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#help-36
forest pagoda
tranquil pine
f(x, y) = \begin{cases}
\frac{3x^3 - y^3}{x^2 + y^2} & \text{when } x \neq 0 \text{ and } y \neq 0 \
0 & \text{when } x = 0 \text{ and } y = 0
\end{cases}
soft zealotBOT
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tranquil pine
f: R^2 -> R
I should find the partial derivates on the point (0,0).
How should I approach such question?
magic sparrow
Try it by the definition
tranquil pine
with definition, it takes a lot of time.
I should be able to solve it in a minute. :)
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scarlet sequoia
**Deleted User 14b4cb6a**
Compile Error! Click the <:errors:504266556054962196> ...
this isn't well defined
what happens when x = 0 and y not 0 and vice versa?
i'm guessing you meant $(x,y)\neq (0,0)$ for the first case, which is not the same thing
soft zealotBOT
rafilou2003
scarlet sequoia
and yes the only feasible way is using the definition $\lim_{(x,y)\to (0,0)}\frac{f(x,y)-f(0,0)}{x-0}$
soft zealotBOT
rafilou2003
scarlet sequoia
(for partial derivative with respect to x, repeat with y)
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south wave
Is this right
candid pulsar
yeah c is right
south wave
Is this right
<@&286206848099549185>
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tranquil pine
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tranquil pine
Why 10^-x ended up being 1/5 and not -5
tranquil pine
" If 10^²x = 25, then 10^-x is equal to?"
I know when a exponent os negative we invert it
Like
10^-x end up becoming (1/10)^x
But I dont understand How the five became 1/5
vivid parcel
They are raising both sides to the power of -1. That gives (10^-x) on the left and 5^(-1)= (1/5) on the right
tranquil pine
They are raising both sides to the power of -1. That gives (10^-x) on the left a...
Is there any exponent rule say this can be done?
vivid parcel
Raising both sides to the same power? Well, if the two sides are equal, then equal things raised to the same integer power are still equal. I don’t know the name of specific rules.
tranquil pine
Let me see
If I'm making an = between two numbers
And I change the exponent of one
I have to do the same about the other?
vivid parcel
Yes. Here you raising both to the -1 power.
You could even write the whole equation as 1/(10^x) = 1/5. In the previous step, you had 10^x=5 so this is still equal
tranquil pine
You could even write the whole equation as 1/(10^x) = 1/5. In the previous step,...
Let me just do a last question
Lets suppose i'm doing
(X^²)^x = 2^²
I cut both exponents
It stay as X^x = 2
Then i wanna know what is the value of negative x and I make the x negative for that
It become X^-x
Is It mandatory for the 2 exponent become negative too? Like X^-x = 2^-1?
vivid parcel
Is It mandatory for the 2 exponent become negative too? Like X^-x = 2^-1?
Yes. You are raising both to the power of -1 (when you make x negative)
low yew
a^(-1)=1/a
tranquil pine
Yes. You are raising both to the power of -1 (when you make x negative)
I finally understand then
vivid parcel
Great!
tranquil pine
Thanks
scenic tree
if $10^{-x}=\frac{1}{5}$, then $\frac{1}{10^x}=\frac{1}{5}$
soft zealotBOT
ـDraedon
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hearty epoch
Given this question, I believe we represent the vectors as matrix A, with [x,y,z] as another matrix, say v, so Av=0, I think we somehow apply gauss jordan elimination to get rref, then that spits out numbers to get our answer, im not sure how to apply gauss jordan elimination in this situation, or if it is the correct way to go about it
final saddleBOT
Given this question, I believe we represent the vectors as matrix A, with [x,y,z...
dusty harness
Do you know how to do row echalon form?
hearty epoch
yeah i do
dusty harness
ok now put them on top of each other, scratch the echalon form
trying to solve th system what do you see first? note: it should be the 0
that 0 means that the first and third row should be synchronized for the 3rd column to be 0 in the end
r u with me?
for the third column to be 0 you multiply the first row with 2
-8 + 8 = 0
now you need to play with the 2nd row to finalize the system
hearty epoch
i think that makes sense
dusty harness
when you multiply the first row with 2 and do the summation, u get 8 10 0
which is a direct multiplication of the second row
hearty epoch
yes
dusty harness
seeing these kind of patterns come with time
well anyways what multiplier do you need for the 2nd row to remove 8 and 10?
firstly it needs to be negative since you are trying to neutralize positive numbers
hearty epoch
if its a direct multiplication of row 1 just r2 -> r2 - 1r1?
dusty harness
we already did 2r1 + r3 + r2
and made the 3rd column 0
now we need a multiplier for 2nd row to make the result of 2r1 + r3 + (multiplier)r2 = 0 0 0
the multiplier is -1
2r1 - r2 + r3 = 0 0 0
actually nevermind its hard to do if u are not familiar
hearty epoch
haha you've lost me, thank you for trying to help, i reckon ill swing by maths help for a few hours to get in person help tomorrow
dusty harness
yea i didnt like linear systems as well when i was in high school
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fiery prism
can someone help me with c
final saddleBOT
can someone help me with c
fathom walrus
Send what uve tried
brave shoal
can i see your work?
fiery prism
brave shoal
wont it be arcsin (-32/160)
atomic cairn
Area=
2
1
×side
1
×side
2
×sin(angle)
atomic cairn
Area=
2
1
×side
1
×side
2
×sin(angle)
you can also use this formula
brave shoal
wont it be arcsin (-32/160)
196=164-160cosx
32=-160cosx
cosx=-32/160
brave shoal
so i think the answee will be 180 minus what you got here
brave shoal
Maybe they're rounding up?
atomic cairn
won't it be 196=164-160.cosC??
then 196=4cosC??
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gloomy locust
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gloomy locust
trying to show that $X \cup {\neg \psi}$ is inconsistent
soft zealotBOT
Waffloid
gloomy locust
as per the hint ive proven that for any valuation of this, the set below is inconsistent
but i dont see what this does for us?
like we can apply deduction theorem and conclude that ${q |q\in Q, v(q) = 1} \cup {\neg q| q\in Q, v(q) = 0} \vdash \neg \phi$
soft zealotBOT
Waffloid
gloomy locust
so we have propositions built out of primitives in Q that prove not-phi, with these propositions coming from a model of X u {not psi}
but so what? if these q were all in X then we'd have a proof of not phi from phi which is a contradiction, but we dont have this...
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onyx sun
I need to find an m x n matrix where it's columns are linearly independent but its not a basis for R^n. I have come up with the 4x3 matrix [[1,0,0],[0,1,0],[0,0,1],[1,1,1]]. afaik the cols are linearly independent, and it cannot be a basis for R^3 since it has 4 elements in each column, not 3. Is that correct?
final saddleBOT
I need to find an m x n matrix where it's columns are linearly independent but i...
hollow iron
Yeah that's correct...
It's meaningless to ask whether the columns are a basis for R^n if m != n
They aren't even vectors in R^n
onyx sun
yeah I was more concerned about the linear independence part.
Now, i also need an m x n matrix where the columns are linearly dependent but that Ax=b has a solution for all b in R^n. normally to check that 2nd condition I would find the determinant and if det != 0, then the equation has a solution for all b. but since its a non square matrix it doesnt have a determinant
im not sure how else to check the second condition
royal gust
det = 0 implies the system has a unique solution.
You don't necessarily need a unique solution for this question, so even matrices with 0 det can fit
Note that the columns being linearly dependent implies the det is 0
This is all if the matrix is square. It doesn't have to be
onyx sun
ah yes sorry i forgot to write but it needs to be a non-square mxn matrix that is where im hitting the wall
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@onyx sun Has your question been resolved?
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severe cove
im so confused
final saddleBOT
im so confused
severe cove
so
(ignore the unselected ones)
so we have learnt to solve this by doing the positive of one absolute, pos of other, etc and solving it
the 4 permutations of each
but when i solve for positive |x(4-x)| > positive |x-1| because its the quad formula, i actually get both the intercepts in the graph of x(x-4)
but one of the intercepts is on the negative |x-1|
why
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jolly tapir
heya, I'm trying to prove this question via induction, but I keep running into unwanted negative signs after I evaluate the convolution. e.g. this is my base case but there shouldn't be a (-1) here. what am i doing wrong?
final saddleBOT
heya, I'm trying to prove this question via induction, but I keep running into u...
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jolly tapir
<@&286206848099549185> 👀
desert mantle
the inputs in the integral have to add to t0
jolly tapir
what do you mean by that? the bounds?
i thought convolution had bounds of -inf and inf
jolly tapir
the inputs in the integral have to add to t0
sorry, could you clarify what you mean?
desert mantle
you have f(t) delta'(t-t0)
the inputs add to 2t-t0
they should add to t0
they should be f(t)delta'(t0-t)
desert mantle
you have f(t) delta'(t-t0)
the t and t-t0
desert mantle
or here tau and t-tau
jolly tapir
ah okay i see. but why add to t0?
what's the reasoning behind swapping it to f(t)delta'(t0-t)
desert mantle
well thats also how it is here
jolly tapir
right but then it would be f'(t0-t)delta(t) no
desert mantle
well doesnt matter in which one of them you place the minus
jolly tapir
i am confused, why would
- f(t)delta'(t-t0)
- f(t)delta'(t0-t)
be interchangeable?
sorry, i might be missing knowledge of some properties here
jolly tapir
ahhh alright i see. but wouldn't the result still have a -1 in front of it?
desert mantle
when you integrate delta'(t0-t) with respect to t, you gain a minus
jolly tapir
ah? so the mistake is trying to simplify delta(t0-t) to delta(0)?
jolly tapir
i thought it was permissible to make t=t0 as that would be where delta(t0-t) is defined
desert mantle
heya, I'm trying to prove this question via induction, but I keep running into u...
this was fine, you were just missing the minus because you had the inputs in the integral wrong
jolly tapir
I have rectified the inputs, but I still do not see where the -1 is coming from
desert mantle
when you integrate delta'(t0-t) with respect to t, you gain a minus
.
I have to go
jolly tapir
ah you are talking about from line 1 to line 2 after ibp
oh, thanks!
i got it hahaha, can't believe i just missed a small chain rule
thank you so much
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brittle nova
Does anyone know why, if I take some $v\in \bC^n$, the summation $$\sum_{j,k}v_j^*v_k$$ always seems to be greater than or equal to $\norm{v}^2$?
soft zealotBOT
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wicked badger
can someone sent me some grade 8 lessons????
final saddleBOT
can someone sent me some grade 8 lessons????
wicked badger
or like notes
native valve
bro what?
wicked badger
our math teacher didnt teach us anything
native valve
bro you can search that yourself in google, why you asking it on discord? searching on google would give you endless resources
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bold coral
What is best/fastest way to check big number is prime in python?
final saddleBOT
What is best/fastest way to check big number is prime in python?
chrome pawn
Miller-Rabin primality test
it may occasionally give wrong answers but is very faster for large numbers and is ubiquitous
bold coral
Hmm
chrome pawn
if you want to check for primes numbers upto a certain limit; you can use Sieve of Eratosthenes
i think i mispelled it but just search for it
native valve
What is best/fastest way to check big number is prime in python?
One popular algorithm for this purpose is the Miller-Rabin primality test, Ive studied it
chrome pawn
Yeah that’s what I suggested 😭
native valve
Yeah that’s what I suggested 😭
im on mobile i completely missed you message lmao
stiff willow
Suppose that g(x) is a function which is 1-1 and that the slope of the tangent to the graph of g(x) at x = 2 is 3. Define the function f(x) by f(x) = g(4cos(x)) for x ∈ (0,π).
a) Find an expression for f −1(x) in terms of g−1(x).
b) What is the value of the derivative f′(π/3)?
final saddleBOT
### Suppose that g(x) is a function which is 1-1 and that the slope of the tange...
west berry
By f-1(x) you mean the inv function right?,
stiff willow
honestly im not sure where to start
i can get it when its just normal questions but when its worded it stumps me
what does a 1-1 function mean?
west berry
We have f(x)= g(4cosx)
west berry
what does a 1-1 function mean?
Each x value has a unique y value
For ex y=x² isn't 1-1
stiff willow
ohh so like the horizontal line test thing?
west berry
But y=x³ is
stiff willow
do i ignore the g when solving ?
im not sure what thats doing there
is it like a composite function ?
wait no
stiff willow
yeha
west berry
But it said in form of g^-1
west berry
well g(arccos(x/4)
How did you reach this again?
west berry
What's y
stiff willow
f(x)
west berry
Oh
stiff willow
like the swapping y and x thingy
stiff willow
i ignored the g and just inversed 4cosx
stiff willow
wdym by taking on both sides
west berry
f^-1 f(x)= f^-1 g(4cosx)
This is the same thing as saying that if f(x)= y then f^-1 (y)= x
By definition
I just used it
stiff willow
so
west berry
So if f(x)=g(4cosx),
west berry
This is the same thing as saying that if f(x)= y then f^-1 (y)= x
The g(4cosx) acts as y here
stiff willow
wouldnt this just the be inverse of g(4cosx)
how would i go about that
cos i thought it would have smt to do arccos(x/4)
which is the inverse of 4cosx
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teal robin
Would anyone be able to verify a few calculus problems? I will send one at a time
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Would anyone be able to verify a few calculus problems? I will send one at a tim...
silver maple
use an online calculator if possible
cyan hawk
you can use mathway, wolfralpha, photomath, etc.
teal robin
you can use mathway, wolfralpha, photomath, etc.
They are word problems so don't think this can work
just need to know if I am interpreting the questions correctly
silver maple
I see
rocky tusk
!da2a
final saddleBOT
No need to ask “Can I ask…?” or “Does anyone know about…?”—it’s faster for everyone if you just ask your question! See https://dontasktoask.com/
teal robin
okay sorry
silver maple
okay sorry
What is your question
rocky tusk
^
teal robin
What is your question
For this quesiton I have to determine the linear function with the level curves
Just want to make sure I did it right
silver maple
For this quesiton I have to determine the linear function with the level curves
Explain your working out
teal robin
Explain your working out
Well for the a and b I know that these are the slopes a being the slope with respect to x, and b being the slope with respect to y. So I chose 2 points that look easy to distinguish and used the slope formula to find the slopes. Then I picked a point on the graph to find the z-intercept
silver maple
It is very hard to see the image clearly
teal robin
I'm really sorry about that. It was sent from someone else so that is all I have : (
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hushed ice
Hey, so I need help with this. It’s said to calculate the derivative of these functions. It is also that said that the answer should be simplified, factored (if possible) and should not have any negative or fractional exponents. I’ve done something but I wanted to do it with someone so I can compare it to my work
final saddleBOT
Hey, so I need help with this. It’s said to calculate the derivative of these f...
rocky tusk
rocky tusk
4th line you forgot the x
hushed ice
4th line you forgot the x
At the very front?
rocky tusk
At the very front?
yes
rocky tusk
yea
hushed ice
Oh you’re right
rocky tusk
why did your 3x^3 become 3x^2
oh that too
hushed ice
9x^2
hushed ice
but on the inside
Yeah but I need to change it to 9x^2
rocky tusk
not 4x
rocky tusk
Yeah but I need to change it to 9x^2
that’s the chain rule
rocky tusk
you multiply by that but the inside function stays the same with a mere power reduction by 1
hushed ice
Ohhhh
rocky tusk
if you multiplied it by x you’d get (3x^3+5x)(3x^2+5)…(3x^2+5)
hushed ice
So would it be (3x^3+5)^6 ?
rocky tusk
you have to fix all the lines below it too but yea
hushed ice
Whoooopsssss
Alright NOW it should be fixed
But what I’m also confused is why do I remove the x in the last two lines
final tangle
you shouldn't
hushed ice
and do I need to keep the (3x^3+5)^6 (4x^2+1)^4 ? If so why
hushed ice
you shouldn't
How come
final tangle
stuff doesn't just disappear
lofty kindle
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lofty kindle
just a quick question but i can just say that by symmetry, the above integral is equal to 0
or is there another term for it?
like, I dont even need to prove it. i can just simply state it
rocky tusk
what symmetry are you using
lofty kindle
oh nvm, its not 0 because its symmetrical
rocky tusk
i mean it will be zero
lofty kindle
yea
rocky tusk
because you’ll get xsinx
lofty kindle
i messed up visualizing it in my head
rocky tusk
and sinx is zero at those two values
and integral of cosx between those two gives zero
because cos(2pi)=cos0
lofty kindle
yea, thank you!
signal patrol
lofty kindle
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slate jasper
I want to visualise something polygon structure if possible
final saddleBOT
I want to visualise something polygon structure if possible
slate jasper
Lowest angle is 120°
And other angles are in difference 5° in a AP
And sides are n=16 and n=9
Can anyone send me a link with it
Desmos
warm python
two different polygons?
slate jasper
Yes
gray agate
could someone help explain this problem to me?
final saddleBOT
could someone help explain this problem to me?
gray agate
Can we just add all the variances up?
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tranquil pine
hello
maiden whale
for the last part i’m having trouble visualising what angle they are talking about. Using inverse tan on the velocity components at t = T is giving me 90 less than the answer smthn like -26 degrees instead of 63
final saddleBOT
for the last part i’m having trouble visualising what angle they are talking abo...
gritty solar
yeah thats fine
angles are directional
depends on where you're measuring it from
your measure might have been with the horizontal normal
maiden whale
would u be able to explain which angle we are actually trying to find?
gritty solar
there's might have been with the wall
gritty solar
would u be able to explain which angle we are actually trying to find?
which one did you find?
maiden whale
i just used inverse tan on the components and got -26 degrees
ig that’s the angle from the normal?
gritty solar
yeah from the normal (to the wall)
its negative because the angle was probably below the normal
ie the ball was descending when it struck the wall
maiden whale
so is this the angle i found
the arrow is my attempt at drawing the velocity vector at the point it hits the wall
gritty solar
yeah
so you can say "64 degrees with the wall" or "26 degrees with the normal to the wall"
both should be accepted
maiden whale
would it be the same thing if the angle from inverse tan was positive
i.e. still with upwards trajectory
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@maiden whale Has your question been resolved?
gritty solar
would it be the same thing if the angle from inverse tan was positive
yes
ruby lake
is there a good way to solve this kind of problems
maby this one is not particularly hard
but should i use a Ven diagram
the part b is particularly confusing
jagged flare
,rccw
soft zealotBOT
jagged flare
assume you can fill in the blank squares with unique prime numbers under 25
and N is a positive integer
if there are x possible values of N, where y is the largest and z is the smallest, what is x+y+z
a.25
b.15
c.26
d.14
e.24
craggy plank
Same
jagged flare
<@337506441634119681> You must check this 🙂
huh?
candid pulsar
assume you can fill in the blank squares with unique prime numbers under 25
it need not be unique right
jagged flare
yes
slim barn
um this seems like a simple brute force program?
candid pulsar
cause there are like only 9 prime numbers lesser than 25
jagged flare
wait hold on, brb for like 10 mins
craggy plank
2 3 5 7 11 13 17 19 23
There are 9 exactly
candid pulsar
nice clue i found
slim barn
1. Create all possible permeutations.
2. Sum the first 7 -> x
3. Sum the other 2 -> y
4. x/y
5. Profit??
there are several different possibilities of N
anyways
slim barn
there are several different possibilities of N
yeah this is just a brute force exhaustive search
it will give all answers
candid pulsar
we know that all prime numbers are odd right
excluding 2
so if the numerator has all odd
it sums up to odd
since there are 7 numbers
but if there's 6 odd numbers and a 2
we'll get even
and denominator is even
and in first case
denominator is even
but odd/even is not a whole number
so 2 is always in the numerator
jagged flare
wait can you just do this
assume the bottom primes are a and b, so $$\frac{100-(a+b)}{a+b}=\frac{100}{a+b}-1$$
soft zealotBOT
Skill_Issue
jagged flare
@candid pulsar
cause the top is just the sum of all the primes up to 25 minus 2 of the primes, and those 2 primes are at the bottom
candid pulsar
cause the top is just the sum of all the primes up to 25 minus 2 of the primes, ...
ohhhhhh smart
so we need to find two primes such that their sum is a factor of 100
jagged flare
we just have to check a+b=1,2,4,5,10,20,25,50,100, since 1,2,4,50,100 is obviously not possible, we have to check 5,10,20,25
for 25 it can be 2+23, that means n=3
for 20 it could be 13+7
for 10 it could be 7+3
for 5 ot could be 2+3, that means n=19
thats it right? 25+5+4=34
which is not in the msq 😭
oh wait
candid pulsar
you need to subtract 1
slim barn
(19, 3, 5, 11, 17, 23, 2) on the top and( 7, 13) gives N=4
(5, 11, 17, 23, 7, 19, 13), on the top and ( 2, 3) gives N=19.0
jagged flare
how about this, you can put +, -, ×, / in those blank squares
slim barn
import itertools
perm = list(set(list(itertools.permutations([2,3,5,7,11, 13, 17, 19, 23,
]))))
for p in perm:
if ((sum(p[:7])/sum(p[-2:]))==int((sum(p[:7])/sum(p[-2:])))):
print(p, (sum(p[:7])/sum(p[-2:])))
but 23 and 2 work aswell for 25?
slim barn
i know, i gave the complete program above
candid pulsar
how about this, you can put +, -, ×, / in those blank squares
what are those numbers
slim barn
i just posted some solutions
jagged flare
what are those numbers
wdym
candid pulsar
in the options
jagged flare
oh, its the possible ways that the symbols can be put and the equation stays true
jagged flare
wair
jagged flare
how about this, you can put +, -, ×, / in those blank squares
all of these options cant lead to an odd number
so it had to be even
wait they dont have to be whole 
shrewd vector
26
jagged flare
erm what the scallop howd you get dat
shrewd vector
for the prime question
jagged flare
oh ok
shrewd vector
3+19+4 = 26 with a bonus point for getting 4 in 2 different ways
jagged flare
how about this, you can put +, -, ×, / in those blank squares
oh omg the () have to be positive integers
ok that simplifies things a lot
each of them has to be even, we just have to find the ways to permute them
though i dont see a way for the first one is either 2, 22, 46, 572 if im not mistaken?
candid pulsar
wait i got one solution
candid pulsar
(2 + 0 * 2 * 4) * (20 - 2 + 4) * (20 + 24 - 2 + 4)
jagged flare
and the left one can be 4 (××+)
jagged flare
(2 + 0 * 2 * 4) * (20 - 2 + 4) * (20 + 24 - 2 + 4)
oh yeah, smort
for the left one could you cange it to ×/, /×, //?
(2×0×2+4)(20-2+4)(20+24/2/4)
(2×0/2+4)(20-2+4)(20+24/2/4)
(2+0/2/4)(20×2+4)(20+24/2/4)
(2+0/2×4)(20×2+4)(20+24/2/4)
(2+0×2/4)(20×2+4)(20+24/2/4)
(2+0×2×4)(20×2+4)(20+24/2/4)
(2+0/2/4)(20-2+4)(20-24+2+4)
(2+0/2×4)(20-2+4)(20-24/2+4)
(2+0×2/4)(20-2+4)(20-24+2+4)
(2+0×2×4)(20-2+4)(20-24+2+4)
vague yew
but you still need the homogenous solution for the complete solution
but in the question it says find a particular solution so isn't it enough the thing that i wrote
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terse siren
but in the question it says find a particular solution so isn't it enough the th...
ig
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viscid stirrup
||for this problem, we have the sum of seven unique prime numbers less than 25 divided by the sum of the other two unique values that happen to equal a positive integer. Notice that the sum of all 9 primes less than 25 is 2+3+5+7+11+13+17+19+23=100. so we have this sum, however, we have to exclude two primes from here since we want the sum of seven primes in the numerator and instead place them as a sum in the denominator. Let us call these primes from the selection of primes less than 25 p_1 and p_2. We thus have that (100-(p_1+p_2))/(p_1+p_2) = N, simplifying further we get (100/(p_1+p_2))-1 = N for N is a positive integer. Notice that 100/(p_1+p_2)>1 since p_1+p_2 must be less than 100 for any selection of primes, thus we don't have to worry about N being less than 1. We also have that p_1+p_2 must add to a divisor of 100 which is 1, 2, 4, 5, 10, 20, 25, 50, 100 for N to be an integer. We notice that p_1+p_2 cannot equal to 1,2,4,50 and 100 hence we eliminate those possibilities. If p_1+p_2=5, we see that one of the primes is 2 and the other is 3. If p_1+p_2=10, one of the primes is 3 and the other is 7. If p_1+p_2=20, we see that one is 17, the other is 3, or one is 13 and the other is 7. If p_1+p_2=25, we see that one of the primes is 23 and the other is 2. All these cases should produce positive integers N. Note that we only care about the sum of p_1+p_2 here. If we had p_1+p_2=5, N=(100-5)/5=19. If p_1+p_2=10, N=(100-10)/10=9, If p_1+p_2=20, N=(100-20)/20=4, if p_1+p_2=25, N=(100-25)/25=3. Thus N could either be 3, 4, 9 or 19. The number of possible values for N are 4, the largest value of N is 19 and the smallest value of N is 3. Thus x+y+z=4+19+3=26||
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candid pulsar
||for this problem, we have the sum of seven unique prime numbers less than 25 d...
already solved it 👍
viscid stirrup
already solved it 👍
Yes, I saw, thank you. My bad
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How do I describe the following area in polar coordinates. (One square = one unit of length.)?
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fourier transform
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cyan kayak
!status
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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.
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low magnet
A uniform electric field of magnitude 1.1×104N/C
is perpendicular to a square sheet with sides 2.0 m long. What is the electric flux through the sheet?
formal trail
slightly ambiguous wording of the problem, but when we talk about the angle between a surface and a field we usually mean the angle between the normal vector of that surface and the field
west berry
theta is the angle between electric field and area vector
The area vector is perpendicular to the plane of the square sheet
Try drawing it out
low magnet
English isn't my first language, I'm having a hard time understanding your instruction sorry
west berry
Ok, let's start with this
θ is the angle between which two vectors according to you?
low magnet
Electric field and surface area?
low magnet
I don't know
hazy hemlock
I don't know
the angle between normal to surface and electric field is theta here
white tiger
perpendicular
low magnet
Electric field is perpendicular to the surface?
white tiger
A uniform electric field of magnitude 1.1×104N/C
is perpendicular to a square...
you said it yourself here
hazy hemlock
Here what do you see? What's the angle between those two lines?
low magnet
none?
delicate fiber
it should be cos 0
delicate fiber
The angle theta is the angle between the area vector and the electric field
low magnet
so it's 44000 nm^2/c?
hazy hemlock
yes if that's what you get with cos0°
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timber pike
a washing machine was available for 15000 rs but was purchased under an instalment plan after paying 2250 as cash down payment followed by two equal half yearly instalments . if interst charged 8% per annum compounded semi annually , find the value of each instalment
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a washing machine was available for 15000 rs but was purchased under an instalme...
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junior storm
a washing machine was available for 15000 rs but was purchased under an instalme...
this is simple interest , find interest for each add up the value and then find the interest for the new value according to the given conditions.... would love to see u doing on ur own, because this has to be very basic and once you do it, i guarentee you that u can skip this chapter for the next, but if u still need an answer will solve it and post the picture... cause it is one of the hardest questions the chapter has to offer
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this is simple interest , find interest for each add up the value and then find ...
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junior storm
<@1219729878669197407> Has your question been resolved?
stupid bot, it was a solution..
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jovial fjord
how would i set out the integration?
white tiger
function above - function below
jovial fjord
oh
wow i really overthought it then
theres areas below the x axis so i thought you had to split it up
final tangle
split if one function isn't always above the other
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candid pulsar
looks pretty tough
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cobalt lodge
i can’t tell if this is incredibly complex or incredibly basic but im leaning towards the latter
i feel there’s very little things you can say other than simple upper/lower bounds 
unless we’re gonna calculate the number of possible sequence pairs again 
ye i remember that lmao
vital surge
what kind of statement do you want in "is there anything we can say about the digit sequences"
wise crescent
vital surge
more or less what it sounds like: "a problem set from a math camp"
ah that explains it too
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covert hornet
i know it may be a weird question but is this form of writing correct?
covert hornet
about the union
rain compass
looks fine, though im not sure what f is
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strong field
Hello! Is anyone available to help
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Hello! Is anyone available to help
tranquil pine
no
lucid marsh
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primal kettle
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primal kettle
could someone help me with this one? not sure if strong induction is the way to go or if it can be done with it normally
and im having trouble setting up number 5
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tranquil pine
loaded question
final saddleBOT
loaded question
tranquil pine
as with most of these starting it is the hardest part
I do notice two triangles forming if that's of any significance
marble agate
isnt this just optimization problem?
tranquil pine
it is
marble agate
what have you tried?
marble agate
ok
soft zealotBOT
Remlis
tranquil pine
$y=(13-\sqrt{x^2+(5-y)^2}-(12-x)$
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Remlis
loud sundial
Hint: Similar triangles
tranquil pine
yeah that idea also popped up for me where if the triangles are matching in size then that would mean the rectangle is as big as it can be
but then I can't incorporate optimizaiton
or maybe I can
since you could either have the smallest or the largest rectangle that way
loud sundial
Using similarity, (5-x)/y = x/(12-y)
which makes your life much easier
b/c now you can just do it in 1 variable
tranquil pine
wait a min realizing something
so in my triangle y is also equal to 5-y?
how would that work
loud sundial
so in my triangle y is also equal to 5-y?
oh I just labeled them differently
the logic is the same if you wanna label it your way
it doesn't rlly matter
tranquil pine
oh I just labeled them differently
yea but
where would we go form here
we don't have a 2nd eqn to solve in terms of a specific variable
loud sundial
like how you would find the max/min of any other single-variable function?
tranquil pine
yeah usually you would need two equations to substitute the other variable in in terms of the first variable
then you'd take the derivative and set = 0
but problem is
we only have 1 equation
$Area= (\sqrt{x^2+(5-y)^2}-(5-y))((13-\sqrt{x^2+(5-y)^2}-(12-x))$
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Remlis
loud sundial
you have $\frac{5-y}{x}=\frac{y}{12-x}$ and you need to maximize $xy$
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Civil Service Pigeon
tranquil pine
I don't get the division
I mean I get that the two sides are equal that way
if the triangles are the same
or is that all you wanted to do with that
loud sundial
sure
soft zealotBOT
Remlis
loud sundial
you can solve for x in terms of y (or vice versa with that)
then substitute the result into the area function
and then you have a single variable function
tranquil pine
**Remlis**
this being the area function?
soft zealotBOT
Remlis
tranquil pine
$\frac{(5-y)(12-x)}{x}=y$
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Remlis
tranquil pine
$\frac{(60-5x-12y+yx)}{x}=y$
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Remlis
tranquil pine
minor problem, idt we can isolate for a variable using this
tranquil pine
$y=\frac{(5-y)(12-x)}{x}$
soft zealotBOT
Remlis
tranquil pine
I don't see what you mean
tranquil pine
$xy=(60-5x-12y+yx)$
soft zealotBOT
Remlis
tranquil pine
$0=(60-5x-12y)$
soft zealotBOT
Remlis
tranquil pine
$12y=(60-5x)$
soft zealotBOT
Remlis
tranquil pine
$y=\frac{(60-5x)}{12}$
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Remlis
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flat furnace
For Calculus, how would I go about optimizing/maximizing the area of a rectangle within a semicircle?
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For Calculus, how would I go about optimizing/maximizing the area of a rectangle...
loud sundial
Did you draw a diagram? If so, can you send it? (I'm too lazy to draw it lol)
flat furnace
My camera on my phone acting up for some reason so I can't post a pic
loud sundial
here's your walmart diagram
I'm going to assume that the semicircle is fixed for convenience, let's call the radius r
flat furnace
Alright
loud sundial
what lengths can we find in the rectangle using this
flat furnace
X and Y, right?
loud sundial
in terms of the radius*
flat furnace
I'm not too sure
loud sundial
draw in some radii
west berry
Ig the angle subtended can be a parameter here
Or no actually assuming them as x and y works too, but you need to write x in terms of y or vice versa
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marsh temple
$\binom nk = \frac{n!}{(n-k)!k!}$
soft zealotBOT
Steakanator
marsh temple
and it's illustrating that the two expressions are equivalent due to symmetry
specifically nCk
marsh temple
and it's illustrating that the two expressions are equivalent due to symmetry
any one of what
the two are equivalent yes
it can be shown pretty succinctly if you know that $\binom nk = \binom{n}{n-k}$
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Steakanator
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Steakanator
marsh temple
which is...
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Ava
marsh temple
eeh, careful
i'd add a couple pairs of parentheses, what you have isn't accurate
you have a { after cdot
you want a (
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Ava
marsh temple
this can be simplified
no
distribute the negative
yes
but what is this thing?
soft zealotBOT
Ava
marsh temple
this new expression
it has a name, one you've said before
$\frac{n!}{k!(n-k)!} = ?$
soft zealotBOT
Steakanator
marsh temple
there we go
marsh temple
**Steakanator**
so we started there and ended up here
the command is in my message above
not afaik since it's less common
short for binomial coefficient
which is exactly the context of the question
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full night
how can I solve this integral?
final saddleBOT
how can I solve this integral?
full night
I rewrote the integral as this
blissful condor
i doubt this has a nice form (don’t quote me on that)
full night
I agree with you on that
tranquil pine
what is F
full night
just some constants for Final and Initial
tranquil pine
,w integrate sqrt(8sin^2(x)+1)
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tranquil pine
blissful condor
I agree with you on that
are you being asked by someone else to do this integral? i take it you’re just trying to find the circumference of an ellipse, which is pretty well-known that doesn’t have a nice closed form
blissful condor
this is sort of what i expected
where WA doesn’t give you a nice answer because there isn’t one
full night
are you being asked by someone else to do this integral? i take it you’re just t...
Its always good to expand my knowledge on this kind of stuff. I dunno what a eliptical integral is tbh
blissful condor
the elliptic integral is defined exactly as what you wrote
full night
but how do I even get any value from it?
blissful condor
like because there isn’t a closed form it just said “let E be the solution to the elliptic integral”
blissful condor
but how do I even get any value from it?
by numerical approximation lol
it’s sort of like asking how to get the value of pi
“well idk we’ll just call it pi and approximate it the best we can, and if it comes up in other calculations that’s pretty cool as well”
full night
the only other option I had in mind was to get the normal vector of the elipse, make a integral for the magnitude of the normal vectors, put the integral inside the limit as the magnitude approaches 0, and then say that is the length
so it would be basically giving the ellipse "thickness" and then making it infinitesimally small thickness
full night
like because there isn’t a closed form it just said “let E be the solution to th...
oh, weird
It might still be the case, though i thought it was a whole different branch of integral calculus I was missing out on
royal gust
Are you being asked to do this, with the expectation that there's a computable final answer?
blissful condor
Are you being asked to do this, with the expectation that there's a computable f...
no of course not
because there isn’t one
also it was said earlier i think
full night
Are you being asked to do this, with the expectation that there's a computable f...
im doing this just because. Moreover I wanna test my math skills
royal gust
Gotcha
blissful condor
It might still be the case, though i thought it was a whole different branch of ...
there is a nice book of integrals that i forgot the name of, it has some really interesting problems
i’ll find it 1 sec
oh yeah it’s called “(almost) impossible integrals, sums, and series”
lol
full night
my idea is basically to get the area of a elipse, then subtract it by the area of the same elipse, but infinitesimally smaller if that makes sense?
that would be one idea but idk how well it would work
vast arrow
pls i need help with this
determine (E) the set of points M with affixes z such that
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pls i need help with this
determine (E) the set of points M with affixes z such ...
tiny gorge
,rotate
soft zealotBOT
tiny gorge
vast arrow
edit we have the rotation R such that Z, = exp(-ipi/4) and the D the image of A with d = isqrt(2) and a = -1+i
vast arrow
I have no idea how to start
tiny gorge
hint: it's the set of all points z that are equidistant from the two points ...
vast arrow
but we can rewrite the equation to: abs(z-a) = abs(z+d)
vast arrow
hint: it's the set of all points z that are equidistant from the two points ...
hmmmm
ok i will try something and come back to you
vast arrow
hint: it's the set of all points z that are equidistant from the two points ...
but in this case its will be one point not a set of point?
tiny gorge
it's a set of points
try a simpler example
say |z-1| = |z+1|
what points z satisfy this?
vast arrow
0
vast arrow
what points z satisfy this?
is their other point?
vast arrow
i don't know if we can say abs(i-1) = i-1
vast arrow
hint: it's the set of all points z that are equidistant from the two points ...
is it like this?
tiny gorge
yep, the whole imaginary axis
now for your problem the answer will be similar
geometrically, it's the perpendicular bisector of the line segment between a and b
vast arrow
I got it now
nocturne vale
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desert mantle
no that doesnt work at all. in the first direction you are just claiming that a has to be positive from nowhere. and for the other direction, how does it matter that 1 in S
olive socket
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random scaffold
I completed the procedure, but I really don't understand what they mean by "no edge in G connects a red vertex and a blue vertex". They right after say to assign red to a and assign blue to c,e,f,g
isn't "a" connected to the 4 of those vertices in G?
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<@&286206848099549185>
Just want to know if i'm misinterpreting that line
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gm