help-43

[devout latch]

seems like it, you could do a re-check of all the condition if you wish

[strange pendant]

,w rank {{1,0,k,2},{2,3,-3,1},{1,0,1,0}} where k = 3

[boreal girder]

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[lime juniper]

okay I have to go sorry for getting confused! gl

[strange pendant]

is fine I appreciate the help

[strange pendant]

i think is cool to check with the cartesian equation of S like that, otherwise we would have had a wrong SnH2

is hard to explain via text tho, all of the ideas I am having

maybe I lack the mathematical maturity

[devout latch]

no imo it’s normal, there is a lot of moving pieces in your goulag type linalg problems

so if that’s it ima call thing is done

!cats

[compact pewter]

[strange pendant]

i appreciate the help, though i hope I can perform well in my exam

so I can get out of this goulag mate

exam was postponed to next week

also, some of this exercises are just too tricky for me

like some of the times you just get stuck and cant continue

either that, or

you find a wrong subspace that doesnt satisfy the requirements

defintely I get the goulag analogy

a friend of mines have done this course 4 times already can you imagine

is just a goulag at that point

[devout latch]

yeah your course is hard for a non-uni lvl course, tbh even my first uni LA course was more chill

but imo you gotta keep going when the day of the exam comes, personally i like to take 5 min at the start of my exams just to read the questions and order them by how quickly i think i am able to gather points on them, but my exams are 2h long so

but this is some unsolicited advice

[strange pendant]

no, I appreciate it, will definetely do it

I will be closing this now, ty for the help, is just surreal and I am just scared I will not be with you guys in the exam

but I think its part of growing up, like not having your hand being held all the time

[devout latch]

yeah imo it’s critical that you can do a mock exam on your own in ish the amount of time given

but i think you already did that

[strange pendant]

i will do that in sunday because i think the exam is on monday

but idk, just will pray to the math gods im in the zone the day of the exam and everything goes smoothly

sometimes u just get an extra hard problem

[devout latch]

good sleep can make a big difference imo, sounds dumb but it’s no joke. However, i also had times when i couldn’t sleep well before exams because of stress so imo take it as it come and continue the grind

[strange pendant]

XD yeah it happens that i just cant sleep the day before because im so stressed

life sucks

.solved

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[devout latch]

(half joking) at this point you could just read FIS and LADR tbh

[strange pendant]

fis and ladr is rigurous, also touches the infinite dimensional topic

[compact pewter]

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[devout latch]

yeah, but it would one metal way to iron out your fondations, though it’s probably not productive to start reading those one week before your exam

[devout latch]

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[strange pendant]

I will skim through them because I am just about to start second midterm

after my exam we should start covering linear transformations and idk, change of basis ig

other stuff

there are some exercises like this for subspaces but with Image of a linear transformation

[devout latch]

imo with books like that skimming them is not the most productive thing to do

but i agree that for the purpose of passing your exam, there is a lot of (for your purpose) superfluous material on space of functions

in LADR for instance

[strange pendant]

no but, I have seen some of the earlier chapters of ladr

like some of his exercises seem very similar for subspaces

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[lime juniper]

what's FIS?

[lime juniper]

oh oops

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[steady latch]

friedberg insel spence (the three authors of a linalg text)

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[urban glen]

why does D have to be true ?

[stone jackal]

The function has two critical values. One must occur on (10, 12) and (12, 14), respectively.

[eternal pulsar]

I guessed this one right on my mock 💀

[urban glen]

why does it have to occur at that two point

i dont get it

[eternal pulsar]

Function goes from decreasing to increasing, vice versa

(Respectively on those 2 intervals)

[urban glen]

oh yea but doesn't it change from dec to inc at 11 and inc to dec at 13 ?

am i trippin

[azure vault]

not exactly there

at least not necessarily

there has to be one critical value in the interval (10,12)

and one in the interval (12,24)

but now what we're sure of

[stone jackal]

Oh, I misread the first column of the table.

[urban glen]

i dont get it how do i solve 😭

[azure vault]

the idea is to use the IVT + MVT (or rolle's theorem)

[native shard]

so observe that it decreases between 10 and 11 to a minimum and increases to a maximum between 13 and 14

[azure vault]

or the minimum could be between 11 and 12

and maximum between 12 and 13

[native shard]

yea but the point is just to notice where it’s increasing/decreasing

[azure vault]

I'll be doing a rigorous justification in the meantime

Since f(10) = 5 and f(11) = 2, f(x1) = 3 for some x1 in (10,11)

So now, IVT (or Rolle) tells us f'(c1) = 0 for some c1 in (x1,12)

that's our first critical point

[urban glen]

so f'=0 when it changes from inc to dec or dec to inc it happenes around 11 and 13 so D cant be true ?

[azure vault]

D is true actually

[native shard]

cant be?

[azure vault]

since we have exactly 2 critical points

[urban glen]

i mean like f'(12) cant be zero

[azure vault]

yes that's correct, because we already found the two critical points elsewhere

Now I'm starting to wonder if D even makes sense if f'(12) isn't defined

[urban glen]

ok i get it

thanks yall 🙂

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[urban glen]

I'm confused on why it's B over A

[stone jackal]

Since f''(0) = -2, we know that the slope should be decreasing around x = 0.

[urban glen]

i thought sign of f'' is for concavity ?

[stone jackal]

$$\sum_{n = 0}^2 \frac{f^{(n)}(0)}{n!}x^n$$

[boreal girder]

@stone jackal

[livid dome]

Yeah just plug in the values to the formula and then you can differentiate answers A and B by solving for f(1)

[native shard]

find the equation of the quadratic

[urban glen]

like vertex

[prisma vapor]

you can write the polynomial in explicit form, and it has a root >1

[native shard]

both A and B are concave down quadratics but you’ll find a problem

[urban glen]

A has vertex around 0 to 1 but it shouldn't be there ?

[native shard]

do this

what do you get

[urban glen]

-2/2! x^2 = -x^2 ?

[native shard]

$f(0) + f’(0)x + \frac{f’’(0)}{2} x^2$

[boreal girder]

knief
Compile Error! Click the reaction for more information.
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[urban glen]

the full one ?

1 + 2x - x^2

[native shard]

yep

so -(x - 1)^2 + 2

vertex is at (1, 2)

but A has a vertex at like (0.5, 1)

[urban glen]

oh yea

thing thing aint even square i have to complete the square 😭

[native shard]

yea it’s light

no fractions

[urban glen]

wait does vertex eq work for non square too ?

[zealous vessel]

It's gonna be (stuff)^2 + or - some number probably

[native shard]

what

it’s vertex form

if it’s a perfect square then the vertex has a y coordinate of 0

that’s all

[urban glen]

i memorized x=b/-2a for sat so im thinking if it works for non perfect square quadratics too

[native shard]

i never even do all that

[urban glen]

ok

js complete square take two points

[native shard]

i factor out the coefficient of x^2 then i saw x^2 - 2x - 1 so -(x - 1)^2 then to get + 1 we need to add 2

[urban glen]

yea i suck at algebra

thanks for helping 🙂

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[urban glen]

for radius of convergence if we have (x-c)^n form and it asked for what value of x does it converge or diverge does the |x-c| <=R means it converge but why does it sometimes not include the ends ? is like similar to interval that we have to check the ends ?

[urban glen]

and why does stuff like |2| converge but |-2| dont ?

[native shard]

the inequality is strict

[native shard]

|2|?

[urban glen]

yea why does same abs value but different sign inside makes it not converge

[native shard]

i don’t know what you mean

[rigid perch]

the ratio test will tell you that it will definitely converge if
|x - c| < R
and will definitely diverge if
|x - c| > R
but is inconclusive about
|x - c| = R
so the endpoints of the interval may or may not converge, or one could converge and the other diverge

[urban glen]

like (x-c)^n is 3 it converge but if its -3 it doesnt converge even if it has same R

[urban glen]

yea i know this

[rigid perch]

so there's no guarantee that the endpoints would have the same behavior

[native shard]

well

if it converges absolutely

[rigid perch]

for example you could have it be an alternating series on one end and not the other

[urban glen]

like for ex we have (x-4)^n and say at x=7 it converge but why doesnt it coverge at x=1 ? isnt it the same if it comes out of | |

[native shard]

no

[rigid perch]

it depends on the specific series you are talking about

[native shard]

it’s conditionally convergent

if it’s only converging at one endpoint then the endpoint it’s converging at is conditionally convergent so at the other endpoint youll lose that

and hence diverge

[urban glen]

wdym lose it at the other endpoint 😭

[native shard]

for example

$\sum_{n = 1}^{\infty} \frac{x^n}{n}$

latex

come on

[boreal girder]

knief

[native shard]

you can show this has radius 1

but it’s only alternating at one endpoint

and since it’s conditionally convergent when x = -1 this means it will be divergent at x = 1

[urban glen]

is this a thing like if its conditionally converge at one end its gonna diverge at the other ?

[native shard]

yes

and if it absolutely converges at one end it absolutely converges at the other

[urban glen]

but why is my series here conditionally converget at x=7

[native shard]

note that if you test one of them and it diverges this means the other one diverges or is conditionally convergent

[native shard]

this isn’t even convergent at 7 so idk what you mean

3^n diverges

you’ll have to give the original series

[urban glen]

huh? the problem had some power series like that and said it converge at x=7 ?

[native shard]

yes but i’m sure the series wasn’t just (x - 4)^n

i’m sure it involved that

[urban glen]

oh dang it had a_n 😭

so cond conv at one end the other end must diverge right ?

[native shard]

yes

[urban glen]

ok

thanks for helping!

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[native shard]

you’re welcome

btw i’m not sure if you’re allowed to say that without proof on the exam

like

if it’s an frq

you might want to go through the calculation anyway

[urban glen]

yea you kinda have to show it

[native shard]

neat trick for mcq though

[urban glen]

yea thanks 🙂

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[gritty geyser]

i got (x-1)(x-2)(x-1)/ x (x-3) (x+3)

am i correct

[kind crane]

Have you tried plotting your answer in desmos

Your answer looks like it has a vertical asymptote at zero unfortunately

[inland nacelle]

because of the x in the denominator

[keen granite]

also a horizontal asymptote of 1

we expect a HA of -1

[sick shard]

,w plot [(x-1)(x-2)(x-1)]/[x(x-3)(x+3)]

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[surreal junco]

what do I do

[plain smelt]

(1/2): i would first rewrite the "1" as--

$\frac{x^2-5x-6}{x^2-5x-6}$

[boreal girder]

hello

[plain smelt]

so you can combine the terms.

[fleet frost]

Factorise the numerator and denominator and cancel any common factors

[surreal junco]

Ohhh

Thank you

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[warm maple]

Hello! 👋
What is a weighted path matrix and how to I calculate it from this table?

[kind viper]

can you show the full question

maybe that will shed some light on what this table is supposed to mean

[deft tangle]

by multiply anddivide by second row sum

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[quartz shuttle]

can someone help me with this question, what is this? im confused how they got this equation.

[plain smelt]

Hi! Could you send an image of A_1 and A_2?

Or what is A_1 and A_2?

[quartz shuttle]

i think A1 refers to Area1 and A2 means Area2

there is no image from the question

a1 is y=1 and a2 is y=x^2

[plain smelt]

$\int_{-1}^{1}1dx$

[boreal girder]

hello

[keen granite]

A=bh

height is 1

and the base is the distance from (1,0) to (-1,0)

[quartz shuttle]

i think i get it now from this

[quartz shuttle]

ooh okay

[boreal girder]

hello

$1\cdot\left(1--1\right)=h\cdot b$

[quartz shuttle]

okay, thank you

[plain smelt]

You can also get the area by this integral: $\int_{-1}^{1}\left(\left(1\right)-\left(x^{2}\right)\right)dx$

[boreal girder]

hello

[quartz shuttle]

or i can just intergrate from this right?

[plain smelt]

Yes, and you can integrate that.

[quartz shuttle]

thanks

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[storm lichen]

can we have a general formula for this

[storm lichen]

lemme ask wolfram alpha rq

,w int from 0 to inf 1/(x^2 + a^2)^b dx

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[storm lichen]

see it's dumb

[kind viper]

quite likely that you can cook up some integration by parts BS and make a recursive formula for it of some kind

[storm lichen]

dont mind the lack of a pi in the green part

[kind viper]

also wait is b meant to be

[storm lichen]

IBP?

integer

[kind viper]

IBP = integration by parts

[storm lichen]

positive

[kind viper]

why not call it n then

[storm lichen]

cause i want to call it b

it doesn't matter

how i've calculated until b = 5 is by using feynman's trick

if i had to predict the b = 6 case

it is most likely pi/a^(11) (63)/(512)

,w int from 0 to inf 1/(x^2 + a^2)^6 dx

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[storm lichen]

yeah

but i'll try ibp

[shut hare]

yes, if you are prepared to use some complex analysis then you can get a formula out quite easily

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[dusk coral]

Hello , why is (-1)^n*(x-n) derivated (-1)^n

[dusk coral]

If N is a constant

[kind viper]

the derivative of $a(x-b)$ is just $a$

[boreal girder]

Ann

[kind viper]

also whats that image though

[dusk coral]

ye now

just dont get why its (-1)^n

it doesnt klick

[kind viper]

(-1)^n is just a constant that's being multiplied by x here

[dusk coral]

for u´

[kind viper]

if u was 69(x-420) and you saw that u' = 69

would you understand

[dusk coral]

ye haha i get that but i think im missing some basic knowledge. So anything * (x-n) derivated = always anything ?

[kind viper]

no

not "anything"

the derivative of a linear function equals its slope

[dusk coral]

ok not anyting but mathematical its always for 69(x-420) derivated = u´=69 like lets say (-1+2+3+4+5)^n*(x-n) = (-1+2+3+4+5)^n

[dusk coral]

ahhhh noo

AX derivated = A of course

so (-1)^n itself is the constant multiplied with X as (x-n) and in that case its just (-1)^n

i get it now

i was just confused cause math noob

thank you very much for your help again by the waay!!!!!

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[timid dune]

How do you do 2.c) and 3.b) c)?

[short ferry]

for 2c, you can use the triangle from 2b

or a very similar triangle

instead of sides in ratio 4 / 5, have sides in ratio x

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[timid dune]

Oh right

[karmic tundra]

are u from india class 12th?

[timid dune]

No

[karmic tundra]

oh i thought that becuase its the kind of math taught there

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[urban glen]

for part (d) why does it check y(t) like what happening to parametric mean I have to check x and what for y ?

[subtle helm]

its in the first quadrant

so y > 0

if u want the particle to move to the y-axis

y value must decrease

thus

y'(t) < 0

[urban glen]

oh that make sens

and since we dont have y(t) we js check where y'(t) = 0 and crosses x-axis right ?

[subtle helm]

we have that y(t) always > 0 over that interval

[subtle helm]

yes

and the answer would be the interval t for which y'(t) < 0

[urban glen]

y'(t) < 0 means y(t) is decreasing and we getting closer to x axis right ?

[subtle helm]

yes

[urban glen]

ok

thank you 🙂

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[forest token]

Let f(x) = 2x^2-x-1 and S={n belongs to Z:|f(n)|<=800}. Then the value of summation f(n) where n belongs to S is equal to:

[forest token]

first i drew the graph and found max possible value of n is x=20 and minimum is x = -19

then i did f(x) + f(-x) = 4x^2 - 2

so the summation will be
4(summation of squares of first 19 natural numbers) - 2(19) + f(20) right?

[eternal pulsar]

Yeah

[forest token]

yeah so im getting
9880 - 38 + 779

which is not correct

[eternal pulsar]

[forest token]

its very close tho

[eternal pulsar]

$\qty(\sum_{n=1}^{19}4n^2-2)+C+f(20)$

Ok I might be stupid

Let me think

[forest token]

im off by 1

[eternal pulsar]

What the hell

OHHHHHHH

Yeah I see

plug in x=0 into f(x)+f(-x) and solve for f(0)

[forest token]

and its a numerical as well so i would have gotten -1 after doing so much calculation 😔

[boreal girder]

;(

[eternal pulsar]

Then try to find C in this equation

[forest token]

WHAT THE FUCK BRO HOW COULD I MISS THAT

😔

got it thank u

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[eternal pulsar]

Its fine bro

:D

I was wondering why you didn't sum from 0 kek

I realized it counted double then saw that

[forest token]

yeah i saw n and immediately thought natural numbers

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[compact pewter]

Please don't occupy multiple help channels.

[kind viper]

did you close prev

[forest token]

yeah it was a glitch

[forest token]

i am not getting option C

and B

[kind viper]

wdym by "not getting"

[forest token]

like not able to do

[kind viper]

not able to check?

[forest token]

yes

[kind viper]

ok

can you say the defn of strictly increasing function

[forest token]

derivative is >=0 and equality should only exist at discrete points

[kind viper]

[medium-volume incorrect buzzer]

no, forget about derivatives

strictly increasing means that for all x1, x2 in the domain you have x1 < x2 => f(x1) < f(x2)

[forest token]

oh ok

so i can have infinite right

[kind viper]

can you name at least one

from your S to your T

which satisfies this

notice btw that any strictly-increasing function is also forced to be injective

(or, in your terms, one-to-one)

[forest token]

(0.1,0)(1.1,1)(2.2,2)(3.3,3)?

[kind viper]

ok can you tell me why this is woefully incomplete

alsoareyouspacephobicorsomethinglikewhydoesyournotationnothaveanyspacesbetweencharactersatall

[forest token]

oh

[forest token]

i dont understand what u mean by naming a function..could u give an example

[kind viper]

naming a function means giving me a rule by which you assign to each point of S a point in T

[forest token]

oh well i cant because the function is not continuous

[left quiver]

How are you even able to talk about continuity here without knowing the topology on T?

[forest token]

what is topology 😔

[kind viper]

if you wanna go for that angle, it's discrete

but also we're pretending the function is real-valued (ie S -> R) except that we also know all its values lie in T

[kind viper]

don't worry about it.

[left quiver]

Ohhhh okay I see

Thank you

[kind viper]

just because a function is not continuous does not automatically mean it admits no description

unless you would cast aside even the signum function as indescribable

or the floor function

like come on

let me try to put it more bluntly

[kind viper]

this list of ordered pairs is not a function S -> T because it fails to tell me where the point 0.69 (which is in S) is supposed to go

[forest token]

oh

[left quiver]

I think maybe they might be under the assumption that it must be one-to-one to be a function. But a function just means that for any given input it gets exactly one output

Think a function is like a machine, you put something in and get something out. Different things going in might result in the same output, but not necessarily

[forest token]

nono i know what types of function there are

wait what..shouldnt every input get exactly one output?

[left quiver]

Yes, but not every output has exactly one input that goes to it

Think of the constant function, it gives the same output no matter what you stick in

[forest token]

yeah

[forest token]

no im not under this assumption

[left quiver]

So then describe a non-constant function from S to T

[forest token]

im not able to

[left quiver]

Okay I'll give one example then

[forest token]

oh i thought u meant increasing

i think i got anns point

eg in (0,1) all points in (0,1) has to be 1,2,3,4

if any 2 are the same it will become not strictly increasing

so 0 functions can be done like that?

[left quiver]

Bingo

[forest token]

ohhh okay

[left quiver]

You'd need an infinite number of points in the output to make a strictly increasing function from (0,1)

[forest token]

oh ok i think i got my mistake..i was thinking of T as an interval rather than points

i think ill be able to do it now

[left quiver]

Ahhhhh yes

[forest token]

thanks a lot

thank u@kind viper

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[left quiver]

👍

[compact pewter]

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[forest token]

i got
f(x) = 1-7x^3 when x>1
f(x) = 1-x^3 when x<=1
g(x) = -2x^3 when x>1
g(x) = 2x^3+2 when x<=1

and im getting option C as correct which is not correct

[kind viper]

wtf is an "into function"

also wow that is some dogshit typesetting

[forest token]

it is just the opposite of an onto(surjective) function

range not equal to codomain

[kind viper]

what is g btw i don't see anything named g

[forest token]

g is psi (sorry forgot to clarify)

[kind viper]

im not sure about your f

[forest token]

2f(x) +f(-x) = -x^3-3
2f(-x) + f(x) = x^3-3
4f(-x) + 2f(x) = 2x^3 - 6
3f(-x) = 3x^3-3
f(-x) =x^3-1
f(x) = -x^3 -1

oh you are right let me calculate it again

yeah i got it

thanks

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[urban glen]

I solved this by finding zeroes and chosing one with odd power cuz that's when it crosses x axis meaning there's sign change but idk if this is correct method

[stone jackal]

That is the correct method.

[plain smelt]

Yes, that's the right method. You can also make a "sign chart" with the zeroes of f''.

If f'' changes sign at a zero, then it's a point of inflection.

[urban glen]

ok thanks 🙂

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[warped oar]

Find all integer solutions to 1/x + 1/y = 1

[timid spoke]

hi

[warped oar]

[timid spoke]

interesting

[warped oar]

[timid spoke]

(2,2)

combined the factor

$\frac{1}{x}+\frac{1}{y}=1$

[boreal girder]

Jon

[timid spoke]

$\frac{x+y}{xy}=1$

[boreal girder]

Jon

[winged lion]

You can solve for y and get y = 1+1/(x-1) now ask yourself when 1/(x-1) is an integer (which means x-1 must be a divisor of 1, and only divisor of 1 is 1, so x-1 = 1 implies x = 2)

[warped oar]

[timid spoke]

$xy-x-y+1-1=0$

[calm trout]

xy - x - y + 1 = (x - 1)(y - 1) = 1

[boreal girder]

Jon

[timid spoke]

$\left(x-1\right)\left(y-1\right)=1$

[boreal girder]

Jon

[timid spoke]

because it is interger

so x-1=1 or y-1=1

x=2,y=2

[calm trout]

(-1)(-1)

[winged lion]

division by 0

[calm trout]

yeah but you gotta check it

[warped oar]

Basically told the answer, but okay.

[zealous vessel]

From here... what would be the harm in doing x+y=xy?

Oh wait im sloww

[timid spoke]

because it is a easy question, and there are not many steps

so yeaaaa

also you can guess the answer from common sense

[warped oar]

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[dawn marsh]

Could somebody show me what the complex fourier series representation looks like?

[compact pewter]

@dawn marsh Has your question been resolved?

[compact pewter]

@dawn marsh Has your question been resolved?

[kind crane]

https://proofwiki.org/wiki/Fourier_Series/Absolute_Value_of_x_over_Minus_Pi_to_Pi

Over -pi to pi instead of 1, but computations are nearly identical

[dawn marsh]

Hum, almost the same yeah

[dawn marsh]

Could you help me verify that what I did is correct

[kind crane]

You really should just show rather than ask to show

[dawn marsh]

even terms of c_n are 0

but not sure how to plug it in the series

sen stands for sine

[kind crane]

You can just write f(x) = c_0 + sum n=1 to inf ( c_n exp(in x))

If your definition of Fourier series has inπx instead of inx, then your work looks fine

Oh your interval is (-1,1) right

[dawn marsh]

that is because we are not working with the classic 2pi periodic function

yes

[kind crane]

,w int 0 to 1 x cos(n pi x) dx

[boreal girder]

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Requested by rie.mann

[dawn marsh]

have not worked with series in the complex before, so not sure how to write it in odd terms

[kind crane]

Integral of an odd function over a symmetric domain is zero

[dawn marsh]

the series

since even terms are 0

[kind crane]

You're done with all the computations, just put it together

[kind crane]

You just plug c_0 and c_n into here. I guess you could simplify c_n a little bit

[kind crane]

Where did 2 come from

(-1)^n -1 = ?

[dawn marsh]

yes

forgot a -

[kind crane]

Everything else looks right

[dawn marsh]

I think there's no other way to simplify it

[kind crane]

You can pair n with -n terms

But the rest is just algebra

[dawn marsh]

How do I convert it into its real form?

[kind crane]

What does that mean

What is "it"

[compact pewter]

@dawn marsh Has your question been resolved?

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[whole hound]

my answer keep comming as root 90

[tiny hamlet]

are u putting the numbers outside of the root back inside?

[whole hound]

yes

mb

i was doing it wrong

[tiny hamlet]

like 3sqrt2 shouldve been left as that not turned into sqrt18

okk

[whole hound]

i was taking power 1/2 common

instead of root 2 common

[tiny hamlet]

i mean power 1/2 is the same thing as root 2

but if u got the right answer np

[whole hound]

no

nv

nvm

how would u solve the question

i got the ansqwer as 11 root 2

[tiny hamlet]

thats right i found the same

i turned everything to square root too

[whole hound]

oh nicee

alr

thx then

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[strange pendant]

is triple scalar product similar to determinant? what would be the diference=

[devout owl]

what is the triple scalar product

[pine osprey]

Yeah define

[strange pendant]

scalar triple product

[pine osprey]

Like scalar product of three vectors ?

[strange pendant]

u.vxw

for u,v,w € R3

(u.v)xw

[strange pendant]

i dont think so?

[pine osprey]

It gives the volume of the parralelepiped with sides u,v,w

[strange pendant]

and volume of paralelepiped with sides u,v,w is the determinant?

[pine osprey]

And yeah it is equivalent

[strange pendant]

how does it work, do you guys have an geometric intuition behind it or no

[strange pendant]

I think I made a mistake

is w.(uxv)

or u.(vxw)

[pine osprey]

I think its just name no ?

[strange pendant]

wdym?

[pine osprey]

Nah nvm

[strange pendant]

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[quiet portal]

i mean it's an easy question, but i just realised I don't know how to answer it

[quiet portal]

how do I construct level curves of functions with only one variable?

like f(x,y) = x^2

normally, I rearrange for y, and just do it that way for different cases of z for the contour plot

[wanton ferry]

you set it equal to a constant and solve?

[quiet portal]

well, yeah, but you'd just get points.

[wanton ferry]

what points

[quiet portal]

take x^2 = 4 or something for the z value

you'd get +- 2

rather than the level curves at z = 4

[wanton ferry]

like:

x^2=c

x=+-sqrt(c)

[quiet portal]

yeah, those are points

[wanton ferry]

no, they are lines

you are working. on. a. plane

[quiet portal]

oh im tripping nvm

yeah

cuz i was lowk chilling with contour plots i just realised this was a misunderstanding

contour plots are useless af

[wanton ferry]

no

in zelda breath of the wild i cant live without them

[quiet portal]

for visualising functions they're not very good i have to say

[wanton ferry]

how else can you visualize them?

they are a good way for compressing

like between the contours for 1 and 2

[quiet portal]

idk how to read maps

[wanton ferry]

you know the function value is between 1 and 2

and more dense contour means...?

a steeper function increase

anyways bye

[quiet portal]

bye

[compact pewter]

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[tiny hamlet]

I got to the 2nd part

let me show where i got

[young raft]

looks correct

i recommend splitting this into two integrals and doing those separately

so $\int_1^4 \frac{e^\sqrt{x}}{\sqrt{x}}dx - \int_1^4 \frac{1}{\sqrt{x}}dx$

[tiny hamlet]

doing sqrtx = u from there i guess?

[boreal girder]

artemetra
Compile Error! Click the reaction for more information.
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[young raft]

yes

[tiny hamlet]

du doesnt work tho

or am i doing it wrong

[young raft]

it should

show

[tiny hamlet]

-1/4xsqrtx

-4x sqrtx du= dx

[young raft]

huh

wut

where did you get 4

and x

[tiny hamlet]

oof wait

okay sorry i got distracted and derived it twice 😭

1/2sqrtx

that makes sense

okay i think i can do it from here thankss

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[karmic stirrup]

Whats the integral of (1-cos(x))/(1+cos(x))

[karmic stirrup]

Idk where to start

Keep in mind this is a choose question and the answer has to be in a specific form

[eternal pulsar]

So show the answer choices

[karmic stirrup]

Its in arabic

[eternal pulsar]

You can start by conjugating the denominator though

[karmic stirrup]

And i dont have wifi to even send anythibg

[eternal pulsar]

That doesn’t matter

How are you accessing Discord then?

[karmic stirrup]

That sounds like a good idea

[karmic stirrup]

Its a metaphor. Pictures will take a A while to load

[eternal pulsar]

That also doesn’t matter

I don’t know what the options are

[karmic stirrup]

Alr ill try to send them

[eternal pulsar]

Hell, it could be some obscure simplification

But you can also check by differentiating each answer choice

[stoic dove]

1-cosx=2sin^2x/2
1+cosx=2cos^2x/2

[karmic stirrup]

Number 11

[stoic dove]

and then the integral becomes simple

[karmic stirrup]

Oh

[karmic stirrup]

Where did you get these from

[bleak dock]

what the hell

[stoic dove]

trig identities

[bleak dock]

I think I see the Arabic numbers 1 and 2

[karmic stirrup]

Whivh are?

[bleak dock]

yeah

[stoic dove]

cos2x=1-2sin^2x=2cos^2x-1

[bleak dock]

lmao even in like Chinese or whatever you'll see sin, cos, tan and actual numbers though

that's just freaky

[floral fjord]

lol

[stoic dove]

you didn't learn about this?

I'm pretty sure they are standard, check your book once

[floral fjord]

PEKKA

[karmic stirrup]

I di

[bleak dock]

I mean they could just have cos 2x = cos^2 x - sin^2 x

[karmic stirrup]

D

I learned them

[karmic stirrup]

But i cant seem to figure out how youve come to this

[stoic dove]

replace x with x/2

and rearrange

[karmic stirrup]

Ooooh

Thats clever

[bleak dock]

yeah so $1 - \cos(2x) = 2 \sin^2 x$: that's a rearrangement

[boreal girder]

south

[bleak dock]

ikr

[bleak dock]

alternatively, $\frac{1 - 2 \cos x + \cos^2 x}{1 - \cos^2 x}$

then it's a term by term integration given $1 - \cos^2 x = \sin^2 x$

[boreal girder]

south

[bleak dock]

oh yeah $\cot^2 x = \csc^2 x - 1$ then integrate

[boreal girder]

south

[karmic stirrup]

Idk what cot and csc are but im working on the problem

Ty tho

[bleak dock]

oh cot = 1/tan and csc = 1/sin

[karmic stirrup]

Srry i was doing smth, Its gonna be tan(1/2 x)^2

Where to now?

Ik the integral of tan(x)^2

[stoic dove]

x/2=t
dx=2dt

[karmic stirrup]

Oooooh i forgot

[karmic stirrup]

Yeah ty thats all i was missing.

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[forest token]

Find the set of real values of x satisfying the equality
[3/x] + [4/x] = 5 where [] denotes greatest integer(floor function)

[forest token]

first i wrote the greatest integer terms in terms of fractional part

so on the one side i got
(3/x) + (4/x) - 5

and on the other side i got sum of the two fractional part terms
so the range of (3/x) + (4/x) - 5 is from [0,2)

im getting the wrong answer using this method

[bleak dock]

probably notice first that x = 1.1, or a number marginally greater than 1 satisfies

cause you would get 2 + 3 = 5

[forest token]

oh ok yeah

[bleak dock]

then you have to find the largest value of x in the set

note that [3/x] and [4/x] are both decreasing functions, so their sum is also decreasing

[forest token]

yeah

[bleak dock]

you should be good from here

you'd need [3/x] and [4/x] to be lower than 2 and 3

when's the earliest that could happen?

[forest token]

x<3/2?

[bleak dock]

that's not the earliest

[forest token]

how do i find the earliest

[bleak dock]

you did 3/x = 2 right

how about the other one

[forest token]

yeah

x<4/3

[bleak dock]

yes!

[forest token]

oh so that is the maximum

and minimum will be x>1

[bleak dock]

yes, so 1 < x

[forest token]

and also does this come with an equality?

[bleak dock]

be careful cause x = 4/3 also satisfies the equality

[forest token]

yeah okay

so the set is (1,4/3]

[bleak dock]

correct

[forest token]

okay thanks a lot

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[carmine garden]

Trying to prove A(B+C)= AB+AC , where A,B,C are matrices .

[carmine garden]

Is there an elegant way, perhaps involving linear maps

[woeful schooner]

What do you have so far?

[carmine garden]

nothing really, I know how to brute force this, but don't want to

The best I can do is

actually nvm

can't do that

was thinking of multiplication as composing functions

but nvm

[flint wing]

wai

[carmine garden]

Wai

[woeful schooner]

Hint: Think about (i, j)th element of $A, B, C; a_{ij}, b_{ij}, c_{ij}$ respectively

[boreal girder]

@woeful schooner

[woeful schooner]

can you relate them to the result

in any meaningful way?

[carmine garden]

that I can yea

a_ij(b_ij+c_ij)

over F, which then distributes

[woeful schooner]

That's not correct

Matrix multiplication doesn't work like that

[carmine garden]

Yea, I'm well aware

as I said , I don't want to perform matrix multiplication here

evenr for one set of elements

want to try and use multiplication as composition

[old mica]

there’s a way to bypass matrix multiplication here, I think

[carmine garden]

oh wait

I think , I can input an arbitrary vector and see where that goes?

[old mica]

consider the action of the left hand side on some arbitrary vector v, and likewise the action of the right hand side on an arbitrary vector v

yeah

you got it

if you think of both sides as functions on column vectors, then they’ll be equal iff they output the same thing for any given input vector

[carmine garden]

yea, that's what I was going for , thanks!

[carmine garden]

sorry for rejecting your idea like that

[old mica]

this is how you’d prove the same equality when A, B, and C are linear maps

[woeful schooner]

It's fine lol

The simplest way to prove is to just expand the definition and show that they're equivalent for both

but if you don't wanna do it, that's aok

[old mica]

it’s not a bad exercise to do this via the matrix multiplication route, tbh

[woeful schooner]

Is it an exercise?

[old mica]

well, I think wai would benefit from checking it explicitly

[carmine garden]

I did that back in 12th 😭

[old mica]

fair enough

[carmine garden]

was horrible

so I have (AB+AC)(v) = AB(v)+ AC(v)

[woeful schooner]

I'll post it since this isn't the solution
$$r_{ij} = \sum_{k = 1}^n a_{ik}(b_{kj} + c_{kj}) = \sum_{k = 1}^n a_{ik}b_{kj} + \sum_{k = 1}^n a_{ik}c_{kj}$$
That's it

[boreal girder]

@woeful schooner

[carmine garden]

I similarly have $A(B+C)(v)= A(B(v)+ C(v))$ by definition, I think I'm doing something wrong though

[old mica]

yeah

[boreal girder]

wai

[old mica]

what do you think you are doing wrong?

[carmine garden]

Because now I'm forced to use distributivity,, am I not

oh righ

nvm

Let B(v)= r, let C(v)= s

A( r+s)= A(r)+ A(s) by linearity of matrices

[old mica]

indeed

[carmine garden]

cool

[slate stag]

How does the under root expression become 25?

[carmine garden]

!occupied

[compact pewter]

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).

[old mica]

you’re not using distributivity of matrix multiplication to do that

just that a matrix is a linear map

[carmine garden]

yup

Awesome !

Thanks!

As for why I chose this method. This is what axler had to say

.close

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[woeful schooner]

Fair enough lol

[carmine garden]

.reopen

[compact pewter]

✅

[carmine garden]

yea?

[woeful schooner]

No I read the paragraph and thought it was hilarious and agreed

nothing else lmao

You can close this if your doubt is solved, I don't have anything to add

.close

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[umbral basin]

Can someone draw me the Electron Flow Direktion from the capacitor, if the Switch is closed

[umbral basin]

Flow the electrons to the - from the Battery or only to the - from the capacitor

[unkempt sail]

My guy

Please translate the question

[warped oar]

ok

[umbral basin]

No answer my question not from the task itself

[compact pewter]

@umbral basin Has your question been resolved?

[grave edge]

What's your question

[compact pewter]

@umbral basin Has your question been resolved?

[umbral basin]

If the switch is closed

How do the electrons flow from the capacitor?

[unkempt sail]

The bulb dissipates heat through the resistance

When capacitor discharges

So charge flows hence current produced

[umbral basin]

.close

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[forest token]

can someone help me calculate the induced charge ?

im unable to understand this

[kind elm]

The inner most shell has a charge Q_1 on its surface

The middle shell has Q_2, but the total charge enclosed by a Gaussian surface woudl be Q_1 + Q_2

So there's - Q_1 on the inner part of the middle shell

And Q_1 + Q_2 on the outer surface

Same story for the outer shell, Q_1 + Q_2 + Q_3 rests on the surface and - (Q_1 + Q_2) on the inner surface

I think if you work with the charge densities you should get 1:3:5?

[compact pewter]

@forest token Has your question been resolved?

[forest token]

yeah correct

ohk i think i understood

thank u

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[bright bobcat]

Let $\vec{F}(x, y) = \left( \frac{-y}{4x^2 + 9y^2} \right) \vec{i} + \left( \frac{x}{4x^2 + 9y^2} \right) \vec{j}$, for $(x, y) \neq (0, 0)$.

Calculate the line integral $\oint_{\gamma} \vec{F} \cdot d\vec{r}$, where $\gamma$ is the circle of radius $2$ centered at $(0, 1)$, oriented in the clockwise direction.

[boreal girder]

pirateking0723

[bright bobcat]

Let $\Omega$ be the domain of $\vec{F}$, Then $\gamma\subset\Omega$

[boreal girder]

pirateking0723

[bright bobcat]

since $(0,0)\notin\gamma$ and $\gamma$ is a simple closed curve, then we can use green's theorem

[boreal girder]

pirateking0723

[bright bobcat]

let $\vec{F}=P\vec{i}+Q\vec{j}$. after calculating $\pdv{Q}{x}$ and $\pdv{P}{y}$ one sees that $\pdv{Q}{x}=\pdv{P}{y}$

[boreal girder]

pirateking0723

[bright bobcat]

so $\oint_{-\gamma}\vec{F}\dot\dd\vec{r}=\iint_B\pdv{Q}{x}-\pdv{P}{y}\dd x\dd y=\iint_B 0\dd x\dd y=0$ where $B$ is the region bounded by $\gamma$

[boreal girder]

pirateking0723

[bright bobcat]

the line integral here is over -γ instead of γ because γ is oriented clockwise and not counter clockwise

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[carmine garden]

So my first thought was to prove that if there exist .... then the rank of A is 1

that makes no intutive sense to me though]

[upper carbon]

what notion of rank are you using

first thing that comes to mind is to take a vector in F^n and show that it maps to a multiple of (c1…cm)

[carmine garden]

I mean both column and row rank are the same,no?

[upper carbon]

yes but what is your idea of what the rank is

like

you want intuition

so what notions do you have surrounding the rank of a matrix (or linear map)

[carmine garden]

It's the numbr of non-zero rows or columns

[upper carbon]

that is not true

the rank of (1 1; 1 1) is 1 for example

[carmine garden]

huh, I see

[upper carbon]

so you do have the idea of gaussian elimination where you get the rank of the matrix is the number of nonzero rows or columns a row-reduced matrix, which you totally could work with if you wanted (denascite’s comment from yesterday comes to mind)

if you’re willing to be more cerebral about it you could also think about the rank as the dimension of the image of A

[carmine garden]

I see, okay

[upper carbon]

the idea of this comment comes from that notion

because one would expect, that since the columns are all multiples of c, the dimension of the image should be 1

[carmine garden]

Oh, I see

yea, makes sense

Thanks!

[upper carbon]

how about the other direction?

not sure if that way was easier for you or not

[carmine garden]

I think the other direction almost follows by defn

right

[upper carbon]

how so?

[carmine garden]

Well, if the rank is 1, then all column vectors but 1 are all zero, with 1 having a 1

[upper carbon]

this is only after gaussian elimination

[carmine garden]

same for row vectors

[carmine garden]

yes

[upper carbon]

okay so why does that tell me anything about A itself?

you’ve now told me something about the rref of A

[carmine garden]

The rref of A is in all charcteristics the same as A

no?

[upper carbon]

the thing that you said here is certainly not true for all matrices A with rank 1 (same counterexample as listed above works, the rref of that is (1 1; 0 0), which does fit what you said, but (1 1; 1 1) does not have that property)

[slate sand]

rref of A is just E_k…E_1A where the E’s are elementary matrices

[carmine garden]

hmm

[slate sand]

Is that where you’re going?

[carmine garden]

no

if the rank od A is 1, then the dimension of the image is 1

right

[upper carbon]

sure yes. i think once you do it using this definition you should also do it using gaussian elimination because both solutions are pretty instructive

[carmine garden]

I'll do that soon, let me get back into my linear algebra groove

[upper carbon]

because the statement you mentioned here, despite my negative comment, is the start of a solution, i just wanted to point out that it was not actually definitionally true like you suggested earlier

[upper carbon]

okay so yes this is true

[carmine garden]

hmm, ookay

[upper carbon]

i feel like i should just let you explore a little bit tbh, there isn’t much that i can really say without saying outright what the proof is.

like you should pick a rank 1 matrix and see why its rank is 1

[carmine garden]

I feel like my brain stops working when I have to deal with matrices

[upper carbon]

so think of it like a linear map then

matrices are linear maps

the columns of a matrix tell you where the basis element (0,…,0,1,0,…,0) in F^n goes

[carmine garden]

in terms of linear maps, isn't this basically true by defn

[upper carbon]

no, because the question is about matrices

and not linear maps

like it sort of is

but you have to draw at least some connection

like you also have to find some (c_1,…,cm) and (d_1,…,d_n) and assert that the conclusion holds for A and those vectors.

[carmine garden]

let me. try again

Let the row vectors be v_1,v_2,\dots v_n

then as the rank is 1, if I take any two row vectorsm they have have to be scalar multiples of one another?

[upper carbon]

yes, but why is that true?

[carmine garden]

hmm

I guess that's a lemma to be proven

[upper carbon]

and it sounds like i am being sort of annoying here, and i am, but there’s a good reason that i am being like this right now - because this is the actual connection between the linear map F^n -> F^m and the matrix A

but okay we can stick a pin in it and just suppose that this is true.

and then prove it later,

[carmine garden]

similarly all the row vectors are scalar multiples of one another

[upper carbon]

that is also true

[upper carbon]

one thing that might be useful is giving the scalars here names, say, if v_1 is nonzero, then v_2 is some multiple x * v_1

[upper carbon]

i purposefully picked a non-suggestive name for the scalar

but you can make it more suggestive if you wish.

[carmine garden]

I really don't think I can do this proof now, I think it's better for me to move on and come bacl to this in a bit

like tomorrow?

[upper carbon]

okay sure. i’ll be around

[carmine garden]

I feel like I'll learn a lot more by studying say isomoprhims

[upper carbon]

up to you. lots of math to do in the world

not enough time to do it all

[carmine garden]

yea, idk why I'm struggliing with matrices so much though

[upper carbon]

it is extremely worthwhile imo.

nearly everyone struggles with it

because there is just so much going on

in some sense

because the notation ended up being so compact to describe what is actually a rather general thing (linear maps), there is a lot of unpacking to do every step, so it is something that you have to get used to, as well as conventions, etc.

[carmine garden]

I think I'll just go with the flow and master this when I really need it : LU factorisation and stuff related to that

[upper carbon]

i think it is worth spending time with this idea later at least, if you do decide to skip it now

because this is truly a fundamental and beautiful part of linear algebra

but you definitely can tour the other parts of LA

and come back later when you’ve built up the gumption and resolve.

what is next in the book?

[carmine garden]

Yea I mean idt axler requires mucht to do with matrices until upper triangular matrices

[carmine garden]

isomorphims and invertibility

[upper carbon]

nice

[carmine garden]

this is what's next

[upper carbon]

is this the latest edition of axler

idr

i never used the book

i have a very intense dislike for ladr