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#help-28
cunning bane
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slender onyx
,w extrema x^7-y^6-z^7 subject to x^2-y^2+z=0
glossy valveBOT
cunning bane
How do I do it mathematically
slender onyx
if wolfram is correct your computation should lead to contradiction
cunning bane
if wolfram is correct your computation should lead to contradiction
But lioke
hoiw do i prove thius
casuse im stuck rn
slender onyx
But lioke
you have two pairs of ors so thats 4 cases to check
you might get smth by plugging each case into the constraint
cunning bane
you have two pairs of ors so thats 4 cases to check
Could u remind me on ors
Are those the cases I listed?
slender onyx
lets say u get this
x=0 or x=1
y=1 or y=2
the four cases are
x=0, y=1
x=0, y=2
x=1, y=1
x=1, y=2
cunning bane
I see
So case 1 x=0, y=0
Case 2 x=0 y = -y^4 + lambda
Case3 = x= 7x^5 - 2lambda y = 0
Case 4 x = 7x^5 - 2lambda y= -y^4 + lambda
slender onyx
y=0 in case 3
cunning bane
y=0 in case 3
What do I do with the case of -7z^6 = lambda
Can I just replace cases 1-4 with that or would that make it more messy?
slender onyx
thats not a case
if ur eqn isnt connected to an or then its always true
so the z eqn and the constraint are always true
btw i think u made an error
cunning bane
With the set up ?
slender onyx
ye
cunning bane
u also should use z eqn
Wdym
cunning bane
I’m kinda confused on how to do that tho since 7z^6 = lambda
Sorry if I’m misunderstanding smt
slender onyx
my point is the constraint and z eqn are always true so u might have to use both in each case to get contradiction
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@cunning bane Has your question been resolved?
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midnight lotus
am i able to ask a simple chem question. would anyone be able to help
thick hedge
Maybe
clear pumice
perhaps
twilit leaf
we would be more likely to confirm if you sent the actual question
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@midnight lotus Has your question been resolved?
fallen violet
I can't understand proofs
a - set is orthagonal
b - vectors are independent
so they want to proof that
a -> b
but they get that
!b -> !a (vectors are NOT independent then set is NOT orthagonal)
how this holds to the first statement? (ignore scribles)
mighty thorn
(a implies b) is generally equivalent to (!b implies !a)
If b is true whenever a is true, it is never the case that b is false but a is true; therefore, if b is false(that is, !b), then a cannot be true(!a)
slender onyx
indeed this fact is known as "any implication is equivalent to its contrapositive"
leading to the method of proof by contrapositive, which sometimes looks a lot like proof by contradiction
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faint karma
Let a b c 3 real numbers, a function f(x)=ab^(x/2)+c
We know that f(2)=5
f(5)=2000
Find f(7)
Anyone know how to solve this ?
onyx glen
faint karma
Honestly i'm skibidi lost as hell
mighty thorn
Let a b c 3 real numbers, a function f(x)=ab^(x/2)+c
We know that f(2)=5
f(5)=20...
do you mean f(x) = a * b^(x/2) + c?
onyx glen
that's a new one
faint karma
do you mean f(x) = a * b^(x/2) + c?
yes
onyx glen
"skibidi lost as hell" is a very ohio turn of phrase
faint karma
yh for reel
onyx glen
i have a feeling there might not be enough info here...?
faint karma
oh really ?
onyx glen
3 unknowns, but only 2 equations
onyx glen
f(x) = a * b^(x/2)?
faint karma
yes
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Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
faint karma
what is this
onyx glen
what is this
one of many "factoids" that we have on this server for invoking responses to common occurrences on here
to put it another way:
show us the original problem
gritty flax
even if not in eng
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torn jolt
I finally had my exam today. It went decently well. Thank you for all the help I got this last week.
Can someone discuss with me if my answers are correct?
torn jolt
https://dontasktoask.com/
One sec
I marked the options out really lightly. Can someone verify if they are correct?
- C
- C
- D
- B
- D
- B
humble spindle
Class 12 maths boards exam?
torn jolt
Tag me please
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@torn jolt Has your question been resolved?
torn jolt
Class 12 maths boards exam?
Yeah
torn jolt
Class 12 maths boards exam?
Would it be possible for you to help?
brittle zodiac
yeah it is correct
humble spindle
Would it be possible for you to help?
Ok. Please wait a moment
brittle zodiac
i just went through it, its correct
dusky girder
wait sorry why isnt 9 C
brittle zodiac
send next? wdym?
dusky girder
i think they have multiple pages lol
torn jolt
send next? wdym?
Next page
brittle zodiac
yeah
torn jolt
wait sorry why isnt 9 C
It’s not definite integrals. It’s area. Which cannot be negative
brittle zodiac
wait sorry why isnt 9 C
take the integral of x^2
torn jolt
- B
- A
- A
- A
- B
torn jolt
- B
- C
- C
torn jolt
- C
- A
- A
brittle zodiac
is it india?
torn jolt
Yes
brittle zodiac
central exam??
torn jolt
Yippee
brittle zodiac
for us most of this was IGCSE, so it was pretty easy
But is this JEE?
well done thou
torn jolt
- D
- B
- B
torn jolt
But is this JEE?
No no
This is just an exam, so that allows you to “pass” high school level mathematics?
torn jolt
noo i need to go, someone else will finnish it off
No worries, thank you so much for discussing so many questions
I’ll close this channel and open another one afresh so that the newer questions get pinned
brittle zodiac
👍🏽
dusky girder
btw r u guys allowed to use a calculator for this
torn jolt
👍🏽
Can I DM you for discussing the longer non-MCQ type answer?
It’s perfectly fine if you don’t want that.
I appreciate the help either way:)
torn jolt
btw r u guys allowed to use a calculator for this
No calculators
brittle zodiac
👍🏽sure
torn jolt
Thank you so much:))
torn jolt
No calculators
But hardly calculation type questions
brittle zodiac
No calculators
that makes it more fun, but i really need to go so bye
torn jolt
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lapis flume
how do i find x
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how do i find x
lapis flume
idk how to find the last angle
maybe i need to do a lil algebra or something
42 + x + (138-x) = 180
i dont think i can do anything with that
lapis flume
yes
neat basin
So angle ebc = 180-42?
little prism
Given $F: [a, b] \rightarrow \mathbb{R}^n$ that is $C^1$ (continuous, differentiable once), with F(a)=A, F(b)=B, prove:
$\int_a^b \vert \vert F'(t)\vert \vert dt \geq\vert \vert A - B \vert \vert$ using Cauchy-Schwarz inequality.
glossy valveBOT
Wild123
little prism
Hi. I'll need some help with this.
onyx glen
btw you should use \Vert or \| to get double bars
compare: $\vert\vert F'(t)\vert\vert$ vs. $\Vert F'(t) \Vert$
glossy valveBOT
Ann
lime ether
$\Vert$
glossy valveBOT
knief
little prism
mm ok. I'll try to remember that.
slender onyx
$\norm{x}$ too
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lime ether
nice
little prism
Given $F: [a, b] \rightarrow \mathbb{R}^n$ that is $C^1$ (continuous, differentiable once), with F(a)=A, F(b)=B, prove:
$\int_a^b \norm{F'(t)} dt \geq \norm{A - B}$ using Cauchy-Schwarz inequality.
glossy valveBOT
Wild123
little prism
Now, how may I employ the cauchy-schwarz? I think it has something to do with $\norm{F'(t)}$, I'll write it explicitly
glossy valveBOT
Wild123
slender onyx
maybe as a first step its good to write the squared ineq?
little prism
maybe as a first step its good to write the squared ineq?
And then I think I have a property that $(\int_a^b f(t) dt)^2 \leq \int_a^b f^2(t) dt$? Or what do with the squared integral
glossy valveBOT
Wild123
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@little prism Has your question been resolved?
slender onyx
And then I think I have a property that $(\int_a^b f(t) dt)^2 \leq \int_a^b f^2(...
i dont think so, plus we're talking about functions to R^n so these things should be inside norm
we can start with $\norm{B-A}=\norm{F(b)-F(a)}=\norm{\int_a^bF'(t)dt}$
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slender onyx
now justify $RHS\le\int\norm{F'}$
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torn jolt
Hello is there anyone that can help me with some grade 8 math(for a competition) if not where can i find someone on this server
slender onyx
@torn jolt #competition-math
torn jolt
<@456226577798135808> <#433980473367330826>
Thanks man
pale steppe
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tight quest
consider the prime factorization of 2023
,, \sum_{k = 1}^n k^3 = \frac 14 n^2 (n + 1)^2
glossy valveBOT
kizzyyy
pale steppe
consider the prime factorization of 2023
7 x 17²
glossy valveBOT
kizzyyy
tight quest
must contain at least one 7 and one 17 right?
pale steppe
Yea
tight quest
do you have any ideas on how that translates?
or what sub-problem you can now work on?
glossy valveBOT
HelpMe
pale steppe
Oh wow I have no idea how to use this bot
,, \sum_{k = 1}^n k^3 = \frac 14 n^2 (n + 1)^2 = 17^2 x 7 x k?
glossy valveBOT
HelpMe
tight quest
well we decided to look only at (n(n+1)/2) for simplicity sake
tight quest
so just 17 * 7 is sufficient
pale steppe
Uh ok 👍
tight quest
yeah n(n+1) = 0 (mod 238) sounds good
tight quest
So uhh n² + n = 238k
it's easier to work with products, no?
anyway do you know sieving? like chinese remainder theorem?
pale steppe
Yes
tight quest
then u should be able to solve the congruence
pale steppe
then u should be able to solve the congruence
So 34?
pale steppe
Oh ok ty
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tight quest
pale steppe
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tight quest
😭
pale steppe
👍 this has been like the 3rd time u help me or smthing
Honestly not my fault I'm js dumb and doing bullshit 4 yrs above my current lv
tight quest
i thought you're helpme not dumb
pale steppe
🔥
pale steppe
i thought you're helpme not dumb
Hm
pale steppe
+ it's kinda like competition math
Yea cus it is competition math xd
This is like 5 years above my current grade
lime ether
This is like 5 years above my current grade
are you in 5th grade?
pale steppe
what grade is this?
Uhhh wdum
tight quest
i mean you said this shit is 5 years above your current grade
so what grade is this meant for?
pale steppe
so what grade is this meant for?
Uhhh how about I js don't say it so u won't know my current grade :>
pale steppe
it’s the pink diamond
What
tight quest
pale steppe
i mean you said this shit is 5 years above your current grade
Also uhh I'm probably like half ur age (if u finish university already or smthing)
lime ether
kizzy i think he likes you
tight quest
kizzy i think he likes you
loll
pale steppe
kizzy i think he likes you
💀 nty not into someone who's probably few thousands of miles away from me
pale steppe
dude what are u like sub 13?
😭 uh yes
So now you know
Ur talking to a kid
Half ur age
pale steppe
discord tos
I'm above 13
tight quest
also you're not half my age
lime ether
👍🏻
pale steppe
I'm almost 14 anyways
tight quest
lol
pale steppe
😭 ykw I shouldn't have said that
lime ether
you’ll get em next time bro
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hoary river
This is from Gilbert Strang's Calculus Volume 2.
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This is from Gilbert Strang's Calculus Volume 2.
hoary river
I have read limits must be finite. If the expression being evaluated approaches infinity, then the limit does not exist.
So,
Is the above incorrect?
And if so, how am I supposed to write this in the exam?
hoary river
the integral diverges
but how would i write this step
how would i show it approaches infinity
Can I say as lim t --> 0+ (1/t^2) approaches infinity, the limit does not exist?
wait no
jolly horizon
ok so this textbook has it wrong if this is where you got this work from
it goes to -infinity
hoary river
Since (1/t^2) approaches infinity as t approaches 0+ , the limit does not exist
jolly horizon
but all that matters when it comes to integrals is if it's finite or not
hoary river
Since (1/t^2) approaches infinity as t approaches 0+ , the limit does not exist
is this a mathematically valid way to write the conclusion?
jolly horizon
yea
hoary river
gotcha, thank you!
jolly horizon
anytime
hoary river
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pale steppe
so just 17 * 7 is sufficient
Btw I wanna ask why can we js take out the 17 fron the equation wouldn't that change the awnser a bit?
tight quest
if 17 * 7 divides (n(n+1)/2) then 17^2 * 7 must divide (n(n+1)/2)
tight quest
Btw I wanna ask why can we js take out the 17 fron the equation wouldn't that ch...
if that's what you're asking
pale steppe
if 17 * 7 divides (n(n+1)/2) then `17^2 * 7` must divide (n(n+1)/2)
Wait so couldn't we js increase it instead like instead of mod 2023 we js make it mod 2023k and it becomes 7 x 17² x k and we just add 7k into this making it 7² x 17² x k² so then it becomes n(n+1)/2 cus from eariler it still was squared
tight quest
Wait so couldn't we js increase it instead like instead of mod 2023 we js make i...
increase what
i'm confused
pale steppe
increase what
Like yk how it's (n(n+1)/2)² = 0 mod 2023 right
pale steppe
increase what
So then uh since 2023 = 7 x 17 can we just increase 2023 by 7 so it becomes 7² x 17²
tight quest
Like yk how it's (n(n+1)/2)² = 0 mod 2023 right
u can pose the question this way, sure
tight quest
So then uh since 2023 = 7 x 17 can we just increase 2023 by 7 so it becomes 7² x...
but why?
for 2023 to divide 1/4 (n)^2(n+1)^2, you want that 1/4 (n^2)(n+1)^2 to contain 2 17's and 1 copy of 7. Thus, you have to pick an n such that you have at least one copy of 7 cuz otherwise n^2 wouldn't contain at least 1 copy of 7
i mean u can "increase" if u like but that seems counterproductive for no apparent gains
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silver pecan
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silver pecan
With CAS I got $n=\frac{-6\pm2\sqrt{3}}{1}$
glossy valveBOT
UCYT5040
silver pecan
-6 and +-2 are not relatively prime
i dont see how I could manipulate this fraction to get relatively prime numbers
i guess its also least value of n so not +-2, just -2
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@silver pecan Has your question been resolved?
silver pecan
so i checked the answer key and yeah -4 is the right answer (which -4=-6-2+3+1)
but is there an error in this question since -6 and 2 are not relatively prime?
or maybe it means that -6, -2, 3, and 1 share no common factors
?
plain harness
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leaden ermine
I dont think the radii are correct
plain harness
😔
leaden ermine
let me help you with a drawing!
plain harness
These two are the radii right?
plain harness
let me help you with a drawing!
That would be awesome!
leaden ermine
plain harness
So big R is the problem
leaden ermine
yes
plain harness
It has to be something + 8, right?
leaden ermine
not quiet
let me try to demonstrate why
or wait
hmm
the issue is
y^(1/2) is at most 2^(1/2) so 2^(1/2)+8 is something like 9.14
at most
but your radius R actually grows till 10 units
leaden ermine
These two are the radii right?
like you already marked it yourself down
it can become up to 10 units long
so you are missing out on a tiny bit volume
plain harness
Oh.
leaden ermine
in order to avoid this you actually thinkg a bit differently
you basically always subtract the function from the line
plain harness
8-y^1/2?
leaden ermine
the line is at x = 10
plain harness
Ye
plain harness
oooooooo I seee
leaden ermine
let me try to make it visually more clear
the red is 10 long and you subtract the blue lengths (the function values) then you end up with the radius, the distance between line of revolution and function
plain harness
Ohhhh yeah that makes it way clearer
Thank you!!
Wait but I just tried it and it says it's wrong
leaden ermine
can you tell vertically when your region starts and ends
plain harness
0 to 4 right?
leaden ermine
yes
plain harness
😔 it's still wrong
leaden ermine
,w pi * Integrate[(10-y^(1/2))^2-(10-2)^2, {y,0,4}]
glossy valveBOT
leaden ermine
did you get 136pi/3?
plain harness
OHHHH I got why I was wrong
I didn't know you had to (10-2)^2
Wait but that's the same as 8^2
plain harness
I didn't - I got 1372pi/2 for some reason
leaden ermine
yea you need to subtract
imagine like
you subtract the smaller region from the bigger one to get the region in between and thus the volume in between
plain harness
That makes sense!
leaden ermine
i can't draw it 😭
leaden ermine
That makes sense!
something like this
plain harness
i can't draw it 😭
No that's okay! I got what you were saying 😅
leaden ermine
ok nice!
leaden ermine
No that's okay! I got what you were saying 😅
but now!
it's like removing a cylinder from a vulcano
plain harness
it's like removing a cylinder from a vulcano
I seeeeee
That metaphor actually makes it make so much more sense 😅
Thank you!! ❤️
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swift bridge
Why are we allowed to just "cancel" common factors when taking a limit? What is the "rigorous" reason? It seems intuitevly correct, but why is it correct rigorsoutly?
glacial pasture
because youre never exactly 'at' x_0 the factor is always non-zero and so can cancel without issue
sand escarp
x_0 is not in the set, thus, we can cancel the terms
rapid rain
Why are we allowed to just "cancel" common factors when taking a limit? What is ...
the limit $\lim_{x\to x_0}$ is always $\lim_{\substack{x\to x_0 \ x\neq x_0}}$
welp I forgot latex
glossy valveBOT
AℤØ
glacial pasture
new trick for me
rapid rain
ah thank you
glacial pasture
that \ between x_0 and x\neq is a double \
glossy valveBOT
rafilou is not not born in 2003
rapid rain
that \ between x_0 and x\neq is a double \\
yeye I just hastily copied
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tidal parrot
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tidal parrot
how does this work im confused
mighty ridge
xln(2x+1) + 1/2 ln(2x+1) = (x+1/2) ln(2x+1) = 1/2 (2x+1) ln(2x+1)
cursive condor
Try to combine like terms.
tidal parrot
lemme work it out on paper one sec\
shrewd stream
Are we deriving this?
tidal parrot
no this is clearly algebra
makes no sense after getting thru the hard part of the question
shrewd stream
xln(2x+1) + 1/2 ln(2x+1) = (x+1/2) ln(2x+1) = 1/2 (2x+1) ln(2x+1)
Yh like this
mighty ridge
expand left side or factor right
cursive condor
Let's only focus on $x\ln(2x+1)+\frac{1}{2}\ln(2x+1)$. Try to factor out $\ln(2x+1)$.
glossy valveBOT
mathisfun
glossy valveBOT
mathisfun
tidal parrot
what steps to take to go from the green portion to the blue
makes no sense whatsoever
cursive condor
what steps to take to go from the green portion to the blue
Well
If we factor the $\ln(2x+1)$ we get $\ln(2x+1)\qty(x+\frac12)$
glossy valveBOT
mathisfun
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raven mauve
i tried doing this question, and i done it wrong, i thought the way i did it would be correct, can someone explain the answer sheet answer pls??
plush egret
i tried doing this question, and i done it wrong, i thought the way i did it wou...
two roots are given
so we can write it in factored form
the rest is not clear on your work to me
plush egret
i tried doing this question, and i done it wrong, i thought the way i did it wou...
you can read the two roots, yea?
raven mauve
the rest is not clear on your work to me
i thought by subbing in 2 points on the graph, then solving simultaneously to find a and b would be
raven mauve
you can read the two roots, yea?
yes
grim sequoia
You plug in the y-intercept into the factored equation
plush egret
i thought by subbing in 2 points on the graph, then solving simultaneously to fi...
sure but the function has 3 unknowns
so you can do 3 simultaneous
if you want
however the solution is easier, we actually can write it in one unknown only
grim sequoia
You plug in the y-intercept into the factored equation
That takes out the x and y values so you can solve for a
raven mauve
sure but the function has 3 unknowns
so id have to do equation 1 - equation 2 then equation 2 - equation 3
to find 2 equations
then solve simultaneously
if i wanted to
but i could also factorise
which is easier ye??
plush egret
so id have to do equation 1 - equation 2 then equation 2 - equation 3
youd have to do something
plush egret
but i could also factorise
as the solution does yes
as long as you believe it is valid to write it this way
a(x+2)(x-8)
raven mauve
youd have to do something
hm??
plush egret
hm??
i mean
there are a lot of things you can do to solve a 3x3 system
they are all harder than the solution you showed
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plain harness
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plain harness
(this is from my determing volume from slicing homework)
I feel like I have to use the washer method
steel solar
(Don't tell me you have to use integration...)
plain harness
Ye
steel solar
(||You're fucking kidding me||)
plain harness
That's what I thought too 😅
The thing is I'm confused about the terminology. It's a ball with a cylinder drilled through it right?
Not a empty round ball inside the ball?
But if I use the washer method, won't I get an empty round ball inside the ball?
normal spoke
But if I use the washer method, won't I get an empty round ball inside the ball?
No
You’d use some line to represent the hole and the half circle formula to get a half circle
Then you washer or shell
Lowkey, this problem is devious. At least you can double check easily. <— (I rescind my previous statement)
spiral vigil
(||You're fucking kidding me||)
it's a good opportunity to make sure you're doing the integration right bc you can check with normal geometry
actually this isn't that easy with normal geometry
plain harness
You’d use some line to represent the hole and the half circle formula to get a h...
Wait but what axis would I be revolving this around?
spiral vigil
it's just revolving this around the x axis
plain harness
Oh okay. But like won't I get like half a ball? Usually when you like revolve it, you get like one plane right? Sorry if you don't get what I'm saying I don't know how to put it into words
Like half a 3d ball
normal spoke
No
slate violet
actually this isn't that easy with normal geometry
hypotenuse = 14 and height = 8 in a right triangle, so base = ?
normal spoke
Oh okay. But like won't I get like half a ball? Usually when you like revolve it...
You get a ball with a hole
normal spoke
hypotenuse = 14 and height = 8 in a right triangle, so base = ?
?
spiral vigil
hypotenuse = 14 and height = 8 in a right triangle, so base = ?
wait i don't see what you mean, you still have a spherical cap to deal with
plain harness
You’d use some line to represent the hole and the half circle formula to get a h...
Wait I get the half circle part but how would the hole be a line?
slate violet
wait i don't see what you mean, you still have a spherical cap to deal with
oh shit right
spiral vigil
Oh okay. But like won't I get like half a ball? Usually when you like revolve it...
normally you revolve it all the way around (2π)
normal spoke
Wait I get the half circle part but how would the hole be a line?
You integrate from line to circle, and ignore everything below line and above y=0
slate violet
I completely forgot thanks Hayley
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median gust
In the following figure, points A, B, C, D, E, F, G are in the plane, so that A, B, E, F are on the same line, as well as C, D, G, F. AB = BC = CA = BD = DE = EG = GF. What is the measure of angle ∠AFC?
slate violet
I haven't really find anything yet, I know that ABC is equilateral
hint: start from letting angle GFE = x
then find all the angles so that you eventually find angle CBA in terms of x
median gust
Great, lemme try that
whole lily
The exterior angle theorem will make it easier
median gust
hint: start from letting angle GFE = x
Nice, I found it
60 = 5x
Thank you very much guys
Have a great day
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stone kraken
can someone help me solve this exercise?
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can someone help me solve this exercise?
cursive condor
,rccw
glossy valveBOT
stone kraken
the instruction is to find the volume pf the area between the functions xy=1, y=0, x=1 and x=2 around the x=-1 axis
cursive condor
Split the integral into $\int_{-1}^1 \frac{1}{x}dx+\int_1^2\frac{1}{x}dx$
glossy valveBOT
mathisfun
stone kraken
i have no idea what the limits in the integral are, whether i should substract or add that -1
stone kraken
Split the integral into $\int_{-1}^1 \frac{1}{x}dx+\int_1^2\frac{1}{x}dx$
wait why
like ik tge equation to find the volume
cursive condor
wait why
Because the area itself will be undefined if you don't
Note that $\int_{-a}^a f(x)dx=0$, for $f(-x)=-f(x)$
glossy valveBOT
mathisfun
stone kraken
which is π* ∫(upper funcion)^2 - (lower function)^2
idk what the upper or lower functions are
stone kraken
Because the area itself will be undefined if you don't
but im not trying to find any area in the negative side of the x axis
cursive condor
What is distance from x=-1 to 1/x for x<0 and x>0?
stone kraken
sry im so lost
cursive condor
but im not trying to find any area in the negative side of the x axis
Then just let f(x)=1/|x| instead
Lots of ways you can go about this
stone kraken
the method i should be using is discs and rings
cursive condor
the method i should be using is discs and rings
Yes
So think about rings (washers) you can use here if you let y=1/|x|
stone kraken
kk i think ill try to solve it from here
tysm
have a great night (or day)
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left trench
trig sub problem
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trig sub problem
left trench
i do not even know how to start
completing the square requires you to add something to the other side of the equation so i have no idea how to do this
cursive condor
Let's just focus on $10x-x^2$. Note that we can rewrite this as $-(x^2-10x)$.
glossy valveBOT
mathisfun
cursive condor
Now, what is the number we should add to make $x^2-10x$ a perfect square?
glossy valveBOT
mathisfun
left trench
- 25
glossy valveBOT
mathisfun
left trench
uh 10x - x^2 on the left side but yes that makes sense
OH
SHIT
I JUST GOT THI
THANK YOU!!!!
let e try
hold on
void widget
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void widget
I think my book's solution for Question no 2 is wrong
The Slope of the tangent at that point is 3 so the slope of normal should be -1/3
But they didn't do that , I think they made a mistake ?
glacial pasture
they have that slope
void widget
Yes
glacial pasture
ah i see
glacial pasture
yeah thats what they have
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thick hedge
is this cengage ?
void widget
is this cengage ?
Yeah
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solemn python
hi im having trouble understanding vector and scalar resolute when one vector is bigger than the other
solemn python
like i know scalar resolute is just the magnitude of the shadow of one vector casted onto the other
like i know magnitude of vector n is the scalar resulte of a on b
but what happens if a is much bigger than b ?
trail pike
you can think of it as the projection a makes onto the line vector b is on
like, extend b out from its tail and tip to go on infinitely, the scalar resolute is the projection a makes on this line
solemn python
Ohhhh, is its like how much vector a goes in the direction of b rather than the vecor b itselff?
solemn python
like, extend b out from its tail and tip to go on infinitely, the scalar resolut...
ohh
trail pike
Ohhhh, is its like how much vector a goes in the direction of b rather than the ...
yeah
solemn python
omgg wait this is good thank you so much
i was thinking about it wrong thank youu
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real silo
Guys, i need help. The mark scheme says the solution is 0, pi/6, pi/2, and 5pi/6
but i only got pi/6 and 5pi/6
real silo
This is the answer to the previous question
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leaden ermine
Guys, i need help. The mark scheme says the solution is 0, pi/6, pi/2, and 5pi/6...
the problem is you divided by the numerator
that is only okay when the numerator is not 0
what happens if the numerator is 0?
then the equation is true since 0 = 0
so you missed out on solutions
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real silo
the problem is you divided by the numerator
ohhh
real silo
let me ping you cause you tend to disappear sometimes
Yes sorry 😭
real silo
Do we solve this to see if the numerator is 0 or not 0?
leaden ermine
if 0
real silo
But how to solve this?
glossy valveBOT
leaden ermine
then you can use
glossy valveBOT
leaden ermine
Then you can square both sides
and solve the quadratic equation in sinθ
oh let me ping the dinosaur again
@real silo
real silo
Ohhh
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sullen jacinth
vivid berry
how do they know
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how do they know
long sky
You're asking how they know the angle highlighted in blue is also 20 right?
long sky
Yea basically
long sky
I mean I haven't seen the full question
vivid berry
oh nvm i see it
long sky
But Fb sin 20 gives the vertical component of the force
And Fb cos 20 gives the horizontal component ig
vivid berry
oh ok one more question
how does total force/mass give u the acceleration
ohh
f = ma
f/m = a
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faint drift
Greetings! I request help to understand an Improper Integral problem.
The problem: (0 to inf) sin^2(x) dx
What I need help with:
At the near end, the problem ends up becoming:
1/2 (inf - 0) - 1/2 ( [sin(2 * inf)/2] - [sin(2 * 0)/2] )
--> (inf/2 - 0/2) - (sin(inf)/4 - sin(0)/4)
--> (inf - 0) - (inf - 0)
--> inf - inf
The problem I am having is with the value of [sin(2 * inf)/2].
The way I see it, I am supposed to change the inf - inf to some 0/0 or inf/inf form to trigger L'Hospital's Rule, but there's... no feasible way for me to do that.
According to a step-by-step helper, the end result would be equal to infinity. But it does not explain how it concludes that fact. According to my classmates, [sin(2 * inf)/2] is considered undefined. I am unsure why, but apparently that would make it (inf - und), therefore making the end result infinity, and concluding that it is divergent.
Can someone help explain what [sin(2 * inf)/2] is supposed to be? What is its value? Thank you 🙂
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Greetings! I request help to understand an Improper Integral problem.
The probl...
@faint drift Has your question been resolved?
violet haven
Greetings! I request help to understand an Improper Integral problem.
The probl...
lim x-> inf sinx doesnt exist but u can say that sin(x) will be something finite as it ranges from -1 to 1 . so inf- (something finite) is still inf
you can also just look at the area under the curve sin^2 x
faint drift
So because sin(inf) has a defined range of -1 to 1, regardless of if I do inf - (1) or inf - (-1), the end result is still infinite? Did I comprehend your answer correctly? o>o
violet haven
So because sin(inf) has a defined range of -1 to 1, regardless of if I do inf - ...
yeah
faint drift
Alright then. And linking that thought, why does sin(inf) not exist?
violet haven
Alright then. And linking that thought, why does sin(inf) not exist?
because it doesnt approach any specific value
it oscillates between -1 and 1
you cant tell which it will be in that interval of [-1,1]
wary pollen
Hello, I am trying to prove that the sum of inverse square is $\frac{\pi^2}{6}$.
For this I am given $f(z)=\frac{cotan(z)}{z^2}$.
We directly see two poles, one at $0$ and one at $n\pi$.
It is straight forward to compute the residue at each poles,
$Res(f(z),0) = -\frac{1}{3}$
$Res(f(z),n\pi) = \frac{1}{n^2\pi^2}$
So, we see that we only need to prove that the contour integral vanishes, for that we can take a square contour of length (2N+1) centered at the origin.
Now my thought was to use the estimation lemma :
$\left| \int_\gamma f(z) dz \right|\leq Ml(\gamma)$
So $l(\gamma)=4(2N+1)$.
But now I'm stuck at finding $M:= sup_{z\in\gamma} \left| f(z) \right| $
Could someone help me figure this out ?
glossy valveBOT
Lucas
sullen hinge
Hello, I am trying to prove that the sum of inverse square is $\frac{\pi^2}{6}$....
parametrise your contour e.g. z = x + ai could parametrise the top and bottom edges with the right a value
then find the sup under each of these contours and then take the largest of those
maybe theres another, perhaps better way, but thats how id try it
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wary pollen
If I parametrised then I could just directly prove that the 4 sides vanishes right ? No need to find the sup then. But the thing is that this takes a while and I assume there must be a faster way to do it somehow ?
sullen hinge
well theres no guarantee that the parametrised integrals would be nice
whereas finding a sup is fairly straightforward
but if you wanna hold out for a better way then be my guest lmao
wary pollen
but how would you find a sup for one of the side ?
the problem I guess is the cotan(z)
sullen hinge
the problem I guess is the cotan(z)
cot(x + iy) = ( cot(x) cot(iy) - 1 ) / (cot(x) + cot(iy)) and cot(iy) = - i coth(y)
so just a bit of an algebra bash to do
wary pollen
Couldn't we just use the fact that when on the contour, it nevers encounters singularities and thus it is bounded ?
sullen hinge
i dont know much CA but if you says thats a thing then sure
but is not your goal to show that M -> 0 as N -> inf
so you need to actually do some computation, no?
oh in fact N M -> 0 i suppose
wary pollen
but is not your goal to show that M -> 0 as N -> inf
Yes
I'm not too sure but If I write |f(z)| as
(For one of the sides)
[
|f(z)| = \left| \frac{e^{iz} + e^{-iz}}{(e^{iz} - e^{-iz}) z^2} \right|
]
glossy valveBOT
Lucas
wary pollen
Where z is some parametrisation like (2N+1) (t-i) with t between -1 and 1
I can say x=(2N+1)t and y=(2N+1)
So
$|f(z)| = \left| \frac{e^{i(x+iy)} + e^{-i(x+iy)}}{(e^{i(x+iy)} - e^{-i(x+iy)}) (x+iy)^2} \right|$
glossy valveBOT
Lucas
wary pollen
Now, If for large N, we will get
$|f(z)| \approx \left| \frac{e^{-ix+y}}{ - e^{-ix+y} (x+iy)^2} \right|$
glossy valveBOT
Lucas
glossy valveBOT
Lucas
wary pollen
If we go back to $\left| \int_\gamma f(z) dz \right|\leq Ml(\gamma)$
glossy valveBOT
Lucas
wary pollen
and take the limit of N to infty on both side it would work I think ?
sullen hinge
i would say you want some more rigour than just saying ~=
if you can chain some <= then yeah
wary pollen
(I'm a physics student x))
sullen hinge
lmao
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stuck loom
torn jolt
what is da question?
stuck loom
Give a pair (a,b) such that a neq = b neq = the expression
Also, a neq = the expression
a, b, c < Z+
I just want a pair, be it using any software.
Now.
unique wagon
okay what is c here
stuck loom
c = the expression
unique wagon
try a = 0 and b = -4 then
i mean really you could make a while loop on vscode with the constraints to find an arbitrary amount of pairs
grizzled pagoda
Could just ask wolfram
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stuck loom
try a = 0 and b = -4 then
I mean 0 is not a positive integer duh.
duh
duh
Neither is -4
LOL<
unique wagon
lesser than Z+???
stuck loom
Oh no
unique wagon
or an element of Z+?
unique wagon
use E next time
stuck loom
Ok
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stone trench
Hello,
Why matrix multiplication is done in that specific way ( multiplying line by column ) ??
hallow walrus
acoustic granite
Hello,
Why matrix multiplication is done in that specific way ( multiplying lin...
convention, more or less. One could define matrix multiplication to be done row by row.
hallow walrus
convention, more or less. One could define matrix multiplication to be done row ...
not at all
it's not just an arbitrary operation we defined, it follows from the fact that matrices represent linear transformations
and multiplication of matrices corresponds to composition of linear transformations
acoustic granite
could that property not be preserved by either changing how we represent matrices or "flipping" the multiplication from row * column to column * row?
hallow walrus
sure, but the operation itself isn't arbitrary
acoustic granite
fair
hallow walrus
that is to say it's not a computation we created for convenience, it falls out naturally
steel solar
Hello,
Why matrix multiplication is done in that specific way ( multiplying lin...
Read any relevant linear algebra textbook and you will see why.
stone trench
Read any relevant linear algebra textbook and you will see why.
can u name one
steel solar
can u name one
Linear Algebra Done Right by Axler. I'm currently reading on it.
hallow walrus
I'd say try Introduction to Linear Algebra by Gilbert Strang, it motivates the ideas much better
stone trench
alright, thank you
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candid tinsel
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candid tinsel
how do i solve this
candid tinsel
not very well
remote zealot
not very well
$\log_a(a^{x}) = x$
glossy valveBOT
donia
candid tinsel
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minor crater
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minor crater
i sketched the domain and figured that i can switch variables:
$=\int_0^1 dy \int_y^1 \arctan(y^2) dx = \int_0^1 (1-y)\arctan(y^2) dy$
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artemetra
minor crater
i can split this into two integrals, and the second one with y*arctan(y^2) is fine
twilit leaf
are you sure thats the right new domain?
twilit leaf
yes
twilit leaf
it makes things so easy
minor crater
i'll need to practice this then
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minor crater
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minor crater
it makes things so easy
i followed through and got a wrong answer 
let me write my solution
twilit leaf
integral of arctan is not 1/(1+x^2)
minor crater
lmao no that i know
twilit leaf
just a guess
minor crater
it's a good guess because i almost made that mistake
starting from the original integral:\
$=\int_0^1 dy \int_0^y \arctan(y^2) dx = \int_0^1 y \arctan(y^2) dy$\
Let $u=y^2$\
$\implies = 2 \int_0^1 \arctan(u) du = 2 \int_0^1 1\cdot \arctan(u) du = 2 \left[u \arctan(u)|_0^1 - \int_0^1 \frac{u}{u^2 + 1} du\right] = 2 \left[\frac{\pi}{4} - \int_0^1 \frac{u}{u^2 + 1} du\right]$
glossy valveBOT
artemetra
minor crater
Let $v=u^2$
$\implies = 2 \left[\frac{\pi}{4} - 2 \int_0^1 \frac{1}{v + 1} dv\right] = 2 \left[\frac{\pi}{4} - 2\ln(1+v)|_0^1] \right]= \frac{\pi}{2} - 4\ln(2)$
glossy valveBOT
artemetra
minor crater
,w pi/2 - 4 ln(2)
glossy valveBOT
minor crater
,w Integrate[arctan(y^2),{y,x,1},{x,0,1}]
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Requested by arte_m_etra
twilit leaf
you did that twice
cursive condor
**artemetra**
Oh yeah, at the start too lmao
minor crater
**artemetra**
so instead i should have $\frac{1}{2} \left[\frac{\pi}{4} - \frac{1}{2}\ln(1+v)|_0^1] \right]= \frac{\pi}{8} - \frac{1}{4}\ln(2)$
glossy valveBOT
artemetra
minor crater
,w pi/8 - 1/4 ln(2)
glossy valveBOT
minor crater
awesome
minor crater
so ydy=du/2
thanks again
twilit leaf
yup
acoustic oriole
Hello, I wasn’t in class for a problem and was wondering how my teacher got this answer on the notes, I just need to know how she got -16 and 36.
onyx glen
so you understand everything up to and including the line x^2 + 20x - 576 = 0?
onyx glen
pulled a factorization out of thin air
or with some trial and error that's not shown here
i probably wouldn't have shyed away from QF here
or completing the square, which is probably better
actually
(x+10)^2 = 676
right from here
recognize 676 as 26^2 and get x+10=±26 instead of all this mess
acoustic oriole
oh i see
so for example on my hw i have 400 = (x+12)^2 it would be (x+30) and (x-10)
?
or -32 and 8 for the positive and negative ?
onyx glen
so for example on my hw i have 400 = (x+12)^2 it would be (x+30) and (x-10)
x+12=±20 is how i would have phrased it.
if you want a factorization for that one, it would look like (x+32)(x-8) = 0, but this is not really necessary to do
acoustic oriole
i think i will have to do the factorization just for my grades sake but thank you i get the concept now, it might be a dumb question but what’s the difference between an extraneous solution and a normal one ?
onyx glen
extraneous means it doesn't make sense for the problem as originally stated
such as when a variable representing a length comes out negative
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stuck fiber
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onyx glen
one of these is very obviously not happening
onyx glen
errrr
wait
no pi at the front
cos(a)=cos(b) means either a = b + 2kpi or a = -b + 2kpi
if you hate negative angles you can instead do a = 2pi-b + 2kpi
but 2pi, not 1pi
foggy shadow
Can i ask for help here ?
onyx glen
Can i ask for help here ?
!occupied
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Can i ask for help here ?
Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).
onyx glen
yes you can but please claim your own channel
foggy shadow
yes you can but please claim your own channel
How
onyx glen
see #❓how-to-get-help for instructions
stuck fiber
See the bots message
foggy shadow
see <#488120190538743810> for instructions
I did click on the i need help
stuck fiber
So read it
onyx glen
foggy shadow
Aha ok
stuck fiber
Was it sin(a)=sin(b) that gives a = b + 2kpi or a = pi - b + 2kpi?
onyx glen
yes
stuck fiber
Okok I see
onyx glen
sin(x) = sin(pi-x)
stuck fiber
Ahh true
Ok well I think that was my issue / reason I didnt get my answer
Thank you!
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gentle sequoia
This right??
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This right??