look at the formula and think what can cause there to be no real solutions
#help-27
wicked turtle
analog yacht
alr im quitting
wicked turtle
happy trails then
devout snowBOT
@analog yacht Has your question been resolved?
devout snowBOT
analog yacht
<@1164994084390318192> Has your question been resolved?
no
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quasi jacinth
hello
quasi jacinth
why did the guy proceed further with |x-y|
isnt +-|x-y| enough to say its limit is 0 so f(x,y) limit is also 0
idk why he said its also less or equal than |x|+|y|
quasi jacinth
we need the limit (x,y)-> (0,0)
the text isnt really useful its just says like we work like this so as a result etc
just look at the math
why is this last step necessary
quasi jacinth
HOLY THATS GREEK
yea why?
patent nexus
isnt +-|x-y| enough to say its limit is 0 so f(x,y) limit is also 0
because denominator also has limit 0
quasi jacinth
because denominator also has limit 0
?
jade bobcat
isnt +-|x-y| enough to say its limit is 0 so f(x,y) limit is also 0
they have taken the definition of a double limit to be when both |x| and |y| are small
so you need to justify why |x|, |y| small implies |x - y| small
jade bobcat
do we not know that lim of |x-y| = 0
again they are showing this
quasi jacinth
we showed that |f|<=|x-y| so -|x-y|<= f <= |x-y|
quasi jacinth
again they are showing this
isnt it trivial
like we can just replacce x,y
|x-y| is continious
quasi jacinth
nws
jade bobcat
why is |x - y| continuous?
quasi jacinth
its not ?
jade bobcat
it is, but the proof is more involved than what they're doing
quasi jacinth
ok so |x-y| <= |x|+|y|
jade bobcat
the fact that taking the limit is the same thing as plugging values in amounts to showing that that function is continuous at 0
quasi jacinth
cant we just
patent nexus
hm, i guess its just shorter to write that its <|x|+|y| than to mention that |x - y| continuous. Just authors choise ig
quasi jacinth
take the limit to 0- 0+ etc
or how do we find lim of |x|+|y|
idk what to do with the absolute value
jade bobcat
you should do it directly by the definition of a limit
quasi jacinth
if i could get rid of it i can just say its a polynomial so continious so i just put 0
how do i show it though
here he just says |x|+|y| -> 0 when x,y -> 0,0
jade bobcat
how do you know if a number is a limit of a two-variable function?
i assume you know about ε-δ
quasi jacinth
i didnt really wanna get into Πε(Po)
this is like second semester and we do mostly calculatively stuff so i rather not go into this cause i dont have the time
quasi jacinth
i assume you know about ε-δ
i do but i havent worked on proofs in the multivariable level
jade bobcat
even if you only care about calculations, on a test you want to be aware of first principles so you don't end up doing something self-referential
quasi jacinth
even if you only care about calculations, on a test you want to be aware of firs...
well what can i do here
i think its a matter of practise
if proffessor jsut shows it as trivial ill take it as is
but do you know how i can tell when to use inequalities and when to take paths to show it ddoesnt exist
or is it a matter of time i get used to it
jade bobcat
there is no other way other than a willingness to understand first principles
quasi jacinth
there is no other way other than a willingness to understand first principles
wdym
what is first principles
jade bobcat
the definition of a limit
you say that a number L is a limit of f: ℝ × ℝ → ℝ at (a, b) if for all ε there are numbers δ, δ' such that |x - a| < δ and |y - b| < δ' implies |f(x, y) - L| < ε
the fact that your professor knows they can stop at |x| + |y| is because they have worked with this definition enough to know that from this point on it is easy
like if you know what you are doing then it is a single line
also the fact that addition +: ℝ × ℝ → ℝ is continuous uses this |x| + |y| → 0 idea
devout snowBOT
@quasi jacinth Has your question been resolved?
devout snowBOT
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fervent helm
How do I solve this? I am not able to figure out the LDE format
devout snowBOT
How do I solve this? I am not able to figure out the LDE format
rare kernel
How do I solve this? I am not able to figure out the LDE format
well y=y(x) so u need dy/dx not dx/dy
and write cot,cosec interms of sin and cos
rare kernel
can u share what u got then
woven radishBOT
Prathmesh
rare kernel
what happened to the e^x
woven radishBOT
Prathmesh
rare kernel
yeah
fervent helm
I substituted cosy =t and differentiated both sides wrt x
rare kernel
yep 👍
devout snowBOT
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daring creek
What I need is the equation or area of mathematics to look for combining a set of percentages (doing this for a fantasy novel). Set One is 1(70%) 2(20%) 3(9%) 4(1%)and Set Two is A(24%) E(24%) F(24%) W(24%) P(1.3%) S(1.3%) B(1.3%). This is in relation to an individual in a set population. A child is born with access to 1 to 4 elements of the listed letters. I want to figure out what any given combo Ala 3, AEF would have to occur for a given individual
devout snowBOT
What I need is the equation or area of mathematics to look for combining a set o...
daring creek
This should be all possible combinations as AA would just be A instead
frozen aurora
What I need is the equation or area of mathematics to look for combining a set o...
combinatorics is your friend!
it has stuff like combinations, permutations, combinations with replacement etc.
daring creek
Thank you!
daring creek
combinatorics is your friend!
Time to go reading i may be back, thank you!
topaz axle
you have to bruteforce with computer
restive oriole
ik i have to do double derivate but am not gettin this
devout snowBOT
ik i have to do double derivate but am not gettin this
green crypt
show your work
robust mango
Lol
Just check the expression in which there is y2, because it is in the expression to be calculated and it first and last appears there
wait
unless y2 refers to something known
onyx torrent
unless y2 refers to something known
it refers to second derivative of y
green crypt
i didnt bother checking what the options were asking for 😭
storm raptor
ik i have to do double derivate but am not gettin this
You'll probably have a better time taking the sin of both sides before differentiating
It'll make the algebra less tedious
devout snowBOT
@restive oriole Has your question been resolved?
restive oriole
i wanna ask for some help-
devout snowBOT
i wanna ask for some help-
north lodge
ik i have to do double derivate but am not gettin this
just differentiate bro you probably did some mistake in differentiating
green crypt
Give up
robust mango
Just integrate bro
graceful cosmos
Willing to help but you'll need to ask specific questions.
If you don't have any specific questions, study until you do
restive oriole
aha am doing only
devout snowBOT
@restive oriole Has your question been resolved?
cedar berry
can anyone teach me mathamatics?🫠 like am cooked fr
a wise redditor once said
"padh le master ji 🙏"
spam pw youtube
one shot op fr
uneven coral
Well just practise a lot, that's all I'll say
You've still got 6 days
Well @restive oriole unless you've got a question in a specific question you encountered (which you would have in some time if you lock in now), please close this channel and open it once/if you're stuck in a question
devout snowBOT
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orchid shuttle
Help
orchid shuttle
Assertion (A): Quotient of √12 and √3 is a natural number.
Reason (R): Division of two irrational numbers is always a natural number
Can some one please help me
real mason
Assertion (A): Quotient of √12 and √3 is a natural number.
Reason (R): Division...
what's the question
orchid shuttle
It's assertion and reason
orchid shuttle
Check if assertion is true and reason is it's correct explanation
real mason
first, is A true?
summer summit
to find if A is true remember the rule $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$
woven radishBOT
Krish
real mason
second, do you think R is true? if not, can you find some example where its false
orchid shuttle
I am not getting it
real mason
!status
devout snowBOT
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
1
real mason
No
ah
how familiar are you with square roots in general
like do you understand what they are
trail eagle
pure cedar
top 10 things that never happened
orchid shuttle
trail eagle
If you tell me answer I will try to find how it is the answer
So if you're given $\frac{\sqrt{12}}{\sqrt{3}}$ you have no idea how to simplify that?
woven radishBOT
Azyrashacorki
trail eagle
Using, for instance, the fact that 12 = 3*4
orchid shuttle
could you share
No you will laugh in my below poverty conditions
orchid shuttle
So if you're given $\frac{\sqrt{12}}{\sqrt{3}}$ you have no idea how to simplify...
No idea
restive river
do you know what is $\sqrt$
woven radishBOT
hail
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
modest bane
Noone gives a shit bro, just send working 
orchid shuttle
do you know what is $\sqrt$
Root of 4 is 2
pure cedar
neuron activation
restive river
Root of 4 is 2
right ok
trail eagle
No idea
And you said you're not familiar with rules like $\sqrt{a\cdot b} = \sqrt{a}\cdot \sqrt{b}$
orchid shuttle
Yeah
woven radishBOT
Azyrashacorki
trail eagle
Is that something you've seen before
orchid shuttle
I have seen
trail eagle
Ok we can work with that
So I have told you that $12 = 3 \cdot 4$.
How can your rewrite $\sqrt{12}$ then?
woven radishBOT
Azyrashacorki
orchid shuttle
How will the quiotent come then if I wrote 4*3
trail eagle
Let's just focus on sqrt(12) for now
orchid shuttle
Yes
trail eagle
Can you complete it?
orchid shuttle
So now I have underroot 3*4
trail eagle
Yes so $\sqrt{12} = \sqrt{3 \cdot 4}$
orchid shuttle
And underoot 3 2 2 means the same too
woven radishBOT
Azyrashacorki
trail eagle
**Azyrashacorki**
What does using this give then?
trail eagle
Yess
orchid shuttle
Now can we divide underroot 3 from underoot 4
trail eagle
Well now you have $\frac{\sqrt{12}}{\sqrt{3}} = \frac{\sqrt{3\cdot 4}}{\sqrt{3}} = \frac{\sqrt{3} \cdot \sqrt{4}}{\sqrt{3}}$
woven radishBOT
Azyrashacorki
trail eagle
Can you simplify something?
orchid shuttle
Yes root 3 and root 3 will be cutted
trail eagle
Good. Now what is left?
trail eagle
Which is
orchid shuttle
So 2 is the answer
trail eagle
Yes. Is 2 a natural number?
orchid shuttle
Yes
trail eagle
So is A true?
orchid shuttle
Yes
trail eagle
Good
orchid shuttle
Assertion (A): Quotient of √12 and √3 is a natural number.
Reason (R): Division of two irrational numbers is always a natural number
So reason is true
pure cedar
😇
trail eagle
A is true. Now considering what we did, can you think of some irrational numbers to divide which DON'T give a natural number?
orchid shuttle
Omg I cant
trail eagle
Here it works out really well since 12 = 3*4 and 4 is a square
What if you choose something like 3 * k where k is not a square?
devout snowBOT
trail eagle
Most numbers aren't squares. If you can just choose some nonsquare number, your favourite one, you will be able to come up with a counter examples to R
devout snowBOT
Available help channel!
haughty acorn
i have a problem with this exercise with the point c) it asks me to Determine the real numbers b and c for which the system of equations is consistent for any real number a
devout snowBOT
i have a problem with this exercise with the point c) it asks me to Determine th...
haughty acorn
from the point b) i know that the det(A(a)) = 0 for a = +-1
so i know that the system is consistent for any a from R/{-1,1} but now the sistem can be consistent for a= -1 or 1 also since it can be Dependent Consistent
but with this information i still dont know how to get b and c
pulsar sand
Plug in a = -1, take it to RREF and see what comes out of it, then repeat for a = 1
haughty acorn
Plug in a = -1, take it to RREF and see what comes out of it, then repeat for a ...
what is PREF
im from romania and terminology is different sorry
haughty acorn
Ahhh so like when u reduce row with row or cloumn with column
haughty acorn
Reduced Row Echelon Form
but how does that help me tho since b and c only appear in the results row of the system
pulsar sand
You know that the system is consistent if a is not equal to -1 or 1. That means if a is say 5, b and c can be anything and it doesn't matter.
The only thing left to do is to study the special cases when a = -1 and 1
haughty acorn
The only thing left to do is to study the special cases when a = -1 and 1
so i plug 1 and -1 into the matrix and what am i looking to get out of it to see the values of b and c
sorry these are dumb questions im not that good with matrix
dapper fable
Let $C_1,C_2,...,C_{45}$ be distinct points along the extension of line $AB$ (no points are between $A$ and $B$). Prove that $\sum_{i=1}^{45}C_iA \neq \sum_{i=1}^{45}C_iB$
devout snowBOT
Let $C_1,C_2,...,C_{45}$ be distinct points along the extension of line $AB$ (no...
woven radishBOT
Copter
dapper fable
isn't this sort of easy? is there any other way to do this
green crypt
That "suppose" is heavenly
dapper fable
just name the points such that C{1},...,C{k} are all on A's side
dapper fable
isn't this sort of easy? is there any other way to do this
^
sand dove
isn't this sort of easy? is there any other way to do this
Not really it kinda joins back your argument. Once you notice that C_kA - C_kB = ±AB for every k, then by parity the difference between sums is an odd integer*AB
dapper fable
okay, thanks!
i was kinda confused cuz this seemed to easy for the level of contest this was in😭
gritty nacelle
That "suppose" is heavenly
wdym it comes preinstalled with Math v4.9
devout snowBOT
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Available help channel!
jovial mural
How to do 13
devout snowBOT
How to do 13
vital edge
Are you familiar with the distance formula for a point and a line
jovial mural
Ye ik the distance formula
scarlet fulcrum
devout snowBOT
scarlet fulcrum
Lowkey stupid question I don't understand the U-Sub in this problem
slate cipher
$$y'(x) = \int \left( \frac{1}{(2-x)^2} + 1 \right) dx$$
woven radishBOT
akeanti💕
slate cipher
the u sub will be used to calculate this integral ig
$$\int \frac{1}{(2-x)^2} dx = \int \frac{1}{u^2} (-du) = -\int u^{-2} du = -(-u^{-1}) = \frac{1}{u} = \frac{1}{2-x}$$
woven radishBOT
akeanti💕
slate cipher
after calculating that integral u will have to solve this one using the value they gave u
$$y'(x) = \frac{1}{2-x} + x + C_1$$
woven radishBOT
akeanti💕
slate cipher
u can find C_1 using the y'(3).... i assume the only question u had was about the u tho and ig it was used to solve the integral i wrote above
scarlet fulcrum
I know how to do the problem
Yeah it was just the U
I didn't know how to get to the du part
slate cipher
I didn't know how to get to the du part
yea i'm pretty sure the U was used to calculate this : #help-27 message
that's probably the only part where the sub is kinda of needed
devout snowBOT
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Available help channel!
drifting bolt
Is this right? Cuz whenever I try plugging in 37 cents to get the year it goes over
devout snowBOT
Is this right? Cuz whenever I try plugging in 37 cents to get the year it goes o...
last parrot
Is this right? Cuz whenever I try plugging in 37 cents to get the year it goes o...
have you tried graph these points on desmos
drifting bolt
We don’t have desmo
last parrot
drifting bolt
But we do got the trust clac
last parrot
its pretty handy
last parrot
is your function base on cent?
drifting bolt
Ye
last parrot
so you used calculator to find this formula automatically
drifting bolt
Does tha mean I typed my thingy wrong
Cuz I keep on getting this on my clac
Calc
last parrot
did you input the points in correctly
drifting bolt
I’m sure I did multiple times
It’s supposed to be like this right
Did I mess up
last parrot
delete line 11 bro
maybe the calculator also take value from redundant line 11
then try again
drifting bolt
I can’t delete line 11 it’s just kinda there
I could ask my teach about it tmr
Maybe my calc is broken yknow
Well goodnight dude
I’ll 👋
.close
devout snowBOT
last parrot
Well goodnight dude
good night, yeah just check it out because the formula aint correct in my calculation
devout snowBOT
Available help channel!
twin magnet
$tan(\phi) = atan(\alpha)$ How do we find the value of $\phi$
devout snowBOT
$tan(\phi) = atan(\alpha)$ How do we find the value of $\phi$
woven radishBOT
glaedr_
void fox
atan meaning arctangent?
void fox
then φ = arctan(a * tan α) + nπ, n ∈ ℤ no?
twin magnet
right but my questions requires it to be $\phi = \alpha$
woven radishBOT
glaedr_
twin magnet
so just was wondering if there's anyway to get to that
fluid stag
this sounds like you need to show the original question
twin magnet
Alright, so
void fox
when a ≠ 1, the solutions to tan x = a tan x are nπ
void fox
this sounds like you need to show the original question
but yeah
twin magnet
$r = ae^{\theta cot(\alpha)}$ is the equation of the curve and we need to prove that $\rho = rcosec(\alpha)$
woven radishBOT
glaedr_
twin magnet
BUT
fluid stag
could you please send a screenshot of the original question statement instead?
twin magnet
i literally did a previous question asking the same thing but the equation of the curve was $r = e^{\theta cot(\alpha})$
woven radishBOT
glaedr_
twin magnet
could you please send a screenshot of the original question statement instead?
Show that the radius of curvature at a point $(r, \theta)$ for the curve $r = ae^{\theta cot(\alpha)}$ is $\rho = r cosec(\alpha)$
woven radishBOT
glaedr_
twin magnet
original question
but there's a previous question in which the curve doesn't have the constant a and that actually proves the required statement
fluid stag
well I was kinda asking for the screenshot or picture of the og question but if that's the exact wording I guess we'll take you out on that
twin magnet
well I was kinda asking for the **screenshot or picture** of the og question but...
it is
fluid stag
**glaedr\_**
usually we'd much prefer screenshots cuz we can see the context and you don't have to type as much
alright I'll step out and let plante continue, sorry for intruding
twin magnet
alright
The previous question was:
Show that the radius of curvature at a point on $r = e^{\theta cot(\alpha)}$ is $\rho = r cosec (\alpha)$
woven radishBOT
glaedr_
twin magnet
like how am i supposed to prove the same thing but with different curves
when we solve the first derivative of this curve and replace the equations we get
void fox
radius of curvature, meaning radius of the osculating circle?
twin magnet
$\text{ Before, }$ \\
$tan(\phi) = tan(\alpha)$ \\
\text{ After, }$ \\
$\tan(\phi) = a tan(\alpha)$
glaedr_
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)
void fox
alright I'll step out and let plante continue, sorry for intruding
nono it's ok
twin magnet
radius of curvature, meaning radius of the osculating circle?
wdym osculating circle
void fox
actually, since this is diffgeo-ish and I am having lunch, you can take over
twin magnet
😭
fluid stag
nah I have other stuff to handle and I'm just here to try to have OP show the og question
sorry
twin magnet
i really don't need help regarding the context of the question
just if $\phi$ can equal $\alpha$
woven radishBOT
glaedr_
twin magnet
@fluid stag can u unpin the og question
void fox
unpinned
twin magnet
**glaedr\_**
for $tan(\phi) = a tan(\alpha)$ where 'a' is a constant
woven radishBOT
glaedr_
fluid stag
alright I'll bite. assuming a \neq 1, it is possible, but only trivially when tan is 0.
twin magnet
whats a \neq 1
fluid stag
a not equal to 1
twin magnet
and wdym trivially when tan is 0
fluid stag
cuz if tan(a) = c tan(b)
then setting a = b and shoving everything to one side gives us (1-c) tan(a) = 0
dividing both sides by (1-c) (only if c not 1) gives us tan(a) = 0
I'll leave finding the possible values of a to you from here
devout snowBOT
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jade vapor
devout snowBOT
ripe grove
,rccw
woven radishBOT
devout snowBOT
@jade vapor Has your question been resolved?
winter prairie
I am not good at complex numbers :/
devout snowBOT
<@&286206848099549185>
Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
solemn hatch
So the lower arc of the circle is not a part of the locus
jade vapor
Yes it is not the part it represent the locus of -3pie/4
last parrot
$\frac{z-z_2}{z-z_1}=\frac{\pi}{4}$
solemn hatch
Yes it is not the part it represent the locus of -3pie/4
Yupp so min value of |z| should just be the extreme points
jade vapor
3,6 and 6,3??
solemn hatch
Ye
woven radishBOT
Minhh
last parrot
use this formula jakshill
jade vapor
$\frac{z-z_2}{z-z_1}=\frac{\pi}{4}$
From that how can i take out minm value??
last parrot
From that how can i take out minm value??
Then the circle would pass points z1 and z2
then find mid point of these points
solemn hatch
But that does not satisfy the circle
last parrot
$M=((\frac{3+6}{2}).(\frac{6+3}{2}))$
jade vapor
This is the diagram
jade vapor
But that does not satisfy the circle
Yes the mid point wont satisfy it
woven radishBOT
Minhh
last parrot
wdym dont satisfy
nimble dragon
You don’t need all that. Since you already found the center, the minimum |z| is just the distance from origin to center minus the radius
solemn hatch
You don’t need all that. Since you already found the center, the minimum |z| is ...
That part is not covered in the locus
jade vapor
nimble dragon
The center is NOT the midpoint
solemn hatch
Min. z should just be 6,3
nimble dragon
It is 6.6 as you pointed out \
jade vapor
Yes yes
nimble dragon
Min. z should just be 6,3
Why?
solemn hatch
The center is NOT the midpoint
When did he say that
last parrot
well if its correct it should be $6\sqrt{2}-3$
nimble dragon
When did he say that
See the. Diagram
nimble dragon
well if its correct it should be $6\sqrt{2}-3$
Yeah this is what I’m saying
solemn hatch
See the. Diagram
Yes it labels the midpoint of the chord as the midpoint
woven radishBOT
Minhh
nimble dragon
Just draw the normal from origin to center and subtract distances
solemn hatch
Oh lord
nimble dragon
Since min distance is always. Along normal
jade vapor
6sqrt2 - 3 is not the answer
solemn hatch
Just draw the normal from origin to center and subtract distances
The z it would hit at is not a part of the licus
jade vapor
You re taking minor arc as locus which is not
solemn hatch
6sqrt2 - 3 is not the answer
@last parrot
nimble dragon
The z it would hit at is not a part of the licus
OHHH I now see what you’re saying
Then you’re right mbmb
solemn hatch
It's most prolly √45
verbal wharf
It's most prolly √45
yeah
solemn hatch
I got 3sqrt5
Yes
last parrot
6sqrt2 - 3 is not the answer
huh isnt it is the origin to the circle subtract radius??
jade vapor
It's most prolly √45
Yes may be i am also thinking that but exactlt root 45 cant be the answer
nimble dragon
Nice, minimum occurs at two points 6,3 and 3,6
solemn hatch
Yes may be i am also thinking that but exactlt root 45 cant be the answer
Well yeah it approaches that answer ig
nimble dragon
Since symmetry about y=x
last parrot
ah i forgot about symmetry yeah true
jade vapor
huh isnt it is the origin to the circle subtract radius??
The minor arc is not the locus and centre minus radius is minor arc
solemn hatch
Well yeah it approaches that answer ig
It's not exactly at 3,6 or 6,3
last parrot
The minor arc is not the locus and centre minus radius is minor arc
i see
nimble dragon
ah i forgot about symmetry yeah true
No the bigger mistake is the locus is not the full circle
jade vapor
It's not exactly at 3,6 or 6,3
Yes
verbal wharf
i did it algebraically thinking
solemn hatch
Fair enough
solemn hatch
i did it algebraically thinking
Exact root(45) would occur at 6,3 or 3,6
verbal wharf
hm
last parrot
then is it 3sqrt5?
last parrot
which is sqrt45
jade vapor
Yess it will make 0°
solemn hatch
So it will approach root(45) ig
Unless the fxn decreases from root(45) and has a minima some where else
Doesn't look like that
solemn hatch
Fair enough
jade vapor
But how will we take out that?
solemn hatch
Parameterise se the circle in terms of theta
6+sin(theta), 6 + cos(theta)
Then you can find |z| as a fxn of theta
Just beware the part where the circle is in the lower arc
solemn hatch
Join centre and the two points
(6,3) and (3,6)
And find the angle the chord subtends on center
That much part is to be excluded
jade vapor
Its 90°
solemn hatch
Yup so a quarter of the circle
jade vapor
Okayy
solemn hatch
You know the co-ordinates
solemn hatch
6+3sin(theta), 6 + 3cos(theta)
.
Of z
Can you find |z|?
Though I think it definitely will be (3,6)
jade vapor
Can you plz explain a little bit more @solemn hatch
solemn hatch
Kinda busy rn
nimble dragon
I saw it diagramatically
Draw rays from origin to the arc you see that the distance function increases from 6,3 till that symmetry point on y=x and then decreases again till 3.6
jade vapor
.close
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fervent helm
since $f'(x)=f(x), \frac{dy}{dx}=y$ therefore $y=e^x \times c$
devout snowBOT
since $f'(x)=f(x), \frac{dy}{dx}=y$ therefore $y=e^x \times c$
zenith jacinth
you mean $C \exp(x)$ ?
woven radishBOT
Herels
fervent helm
you mean $C \exp(x)$ ?
yea
fervent helm
yea
fervent helm
but f(0) =0 so if we put it in the equation then c=0
fervent helm
write c \times e^x
alr
woven radishBOT
Prathmesh
zenith jacinth
y = f(x) right ?
fervent helm
y = f(x) right ?
yup
zenith jacinth
so, f(x) = C exp(x)
fervent helm
right
zenith jacinth
we need to find C
fervent helm
yup
zenith jacinth
we have f(0) = 0 and f'(0) = 3
fervent helm
yup
final spade
Why not just take a cubic polynomial
fervent helm
Why not just take a cubic polynomial
but what about f'(x) = f(x) ?
zenith jacinth
I dont get it, why we have f(0) = 0
but with f'(x) = f(x), we have f'(0) = f(0) = 3
final spade
but what about f'(x) = f(x) ?
Yea take it as ax^3 + bx^2 + Cx + D
And compare coeff
zenith jacinth
Yea take it as ax^3 + bx^2 + Cx + D
And compare coeff
oh yea good point
final spade
Find the actual polynomial
fervent helm
ohh
zenith jacinth
How do you get that physics community badge?
you mean the tag in front of his name ?
final spade
Yeah
fervent helm
you see the server tag right there?
final spade
I mean that
final spade
you see the server tag right there?
Yeah
fervent helm
that's it
fervent helm
welcome
fervent helm
all
verbal wharf
you mean the tag in front of his name ?
not relevant but just want to say you have based mutual servers
green crypt
Can u guys solve it?
!occupied
devout snowBOT
Can u guys solve it?
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).
final spade
Oh sorry
rare kernel
since $f'(x)=f(x), \frac{dy}{dx}=y$ therefore $y=e^x \times c$
Is it just me or does this question not make sense
rare kernel
guys I am sorry this question was wrong. it should be f''(x) = f(x)
Yeah exaclty
fervent helm
Yeah exaclty
😭
rare kernel
Do u know what to do ?
fervent helm
just a sec let me think
rare kernel
Alr alr take ur time
fervent helm
I can integrate both sides by multiplying f'(x)
zenith jacinth
not relevant but just want to say you have based mutual servers
rare kernel
I can integrate both sides by multiplying f'(x)
Yep
fervent helm
Alr alr take ur time
do you have a better way to solve this question?
zenith jacinth
guys I am sorry this question was wrong. it should be f''(x) = f(x)
NOW it makes sense xDD
,wolf y'' - y = 0, y(0) = 0, y'(0) = 3
woven radishBOT
fervent helm
NOW it makes sense xDD
not my fault dawg this is a jee pyq and they printed this shi wrong. I was stuck on this for 15 minutes 😭
zenith jacinth
not my fault dawg this is a jee pyq and they printed this shi wrong. I was stuck...
Thats fair xD
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tawdry meteor
Need help with trig proofs!
devout snowBOT
Need help with trig proofs!
tawdry meteor
I’m only allowed to use these properties
abstract zodiac
Need help with trig proofs!
In the first step take a common numerator of the two terms on left
tawdry meteor
If you look down the page- I did that
abstract zodiac
That’s overcomplicating the term, you need to do it on the first step and you get something much simpler
winter patrol
what's going on here
tawdry meteor
Well then how do you suggest I do that?
tawdry meteor
what's going on here
I don’t- even know- I don’t think the person helping me knew what they were doing
Let me erase
abstract zodiac
Well then how do you suggest I do that?
After first step you should have something like
( (cosx)^2 + (1 + sinx)^2 ) / (cosx + cosx.sinx)
I wrote X instead of theta as I’m on phone it’s kinda hard to type
Simplify the numerator what do you get ?
winter patrol
if there was an = intended in between, the multiplication wasn't done correctly
(you'd want to distribute to all terms, not just whatever is closest)
the flow of work is also horrid if you don't clearly indicate your doing one component at a time / and which component you're working on.
anyway for the most efficient approach,
you'd want to use conjugates/apply pythagorean trig identity on the first fraction
i.e. multiply that first fraction by (1-sin(t))/(1-sin(t))
a lot of these types of trig proof questions involve conjugates / pythaogrean trig identites in some way
spice token
Need help with trig proofs!
nah bro you think too hard
tawdry meteor
After first step you should have something like
( (cosx)^2 + (1 + sinx)^2 ) / (...
Can you put this in the whatever program?
spice token
convert it to o, a, h(opposite,adjacent,hypotenuse) can solve it
tawdry meteor
I can’t use the pythag identity because it isn’t squared
winter patrol
I can’t use the pythag identity because it isn’t squared
you can use conjugates
to get something so that you can apply the pythag identity
try:
$$LHS = \frac{\cos(\theta)}{1+\sin(\theta)}\red{\cdot \frac{1-\sin(\theta)}{1-\sin(\theta)}} + \frac{1+\sin(\theta)}{\cos(\theta)}$$
tawdry meteor
THATS WHAT I TOLD THEM IN THE FIRST PLACEEEEE
woven radishBOT
ραμOmeganato5
tawdry meteor
I got gaslighted out of the easy way
winter patrol
btw, don't expand the cos(t) * (1-sin(t))
tawdry meteor
**ραμOmeganato5**
Wait how does that just become sin on top and just cosine on bottom
abstract zodiac
Someone gotta do it , I’m not good with latex
Next steps would look something like
((cosx)^2 + 1 + (sinx)^2 + 2sinx) / (cosx (1 + sinx))
See the Pythagorean identity in the numerator? Now it becomes:
(2 + 2sinx) / (cosx (1 + sinx))
Take the 2 out of it
2 (1 + sinx) / cosx (1 + sinx)
2 / cosx
2 secx
Done
Other methods might be lengthier
winter patrol
Wait how does that just become sin on top and just cosine on bottom
take it one step at a time,
the denom is a difference of two squares identity,
what do you get when you multiply those?>
tawdry meteor
Someone gotta do it , I’m not good with latex
Next steps would look something l...
Can you write this down on paper?
tawdry meteor
take it one step at a time,
the denom is a difference of two squares identity,
w...
I don’t think I can use that identity-
winter patrol
why not?
tawdry meteor
The sheet I sent is all I’m allowed to use
winter patrol
those are the trig identities you're allowed
tawdry meteor
Yes
abstract zodiac
Can you write this down on paper?
Broo, I’m in a class right now, 😭, it was already hard for me to type this out
winter patrol
this is a basic algebraic identity
tawdry meteor
Broo, I’m in a class right now, 😭, it was already hard for me to type this out
IM SORRYYYY
winter patrol
the end result of simplification after distributive property
you shouldmost definitely be allowed to apply that
tawdry meteor
Wouldn’t it be just…. Sin^2x on bottom and 1- sinxcosx on top?
winter patrol
no
tawdry meteor
Explain how
tawdry meteor
By multiplying….
winter patrol
how exactly are you multiplying
spice token
ans
tawdry meteor
how exactly are you multiplying
Straight across…
winter patrol
wdym by straight across
tawdry meteor
Cos • 1-sinx
winter patrol
can you show the exact calculations you're performing
tawdry meteor
1+sinx • 1-sinx
winter patrol
missing ()
tawdry meteor
Huh….
spice token
overcomplicating
im not good at trigonometry....
winter patrol
cos(x) * **(**1 - sin (x) )
the cos(x) distributes to both terms
not whichever one you feel like
abstract zodiac
Lets see , I told gpt to express it in latex
After first step you should have something like:
$$
\frac{\cos^2 x + (1 + \sin x)^2}{\cos x + \cos x \sin x}
$$
Next step:
$$
\frac{\cos^2 x + 1 + \sin^2 x + 2\sin x}{\cos x (1 + \sin x)}
$$
Using the Pythagorean identity:
$$
\sin^2 x + \cos^2 x = 1
$$
So the numerator becomes:
$$
\frac{2 + 2\sin x}{\cos x (1 + \sin x)}
$$
Factor out 2:
$$
\frac{2(1 + \sin x)}{\cos x (1 + \sin x)}
$$
Cancel (1 + \sin x):
$$
\frac{2}{\cos x}
$$
Final answer:
$$
2\sec x
$$
woven radishBOT
K1ng_ftw
tawdry meteor
im not good at trigonometry....
No offense but- why would you offer to help if you weren’t…good at it?
winter patrol
also I mentioned earlier that you don'treally want to distibute in the numerator
that works against you, you'll see after expanding the denominator properly
tawdry meteor
**K1ng\_ftw**
How did you- get the random extra cosines?
winter patrol
focusing on the denominator
$$(1 - \sin(\theta))(1 + \sin(\theta))$$
woven radishBOT
ραμOmeganato5
warm ibex
nothing is ever random in math
abstract zodiac
How did you- get the random extra cosines?
It’s when you take the common denominator
winter patrol
are you familiar with what
$$(a-b)(a+b)$$
expands out to?
woven radishBOT
ραμOmeganato5
abstract zodiac
By Cross multiplying
spice token
No offense but- why would you offer to help if you weren’t…good at it?
i dont need to be gauss for this question,
winter patrol
cross multiply not quite the correct term,
what you're describing would be butterfly
which i actually hate. as in many cases reduces efficiency.
nimble dragon
cross multiply not quite the correct term,
what you're describing would be butte...
Could you give an example?
Ok I see what you’re saying nvm
If gcd of numbers in denominator isn’t 1
You’ll be doing unnecessary multiplications
winter patrol
in many cases you'd ultimately end up applying it,
but it shouldn't be something you do immediately/blindly
devout snowBOT
@tawdry meteor Has your question been resolved?
tawdry meteor
I’m..a little lost
But im eating so give me 2 seconds to like
Finish and lock in
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viral kernel
but in the study of maximums and minimums, what is the horizontal tangent?
viral kernel
what f'(x)=0 mean
ebon coyote
An equation to which the answers are the x-coordinates of all the points on the graph where the tangent is horizontal
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final scarab
what would the correct answer here be? I keep getting 3/4 but its 3/8 apparently?
devout snowBOT
what would the correct answer here be? I keep getting 3/4 but its 3/8 apparently...
final scarab
My solution assumed that when the man lies he randomly picks one of the 5 remaining numbers adding a factor of 1/5 somewhere. The official solution didnt have this factor
fluid stag
oh but the thing is you know he's reporting a 6
so he can't be picking numbers
he already picked a 6
so it's either he's telling the truth and it IS a 6, or he's a damn liar and the die is not showing a 6.
it's not that he's telling you some other number. he's telling you he saw a 6.
so your attention should be whether the die is showing some other value, not whether the guy lied by picking another number to say
devout snowBOT
@final scarab Has your question been resolved?
restive river
what would the correct answer here be? I keep getting 3/4 but its 3/8 apparently...
Do you know the formula for two independent events?
restive river
he has to pick which lie to tell though
You're using logical thinking?
restive river
it's not a straightforward question
well using formula it is just that easy
Oh wait
I am wronglol
sorry for misunderstanding the question oof
topaz axle
i'm getting 18/43
i miscalculated
3/4 is right lol
sometimes he gets a 6 and decides to say that, p=0.125
sometimes he gets non 6, decides to lie, and then decides to say 6
,calc (5/6)(1/4)(1/5)
woven radishBOT
Result:
0.041666666666667
,calc 0.125 / (0.125+0.041666666666667)
woven radishBOT
Result:
0.75
it makes sense to say 6 is a standard lie, like it's the best number so why lie something else
but it's not in the text
serene forge
Hihi why can you scratch these?
devout snowBOT
Hihi why can you scratch these?
serene forge
√3
green crypt
you mean cancelling the sqroot 3?
serene forge
you mean cancelling the sqroot 3?
Yes
hollow ice
number - same number = 0
green crypt
if i say walk 2 steps left and then 2 steps right, you're at where you started
serene forge
Oh we count root with root
serene forge
if i say walk 2 steps left and then 2 steps right, you're at where you started
Oh okayy thank youuy
serene forge
number - same number = 0
Oh okayy thank youu
devout snowBOT
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orchid wasp
To find rref fast do u always make first column 1 0 0 first and then u keep going left to right column doing that
devout snowBOT
To find rref fast do u always make first column 1 0 0 first and then u keep goin...
orchid wasp
For linear algebra
green crypt
you want the pivots in the successive rows to be right of the previous row's pivot (need to be 1)
god i worded that so horribly
orchid wasp
Yeah ik but how u do that fast
Cus after I get first column as 1 0 0 if i do row operations on second column to get 0 1 0 it messes up the first column
brave plaza
after you got first column as 1 0 0
the second row starts with 0 a1, a2, ...
so if say a1 != 0 you can kill all below
and you can continue doing this
if that makes sense
orchid wasp
the second row starts with 0 a1, a2, ...
So for each column I should start at the top and make it the needed 0 or 1 first?
ionic harness
the second row starts with 0 a1, a2, ...
(The fact that your first column has 0's should mean working in he second cannot mess up the first, unless I suppose you try and use the first row which you shouldn't)
brave plaza
So for each column I should start at the top and make it the needed 0 or 1 first...
you start one row lower each time
orchid wasp
<:this:489575168314966026> (The fact that your first column has 0's should mean ...
This is what I did so what should I have done
brave plaza
look at the second row you have 0 1 12 39
orchid wasp
you start one row lower each time
So I always start column form left to right, and then row where the 1 should be?
brave plaza
you can kill the 16 below
orchid wasp
you can kill the 16 below
How
brave plaza
R3 = R3 - 16R2
orchid wasp
R3 = R3 - 16R2
What if I continued with mine so I need to restart column 1 again but that would change column 2s 0s too
orchid wasp
R3 = R3 - 16R2
And these numbers r too big cus I can’t use a calculator during the exam
brave plaza
you cant multiply 16 * 39?
orchid wasp
I can but then u gotta do those to a buncha numbers later on and its gonna time
brave plaza
What if I continued with mine so I need to restart column 1 again but that would...
idk what you mean by this
orchid wasp
idk what you mean by this
If I continued with what I have to the right side of the pic
brave plaza
If I continued with what I have to the right side of the pic
you can try if you want, but this is not what you are supposed to do and it will likely take more time than multiplying some two digit numbers
orchid wasp
you can try if you want, but this is not what you are supposed to do and it will...
Yeah so I’m asking what’s the best and fastest way to do it
brave plaza
I told you
orchid wasp
No I meant so I go from columns left to right always and in each column start from the row where there needs to be a 1?
brave plaza
this is the idea yes
devout snowBOT
@orchid wasp Has your question been resolved?
silver viper
this is an sat question how do i do this using desmos 😢
Both of these lines r the same so when simplified
y= -7x/6 + 5/6
how can i find which one it is using regression
silver viper
they the exact same
infnite solutions
bro can u pls help 😭
what r u on about
show me masta
oops
i did
y= -7t/6 + 5/6
ill use t
because r is radius
mortal tendon
for every x=t, y=-7t/6+5/6
(t,-7t/6+5/6) is a family of infinite solutions
mortal tendon
because r is radius
t is time, cant use that for position
silver viper
t is time, cant use that for position
im using desmos
this is a sat question
thats why i refrained from using r
💀
who hired u guys
mortal tendon
he's a professional
pure cedar
with no thx to both of you 😭
ouchie
mortal tendon
or set y=r and solve for x just like he said
pure cedar
i wrote all coordinates separate until it hit the graph
you wont get that much time in SAT dude
mortal tendon
wouldve saved you the time
silver viper
you wont get that much time in SAT dude
wth do i do then bro
mortal tendon
wth do i do then bro
literally what he said 😭
plot (t, that thing) it'll land on the line
silver viper
literally what he said 😭
idk what ur saying
pure cedar
wth do i do then bro
simple observation, the lines are coincidental, hence they are true for every x and y
silver viper
plot (t, that thing) it'll land on the line
i need to find r
mortal tendon
not even plot, its just an equation
mortal tendon
i need to find r
just like how t,-7t/6+5/6 is a family of infinite solutions, so is -6r/7+5/7,r
silver viper
just like how t,-7t/6+5/6 is a family of infinite solutions, so is -6r/7+5/7,r
oh shit
mortal tendon
idk what u mean by not plotting
for some y=r, there will exist an x satisfying that line
silver viper
how dyk that tho
mortal tendon
because the line exists
its a given line
r is a random parameter it shouldn't affect anything
if there is an x=r on the line
theres a y satisfying the equation of the line
same goes for if theres a y=r
silver viper
r is a random parameter it shouldn't affect anything
yeah ur right i just tested
wow
mortal tendon
option 4 is basically setting y=r and solving for x
x came from the equation itself, it should satisfy the line
silver viper
option 4 is basically setting y=r and solving for x
as in r = -7/6x + 5/6 ?
mortal tendon
yes thats a relation between r and x
plug in some r and you'll get some x
you can algebraically get x to be -6r/7+5/7
multiply by 6, subtract 5, divide by -7
mortal tendon
💔
we have the equation of the line
we know that if we plug in a y, we'll get an x
like for y=1, x would be -1/7
silver viper
ima just stick to this cuz i dont get what ur saying without like visualizing
mortal tendon
for y=2, x would be -1
but we dont want a particular value of y or x
we want a general value
r is a general parameter
silver viper
yeah theres no point i dont get this 😭
mortal tendon
🥀
devout snowBOT
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modest minnow
Is this like a bot channel?
devout snowBOT
Is this like a bot channel?
modest minnow
.close
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mint cairn
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junior flax
There has to be a typo with the transformation labels on this
devout snowBOT
There has to be a typo with the transformation labels on this
junior flax
can somebody give a second opinion so I don't feel like I'm tripping, thanks
?
this is the offical lecture notes given out so I want to confirm there isn't a super strange reason why the above statement is actually true or that there's weird notational quirks that I'm unfamiliar with
acoustic leaf
how do the notes define what the basis labels mean?
junior flax
how do the notes define what the basis labels mean?
$\alpha,\beta,\gamma$ are ordered basis for V, W, Z respectively and the bottom one shows the domain of the transformation and top shows the codomain
acoustic leaf
\gamma
woven radishBOT
greaterthan.333
junior flax
so T would be a transformation from $\beta \rightarrow \gamma$
woven radishBOT
greaterthan.333
acoustic leaf
hmm yeah the transformation labels seem to be swapped between U and T
junior flax
Yeah
I was sus of it so I had chat double check it, but I don't trust chat enough
thanks guys,
I'ma email my prof this
.close
devout snowBOT
void fox
a choice of basis B = (b1, b2, ..., bn) is an isomorphism ϕ_B: F^n -> V sending e_i to b_i
the most important property of [T]^β_α is that it is the unique matrix making this diagram commute
void fox
this is the offical lecture notes given out so I want to confirm there isn't a s...
you can now prove the given statement by gluing together diagrams and using uniqueness
I find it a lot clearer to think about base change this way, hope that helps
woven radishBOT
greaterthan.333
void fox
yes
junior flax
.reopen
void fox
represented canonically by a matrix
devout snowBOT
✅ Original question: #help-27 message
junior flax
would that mean the repersentation of $[T]_b^a$ in numbers would tak some $x \in F^m$ and map it to $x \in F^n$?
woven radishBOT
greaterthan.333
void fox
yes, what this means is you take some vector x in V, interpret it as a list of numbers according to your choice of basis α, and then send it to F^n ≅ W through this matrix [T]^beta_alpha
and then you reinterpret that list of numbers as a vector in W
the important property here is that [T]^beta_alpha is the unique matrix so that this translation process works
junior flax
yeah that makes sense
we're repersenting elements of V and W as a linear combination takings the coefficents to be a vector in $F^m$ and $F^n$ respectively
woven radishBOT
greaterthan.333
junior flax
so $[T]_b^a$ is really a matrix transforming this repersentation
woven radishBOT
greaterthan.333
void fox
yes
junior flax
man, naming things is difficult
void fox
lol
junior flax
like I gotta forget what numbers mean in their classic sense to understand this
Thanks for the explination
It lowkey took me from doing linalg in highschool till now to understand what a vector is. The way they're introduced just brings a lot of bias
.close
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void fox
It lowkey took me from doing linalg in highschool till now to understand what a ...
the highschool way of introducing a vector is very damaging if you do maths later on
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junior flax
the highschool way of introducing a vector is very damaging if you do maths late...
no wonder ppl always struggle with their 1st proofs course
devout snowBOT
no wonder ppl always struggle with their 1st proofs course
junior flax
.close
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thick ledge
Prove f(x) = x^2(x-1)^2(x-2)^2...(x-2023)^2 + 1 can't be factorized into 2 polynomials with integer coefficient
Im tryna prove by contradiction where F(x) = K(x)G(x), and so K(m)G(m) = 1 for m = 0 -> 2023. So K(m) and G(m) have to be 1 or -1 for m = 0 -> 2023
devout snowBOT
Prove f(x) = x^2(x-1)^2(x-2)^2...(x-2023)^2 + 1 can't be factorized into 2 polyn...
thick ledge
Prove f(x) = x^2(x-1)^2(x-2)^2...(x-2023)^2 + 1 can't be factorized into 2 polyn...
Also K(m) = G(m) for m = 0->2023
Maybe you can prove K(x) = G(x) in general
devout snowBOT
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thick ledge
Maybe you can prove K(x) = G(x) in general
<@&286206848099549185>
devout snowBOT
@thick ledge Has your question been resolved?
arctic mountain
I have mo clue how to start
devout snowBOT
I have mo clue how to start
lunar harbor
Hint: Rewrite $9$ and $27$ as powers of $3$.
woven radishBOT
Civil Service Pigeon
arctic mountain
$\frac{3^{2x}}{3^{3y}}$?
twilit comet
woven radishBOT
solomoncyj
arctic mountain
oh ok thabks
devout snowBOT
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arctic mountain
.close
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pure cedar
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pure cedar
is this differential eqn solvable
i tried taking y/x = t
but that still isnt seperable
topaz axle
wolfram solves deqs
versed juniper
wolfram isn't on ur test
stone stump
it doesnt solve this one
if it was possible to solve it you would probably be asked to do that
green crypt
hmmm
pure cedar
i can prove that fx is greater than 1 for every x greater than 1
what i dont get is how to prove the upper bound
dapper fable
bound the derivative
pure cedar
how
dapper fable
it is clear that f is increasing
pure cedar
yes
dapper fable
so f(x) >= 1 for all x
pure cedar
yes
dapper fable
1/(x^2 + f^2(x)) ≤ 1/(x^2 + 1)
dapper fable
integrate with bounds 1 to x
pure cedar
integrate with bounds 1 to x
integrate what
dapper fable
f(x) - f(1) = ∫ f'(x) dx ≤ ∫ 1/(1+x^2) dx
stone stump
$f(x)-f(1) = \int_1^x f'(x) dx$
woven radishBOT
Denascite
dapper fable
f(x) - f(1) = ∫ f'(x) dx ≤ ∫ 1/(1+x^2) dx
this integral is where the π comes from, i think
pure cedar
so you are saying , i integrate $1/x^2 + 1$ instead of solving the DE