help-33

[fossil perch]

How would I solve a problem for the exterior angle theorem?

[balmy pumice]

do you have any problem in mind or like a type of problems?

[fossil perch]

uh not at the top of my head

I just want an idea of how to do it

[balmy pumice]

i mean you just apply the theorem

[fossil perch]

I don't understand it tho

[balmy pumice]

the sum of any 2 interior angles (like <CAB and <ABC in this case) is equal to the exterior angle of the opposite angle (<BCD in this case)

[summer trench]

bruh what have u done

[balmy pumice]

this is a new account i just verified the number, idk what discord has

[balmy pumice]

want to try and solve some problems with it?

[fossil perch]

sure

[balmy pumice]

if you can solve x then you understand it

[sacred kestrel]

https://math.stackexchange.com/questions/4096423/proof-of-the-exterior-angle-theorem

[fossil perch]

would it be 49??

[balmy pumice]

yes\

[fossil perch]

ok i just wanted to know how to do it

thank you

[balmy pumice]

no problem

[fossil perch]

.close

[marsh citrus]

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[tough abyss]

can i prove this by mathematical induction?

i find it confusing since i have N and k

[marsh citrus]

@tough abyss Has your question been resolved?

[marsh citrus]

@tough abyss Has your question been resolved?

[marsh citrus]

@tough abyss Has your question been resolved?

[obtuse inlet]

YEP

SIMPLEST CASE OF K>N IS N=||1|| & K=||2||
GENERAL CASE TO ASSUME TRUE: N=||R|| & K=||R+1||
IMMEDIATE SUCCESSOR OF GENERAL CASE : ||N=R+1 & K= R+2||

ALTHOUGH POSSIBLE, YOU DONT REALLY NEED MATHEMATICAL INDUCTION FOR THIS. A BUNCH OF STATEMENTS WOULD DO

[marsh citrus]

@tough abyss Has your question been resolved?

[desert dirge]

what on earth

[marsh citrus]

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[marsh citrus]

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[carmine scroll]

hi

[drowsy breach]

hi

[carmine scroll]

i watched that this is true because

.close

[marsh citrus]

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[desert dirge]

hm

[marsh citrus]

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[fossil grail]

Quick question, in discrete mathematics, does A\B translate to (not)A(or)B? Or: ¬A∨B?

[keen tulip]

to prove (or disprove) that two sets are equal you may take an element x from the first set and prove that its in the second set
take any x that belongs to A\B, thus x belongs to A and does not belong to B
consider (not)A(or)B
x should be either in (not)A or B
x is in neither of those, so the two sets are not equal

[fossil grail]

it should equal A only?

so it's true only when B is false?

and A is true

1 0

yes?

<@&286206848099549185>

[cunning basin]

as Davit said, if x belongs to A\B, then x belongs to A and does not belong to B

[fossil grail]

right, so A is true, and B is false

for that to be true

?

[cunning basin]

yes

[fossil grail]

if it's 1:1 it's no again? 0:0 too?

[cunning basin]

we get A and not B

[fossil grail]

oh

right

[cunning basin]

?

[cunning basin]

what

[fossil grail]

oh so A∧¬B?

[cunning basin]

yes

[fossil grail]

tysm

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[hard schooner]

hi

[hard schooner]

$\lim_{x\to0}{ (\log(1+x)(\frac{1}{\sin{x}}-\floor{\frac{1}{\sin{x}}}}))}$

[elfin berry]

JoeTheLazy1
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[hard schooner]

I've been told that if a limit with is euqal to zero is multiplied by another limit which is += ∞ it's indeterminate form

I guess in this case we can rewrite the whole thing like this
log(x+1) (t-[t]) and t-[t] is between 0 and 1 so this limit is 0

is this correct?

[marsh citrus]

@hard schooner Has your question been resolved?

[hard schooner]

<@&286206848099549185>

[wet holly]

yeah can prove it with squeeze thereom

E. G. Lower bound being 0,upper bound being log(1 + x)

[hard schooner]

like this?
0<= (1/sinx - [1/sinx]) <= log(x+1)

and then the sin thing equals 0?

[wet holly]

Like 0 <= log(1 +x) (1/sinx - [1/sinx] <= log(1 + x), although now that I think about it you might need a diff bound , or like calculate the left and right handed limits separately and show they're both 0

Cus the bounds don't work when log(1 + x) is negative

but it would if you just do it separately for right handed limit and also left handed limit

Well would have to do diff bounds for left handed limit but ye

[hard schooner]

got it

ty

[wet holly]

np

[hard schooner]

I just have a little question

$\lim_{x\to0}{ (x)(\frac{1}{\sin{\frac{1}{x}}}-\floor{\frac{1}{\sin{\frac{1}{x}}}}}))}$

[elfin berry]

JoeTheLazy1
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[hard schooner]

it works for this one as well, right?

[wet holly]

yeah

[hard schooner]

I am worried about the 1/x

[wet holly]

the "x" makes up for it

Intuitively it makes the amplitude go to 0

Rigorously

U can p much do the same trick as before, where the function inside the parentheses is between 0 and 1

And use squeeze theorem on that

[hard schooner]

can we solve this buy using mac laurin series as well?

[wet holly]

hmmm idk how'd you get that to get out of the floor function tho

[hard schooner]

cuz the 1/sin(1/x) gives us a beautiful x at the end

but imma do it with squeeze theorem like the other1

thank you so much man

[wet holly]

np

[hard schooner]

.close

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[still temple]

Guys

[still temple]

Help

Me

Is it 12cm and 2.1m?

Guys

<@&286206848099549185>

.close

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[red plover]

hello so i've completed this question

however i dont understand this question

its connected to it

[paper fiber]

think about what changes about the numbers when you round them for the estimate

do they go up, down and are they in the numerator or denominator?

[red plover]

i change the denominator and

i round 8.34 to 8

which is down

and 10.21 to 10 which is also down

so would my expanation be acceptable if i said "the numbers that i change are in the denominator, and i round the numbers down"

@paper fiber

[paper fiber]

you should explain why that results in a higher value

and explicitly say the numerator stays the same

[red plover]

i dont get it

[paper fiber]

what don't you get

[red plover]

so rounding the numbers down (for the estimate) would get me a higher answer

[paper fiber]

yes

[red plover]

the numerator stays the same

so like

[paper fiber]

take this example

1/4 = 0.25

let's make the denominator smaller

1/2 = 0.5

we get a bigger number

when you round down the numbers

you are getting a smaller denominator

[red plover]

ohh

[paper fiber]

which means you are dividing 1 into a smaller amount of groups than the exact valueb

since you have less groups then the group size will be bigger

[red plover]

the numerator stays the same, and i'm rounding the numbers to the nearest significent figure, which means im rounding it down, which means the denominator is becoming smaller, which means the estimate is going to be more than the actual answer

would that get me the mark

[paper fiber]

is this an exam?

[red plover]

no

its a practise thing

that we take home to practice

[paper fiber]

okay

[red plover]

would that be correct tho

[paper fiber]

it might be best to say how you are dividing by a smaller number

the same quantity by a smaller number

that being 1

so since it isn't 'divided as much' the result is larger

[red plover]

the numerator stays the same, and i'm rounding the numbers to the nearest significent figure, which means im rounding it down, which means the denominator is becoming smaller, so im dividing the same number which is 1 by a smaller number which results in a larger answer

is that any better

[paper fiber]

that's good

as long as you understand what you are saying

[red plover]

.close

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[vestal sandal]

is there any way for me to say that if the Trace of a matrix A is 0, then said matrix A has no inverse?

[vestal sandal]

to be more precise, my problem states: Prove that if AB-BA=A, where A and B are square n x n matrices, then A has no inverse

since the whole question is using traces, i suppose that this sub-question will be no different, and ive ended up quite easily at the remark that Tr(A)=0

[whole sleet]

[1 0]
[0 -1]
Is a matrix with trace 0, but has an inverse

[vestal sandal]

yeah thats right...so i assume that i have to work with determinants, because the only way i know to prove that a matrix has no inverse is to prove that detA=0

[iron meadow]

you could also talk about its kernel or rank

[vestal sandal]

not sure what those are, havent discussed those in class yet...

[marsh peak]

Big hint: ||Move BA to the right hand side and take determinants||

[iron meadow]

is it a linear algebra class? maybe it's a class that's just purely about matrices without going into vector space ideas, dunno

[vestal sandal]

it is but its kinda strange, its linear algebra and analytical geometry but two professors teach us, and one seems to do a loooot more hours than the other

so while we are completely finished with matrices and determinants by now, heading into other stuff, in analytical geometry we've barely done anything

[vestal sandal]

thank you, ill try to work from there and ill let you know how it goes

oh yeah that went easily, if detA is different than 0 then you end up with detB=detB+1

which is clearly false

thank you bean!

and thank you to the rest who enlightened me on the matter, of course

.close

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[untold mountain]

Do all free falling objects have an initial velocity of 0 m/s ?

[whole sleet]

No

[iron meadow]

relative to what?

[untold mountain]

earth

[iron meadow]

well if you throw a ball up into the air as far as you can, it has an initial velocity leaving your hand right? but as soon as it leaves your hand, the only force acting on it is gravity (ok say you are doing this experiment in a big vacuum chamber so no air drag) so it's in freefall

or a parachuter jumping out of a plane might have some horizontal velocity simply from jumping and pushing off the plane

or a spaceship that is in orbit (which is also in freefall) has a lot of horizontal velocity, enough that it keeps completely missing the planet over and over as it is falling

does that answer your question or is there something else you're trying to express \ ask with it and i'm missing the point?

[untold mountain]

I don't see a way to answer this without assuming that its initial velocity is 0.

[iron meadow]

yeah you can use the idea that its initial downwards velocity was 0 at the beginning of its fall

in this case any horizontal velocity is irrelevant since it doesn't affect how long it takes to fall down (assuming it's small and not planetary scale like a ship 😬 )

[untold mountain]

thanks

.close

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[terse dome]

is this the correct inverse?

[mystic heath]

,w inverse of 4*(5/3)^(2-3x) - 1

[elfin berry]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by keto11

[terse dome]

any solution?

,w inverse of 2*(4)^(x) - 10

[elfin berry]

Results provided by WolframAlpha

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Upgrade to WolframAlpha Pro!

Requested by marcoo_j

[marsh citrus]

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[warped ether]

What does it mean to be closed under addition / closed under scalar?

[desert dirge]

if you add two elements of W, you get an element of W

if you multiply an element of W by some scalar from the field, then that result is also in W

[warped ether]

Ohhhhh

So closed meaning it exists in W

and if it exists outside w, it is not a subspace

[desert dirge]

means doing those operations wont give you something not in W

[warped ether]

Ohhh okay, and how about non empty?

I had notes that referred to zero vector

[iron meadow]

example: whole numbers Z are closed under addition, if you add 2 whole numbers you also get a whole number
but they are not closed under division, because for example 2/3 is not a whole number

[warped ether]

Oh ok that makes sense!

So when it says non empty

does that mean it contains a zero vector?

[iron meadow]

nonempty is describing a set, i.e. it not the empty set { } that contains no elements

so there is at least one element in that set

[warped ether]

In my text it says this

So thats why i am thrown for a loop

[iron meadow]

why is that confusing?

[warped ether]

So if there was no zero vector, it would mean it is a empty set?

[iron meadow]

the set W contains some element, therefore it is non empty

[iron meadow]

empty set is just a bag that has nothing in it

if there is literally anything in the bag, it is non empty

[warped ether]

💀 wow i kept thinking it had smt to do with zero vectors

[iron meadow]

that anything could be either the 0 vector or any element

[warped ether]

i get it now

its so straightforward i over complicated it

ty soosh

[iron meadow]

we do that a lot in math lol

[warped ether]

haha

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[winged obsidian]

does anyone have experience when it comes to N-bit shift registers (bi-directional), its not really math related but logic related

[winged obsidian]

the question is to Design a 4 bit bi-directional shift register

[marsh citrus]

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@winged obsidian Has your question been resolved?

[marsh citrus]

@winged obsidian Has your question been resolved?

[marsh citrus]

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[dry tulip]

In math currently (sorry its in french) we need to learn how to graph functions by transforming the default function into the given equation using a transformation table known as BDX AMY HAX KAY(B divides x, H adds to y, a multiplies y, k adds to y) I dont exactly know how to use this table and we HAVE to use the table in order to complete the test

[dry tulip]

(sorry for the bad handwriting)

<@&286206848099549185>

Other option is to write a full paragraph on why this is studpid and I shouldnt use that. Could someone give me some solid reasons on why not to do it?

[marsh citrus]

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[dry tulip]

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[limpid kindle]

HELP

[proud ice]

You have to say what you want

[fleet fossil]

Does this look like a help channel to you??

[marsh citrus]

@limpid kindle Has your question been resolved?

[marsh citrus]

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[vestal mist]

how can i find the antiderivative of f(x)=(5sin(t))/cos^2(t)?

[granite knot]

take u = cos(t)

[late geode]

consider
sin(x)/(cos^2(x)) = sec(x)tan(x)

[boreal rose]

Rewrite it as 5 tan(t )sec(t)

And it’s a common derivative

[late geode]

wait, why is there x and t ?

[boreal rose]

Oh yeah I should be T

Ty R

[late geode]

f(x or t)?

[boreal rose]

Oh yeah I think they messed up

I’m assuming f(t)

[marsh citrus]

@vestal mist Has your question been resolved?

[vestal mist]

Oh yeah that makes sense

[vestal mist]

F(t)

My bad lol

.close

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[cobalt yacht]

Can someone explain this (I just saw on TikTok so maybe it’s not true.)?

[cobalt yacht]

Why is it 2-

And I thought you could only bring the limit inside if the function is continuous

[opal steeple]

it's 2^- because it's approaching 2 from the left hand side

[cobalt yacht]

I’m not seeing how it’s approaching from the left hand side

[late geode]

from below (values less than 2)

[opal steeple]

like if you turned it 90 degrees you can see that as x approaches 1 it goes towards the left

i misspoke but that's what i meant

[frosty tundra]

how's the limit of h(x) as x->1 equal to 2 from the left hand side

[cobalt yacht]

Oh so like as x goes to one h(x) goes to 2 from below?

[late geode]

follow the curve

[cobalt yacht]

Like it’s slightly less than 2

[cobalt yacht]

I think I get it

Yeah

Never seen this before

Okay thanks 🙏

.close

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[gilded peak]

https://gyazo.com/a8d048d844b4796690b08463c5004b08

[marsh citrus]

@gilded peak Has your question been resolved?

[gilded peak]

<@&286206848099549185>

[wary condor]

!show

[marsh citrus]

Show your work, and if possible, explain where you are stuck.

[gilded peak]

im not sure where to start because theres no standard deviation or mean given, and i dont know how to use the probability given to calculate it

@wary condor do u have any ideas?

[wary condor]

Yeah, you should start by relating that probabilty to the CDF for a normal r.v. variable

For 2009, we are given that $P(6 < x_{2009} < 12) = 0.8$

[elfin berry]

Xenophon

[wary condor]

right?

[gilded peak]

yeah

[wary condor]

Can you write that probability using the CDF of a normal r.v.?

[gilded peak]

like Fx (x) = 0.8?

[wary condor]

It won't look exactly like that

[gilded peak]

ok let me see how use it

[wary condor]

Seen this before?

[gilded peak]

no i havnt

[wary condor]

Mhmm, ok. You should review CDFs.

what about this?

[gilded peak]

yes i have seen that one

[wary condor]

Ok, have you heard of the standard normal distribution

sometimes related to z-score

[gilded peak]

yes i have

[wary condor]

Okay

[gilded peak]

i know how to use the zscore, but i dont know how to find it here

[wary condor]

I'm going to make the assumption that the mean for each event is at the center of the given range

I think that's reasonable given the context

[gilded peak]

yeah

[wary condor]

now, using that assumption, write this in terms of the standard normal distribution

And you will be able to solve for the standard deviation

[gilded peak]

so it would mean the area between the mean and 12 and the area between 6 and the mean is 0.4 each?

[wary condor]

yes

[gilded peak]

how would i use the cdf formula here or is looking up the z scores all i need to do?

[wary condor]

Z scores are related to the CDF of the standard normal dist.

[gilded peak]

oh okay

thanks

for this , would it be able to assume the mean is 9 because its in the middle of 6 and 12?

or thats only determined after i find sd

[wary condor]

yes

I don't think the problem can be solved if you try to assume that any mean is possible

[gilded peak]

oh okay

the thing is if i assume the mean is 9 therefore a sd of 3 , then what relevance is the given probability?

[wary condor]

The sd. need not be 3 just because the mean is 9

The given probability gives you the width of the distribution

[gilded peak]

ohh right

[wary condor]

i.e. what z-score gives you an area of 0.4?

[gilded peak]

1.29

this is the standard deviation?

i mean 1.28

actually my mistake thats just the z score of 9

[wary condor]

You want the z-scores of the boundaries

[gilded peak]

would the area of the zscore of 12 be 0.9? i really dont know what to find

[wary condor]

No

[winged obsidian]

@wary condor U copy cat my name

Can u change it or na ?

[wary condor]

You want $\phi(z_{u}) - \phi(z_l) = 0.8$

[elfin berry]

Xenophon

[wary condor]

These are the cdfs I mentioned

$z_u$ comes from 12 and so forth

[elfin berry]

Xenophon

[gilded peak]

so zu is the right of 12?

zl is left or 6?

of*

[wary condor]

$z_u$ is the z score you get when x=12

[elfin berry]

Xenophon

[gilded peak]

oh okay

[wary condor]

u for upper

[wary condor]

Since the normal is symmetric, this is equivalent to $2\phi(z_u)=0.8$

[elfin berry]

Xenophon

[wary condor]

And since mean is in the middle of the range

[gilded peak]

i found the zscore of 0.4 its 0.25 right? how do i use that

[wary condor]

show your work

[gilded peak]

i just looked up 0.4 on the zscore table

[wary condor]

You read it wrong I think

You can see that the z score is at least 1.2

[gilded peak]

oh okay 1.28

https://gyazo.com/e77c4c6319fb73013372c229e715a0c4 why wouldnt this one work?

[wary condor]

I don't have context for what that table is talking about

[gilded peak]

oh alright

[wary condor]

The table I sent is from 0 to z

you might be going from -inf to z

[gilded peak]

oh i see

with 1.28 how do i go from there?

[wary condor]

What is the definition of the z variable?

z=?

[gilded peak]

(x - u )/ sd

[wary condor]

yep

[gilded peak]

i thought to plug those values but the 1.28 isnt the real z score of 12 so is it possible to do it?

[wary condor]

You know x and u

Solve $\frac{12-9}{sd}=1.28$

uh, one sec

[elfin berry]

Xenophon

[wary condor]

here

[gilded peak]

okay thanks, this works even though 1.28 is not the real zscore of 12 but the area between 12 and 9?

[wary condor]

You're confusing a few different things

The area between 12 and 9 is 0.4

The "z-score" is the argument of the standard normal cdf

So P(-1.28<z<1.28)=0.8

[gilded peak]

oh okay that makes sense

[wary condor]

This implies that $P(-1.28< (x-\mu)/\sigma^2 < 1.28 ) =0.8$

[elfin berry]

Xenophon

[wary condor]

As required by the problem

[gilded peak]

i see now

thanks

.close

[marsh citrus]

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[slate arch]

would this not be -u(t-2pi)sin(t-2pi) or am i missing something?

[marsh citrus]

@slate arch Has your question been resolved?

[marsh citrus]

@slate arch Has your question been resolved?

[marsh citrus]

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[raw tusk]

Let a∈R and f be a function defined on all real numbers. If there exists a function g, defined on all real numbers, such that f(x) - f(a) = g(x)(x-a), then f(x) is differentiable at x = a.

Is this true or false?

[quartz plank]

think about what (f(x)-f(a))/(x-a) means in the context of derivatives

[raw tusk]

well its the definition of the derivative if lim x--> a of that exists

[quartz plank]

yeah

and if g(x) is defined on all real numbers

what is it saying about that limit

because that also means g(x) is defined at a

[raw tusk]

erm

[quartz plank]

whats the last word here

[raw tusk]

exists...

[quartz plank]

yeah

[raw tusk]

er i guess the issue i had here

is this "counterexample" (or so i hope)

f(x) = 1, x ≠ 0

f(x) = 0, x = 0

g(x) = 0

if i say f(x) - f(0) = g(x)(x-0)

then i get f(x) = g(x)(x)

the right would be 0 but the left would be...?

i dont know if f(x) is 0 or 1 here

[quartz plank]

why would g(x) be 0

are you saying g(x) is always 0?

[raw tusk]

yeah

im not sure if this is a valid counterexample

[quartz plank]

it is

because f has to be continuous at least

cuz dont think about g for a second

its impossible for f to be discontinuous and differentiable at the same place

[raw tusk]

yeah

im sorry, i dont see how that helps me

[quartz plank]

well you know that any discontuous function is a counter example

so i think generally the statement is false

[raw tusk]

but i need it to satisfy both conditions for it to be a valid counterexample no?

it needs to satisfy f(x) - f(a) = g(x)(x-a) as well as being not differentiable at x = a

ah i just realized i didnt write it as an if-then statement in my original question, my apologies

[quartz plank]

wait

i think its true then

mb

if its not differentiable at a

[raw tusk]

i originally thought it was true too- but then i dont know what to make of the "counterexample" i wrote above

[quartz plank]

then lim x->a (f(x)-f(a))/(x-a) doesnt exist

so g(x) doesnt exist

so its impossible for both requirements to be true

[quartz plank]

the thing with your counter example

is that like

ok so you have (f(x)-f(0))/(x-0)=g(x)

[raw tusk]

yes

[quartz plank]

what should g(x) output when x =0

f'(0) right

[raw tusk]

yeah..?

[quartz plank]

but because its not continuous at 0

f'(0) doesnt exist

and g(x) has to have a value for all reals

and zero is a real

[raw tusk]

ah wait i just realized you're doing the contrapositive right

[quartz plank]

yeah basically

[raw tusk]

wait so if f'(a) doesnt exist

[quartz plank]

then g(a) doesnt exist

which is a true contrapositive

so we know the original statement is correct

[raw tusk]

wait im sorry how did you reach that conclusion

[quartz plank]

we have if g(x) exists on all reals st g(x)=(f(x)-f(a))/(x-a) -> f is diff at x=a (same as f'(a) exists)

with constrapositive we have

if f'(a) DNE -> there doesnt exist a function g(x) defined on all reals st g(x)=(f(x)-f(a))/(x-a)

we know g(a) is equivalent to f'(a)

[raw tusk]

yes

[quartz plank]

If g(a) DNE -> there doesnt exist a function g(x) defined on all reals

which is a true statement

cuz a is real

[quartz plank]

for this line i just substituted g(a) in for f'(a)

[raw tusk]

um

how can we do that

[quartz plank]

substitution property of equality

[raw tusk]

isnt f'(a) defined as the limit

[quartz plank]

we know g(a) = f'(a)

[raw tusk]

whereas g(a) isnt a limit

[quartz plank]

g(a) is just the value of the limit

is f'(a) Does not exist then theres no value to the limit

so theres nothing for g(a) to be

[raw tusk]

ah wait i forgot

f'(a) is just a number

[quartz plank]

yes

[raw tusk]

okay

then one more question i guess

why is the contrapositive f'(a) DNE --> there doesnt exist a function g(x)

[quartz plank]

ok

[raw tusk]

shouldnt it be f'(a) DNE --> every function g(x)?

[quartz plank]

wdym

the opposite of there existing is there not existing

[raw tusk]

because the negation of "there exists a function g(x)" would be "for all functions g(x)"...?

is that wrong

[quartz plank]

yes

The opposite of g(x) exists is g(x) does not exist

or i mean

yes that is wrong

[raw tusk]

i was taught that the negation of "there exists" should be "for every" though, why is that incorrect here

[quartz plank]

ok think about a different example

If an animal is a dog then theres a person that exists that has petted it

The contrapositive according to you would be

For every person that has petted an animal it was not a dog

That doesnt make sense

[raw tusk]

wouldnt it be "if every person did not pet an animal, then the animal was not a dog"?

like a statement such as "if there is a math class being held, then there exists a student that will not fail". Then the contrapositive would be "If every student will fail, then there is not a math class being held".

[quartz plank]

every student failing is equivalent to there not existing a student that will not fail

so yeah ig you could think about it like that

idk for me it just makes more sense to go from there exists -> there does not exist

[raw tusk]

okay okay thats fine

but what does it even mean that every function g(x) satisfies those conditions

nvm

i got it

i need to negate the conditon too

my bad

every function g(x) that is defined on all reals such that f(x) - f(a) ≠ g(x)(x-a)

ok im sorry for taking up your time

i am good now

thank you very much

[quartz plank]

np

[raw tusk]

.close

[marsh citrus]

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[lyric bay]

Hello

[lyric bay]

$\int_{0}^{\pi}\cos\left(x\right)dx$

[elfin berry]

LE SSERAFIM

[lyric bay]

How do I represent this as Riemann's sum?

[vast wing]

hi

left or right?

and do you have specific intervals

or no

[lyric bay]

wdym?

[vast wing]

for example does it tell you to make a reimanns sum of pi/2 or pi/4?

or 3 sub intervals

or 4 sub intervals

!original

[marsh citrus]

Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.

[vast wing]

hm

this seems like khan academy

[lyric bay]

Yeah it is

[vast wing]

the riemann's sum can be approximated by evaluating it for a finite number of subintervals. for example the following is a left riemann sum approximation for the definite integral above using 4 subintervals

1^2 Δx + (1 + Δx)^2 Δx + (1 + 2Δx)^2 Δx + (1 + 3Δx)^2 Δx

[lyric bay]

How

?

[vast wing]

divide the interval of integration [1, 2] into 4 subintervals of equal width Δx

do you know how to do that?

sorry i dont know what i talking about

goodbye

i tried

[lyric bay]

$\frac{\pi-0}{n}$

[elfin berry]

LE SSERAFIM

[lyric bay]

.close

[marsh citrus]

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[marsh citrus]

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[azure trench]

Don’t understand why the transpose part is there

[proper zodiac]

$||x||^2 = x^Tx$

[elfin berry]

ΣΑCu

[proper zodiac]

replace x with lambda*a + mu*b and expand

[azure trench]

I’m getting ab^t + ba^t as part

The rest I can derive except that bit

[marsh citrus]

@azure trench Has your question been resolved?

[limpid igloo]

important to note that ab^t is not the same as b^t a

you're probably getting b^t a + a^t b

[limpid igloo]

if you're using this equation

[azure trench]

Yh did it again and got this

Oh and a^T * b = b^T*a?

.close

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[viscid jetty]

I need help

[viscid jetty]

I don't understand this question

I don't know what to do

[fringe vault]

Do you remember what tangent means?

[jade edge]

Find theta and then use tangent:)

[viscid jetty]

no i don't

[viscid jetty]

How do i do that>?

[late geode]

are you familiar with what tan does in a right triangle / sohcahtoa?

[viscid jetty]

yes

opp/hyp = tan

[late geode]

no

[viscid jetty]

nvm wrong one

opp/alt= tan

[late geode]

adj, not alt

[viscid jetty]

yes my brain is funky rn

[late geode]

applying that, focussing on just the triangle,
what is tan(theta)?

[viscid jetty]

4

[late geode]

focussing on just the triangle,

[viscid jetty]

hmm

20/x

[late geode]

yes

[viscid jetty]

tan θ = 20/x

[late geode]

and that has the numerical value of 4

[viscid jetty]

ok

[late geode]

so to find x,
solve
20/x = 4

[viscid jetty]

5?

yes 5

[late geode]

yes

[viscid jetty]

thank you so much know i understand 🙂

.close

[marsh citrus]

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[azure trench]

Stuck

[silent rampart]

If A is n×n orthogonal matrix,then A(A^T)=n×n identity matrix
(A^T is the transpose of A)

Use this characteristic to prove

[azure trench]

What just pick an example and do that

I like method but they’ve done it some other way

I don’t really see how they get to the conclusion just from that

[silent rampart]

It's the same with what I mentioned above
A(A^T)=I→A and A^T are the inverse to each other
i.e. A^T=A^(-1)
Then you can prove it like the answer

orthogonal matrix is invertible so you can prove by this way

[silent rampart]

If use this to prove

det(AA^T)=det(I)
→det(A)det(A^T)=1
Because det(A)=det(A^T) for any square matrix
We can rewrite
[det(A)]²=1→det(A)=±1

[azure trench]

Oh i think i kind of get it

Oh shit

So ur saying that determinant of identity matrix is where the 1 comes from

[marsh citrus]

@azure trench Has your question been resolved?

[marsh citrus]

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[silent marten]

Hii guys

[silent marten]

I have a question in my real analysis homework , ( English is not my main languague so i used chat gpt to translate the question )

[desert dirge]

one of the first azo supported uses of gpt on here

[silent marten]

I have this question and i already showed the Lagrange Theorem

But i dont understand the next part of the question

Cause if f has a non-finite derivative at some point lets say a point c ( that is between a and b) , it means that the function is not differenciable there so you cant apply the Lagrange Theorem right?

And the same thing with the second condition , if you dont have a point (c) in the intervel a b , it means that you cant apply the theorem so you cant conclude nothing

[desert dirge]

yeah nvm, your answer is better

i concur

[silent marten]

Thank you for your attention, but still i am not 100% sure , cause did I really answer the question? Perhaps this is more a matter of interpretation, but the question asks for the conclusions that I can draw from the theorem considering these conditions, and I merely stated that it is not applicable under these conditions.

So i didnt conclude nothing

But i dont know , cause for me the question doesent even make sense in the first place

cause like we discuss the theorem cant be applied

[desert dirge]

thats pretty much it, the conditions arent fulfilled so you cant really accept the outcome of the theorem
ig its not impossible that such a derivative exists in that region, but the theorem also cant be used for it

[silent marten]

Ok thank you so much!!!

.close

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[elfin berry]

Soosh
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[marsh citrus]

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[iron meadow]

derp

.reopen

[high pine]

give an example of an integral with such a u-sub if possible

[iron meadow]

$\int\frac{dx}{x^3\sqrt{x^2-9}}$

[elfin berry]

Soosh

[main idol]

!msgdel

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[boreal rose]

!msgdel

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[high pine]

and here you propose u = 3tan(theta), right

[iron meadow]

#help-30

[marsh citrus]

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[hollow spear]

Not sure what to do after this

[rigid sequoia]

You know the integration of 1/x²+a²?

[hollow spear]

is it x^-1/-1 + a^-1/-1

[rigid sequoia]

No

Ok do you know integration of 1/(x²+1)

I dont think I asked you

[undone summit]

sorry

[hollow spear]

tan^-1(x)?

[rigid sequoia]

Yes

Then for 1/(x²+a²) put x=au

In your case a² is 25

[hollow spear]

so -5tan^-1(x+5)

.close

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[leaden spade]

I dont know what exactly this is asking, no one on chegg has been able to do it. No radian value appears to work

[oak void]

i think they want the angle of the line to the horizontal line

[leaden spade]

so the tangent and inverse tangent?

[oak void]

this angle i think

oh wait

is it asking how many of the "red" circle can fit in "blue" circle

[leaden spade]

i dont know exactly, it wants me to use inverses though

[oak void]

<@&286206848099549185>

[signal fjord]

But here do we agree that teta = Pi/3 and 2pi/3 in order to get h=0.5 ?

[oak void]

@leaden spade

[leaden spade]

I think? it might if were measuring by the unit circle

[marsh citrus]

@leaden spade Has your question been resolved?

[leaden spade]

.close

[marsh citrus]

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[marsh citrus]

@still temple Has your question been resolved?

[marsh citrus]

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[jovial cobalt]

if you have a matrix A and you multiply it by its eigen vector, why do you get the eigenvector back

[copper raven]

wdym ?

that's false for eigenvectors in general

@jovial cobalt

[jovial cobalt]

okay this is bad

i dont think my intuition of eigenvectors is good

i know how to get the stuff

but like

does it just scale a vector

[copper raven]

A just scales its eigenvectors yes

scaling != "getting the vector back" in my book

you can scale with factors which are not 1

[jovial cobalt]

also i want to use my intuition to understand why an inverse matrix shares the same eigen vectors but reciprocal eigenvalues

I also want to understand why you can just raise the eigenvalues to powers if you raise some matrix to the power

i am finding eigenvaleus difficult to measure

visualize

idk

[copper raven]

well the inverse matrix reverses the action that was done by the original matrix

so if A scales the vector x by a factor lambda, then A^-1 will scale back that vector by a factor 1/lambda

[jovial cobalt]

so for inverse its always best to think of undoing

essentually

or flipping

[copper raven]

well A^k x amounts to scaling by lambda, scaling by lambda .... k times, so in total x would be scaled by lambda * lambda .... k times : the total scaling factor becomes lambda^k

[copper raven]

yea

[copper raven]

honestly just computing the result is easier to remember in this case

[jovial cobalt]

also can you explain what the point of SVD is

[copper raven]

one way to understand SVD is that it's a way to generalize diagonalization to rectangular matrices

with singular values playing the role of the eigenvalues

it allows you to see which directions are the most important in a transformation : the greater the singular value associated to a singular vector, the more the singular vector contributes to the final result

[jovial cobalt]

also whats the importance of similar matricies

[copper raven]

you can create approximations of matrices (by only keeping the most important directions), which are in some way the best you can do

[lone heart]

I do know it's very useful in image compression, since the diagonal matrix is from largest to smallest values so you can ignore the smaller ones

[jovial cobalt]

i just know if we have A and C and A = B C B-1

but like why is that important

[copper raven]

A and C represent the same transformation in different bases

B C B^-1 is just a change of basis

[jovial cobalt]

thats another thing i dont get

a change of basis

i watched a video of it

by 3blue1brown

but i still didnt get it

[copper raven]

you want to do all linear algebra lord

[jovial cobalt]

like why are you translating the basis

[jovial cobalt]

idk my class doesnt really teach why, they just show the formula

[copper raven]

gotta find a good book/video if your class sucks then

we can help to some degree

but a full class is too high of a bar

[jovial cobalt]

ahhhhh

true

thanks for helping

[copper raven]

there's some recommendations in #books

[jovial cobalt]

basis are kinda cool

its like finding all the possible lines in some dimension, but like without the fact that you can duplicate other vectors

so its like all the unique vectors that exist

[copper raven]

one book I like is "linear algebra and learning from data" by gilbert strang

and since you want to do ML at some point it's prolly a good mix

[jovial cobalt]

nooo

im not doing ml anymore

[copper raven]

you switched quickly lol

it's still a v good book anyway

[jovial cobalt]

i quit research

i wasnt happy

[copper raven]

aight

[jovial cobalt]

i want to take more classes

that are interesting

.close

[marsh citrus]

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[timber hull]

what do i do to prove this

do i just use how its defined

like

it seems so obvious but idk how to properly prove it

[proper zodiac]

Are you using ε-N definition of a limit?

[timber hull]

uh

we use

|sn-l|

is less than epsilon

[proper zodiac]

Yeah so unpack that definition for sn and tn

And then try and find the right N to prove it for rn

[timber hull]

yeah i did that but idk how to find the right N

[proper zodiac]

Have a think about what you need N to be bigger than to get what you want

Write out in detail what you need to be true and what you know

[marsh citrus]

@timber hull Has your question been resolved?

[timber hull]

ima be honest i have no idea how to go about this

<@&286206848099549185> someone else or something more useful pls

[spark siren]

you have an N for s_n, and another N for t_n, so wihich one of this two N could you use for r_n?

[timber hull]

what makes the difference

from choosing the other

[spark siren]

i dont know what your question is. if you think it makes no difference, then you can choose any.

[timber hull]

then how does it make a difference

[spark siren]

ok. lets go back. you asked "but idk how to find the right N", so when is an N the right N and when it is the false N?

[timber hull]

the N is > epsilon?

[spark siren]

well, then you have the answer for "how does it make a difference".

[marsh citrus]

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[mental phoenix]

what is the equation of the following function in the form y = ab^x +k where the coordinates are (-2,8),(0,24),(2,72),(4,216).

[frail orbit]

,r

,rccw

[old dagger]

Sub

0,24

[mental phoenix]

a and k both need to be determined though

[old dagger]

Is that -6 asymptote a different question

[mental phoenix]

yeah, but same format though

[finite glacier]

you could just notice the pattern that every time x changes by 2, you multiply by 3

[mental phoenix]

ik. that would probably be truee, but im not sure if my teacher would dock marks for not showing work

[finite glacier]

you can subtract your 2nd equation from your 3rd and the first from the second to cancel out the k

they you just have a system of 2 euations

[mental phoenix]

k, let me try that outt

doing so gives, -16 = ab^-2 - a and -48=a-ab^2

not to sure where to go from here

[marsh citrus]

@mental phoenix Has your question been resolved?

[marsh citrus]

@mental phoenix Has your question been resolved?

[marsh citrus]

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[frail lodge]

how do you prove this?

b=ka for integer k

I want to show b=t(-a) right?

for integer k

so pick k= -t

does this work?

[hasty ruin]

thats all

[marsh citrus]

@frail lodge Has your question been resolved?

[marsh citrus]

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[spark swallow]

.reopen

how do i solve this problem?

[boreal rose]

omg

no way

holy shit

calculate $w'(t)$, and then apply the formula $v'(t) * w(t) + w'(t) * v(t)$

[elfin berry]

nosqldb

[boreal rose]

and then take a nap ig

[timber hull]

wtf

take a nap

solid advice

[marsh citrus]

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[severe island]

Sorry, what's happening here?

[mystic heath]

what topic is this?

[severe island]

i have no clue lmao

i think descriptive geometry?

I mean i don't see any symmetry

these are supposed to be cases where mean = median btw

if that gives any insights but idk anything beyond that

<@&286206848099549185> , does anyone have any clue?

[proper sleet]

maybe has to do with frequency. there are two modes (repeating values), 1 and 13, both on either side of 9 and are max min values

[marsh citrus]

@severe island Has your question been resolved?

[severe island]

could you elaborate

honestly just based on intuition this doesn't even look symmetric to me lol

[severe island]

this makes sense yeah but idk the whole idea of having 4 on one side and 3 doesn't make it symmetric?

mode is 1 and 13 on the on the let and right

but then you have 2 and 5 on the left but only 12 on the right

[proper sleet]

if there are multiple types of symmetric properties then the symmetry you mention, the count to left and the right of center, is not the only type of symmetry that can describe the set

[severe island]

okay then what type of symmetry is this

and where do i even read about this

[proper sleet]

also the medians of the modes and the median are the same 1+13/2=7 and sum of the set/count of the set =7

no reference to give you, just reasoning and definition of symmetrical distribution

[severe island]

lol

so do i just

okay i kinda can see

[severe island]

is an evenly spaced set a type of symmetry?

[proper sleet]

all of my responses should be prefaced by a big maybe because i am not certain about this question but am tryiing to think it through

is there a part to the question above the text "Two possible cases..."

[severe island]

nothing

it's a "flashcard" lol

WHen i click "when mean = median"

[severe island]

this is what you'll see

There's really nothing beyond that

[proper sleet]

i think this is a typo

did you calculate the mean and the median of the set and then check that they equal?

because they don't

[severe island]

yes they are

,w median {1,1,2,5,9,12,13,13}

[proper sleet]

you're right

[elfin berry]

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[severe island]

,w mean {1,1,2,5,9,12,13,13}

[elfin berry]

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[severe island]

Weird right lol

[proper sleet]

well that is the within the definition of a symmetric distribution

mean=median and sometime mean=median=mode as well

[severe island]

So they just constructed one?

And technically i wouldn't be able to spot just on cursory glance right?

Cuz they look like random numbers to me

[proper sleet]

they are giving you an example

[severe island]

Yeah but they're giving me two cases, evenly spaced and symmetric

it's trivial to spot an evenly spaced case

can i spot a symmetric though?

I could if it was continuous cuz it'd resemble something of a symmetric gaussian curve

If i don't require to be able to spot a set = symmetric then ig i won't worry about it too much

the fact that they mentioned "spot symmetry around 9" kinda feels like they're suggesting that a pattern exists for me to instantly spot a symmetric set (had another one popped up)

[proper sleet]

well with an evenly spaced set the mean=median always

[severe island]

Yes but i can spot ane venly spaced set right

{-2,0,2,4} okay boom evenly spaced set

now i can leverage mean = median

[proper sleet]

calculating the mean didn't take too long did it?

or the median

[severe island]

oh yeah for that question it wouldn't, but what if they give me a huge set?

[proper sleet]

excel!

[severe island]

this is in preparation of an exam lol

😭

i have 1 minute too

[proper sleet]

per question

[severe island]

yeah

or 2

[proper sleet]

i see

[severe island]

it's still easy for evenly spaced

but symmetric is not so easy cuz where's the pattern

[severe island]

even without knowing mean = median

that's why i was curious

[proper sleet]

calculator allowed?

[severe island]

also no fancy calculator

yep no lol

[proper sleet]

basic calculator?

[severe island]

it's worse than the calculator you get on windows

but somewhat same

[severe island]

basic yes

[proper sleet]

my guess is the test takers would not give you an impossible task

so it would be a set of reasonable size

[severe island]

no my issue is, they could give me the mean of a set (let's say one that's "symmetric") and ask me to find the median, but if i can't even identify it's symmetric then i wouldn't be able to give the set's median right?

[severe island]

hmmm

they're not going to ask computation heavy stuffs yeah

that's right

[proper sleet]

median is the center point of the set

do you know how to find that?

[severe island]

yeah but that'll take forever?

[proper sleet]

its the mid-value

[severe island]

like if they give me a set of even numbers between 101 and 901 i can solve for the mean

Imagine an analogous version but it's "symmetric"

{1,1,2,5,9,12,13,13,...} like 400 elements of this 😭

[proper sleet]

they wouldn't do that because its not reasonable to count up to 200 and 201

why do you think the set would be that large

[severe island]

Okay so your proposal is that the symmetric thing isn’t a big deal and it’s just giving an example and I shouldn’t worry too much about it right?

[severe island]

idk how they can construct the set

but i can see them being able to construct one

definitely a mathematician can ig

And then I’d be expected to realize it’s symmetric but I don’t see any symmetry lmao then I’m doomed

If it’s supposed to be something I shouldn’t be able to spot instantly then okay, just making sure I’m not missing anything

[proper sleet]

okay, i see like if they give a set that is constructed from a sequence

[severe island]

anything that'll give something like this ig

[proper sleet]

all the values are not explicitly listed, but you could find the center value of the set by the index of the sequence

[severe island]

wha tabout mean?

[proper sleet]

thinking you'd need every value of the set to calculate the mean if it was not given

[severe island]

okay i'll just assume this is something on the dark side and not venture more into this

the symmetric thing

[proper sleet]

or you could do sum of a series and divide by index amount would give you mean