has your question been resolved?
full forumBOT
@gray pagoda Has your question been resolved?
torn jolt
full forumBOT
torn jolt
Can the 5 become the coefficient here
smoky wing
torn jolt
How can I expand further
torn jolt
Oh ok
torn jolt
Does the 5 go on both the 4 and m
vivid aurora
no
torn jolt
Ok
torn jolt
Trying to find the inverse z-transform of
[
\log\frac{z}{z+1}
]
glossy valveBOT
♡LexQa♡
onyx glen
does that even exist?
torn jolt
I substituted $(\frac{1}{y})$ to get
[
\frac{(-1)^n}{n}
]
glossy valveBOT
♡LexQa♡
onyx glen
based on this, it would appear that a function is only the z-transform of something if it's defined at 0
which yours isn't
torn jolt
I managed to get a) but I am not sure if my method is correct anymore 
onyx glen
,w z-transform (-1)^n/n
glossy valveBOT
Requested by Ann#0413
onyx glen
short siren
Man
You out here doing cool math and while I am taking basic arithmetic for designing values
Anyways I think
torn jolt
You out here doing cool math and while I am taking basic arithmetic for designin...
Haha I'm gonna do Signal processing and digital systems next semester, I am trying to get the math down first
short siren
$$\frac{z}{z-a} = a^n u(n)$$
glossy valveBOT
VulcanOne
short siren
Haha I'm gonna do Signal processing and digital systems next semester, I am tryi...
The math behind signal processing is super cool. I wanted to learn more but my field long stopped after learning Fourier Series lol.
torn jolt
Wait it is like Laplace
Yeah I think
torn jolt
The math behind signal processing is super cool. I wanted to learn more but my f...
Lmaoo my entire major revolves around Fourier
short siren
Lmaoo my entire major revolves around Fourier
It's a great math tool but too bad I can't learn it properly now
Hopefully I do in the future
Gonna use it in some Civil Engineering Structure 
Anyways
I think z/(z+1) can be rewritten as
1/(1+z^{-1})
$$\frac{1}{1+z^{-1}}
short siren
$$\frac{1}{1+z^{-1}}$$
glossy valveBOT
VulcanOne
royal kraken
Hi guys, who can help me ?
royal kraken
torn jolt
But carry on @short siren
short siren
Then we use some transformation for log
I got stuck at the log part
I think it involves some kind of summation
torn jolt
Yeah my method was basically
torn jolt
Time to speedrun latex on phone
brave blaze
torn jolt
Putting $z = \frac{1}{y};$
[
F(z) = \log(\frac{\frac{1}{y}}{\frac{1}{y} +1})
]
[-\log(1+y)]
[\to -y + \frac{y^2}{2} + ... ]
[
-z^{-1} + \frac{z^{-2}}{2} -....+ \frac{(-1)^{n}}{n} z^{-n}
]
Pain
short siren
[-\log(1+y]
short siren
$$-\log(1+y) \to -y + \frac{y^2}{2} + ...$$
glossy valveBOT
VulcanOne
short siren
$$z^{-1} + \frac{z^{-2}}{2} -.... + \frac{(-1)^{n}}{n}$$
glossy valveBOT
VulcanOne
short siren
Hmm
short siren
Damn hopefully you find the answer soon
Gtg
Imma keep my eyes on this for later
👀
glossy valveBOT
♡LexQa♡
glossy fern
why doesnt tan(x+y) identify hold if x,y are odd multiples of π/2
stable heart
because tan(pi/2) is undefined as its essentially dividing by zero
$\tan\frac{\pi}{2}=\frac{\sin\frac{\pi}{2}}{\cos\frac{\pi}{2}}=\frac{1}{0}$
glossy valveBOT
Duh Hello
full forumBOT
@glossy fern Has your question been resolved?
marble whale
Any help pls
full forumBOT
Any help pls
marble whale
A deffo has to be 1 and f has to be 5 and C, 4
But if u use that I don't get the right answer
Any ideas anyone?
full forumBOT
@marble whale Has your question been resolved?
full forumBOT
@marble whale Has your question been resolved?
marble whale
<@&286206848099549185>?
nocturne creek
A deffo has to be 1 and f has to be 5 and C, 4
I agree 🤔
I also think d should be 2? Maybe 3 but not 6
nocturne creek
even my teacher said there was no solution
I tried all the possibilities and couldnt find one that matches
marble whale
and he came from cam so idek anymore😭
nocturne creek
Like if nothing works for d=2, no point trying for d=3 or 6
marble whale
Like if nothing works for d=2, no point trying for d=3 or 6
yea
worthy tree
GCD( b , c+1) = 1
GCD (d , f) = 1
marble whale
A deffo has to be 1 and f has to be 5 and C, 4
maybe this is wrong
nocturne creek
a,has to be one anything else would give zero or negative output
marble whale
a,has to be one anything else would give zero or negative output
yeah
unless the answer is negative and they ignored it
highly doubt it
nocturne creek
unless the answer is negative and they ignored it
Wdym?
The right hand side cant be negative
worthy tree
The right hand side cant be negative
so the left side must be positive
marble whale
I'm prolly waffling😭
worthy tree
that means that a < 2
nocturne creek
so the left side must be positive
Why is why a=1
nocturne creek
Why is why a=1
Which is*
nocturne creek
Yk when ur adding areas right, if u have a negative area u ignore the negative
I have no idea what youre talking about🗿, when i do areas i use absolute value symbol
worthy tree
$\int_{a}^2 bx^c + 2ex dx = (2^{c+1} - a^{c+1}) \frac{b}{c+1} + (4-a²)e$
marble whale
I have no idea what youre talking about🗿, when i do areas i use absolute value ...
😭😭
Yes i got this
nocturne creek
c being 4 males some sense tho, cuz the other side has 5 in the denom
And f cant be anything but 5
glossy valveBOT
Mehdi_Moulati
marble whale
c being 4 males some sense tho, cuz the other side has 5 in the denom
Shit, true
Is there like a program we can run
nocturne creek
Shit, true
But it might not be im not sure
marble whale
Is there like a program we can run
That will brute force all numbers into the eqn
worthy tree
Why is why a=1
$(2^{c+1} - a^{c+1}) \frac{b}{c+1} + (4-a²)e$ must be positive so 2² - a² and $2^{c+1} - a^{c+1}$ must be positive
glossy valveBOT
Mehdi_Moulati
marble whale
Yes that's true
glossy valveBOT
Mehdi_Moulati
nocturne creek
Try c something other than 4, becuase 4 dont work, so if this question isnt wrong c might be something else
marble whale
Exactly
I was thinking we could do something like this
But I'm not good at python 😭
marble whale
Nah that's pseudocode
worthy tree
**Mehdi\_Moulati**
the fraction of b and c+1 mustn't have common factors
marble whale
Yep
marble whale
Looks so different from what i write💀
U think u can make a code similar to that?
nocturne creek
b shouldnt be divisible by c+1, or do they need to have 0 common factors?
nocturne creek
Thats why b shouldnt be divisible by c+1, cuz the thing on the right is not an integer
marble whale
@worthy tree @nocturne creek you guys gotten anywhere?😭
worthy tree
Is there like a program we can run
i have one in python
values = [1,2,3,4,5,6]
def equation(A):
a,b,c,d,e,f = A
return ( 2 ** (c+1) - a **(c+1)) * b / (c+1) + (4-a*a)*e == 144 * d/f
def solve(vars=[],depth=6):
if depth <= 0 :
if equation(vars):
print(vars)
else:
for var in values :
if vars.count(var) == 0:
P = list.copy(vars)
P.append(var)
solve(P,depth -1)
solve()
he found that :
a = 1
b = 6
c = 5
d = 2
e = 3
f = 4
worthy tree
I'll try this tysm
anytime brother
the answer is :
$\int _1^2 6x^5 + 2\cdot 3 * x dx = 144 * \frac{2}{4} = 72$
glossy valveBOT
Mehdi_Moulati
worthy tree
,w integral 6x^5 + 2*3x dx from 1 to 2
glossy valveBOT
marble whale
he found that :
a = 1
b = 6
c = 5
d = 2
e = 3
f = 4
You sure this is right?
It's wrong for some reason
worthy tree
this is the integral
marble whale
**Mehdi\_Moulati**
The final answer shouldn't be divisible
So it should be a fraction
worthy tree
he found that :
a = 1
b = 6
c = 5
d = 2
e = 3
f = 4
my program found only one possible solution
marble whale
😭😭😭
Wait
,w integral 6x^4 + 2*3x dx from 1 to 2
,w integral 3x^5 + 2*6x dx from 1 to 2
😭😭
worthy tree
$\int_{a}^2 bx^c + 2ex dx = 144 * \frac{d}{f}$
glossy valveBOT
Mehdi_Moulati
worthy tree
**Mehdi\_Moulati**
the integral like this Right ?
marble whale
D/f can't be cancelled down any further
Yeah
But ur program had 2/4 which can be simplified
worthy tree
But ur program had 2/4 which can be simplified
,w integral bx^c + 2ex from 1 to 2
worthy tree
where did you make ur program btw?
IDE ?
marble whale
ahh right
worthy tree
ahh right
Visual studio code
values = [2,3,4,5,6]
def equation(A):
b,c,d,e,f = A
return b * (2 ** (c+1) - 1)/ (c + 1) + 3*e == 144 * d/f
def solve(vars=[],depth=5):
if depth <= 0 :
if equation(vars):
print(vars)
else:
for var in values :
if vars.count(var) == 0:
P = list.copy(vars)
P.append(var)
solve(P,depth -1)
solve()
same result
b= 6
c= 5
d= 2
e= 3
f= 4
worthy tree
yes
marble whale
oh
worthy tree
GCD(d,f) = 1
worthy tree
def GCD(a,b):
while(a!=0 and b !=0 ):
if a>b : a %=b
else: b %=a
return a+b
values = [2,3,4,5,6]
def GCD(a,b):
while(a!=0 and b !=0 ):
if a>b : a %=b
else: b %=a
return a+b
def equation(A):
b,c,d,e,f = A
return GCD(d,f) == 1 and b * (2 ** (c+1) - 1)/ (c + 1) + 3*e == 144 * d/f
def solve(vars=[],depth=5):
if depth <= 0 :
if equation(vars):
print(vars)
else:
for var in values :
if vars.count(var) == 0:
P = list.copy(vars)
P.append(var)
solve(P,depth -1)
solve()
but he didn't find any solutions
worthy tree
may be there is an error in the problem
worthy tree
It's not a b
a and b are just variable ,you can name them whatever you want
look at the equation function
you will find GCD(d,f)
marble whale
oh yeah mb
hmm this is so weird
is there a way where you can just brute force it?
like all the possibilities
wait
worthy tree
b= 6
c= 5
d= 2
e= 3
f= 4
this is the only solution
worthy tree
```python
values = [2,3,4,5,6]
def GCD(a,b):
while(a!=0 and b !=0 ):
...
this code check for all the possibilities for b,c,d,e,f
marble whale
oh damn it does
😭
wait let me think
@worthy tree don't restrict it to > 0
so A isnt limited to just 1
maybe that will do something
worthy tree
so A isnt limited to just 1
let me verifier
marble whale
Alrr
worthy tree
i think i know the problem :
the integer well be negative if a > 2
marble whale
i think i know the problem :
the integer well be negative if a > 2
yes but hopefully it will give us something we want
since the only solution when a is 1 isnt what we want
so thats out only choice here, and hopefully it works😭
worthy tree
two solution :
[a=4 , b=6 , c=1 , d=2 , e=5 , f=3]
[a=6 , b=2 , c=1 , d=4 , e=5 , f=3]
marble whale
great i try these ones
worthy tree
,w integral 6x^1 + 25x from 4 to 2
glossy valveBOT
marble whale
☠️
that dont look right lol
they both wrong😭
but howww
ok ok
i have an idea
so A is most certainly 1
can you make f, 5?
so we only need to add 2, 3, 4 and 6
and we dont need a and F anymore
marble whale
yeaa
worthy tree
and $(2^{c+1} - a^{c+1} )* \frac{b}{c+1} + (4 - a²) e = 144 \frac{d}{f}$
glossy valveBOT
Mehdi_Moulati
glossy valveBOT
worthy tree
even the program didnt work
as I told you , there is only one solution :
a = 1
b= 6
c= 5
d= 2
e= 3
f= 4
the program test 720 possibilities , and he find one solution
marble whale
the program test 720 possibilities , and he find one solution
720!!😭
wait what exactly does this program do?
it tests all the numbers from 1 to 6 to see if the left hand side matches with the right hand side?
worthy tree
it tests all the numbers from 1 to 6 to see if the left hand side matches with t...
yes
marble whale
hmmm
worthy tree
and without repeating the numbers
marble whale
oh yeah that too
marble whale
@worthy tree ill leave it for now and come back to it later, ty tho
you think u can help with another one?
worthy tree
of course
worthy tree
@marble whale
the turnings point on the curve are :
A = (0,a^4) ,B = (a,0) and C = (-a,0)
to find them just solve this :
y '(x) = 0
those three point form equilateral traingle when :
AB = BC =CA
marble whale
those three point form equilateral traingle when :
AB = BC =CA
ohhhh
worthy tree
so just calculate the distance between each two point
glossy valveBOT
Mehdi_Moulati
worthy tree
anytime
marble whale
WWW
worthy tree
did you solve the first one ?
worthy tree
this ?
worthy tree
Nah not yet😭
where , i will help
marble whale
where , i will help
Nah its fine I can take it from there, if I get stuck I'll ask for help
Huh how??
It's deffo a decimal
marble whale
This one
Also for this a-b = 1 and a + b =17
And u combine them together to get 117
So the a-b part is always first
worthy tree
c is prime and the last digits made by a+b
lament shard
Yes
worthy tree
a+b != 2
lament shard
First 2 made by a-b
Last 3 by a+b
And we need to find a + b
For the lowest 5 digit prime
worthy tree
so a+b must be odd
worthy tree
which means that :
a is even and b is odd
or
b is even and a is odd
lament shard
Yes
worthy tree
And a-b has to equal as close to but over or equal to 10
depend on how much digits in a+b
if a+b have four digits
worthy tree
since a+b is odd
lament shard
No but we can have a-b even
worthy tree
nope
worthy tree
a+b is odd right ?
lament shard
Yes
worthy tree
then a-b is also odd
lament shard
That’s true
worthy tree
number of digits of a+b must be bigger or equal to 3 and less than 5
worthy tree
so a+b > 100
worthy tree
the answer is a = (3)^(1/6)
marble whale
the answer is a = (3)^(1/6)
Oh nice I'll try this ty
worthy tree
do you know how to get the answer ?
worthy tree
okay
marble whale
,w solve a^2 + a^8 =4a^2
glossy valveBOT
marble whale
@worthy tree did you get these solutions too?
worthy tree
,w solve a^8-3a^2=0
glossy valveBOT
marble whale
so a = 3^(1/6)
Yeahh
marble whale
This was wrong btw😭
There's probably one smaller than this
also we got one more
@worthy tree
This one is quite wordy
marble whale
so a = 3^(1/6)
Yep this is right, nice
marble whale
<@942843959783333890>
A pointer primes add the digits of the prime number together to get the next a pointer prime
Like 11 + 2 = 13
But 13 + 4 = 17
However 17 + 8 = 25
Which isn't prime so 17 isn't an a pointer prime
Same applies for the M pointer primes I assume
marble whale
<@942843959783333890>
just the integral and this one
full forumBOT
@marble whale Has your question been resolved?
worthy tree
the smallest c can be : c = 11003 where :
a = 502
b = 501
a-b = 1
a+b = 1003
@marble whale
marble whale
the smallest **c** can be : *c = 11003* where :
**a = 502**
**b = 501**
**a-...
Oh yeah we got the answer for that
Wait
Yep ur right
11003
Now we just need 2 more
marble whale
<@942843959783333890>
For this the only method I can see is trial and error
U got any ideas? @worthy tree
Like a quicker way
worthy tree
Like a quicker way
i don't think so
marble whale
@worthy tree done
Now we just gotta do that integral
There's definitely a solution
Other than the one u found
two solution :
[a=4 , b=6 , c=1 , d=2 , e=5 , f=3]
[a=6 , b=2 , c=1 , d=4 , e=5 , f=3]
Btw r u sure ur code is right?
Maybe there's something wrong
worthy tree
there is no solution for this
final answer
marble whale
Damn, why would they give us a question that has no solutions
It's so weird
Thanks @worthy tree for the help tho
worthy tree
did you find an answer for this ?
worthy tree
Thanks <@942843959783333890> for the help tho
anytime bro
marble whale
alrr
worthy tree
by programming , right ?
marble whale
by programming , right ?
we cant really program 🥲
worthy tree
That the number had to contain all digits except for 1 being 1
i'm still confused how he did it , i want to know how (if you want @lament shard of course)
lament shard
Well
With the product of number above 100 being increasingly high
And the gap between primes always lower than 30
worthy tree
And the gap between primes always lower than 30
is it always ?
lament shard
The sum + product has to be smaller than 30
Yes from what I could see
Unless u had products that far exceeded the gap in between primes
worthy tree
the product mustn't be equal to 0
lament shard
Yes
So it must be single digit so
By this
I mean that there cannot be many digits that does not equal 0
And if there is they must be low
Like 3 or below
worthy tree
And the gap between primes always lower than 30
this is not true for all the numbers
for example :
15683 is a prime
the next one is 15727
the gap between them is 44>30
marble whale
a = 1
b= 6
c= 5
d= 2
e= 3
f= 4
worthy tree
a = 1
b= 6
c= 5
d= 2
e= 3
f= 4
if d and f have a common divisor
marble whale
if d and f have a common divisor
oh yeh
lament shard
F has to be 5 and a 1
worthy tree
F has to be 5 and a 1
if F = 5 then C = 4
marble whale
F has to be 5 and a 1
if thats the case we dont get any solutions
lament shard
Why does c have to be 4
lament shard
Ahh okay
worthy tree
$(2^{c+1} - a^{c+1} )\cdot \frac{b}{c+1} + (4 - a²) e = \frac{144 \cdot d}{f}$
glossy valveBOT
Mehdi_Moulati
marble whale
loll
marble whale
but those when a,b,c,d,e,f in R(reel numbers) and the repetition is allowed
in fact
try your program with repetition
worthy tree
try your program with repetition
i already did
there is solutions
but in the problem they said "no repetition"
marble whale
and there were repeats in the solution?
worthy tree
and there were repeats in the solution?
when i allowed the repetition i found 9 solutions
worthy tree
a = 1
b= 6
c= 5
d= 2
e= 3
f= 4
if the repetition isn't allowed
worthy tree
if the repetition is allowed :
a = 1 , b = 2 , c = 5 , d = 1 , e = 5 , f = 4 ,
a = 1 , b = 4 , c = 1 , d = 1 , e = 6 , f = 6 ,
a = 1 , b = 4 , c = 3 , d = 1 , e = 3 , f = 6 ,
a = 1 , b = 4 , c = 5 , d = 1 , e = 2 , f = 3 ,
a = 1 , b = 4 , c = 5 , d = 2 , e = 2 , f = 6 ,
a = 1 , b = 6 , c = 1 , d = 1 , e = 5 , f = 6 ,
a = 1 , b = 6 , c = 5 , d = 1 , e = 3 , f = 2 ,
a = 1 , b = 6 , c = 5 , d = 2 , e = 3 , f = 4 ,
a = 1 , b = 6 , c = 5 , d = 3 , e = 3 , f = 6 ,
if not :
a = 1 , b = 6 , c = 5 , d = 2 , e = 3 , f = 4 ,
worthy tree
so the problem doesn't have any solution
worthy tree
if a = 1 and f = 5
then c must be 4
if we replace those number we will get :
$(2^5 - 1 )\cdot \frac{b}{5} + 3 e = \frac{144 \cdot d}{5}$
glossy valveBOT
Mehdi_Moulati
glossy valveBOT
Mehdi_Moulati
worthy tree
$31b + 15 e =144d$
glossy valveBOT
Mehdi_Moulati
worthy tree
b and e must be
both odd or both even
marble whale
Yeahh
marble whale
Yeah I remember my teacher got to this part
worthy tree
31 * b + 15 * e = 432
marble whale
And that's when he said there wasn't a solution
Maybe there really isn't a solution
But there has to be
How else can we answer to question
It doesn't make sense
marble whale
Damn it's a pretty big one then😭
worthy tree
Damn it's a pretty big one then😭
i guess they want to say that
the nom and denom have a common factor
maybe
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@marble whale Has your question been resolved?
hollow herald
Unless u had products that far exceeded the gap in between primes
there are ... there's consecutive primes that have a gap of upto 700
atomic elm
Hi
atomic elm
How do you solve geometry proofs ?
any tips for that ?
and also
how do you solve this question?
If x^2 + 1/x^2 = 98, then find the value of x^3 + 1/x^3
torn jolt
x^2 + 1/x^2 + 2 = 98 + 2
=> ( (x)^2 + (1/x)^2 + 2 * x * (1/x) ) = 100
=> (x+1/x)^2 = 10^2
proceed from here
atomic elm
thank you
astral temple
Can anyone tell me why the equality highlighted in yellow holds?
stiff musk
try writing $|x+y|^2 = \langle x+y, x+y \rangle$ and expanding the latter, same with $|x-y|^2$
glossy valveBOT
Bungo
astral temple
try writing $\|x+y\|^2 = \langle x+y, x+y \rangle$ and expanding the latter, sam...
Umm but I only get
$$||x+y||^2 = \langle x+y, x+y \rangle=||x||^2+2\langle x,y\rangle +||y||^2$$
So I don't know how the "Re" comes out.
glossy valveBOT
Trenton
gritty rose
**Trenton**
That's only true for real values x and y
Write it down in terms of the dot product as a sum
Complex conjugate should be involved
astral temple
ahh
So it should be
$$||x+y||^2 = \langle x+y, x+y \rangle=||x||^2+\langle x,y\rangle +\overline{\langle x,y\rangle}+||y||^2$$
glossy valveBOT
Trenton
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tame bobcat
I need some help with linear algebra, question follows
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I need some help with linear algebra, question follows
tame bobcat
Let A be a Matrix in R^{n, n} and A = v*transposed(v) for any vector v in R^n
|| v || = 1 and n>2
Now the question I'm trying to answer is whether or not v is an Eigenvektor of A .
My intuition tells me thats wrong because v could be any possible vector and when doing the Eigenzerlegung we get V * D * transposed(V) where V is an orthogonal matrix.
I am a bit lost on where to start, would appreciate any help!
fast peak
well what is A*v
tame bobcat
well what is A*v
I am sorry I cant really answer that question other than guessing
fast peak
$Av = (vv^T)v = ?$
glossy valveBOT
Denascite
fast peak
that shouldn't even be a guess. this so far is literally just plugging in the definition of A
tame bobcat
yes of course
fast peak
you shouldn't have to guess to do that
tame bobcat
I just wasnt sure what you wanted to point out
tame bobcat
you shouldn't have to guess to do that
I know, it just wasnt that I didnt know where I need to go
tame bobcat
oh yeah
because these are vectors I am able to do so
sorry I am really not that fit in linear algebra
you know the thing is
I cant really see where this should lead
the rearrangement is plausible
but why I should consider doing that isnt clear to me
fast peak
well you take the brackets away and see you have v v^T v and you notice that the term v^Tv is familiar
tame bobcat
give me a second
ok i think i got it
now v is a n by 1 vector
if I multiply v^T v I get a number
that number is the eigenvalue
I think
and therefore v is the eigenvektor ?
because Av = lambda v
fast peak
therefore v is an eigenvector but yes
tame bobcat
I thought it was written completely in german
fast peak
with lambda=v^Tv = norm(v)^2 = 1
tame bobcat
I am from germany sorry
fast peak
the point I wanted to make is an vs the
tame bobcat
ohh
I'm sorry
oh yeah sure of course that makes sense
why is it so important that norm(v)^2 = 1 in this case ?
oh wait by ^2 you mean the square or the Norm 2
?
proper stream
How can I solve this DE?
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@proper stream Has your question been resolved?
serene niche
sup
serene niche
Just a moment
my device lagging
I do have a question dw
Part 59 option b
my working too
It says it's not derivable at x 0
molten trench
intergrate all of them
serene niche
intergrate all of them
I can't I don't know how this is differential calc
molten trench
boring
serene niche
I am inclined to agree
molten trench
when it says u can’t derive at 0 what does it mean
serene niche
Slope not defined?
molten trench
the function is not continuous over 0
serene niche
but where is my working wrong when you plug in zero it gives a slope right
molten trench
look at ur working
serene niche
thats what I wrote?
serene niche
what do you mean what is 2 squared
molten trench
4…
wait what year level are you in
this is literally basic index
wait there’s no way
you are doing
2^2x
and ur squaring the 2
my eyes
molten trench
i think you divide both sides by sine then derive it with respect to 0 you should get your answer
serene niche
i think you divide both sides by sine then derive it with respect to 0 you shou...
why would it be different is my working flawed somewhere I skimmed it over again and I really don't think they're anything wrong
the book has the standard differentiation of 1/1+x^2 but idk how long that would take I wasn't going to try
it's clearly meant to have substitution when compared with sin 2theta formula
serene niche
ah shittt
molten trench
you’ve manipulated it correctly to get an identity but you’ve used sin 2theta instead of tan 2 theta
molten trench
i was trolling u before
then i felt bad :(
have fun but is this calculator allowed
serene niche
it's ok we all do a little trolling
molten trench
you can make this much easier on your self by just graphing each function
serene niche
have fun but is this calculator allowed
no
serene niche
you can make this much easier on your self by just graphing each function
haha I wish
molten trench
whatever isn’t continuous over 0 can’t be derived at 0
do you know how to work out maximal domains
serene niche
are you guys allowed calculator for this? What's the point of asking the question then
serene niche
do you know how to work out maximal domains
I'm in grade 11 so let's just say uh no
molten trench
you should know by y11 but
serene niche
we would be allowed cas
No but like the sharp point is where it can't be differentiated right
the calc would tell you that?
serene niche
you should know by y11 but
damn not here
maybe we call it something else
what is it
molten trench
No but like the sharp point is where it can't be differentiated right
wdym the sharp point
serene niche
wdym the sharp point
a graphing calc shows the graph right
of the function
the point where the graph takes a sharp turn
it can't be differentiated
We had that for continuity and differentiability
molten trench
is the answer B
serene niche
do you know how to work out maximal domains
Just checked we just call it domain here maximal domain is hella fancy
serene niche
is the answer B
Along with c and d yes
But I'll check c and d myself later
can you show the work for b
molten trench
i just did it in my head
you can figure out the general shape of the graph
by spilltitjg apart the equation and then thinking about the different aspects of it
and how they would affect the graph
molten trench
i’m very visual with the way i think so i always go with graphs
serene niche
Is this even realistically possible to solve
serene niche
I mean sure graphing works but I have no idea how to do that
molten trench
which i’m presuming is derivknf each one
then subbing in 0 for the derivatw
is that correct
serene niche
the way ur doing it
I was wrong though it's supposed to be 1 minus in the denominator
serene niche
is that correct
yes but how do you derive this they surely can't expect us to use tan inverse derivative
if they do
well
molten trench
but is it not required for u to just know it’s 1/(1+x^2)
even if it isn’t i heavily suggest you learn that
serene niche
but is it not required for u to just know it’s 1/(1+x^2)
you need to sub the entire value there
molten trench
because once you get to intergration
serene niche
even if it isn’t i heavily suggest you learn that
I know what it is lol I mentioned it earlier
wait why did I say tan inverse
Sorry
Sin inverse
molten trench
do you knew what u sub is
serene niche
doesn't make it easier anyways
molten trench
why
molten trench
serene niche
This is wrong
molten trench
So it becomes
serene niche
it isn't sin 2 theta right no point using that
molten trench
i’m going to cry
serene niche
me every day
molten trench
do you know what u sub is
serene niche
what
molten trench
yep thought so
serene niche
I don't
molten trench
gl i’m gonna rub one out
molten trench
chain rule?
molten trench
chain rule sin^-1(tan(2x))
please tell me you can do that
when that’s fine
done
u got ur deriv
sub in 0
if it’s undefined u got ur answer
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@serene niche Has your question been resolved?
north nacelle
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north nacelle
does anyone know how to solve this?
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@north nacelle Has your question been resolved?
vast fossil
Notice that the angles 4x + 8 and 6x - 16 share the same arc, meaning they are equal
vast fossil
So you need to just solve 4x + 8 = 6x - 16
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@north nacelle Has your question been resolved?
sleek brook
Can i do this?
sleek brook
multiply the outer with the inner while its in power?
sleek brook
how can i simplify x^6(1/x - 4)^6?
clear lily
$a^6b^6 = (ab)^6$
glossy valveBOT
Gijs
clear lily
np
sleek brook
.close
torn jolt
help
echo lance
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echo mulch
Hi
echo mulch
I'm trying to find the value of that arc using line integral, could someone help me please?
the integral expression is what matters (i know its wrong because computer calcule a diferent value, and not 6 which is the correct answer)
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fair iris
i dont know how to solve this
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i dont know how to solve this
glass crystal
you should just compute the coefficients with the usual formulas
but for a constant functionn they should all equal 0 except $a_0$
glossy valveBOT
Benjamin
robust depot
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robust depot
uhm i mean i can understand why x dot and y dot are like that but
what happend when they differentiated a second time
i legit have no clue how they got there
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@robust depot Has your question been resolved?
robust depot
<@&286206848099549185>
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@robust depot Has your question been resolved?
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rough inlet
Does a one sided limit always equate to a infinity
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Does a one sided limit always equate to a infinity
rough inlet
-infinity or +infinity?
stable plover
no
take any finite limit
then the left and right limits must be equal, and equal that finite value
rough inlet
For this example, do I plug in 7 and 9?
Or better question is that the right approach?
stable plover
i would graph the function
you can note that the function has a vertical asymptote at x = 0 and x = 8
since you are approaching 8 from the left, you can certainly try substituting values like 7
rough inlet
since you are approaching 8 from the left, you can certainly try substituting va...
That gives me one
Which doesn’t work because that wouldn’t be my answer
stable plover
you just want to observe whether the function is positive or negative close to the left of x = 8
since you already know it has a vertical asymptote at x = 8
since this is a zero of the denominator
rough inlet
How come some times your answer can be a integer
Wait so when we are only given a one sided limit as a ‘problem’
We are expected to just observe the behavior of the function and explain notation wise what we observe?
stable plover
,w graph |x|/x
glossy valveBOT
stable plover
the one sided limits at 0 are -1 from the left and +1 from the right
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light scaffold
i am stuck with this problem 😦
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i am stuck with this problem 😦
light scaffold
idk what theorem should i use
i use the Pythagorean theorem
but i think its wrong
light scaffold
i am stuck with this problem 😦
find x
devout valley
but i think its wrong
Why do you think it's wrong?
light scaffold
i think that is not the way to do it
hot herald
show work
devout valley
Can you show me where "b" is on that diagram?
idle willow
b is part of the pythagorean theorem its aa variable you apply to a right angle triangle
hot herald
what did you do after that
light scaffold
its the height
hot herald
no
devout valley
i am stuck with this problem 😦
I mean, on here, where is "b"
light scaffold
I mean, on here, where is "b"
ohh
hot herald
can you highlight the segment you think b represents
in what you just did
15.1^2 = 12.8^2 + b^2
= 8.01
its also bad to write that,
you're implying that 15.1^2 = 8.01 (which it is not)
light scaffold
light scaffold
ohhh
hot herald
and in that triangle, again try indicating what b represented
light scaffold
ohhh is the unknown the hypotenuse?
hot herald
no
light scaffold
ok lemme try again
devout valley
ohhh is the unknown the hypotenuse?
Remember you did 15.1^2 = 12.8^2 + b^2
hot herald
swapping a,b doesn't really matter if you're being consistent
a represents one leg of your right triangle and
b the other
light scaffold
i am confused on finding the c