help-33

[ionic owl]

,calc 15277^2 - 4153^2

[elfin berry]

Result:

2.1613932e+8

The following error occured while calculating:
Error: Syntax error in part "\sqrt{9}" (char 1)

[ionic owl]

Where did they get 14649 from?

[sleek lake]

,calc sqrt(15277^2 - 4153^2)

[elfin berry]

Result:

14701.677455311

[sleek lake]

it's 15,227

[marsh citrus]

@ionic owl Has your question been resolved?

[marsh citrus]

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[lost hearth]

Say I have g(2x) and I derive it, does it become g'(2x) or 2g'(2x)?

[proven timber]

The latter

[marsh peak]

Also, differentiate*

[lean flame]

Can you explain the difference?

[marsh peak]

Derivation is like a proof of an equation

Showing where it came from

And differentiation is different

[lost hearth]

And it's the latter because of chain rule, correct?

[marsh peak]

Yes

[lean flame]

So "derive a function" is wrong?

For differentiating a function

[proven timber]

The verb for "to take a derivative" is "differentiate," not "derive." Although, it's a very common mistake

[marsh citrus]

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[marsh citrus]

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[eager hound]

How do I find the differential equation of this family of curves

[eager hound]

?

[marsh peak]

Even differentiating it would be enough

But don't forget to give the initial conditions

[eager hound]

it dosen't have initial conditions

but when i take the derivate

and replace the unknown

is y'=1

but it's wrong

[marsh peak]

You should have y' = ae^x = y

[eager hound]

look

I think the problem is that i don't have to take the implicit derivate right?

.close

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[ivory laurel]

Why is the supremum of the empty set -infinity

[sullen lichen]

it's most likely one of those things that when they define what supremum of a set is they realise that supremum and infimum of the emptyset doesn't make sense, so they just define it as sup(E)=infinity and inf(E)=-infinity.

[devout mauve]

its a pretty sensible definition tho

-infinity is bigger than all elements in the empty set, so it is an upper bound

and it clearly is also the least upper bound, cause there is no smaller number

[marsh citrus]

@ivory laurel Has your question been resolved?

[tight furnace]

sup is -infinity and inf is +infinity

[marsh citrus]

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[wet zenith]

i understand why D is correct, but it also says B is correct, and i dont understand why

i think it should be A and D

[opal needle]

b

[wet zenith]

huh?

My work

[tight furnace]

You could try sketching the graph

or imagining what it would look like

as an example for why it's not differentiable just because it's the integral of something

[wet zenith]

im seeing it on screen

[tight furnace]

the sign function integrates to |x|

[tight furnace]

You put constant bounds on your integral

[wet zenith]

Theres a sign function here?

[tight furnace]

so the graph you're looking at is just whatever the integral happens to be from a to b

[tight furnace]

not here it was just an example

[wet zenith]

Im using the desmos prebuilt integration stuff

[tight furnace]

do you understand why the graph shows a constant function?

[wet zenith]

But there are example where the integral of something is also differentiable like e^x is the integral of e^x, which is still differentiable?

[tight furnace]

yes

[wet zenith]

Its the same constant as the one i have got

[tight furnace]

the integral of something can be differentiable or non-differentiable

but you understand why your graph isn't the f(x) from the problem

the orange one

[wet zenith]

Nope

I do not understand

[tight furnace]

there's no x anywhere in it

it's just a definite integral

it doesn't depend on x

that's not true of f(x) from the problem

[wet zenith]

Are you talking abt inside desmos?

[tight furnace]

yes

[wet zenith]

Oh yea i fixed it now

[tight furnace]

ah

[wet zenith]

Its still the same tho, cuz the f(x) or the f(u) doesnt matter, theyre dummy variables right?

[tight furnace]

yes

why choose those bounds for the integral

[wet zenith]

U mean 0 to 3?

[tight furnace]

yes

[wet zenith]

Well thats the question

Im interested abt the f(x) in the question from x belong to (0,3]

[tight furnace]

you're right that the interval (0,3] appears in the problem description

but why take the integral on that

what's the reasoning

[wet zenith]

Look at this, the 4th last line

Since t is in the differential, therefore the bounds are from t=0 to t=x, and ive taken x belongs to [0,4), now i can divide the integrals accordingly to simplify for fractional part (t), like bounds from 0 to 1, make sure {t}=t

Allowing x belongs to [0,4), covers the case for (0,3] which is what the question wants

[tight furnace]

ok well this is no longer about the graph

[wet zenith]

I get f(x)= constant , for x belong to (0,3]

So i dont understand why solution says option B is correct

[tight furnace]

f(x) is not constant

[wet zenith]

Alright

[tight furnace]

you just end up evaluating the integral from 0 to 3

but f(x) isn't the integral evaluated from 0 to 3

f(x) is the integral evaluated from 0 to x

[wet zenith]

But isnt it of our interest that x belong to (0,3]?

[tight furnace]

it is "of interest", yes

[wet zenith]

Okay let me think

Yep i am probably wrong

Would it be like this?

I did find out a good way of evaluating f(x) where x is an integer💀

Okay i think f(x) is going to be a piecewise function @tight furnace , but i cannot justify this any other way without looking at the graph

[tight furnace]

kind of piecewise

[wet zenith]

The graph is something drawable without desmos

[tight furnace]

it depends on frac(x) and floor(x)

would be another way to write it

otherwise you get an infinite number of pieces

[wet zenith]

This is what i could write, but it was only after thinking in terms of the graph of e^{t} vs t

Not differentiable at the points where the definition changes

So B is correct

😮

[wet zenith]

How would you write it

[tight furnace]

e^(frac(x))+(e-1)floor(x) i think

[wet zenith]

but if x from [0,1), then what uve written doesnt match with what ive got, urs would be just e^x

[tight furnace]

oh

subtract one from the entire thing then

[wet zenith]

Then it does work, but how did u get this

[tight furnace]

it doesn't really matter

i knew there had to be a part that repeats every 1 and a part that steps up by the same amount every 1

the part that steps up steps up by e-1 each time

since f(1) = e-1

f(2) = 2e-2

etc

and the part that repeats is indefinite integral of e^x from 0 to frac(x)

[marsh citrus]

@wet zenith Has your question been resolved?

[wet zenith]

I see

[marsh citrus]

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[cinder galleon]

How do you calculate 4 c)

[marsh citrus]

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[cinder galleon]

.close

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[sleek lake]

it says b mod 3 = 2 and b mod 5 = 3
supposedly you can sorta combine them into one

--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|
---|----|----|----|----|----|----|----|----|----|----|
```this one you can clearly combine into 8 (mod 15)

i don't know the math for it

well i know that you can check 2+3n until it's 3(mod 5) wikipedia calls it sieving but i doubt it's the intended way

maybe it is

basically you find the first, smallest number that satisfies the conditions

ignore that

you can start with 2 because it satisfies #1
increase it by 3 until it satisfies #2, that happens when it becomes 8 after you added 3 twice

or you can start with 3 and increase by 5

then you'll find 8 after one increase

it works with any amount of conditions done in any order (but it has to be possible in the first place)

[marsh citrus]

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[compact valley]

https://gyazo.com/1b18cef340ead3b7887d96cc87a3a443

[compact valley]

how do i get this

this is my derivative

https://gyazo.com/9132a9c8820e8bb1021fa6636b8a97a4

[hollow sparrow]

do you know the sign test?

or a sign table?

[compact valley]

no

[hollow sparrow]

so basically your f'(x) is the slope at any given point right?

[compact valley]

yeah

[compact valley]

do i need a sign table?

or could i solve the top and bottom?

[hollow sparrow]

well a sign table will be easier

basically we can say that when we have f'(t) = 0, then we have a slope of 0 in our graph

well at slopes of 0, the slope potentially changes from a negative to a postive or a positive from a negative

[compact valley]

ok

[compact valley]

then what

how would u write it

[hollow sparrow]

well

find where f'(t) = 0

[compact valley]

at 4

at 2*

i mean

[compact valley]

2 is when it = 0

[hollow sparrow]

well one more

t^2 - 4 is a difference of squares

so then it would be (t - 2)(t + 2) = 0

[compact valley]

ok yes

[hollow sparrow]

so we have t = -2, and t = 2 is where our slope is 0 🙂

okay so for the sign chart

here's what you do

[compact valley]

well u already solved it for me rpetty much right

why would i need a sign chart

[hollow sparrow]

okay well then tell me, where would f(t) be increasing and decreasing here?

[compact valley]

ok

[hollow sparrow]

the point of the sign chart is that sometimes it's not so obvious where it's increasing or decreasing

[compact valley]

https://gyazo.com/10e731d69fe097f9d7c9c157e946ceb4

it should be like this

cause since ist to the power of 1 the sign will change yes

its*

[hollow sparrow]

yep perfect

so the negatives are where it's decreasing

and the positives is where it's increasing

[compact valley]

is this a sign chart?

[hollow sparrow]

yep!

[compact valley]

oh i didnt need a sign chart i just wanna know how to do this

like how would u find the critical points

or where would u start*

sorry

[hollow sparrow]

oh well the critical points is where f'(t) = 0!

so we did that 🙂

the critical points are where the slope of f(t) is 0

[compact valley]

so you always do that when the f'(t)=0?

[hollow sparrow]

yep that's the definition

[proven timber]

There's nothing critical about when f'(t) = 1 /s

[compact valley]

https://gyazo.com/36c9f106ca558440fc94f95393eb2d55

what about this

cause f'(t) never reaches 0

[hollow sparrow]

yep! So we can just pick a random point that's not at the asymptote and check out it's sign with f'(t). In this case, f'(t) = - 3/(x-1)^2, so no matter what you plug in for x, you'll get a negative value

so it's always decreasing

which, looking at the graph

make sense 🙂

[compact valley]

huhhhhhhhhhhhhhhhhh

hmmmmmmmmmmmmmmmmmmmm ok lemme recheck

[hollow sparrow]

hint the reason is because (x-1)^2 is always positive (except for where x = 1, which we can't use)

[compact valley]

ok i know that but basically a critical point is where an equation should NOT equate to zero correct

[hollow sparrow]

the original equation? No. The derivative? yes

[compact valley]

or a expression should NOT equate to zero

[hollow sparrow]

the CP is the point where f'(t) = 0

[compact valley]

i hope that words it better

[compact valley]

ok the derivative should equate to 0?

[hollow sparrow]

well there's 2 stipulations
we either say that c is a critical point of f(t) if f(c) exists and it holds one of two properties:

1. f'(c) = 0
2. f'(c) does not exist

so in your example above, we have no critical point since there's not a spot where f'(c) = 0. So the slope is always 1 sign

[compact valley]

critical points are where it changes the max and min right

or negative to positive and positive to negative

[hollow sparrow]

yes. CPs are the points on the graph where the function's rate of change is altered (+ to - or - to +) and it's also the spot where you'll have local extrema (local min and local max) if the critical point exists

[compact valley]

OHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH OK I GET IT

OHHHH

ty sensei llama

[hollow sparrow]

yeah of course!

seems like you got this 🙂

[marsh citrus]

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[still temple]

Hi i need some help with some geometry that i dont really understand

[wise jackal]

180(n-2)

that’s the formula for the total degrees of a shape’s angles

n = number of sides

and then you can add up all the angles and set it to that number

[still temple]

yes i understand that my problem is finding the x it doesnt really leads me to an answer when i try

[marsh citrus]

@still temple Has your question been resolved?

[marsh citrus]

@still temple Has your question been resolved?

[still temple]

.close

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[clear plume]

How do I go about writing 34 in a(x-r)(x-s) form? FOIL?

[clear plume]

or should i just plug into a graphing calc and then find the vertex using calc > min/max, and then insert the found vertex into factored form?

or do i need to find the x-intercepts to insert into factored form?

definitely unsure of how to solve this

[echo bough]

factorize by the things in front of the xs

inside each parenthesis

factorize by -8 in the first parenthesis

and by 5

in the second one

[clear plume]

So when I factored it I got -3[4(-2x)5(x+2)]

did i factor that correctly?

or would it be -3[8(0.5-x)5(x+2)]

or am i not doing that right at all?

[marsh citrus]

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[agile phoenix]

Hello guys

[agile phoenix]

I have a question on how to find the slope on a line

I watched this video and the guy did

Y1-y1/x1-x2

Shouldn’t it be the opposite such as y2-y1/x2-x1?

[eternal tundra]

doesnt matter

[agile phoenix]

Really?

Wouldn’t that effect the whole slope

[sly plume]

Equation of a line is y=mx+c isolate y and x coeff is slope

[woeful sluice]

doesnt

[eternal tundra]

try multiplying it with (-1)/(-1)

[agile phoenix]

Oh

[sly plume]

@agile phoenix can u tell me which lines slope are u trying to find?

[agile phoenix]

I was watching this video

And he did y1-y2

And I was taught y2 came first

[eternal tundra]

(y1-y2)/(x1-x2)
=(-1)(y1-y2)/(-1)(x1-x2)
=(-y1+y2)/(-x1+x2)
=(y2-y1)/(x2-x1)

[agile phoenix]

So it will eventually be the same

[sly plume]

That's just notation

Because if u get negative it will cancel

[agile phoenix]

Don’t matter what side side u start from

[sly plume]

Yes

[agile phoenix]

Oh thx

[agile phoenix]

By the way the quote on your bio actually hit me lol

So inspirational

[sly plume]

Lol

[agile phoenix]

.close

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[woeful sluice]

how do you find the missing value?

[sly plume]

@woeful sluice arrange in ascending order whatever is the middle term

[marsh citrus]

@woeful sluice Has your question been resolved?

[woeful sluice]

how exactly

[marsh citrus]

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[marsh citrus]

@livid maple Has your question been resolved?

[livid maple]

<@&286206848099549185>

[vast charm]

Keep track of your signs

Y int is (0,-4)

[livid maple]

Oh

[marsh citrus]

@livid maple Has your question been resolved?

[livid maple]

<@&286206848099549185>

.close

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[edgy flint]

( translated ) Study the injectivity, surjectivity and bijectivity of the functions :

[bitter dawn]

What is the problem?

[edgy flint]

can you tell me when the function f is injective?

[marsh citrus]

@edgy flint Has your question been resolved?

[steel basin]

What is injectivity property

[edgy flint]

If there's a function f:A->B then the function is injective if every x from A has a unique value y from B

the parameters confuse me

in this case i should prove that the function is strictly increasing or decreasing to show that it' s injective

but for f it is strictly increasing

[steel basin]

Use derivatives ig

[edgy flint]

2x-1<=x+a right?

that would be a condition

[edgy flint]

idk if they are useful here

[steel basin]

That isn't necessary the function is continuos

And linear

With derivative>0

So a is injective

[edgy flint]

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[cobalt sentinel]

how do i do this

[marsh citrus]

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[cobalt sentinel]

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[fierce beacon]

i dont really understand this

[fallen moss]

Which one

For this you just get the coefficient of x on both sides to the same number

And see if the constant is the same on both side

[lean flame]

I doubt

[fallen moss]

Wdym

Oh nvm I misread the question xd

Just solve for each equation and see what x equal to

[marsh citrus]

@fierce beacon Has your question been resolved?

[fierce beacon]

the prompt in general 😨

[fierce beacon]

but it says dont depend on solving

[fallen moss]

Maybe look at which ones are multiples of each other?

Like u can factor out the 3 for equation V

And u will get equation I

[fierce beacon]

im a little dumb can you give example of what factoring 3 out looks like

[marsh citrus]

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[fickle fractal]

welp

[still temple]

huh

#help-18 message

ig someone is doing the same course as you lmao

[fickle fractal]

yes, we in vc together, but we still don't get it

[still temple]

ahh ic

which part dont you get?

do you understand what an equivalence class is

[fickle fractal]

yes but in a vague setting like this, is confusing

[still temple]

so do you understand that all elements in a class [S] are all bijections of one another?

[fickle fractal]

so they link together? one-by-one?

[still temple]

yes, all elements of U in [S] can be mapped one-to one and onto elements of T in [S]

[fickle fractal]

ok so are we saying the representative set is an element in [S]_R

and we can say that because [S]_R is bijective

[still temple]

not sure what you mean there

[fickle fractal]

me neither 😭

[still temple]

Let's work with examples. Do you see why {1,2,3} and {5,6,7} are bijective sets?

[fickle fractal]

not particularly

oh so

[still temple]

$1\mapsto 5$, $2\mapsto 6$, $3\mapsto 7$

[elfin berry]

Toby

[fickle fractal]

right I see now

[still temple]

they dont even have to be numbers, it can be anything

[fickle fractal]

ok

[still temple]

so like {sheep, cow, dog} is bijective to {1,2,3}

and notice they all have 3 elements

[fickle fractal]

and sheep 1, cow 2, dog 3

yh

we saying they equal

[still temple]

yeah, they are equal under the equivalence relation

[fickle fractal]

in carnality

[still temple]

yup

[fickle fractal]

ahhhh

[still temple]

so all sets bijective to {1,2,3} have the same cardinality (namely all have 3 elemensts)

[fickle fractal]

yes

so how are we using the representative class to extend counting to infinite?

[still temple]

yeah exactly

[fickle fractal]

natural numbers?

[still temple]

it seems that you are trying to show that natural numbers have "infinite" cardinality

and to formally describe the notion of "infinite"

(countably infinite)

theres a few nice pics here https://en.wikipedia.org/wiki/Cardinality#Comparing_sets

[fickle fractal]

i've almost got this

so equiv. classes are bijective, and the cardinality is equal between the equiv. classes

[still temple]

yup, thats precisely what's happening here

(you probably meant "members of equivalent classes are bijective", where these members are sets)

[fickle fractal]

so members of equiv. classes are bijective, and the cardinality is equal between the equiv. classes

right?

[still temple]

cardinality is equal between members of the equiv. classes

equivalence classes are groups of sets

equivalence tells us things about the members of the sets

[fickle fractal]

so like this? equiv. class = {(1,1), (2,2), (3,4), (5,4)}

[still temple]

yup

[fickle fractal]

so a representative set is just {(3,4)}

as an example

[still temple]

yeah

[fickle fractal]

and we can basically add this to other rep. sets in the equiv. class to make it infinitely countable

[still temple]

yup exactly

[fickle fractal]

ahhhh this makes much more sense

I think I can solve this now

[still temple]

gl :)

[fickle fractal]

thanks :)

.close

[marsh citrus]

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[austere berry]

been stuck on this one for a bit, cant solve for a or b on their own, just doesnt feel like theres enough information, even when i know the correct answer already i just cant get to it (doing question b)

[austere berry]

heres the mark scheme, not sure how they obtained an equation without b in it

[late geode]

equate respective log and non-log components

note that ln(1) = 0

[austere berry]

like this? or am i misunderstanding or starting in the wrong place

[late geode]

set ln(81) = ln(b)

[austere berry]

could you please explain how thats possible to drop the 2a - 10?

[late geode]

since you're told b is an integer

that would be the simplest case

[austere berry]

ahh so is it that there isnt a possible scenario where:
ln(b) = ln(81) + x
wouldnt equal an integer?

wouldnt solve for b as an integer*

is there another method to solve for a first?

[late geode]

ideally you set up the equations at the same time

ln(81) = ln(b), 2a-10 = 0

[austere berry]

is that a rule with logs? wasnt aware of that

[late geode]

well you're also told that a is an integer

[austere berry]

alright, i think i got it

thanks

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[late geode]

were you doing this with a calculator

try to do it without one

what exactly were you doing with your calculator

[lucid cobalt]

wait, i will show you

[late geode]

these are nice trigonometric ratios

you shouldn't need a calculator

[late geode]

yes

yes, that'll give you one solution

[marsh citrus]

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[fading raptor]

Can someone walk me through this solution please. I don't really get it.

[fading raptor]

In particular, where exactly did the value 3.57 come from?

[marsh citrus]

@fading raptor Has your question been resolved?

[fading raptor]

<@&286206848099549185> 👋 🙏

[trim quest]

,calc sqrt(2*6371/1000)

[elfin berry]

Result:

3.5695938144276

[trim quest]

@fading raptor

[fading raptor]

But why?

[trim quest]

Are you asking where the expression $d = \sqrt{2 \times 6371 \times \frac{h}{1000}}$ came from?

[elfin berry]

tatpoj

[fading raptor]

I don't get how that equation simplifies down to 3.57*h^(1/2)

I see the calculation, but how does that get simplified down?

[trim quest]

Exponents are distributive over multiplication

So

$$\sqrt{2 \times 6371 \times \frac{h}{1000}}$$
$$=\sqrt{2} \times \sqrt{6371} \times \sqrt{\frac{1}{1000}} \times \sqrt{h}$$
$$=\sqrt{2 \times 3671 \times \frac{1}{1000}} \times \sqrt{h}$$
$$=3.57 \times \sqrt{h}$$

Apparently there's a compile error but it's still showing what I wanted so whatever

@fading raptor Does that help?

[fading raptor]

Sorry lemme read through quickly

[fading raptor]

Also pretty sure this is the compile error

[trim quest]

oh lol

[fading raptor]

If it was bugging you

[elfin berry]

tatpoj

[trim quest]

Much better

[fading raptor]

There we go

[trim quest]

Basically you can distribute exponents (including roots) over multiplication

They just evaluated the square root of all the constant factors to 3.57 and left the sqrt(h) out of it

[fading raptor]

I just don't get how it goes to *h when it's +h in the original equation

[trim quest]

Well from part (i) we had $d = \sqrt{h}\sqrt{2R+h}$

omg I suck at latex

[elfin berry]

tatpoj

[trim quest]

Then in part (ii) it says we that if $h$ is small, we can approximate $d$ by $d = \sqrt{2Rh}$

[elfin berry]

tatpoj

[trim quest]

Do you see why that approximation works?

[fading raptor]

Yes

Since it's only going to be off by the magnitude of * h^(1/2) , right?

[trim quest]

Something like that

Basically they just dropped the +h from 2R+h

[trim quest]

d = sqrt{2Rh} is the formula we use to get here

[fading raptor]

Okay

shit I get it now

I was trying to find a using the original equation

[fading raptor]

So I just need to state why it should be a somewhat accurate approximation if h is small

And then work through the approximation equation to reach 3.57*h^(1/2)

Right...?

[fading raptor]

Agh but where does the /1000 come from?

so h is in metres and d is in kilometres, I'm guessing that's why h is divided by 1000

but in that case, why isn't the h in 3.57 * h^(1/2) divided by 1000 too?

ok wait

nvm

[trim quest]

Yes, that's right

That's why h was replaced by h/1000

Once we do that, it's already taken care of, you don't need to divide by 1000 again

[trim quest]

It's the same h

[fading raptor]

Yeah, realised quickly after 🤦‍♂️

thanks 🙏

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[marsh citrus]

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[marsh citrus]

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[marsh citrus]

@hollow coyote Has your question been resolved?

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[neat plover]

Using a double-angle or half-angle formula to simplify the given expressions, If cos^2(28deg)-sin^2(28deg)=cos(Adeg), then A = _____________ degrees

[neat plover]

find A = ? degrees

[marsh citrus]

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[neat plover]

<@&286206848099549185>

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[still temple]

For a graph like this, do we say it is increasing from 0 to 60 in parenthases or brackets?

[0,60] or (0,60)

[calm anvil]

looks like its increasing from -infinity

[still temple]

I didn't mention the problem sorry but it involes the price at which something is sold and the revenue made

x is the price

and y is the revenue

[calm anvil]

if the domain is [0,120] then it would be [0,60)

Since 60 is not increasing, it's flat lined

[still temple]

so its increasing from 0 up to 60

but at 60 nothing happens

is what you are implying

[still temple]

would it be (60,120] for decreasing?

[calm anvil]

yeah

[still temple]

ahhh i see

thank you my friend

[calm anvil]

Not implying I'm stating

[still temple]

yes

yeah lol

[calm anvil]

no problem

[still temple]

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[hearty herald]

What is the power set of {2,{0,3}}

[hearty herald]

not sure how to write the 0 and 3

[hollow sparrow]

we don't consider {0, 3} separately. The power set is just the 2 elements

[hearty herald]

so { {empty set}, {2}, {0,3}} ?

[hollow sparrow]

yep!

[trim quest]

Wait

And, the set itself, right?

[hollow sparrow]

oh yes you're right

[hearty herald]

wdym

[trim quest]

{ {}, {2}, {{0,3}}, {2, {0,3}} }

[hearty herald]

ohhh ok

oh yeah i forgot the set is a subset lol thank you

[trim quest]

👍

[marsh citrus]

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[still temple]

hi

[still temple]

this is what they mean by defined inductively

[stoic saddle]

well, what is d(λ)? and for a symbol a and word w, what is d(aw)?

[marsh citrus]

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[marsh citrus]

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[still temple]

.reopen

[marsh citrus]

✅

[still temple]

Sorry i didnt see this

d(lambda) is lambda

D(aw) would be aa • d(w)

[stoic saddle]

well there you have it.

[still temple]

Oh

Is that it?

Im still a bit confused

[still temple]

How did you know that a is a symbol in the reverse definition?

[stoic saddle]

dunno, maybe it wasnt. and it isnt really important.

so long as you simply make yourself aware of what means what.

[still temple]

I see

[still temple]

do you think you could help me out with (b) as well?

im having trouble figuring out what the base case is meant to be

[stoic saddle]

the base case is the empty string

[still temple]

hmm

for reverse or double?

i rly like ur pfp btw :D

rly cute

[still temple]

rn i have (w)R = w

and i proved it using (lambda)R = lambda

because for this, the base case seems to be way more complex

[stoic saddle]

thank you

sorry, i need to disappear for a phone call.

[still temple]

ofc

no problem

might someone else be able to help?

<@&286206848099549185>

[marsh citrus]

@still temple Has your question been resolved?

[still temple]

<@&286206848099549185>

[marsh citrus]

@still temple Has your question been resolved?

[still temple]

<@&286206848099549185>

[marsh citrus]

@still temple Has your question been resolved?

[still temple]

<@&286206848099549185>

[marsh citrus]

@still temple Has your question been resolved?

[marsh citrus]

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[still temple]

what is a contour

[bitter dawn]

In complex analysis? In differential geometry?

[marsh citrus]

@still temple Has your question been resolved?

[still temple]

Both?

im not sure i couldnt find anything good online

[marsh citrus]

@still temple Has your question been resolved?

[still temple]

<@&286206848099549185>

.close

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[still temple]

Complex functions are generally supposed to have a domain that contains a nonempty open subset of the complex plane.

Its from wikipedia, what does it mean?

[bitter dawn]

When you do complex analysis

You need to be able to derivate

To derivate you need to compute a limit

To do so you need an open set surrounding your point

[still temple]

open set surrounding my point??

[bitter dawn]

Yes. Imagine you want to compute the derivative of $f$ in $z$. It is defined as $\underset{z_{0} \to z}{\lim} \frac{f(z)-f(z_{0})}{z-z_{0}}$. For it to work you need $f$ to be defined in $z_{0}$ "around" $z$ to compute the limit.

[elfin berry]

black_couscous

[still temple]

makes sense!

when is it not the case?

[bitter dawn]

Imagine $f: \mathbb{N} \longrightarrow \mathbb{C}$. It's difficult to define a notion of derivative for this

[elfin berry]

black_couscous

[still temple]

hmm

[bitter dawn]

You cannot get close to each number

So you cannot define a limit

[still temple]

hmm??

oh

[bitter dawn]

Let's take $f: n \in \mathbb{N} \longrightarrow n \in \mathbb{C}$. Can you compute $\underset{z_{0} \to z}{\lim} \frac{f(z)-f(z{0})}{z-z{0}}$?

[elfin berry]

black_couscous

[still temple]

N is just 1,2,3,4 right

so we could? but like not accurate?

im not sure

[bitter dawn]

No you cannot define a limit

Because to compute a limit you would need to take $z_{0}$ very close to $z$

[elfin berry]

black_couscous

[bitter dawn]

Here the distance is either $0$ either superior to $1$

[elfin berry]

black_couscous

[still temple]

yup

[marsh citrus]

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[hollow sun]

how do i explain that this is fals

[stoic saddle]

a counterexample ought to do

[hollow sun]

i cant seem to get one ://

[devout mauve]

dont think too complicated

what is a simple subspace in R^2 for which you can easily see orthogonal projections for example

[hollow sun]

uhh

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[charred edge]

Can someone help me understand inverse equations please

[charred edge]

🥲

[main idol]

[charred edge]

I am just asking

I don't have an example in specific, I want to know in general how to find an inverse function

Because the guides online are really confusing for my pea size high school brain

Uhhh for example 5x-4y=18 I guess

[main idol]

use rules of algebra to solve for x

https://en.wikibooks.org/wiki/Intermediate_Algebra/Algebraic_Axioms

[charred edge]

I think I'm missing something

Because I got x=(4y/5) + (18/5)

And it feels like I didn't really solve it

[main idol]

then you swap back x and y variables

y = 4x/5 + 18/5 is the inverse

[charred edge]

Okay so the inverse is the opposite of the solved equation

[main idol]

"opposite" is vague

definition of inverse of a linear function $f(x) = ax + b$ is the function $g(x) = cx + d$ such that $f(g(x)) = x$

[elfin berry]

riemann

[main idol]

So you can always check your answer that way

$f(g(x)) = a (cx + d) + b = x$

[elfin berry]

riemann

[charred edge]

So an inverse function multiplied by your original function will always give a product of x

[main idol]

so here $y = f(x) = (5x - 18)/4$

[elfin berry]

riemann

[main idol]

multiplication is not the same as composition

f(x) * g(x) != f(g(x))

composition is "plugging in"

[main idol]

if you read this carefully, it is not (ax+b) * (cx + d)

[main idol]

check the answer you got $g(x) = 4x/5 + 18/5$ is the inverse of $f(x)$

[elfin berry]

riemann

[marsh citrus]

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[hazy ruin]

To find the position vector d, why can I not subtract -2 from -1 and 1 from -2 and 5 from 2?

[marsh citrus]

@hazy ruin Has your question been resolved?

[hazy ruin]

<@&286206848099549185>

[glass perch]

If you want to go from P to Q you calculate q1 - p1, q2-p2, q3-p3

[hazy ruin]

.close

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[tight obsidian]

I don't know how to use the fact that the two vectors are independent
It's the very first week of the course so I'm a noob 🙂

[tight obsidian]

How do I start this?

[marsh citrus]

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[tight obsidian]

<@&286206848099549185>, Should I add something more?

[marsh citrus]

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[marsh citrus]

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[still temple]

Is this correct? 16 not 18 mistype

[still temple]

think I did it entirely wrong

[raven olive]

If 8+8 = YX how did you get 18

[marsh citrus]

Please don't occupy multiple help channels.

[still temple]

I meant to type 16

mistyped

[raven olive]

Also what confuses me is how do we know that the sides are equal

[still temple]

didn't realize myb

[still temple]

idk

I think its wrong

[eternal tundra]

i guess we can assume so, since there is no more information given

if they are not equal i dont think we can solve

[raven olive]

Yeah that just seems kind of wrong

[eternal tundra]

8+8=16 btw

[still temple]

Ik 😂

I mistyped

[eternal tundra]

oh yeah

didnt read haha

did 16 work?

[raven olive]

But yeah everything looks right

[still temple]

So then Just leave at this I guess?

[raven olive]

Ye

[still temple]

Alright

what about these

do they seem correct

[raven olive]

For number 3 make sure to fix your sqrts

but yeah the answers seem correct

[still temple]

ah yeah

can you help me out with this one @raven olive

How do i find AC from the given lengths

[raven olive]

Well start off by writing down what you do know

[still temple]

I would assume PC = 10 also if its equal

[raven olive]

Ye

Now just solve like normal with pythag

[marsh citrus]

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[marsh citrus]

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[vagrant minnow]

How do I cancel out the 2 variables?

[marsh citrus]

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[marsh citrus]

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[still temple]

can i get help on 7 and 8

[marsh citrus]

@still temple Has your question been resolved?

[still temple]

<@&286206848099549185>

[spark berry]

8. is just basic derivative
   Just use the power rule
   f'(x) = 6x - 5
   Substitute x = 10
   7 is just the sum and product of a quadratic equation

[marsh citrus]

@still temple Has your question been resolved?

[still temple]

oh alright

[marsh citrus]

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[marsh citrus]

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[tender latch]

i dont understand how the answer is log ((x+1)^2/3)/x^1/3(x-3)^1/3)). how does the (x-3)^1/3 get to the denominator? shouldnt it be in the numerator?

[main idol]

use log properties

[tender latch]

i did but wouldnt a/b/c mean c moves to where a is?

[main idol]

show what you tried

[late geode]

$\frac{\br{\frac ab}}{c} \neq \frac{a}{\br{\frac bc}}$

[elfin berry]

ℝamonov

[late geode]

order of operations are important

[tender latch]

srry if it isnt that clear

[late geode]

a/(b/c) = ac/b
but that isn't what you should have here

,rotate

[late geode]

it'd be clearer if you didn't use skyscraper fractions
$$=\frac13 \bigg[ \log\br{(x+1)^2 \cdot \frac 1x \cdot \frac{1}{x-3}}\bigg]$$

[elfin berry]

ℝamonov

[tender latch]

oh

yea that makes more sense now

thx

[marsh citrus]

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[uncut ferry]

https://cdn.discordapp.com/attachments/958319236722339863/1035153243858280550/IMG_8809.jpg

[uncut ferry]

how

:)

[woeful sluice]

my eyes are killing me

its purpl

e

[uncut ferry]

yea thats my room lighting

[burnt abyss]

hi

where are you stuck?

[uncut ferry]

hello

im stuck at reading the diagram and finding out the angle pair, ( corresponding, interior, alternate interior )

[burnt abyss]

ok

at a)

you are correct

4 and 5 are equal

now

[uncut ferry]

how to figure out the angle pair ?

[burnt abyss]

what kind of angles are 1 and 3?

corresponding , alternate internal, alternate external?

[uncut ferry]

alternate?

[burnt abyss]

no

they cant be alternate

if they are on the same side

[uncut ferry]

okay

would it be corresponding then

[burnt abyss]

yes

remember

2 angles cant be alternate if they are on the same side

they are only on opposite sides

ok?

[uncut ferry]

yes got it

[burnt abyss]

good

if 2 angles are corresponding, how are they?

look at the answer choises

you can look at the drawing if you don't know the answer

[uncut ferry]

im lost ngl

lmfao

[burnt abyss]

ok

look at the drawing

[uncut ferry]

yes

[burnt abyss]

at 1 and 3

how are they?

how do they look?

@uncut ferry

[uncut ferry]

theyre on the same slope

idk

[burnt abyss]

ok

i ll tell you

if 2 angles are corresponding, they are ALWAYS equal

got it?

[uncut ferry]

yes

[burnt abyss]

good

you can look at the drawing as i said

dont the angles look equal?

[uncut ferry]

yes they do

makes sense now

[burnt abyss]

nice

now to angles 3 and 5

[uncut ferry]

i completely missed this lesson so im rlly lost ty for help

[burnt abyss]

np

so

look at the drawing

https://cdn.discordapp.com/attachments/958319236722339863/1035153243858280550/IMG_8809.jpg

how many degrees do you think angle 5 has?

[uncut ferry]

ok this is where i get stumped

90?

[burnt abyss]

ye

s

does angle 3 appear to have more than 90 or less than 90 degrees?

[uncut ferry]

more

?

[burnt abyss]

yes

so

are the 2 angles equal?

[uncut ferry]

no

[burnt abyss]

good

what does supplementary mean?

[uncut ferry]

im not sure

when 2 angles add up to 180 degrees right?

[burnt abyss]

yes

can they be supplementary?

[uncut ferry]

i think for 3 and 5 i would put neither

[burnt abyss]

?

[uncut ferry]

im not sure but i think so

yes

?

[burnt abyss]

no

if 1 angle is 90

and the other is let's say 91

because it's more than the first one

90+91=181

[uncut ferry]

ok yes

that makes sens

[burnt abyss]

so they will never equal 180

[uncut ferry]

just had a brain rot moment tbh

i tend to overthink questions

so would i be correct with 3 and 5 putting alternative interior and circling neither?

[burnt abyss]

no

if 2 angles are alternate interior then they are equal

and ours are not equal

so they cant be alternate internal

[uncut ferry]

okok

it cant be interior either since 3 and 5 dont add up to 180

[burnt abyss]

it cant be interior becaus they are not interior

[uncut ferry]

ok so its corresponding

ive put down corresponding for a b and c now though

[burnt abyss]

4 and 5 arent corresponding

[uncut ferry]

interior then

?

[burnt abyss]

yes

[uncut ferry]

kk

[burnt abyss]

3 and 5 arent corresponding

[uncut ferry]

would 3 and 5 be alternate interior?

[burnt abyss]

no

[burnt abyss]

.

[uncut ferry]

ok so interior

4 and 5 interior, equal. 1 and 3 corresponding equal, 3 and 5 interior, neither is what i have down

[burnt abyss]

ok

it's correct

[marsh citrus]

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[marsh citrus]

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[unreal marsh]

Why do we use log base e to solve exponent problems instead of log base x

[distant peak]

can you give an example

[unreal marsh]

Sure!

This a YouTube short I found while scrolling

The way they solved for x

Why did they use the natural log

Instead of log base x

Why would ln work

[distant peak]

I mean, it should work, but we dont know really know the domain i guess

[unreal marsh]

Wdym domain

[distant peak]

log_x (a) = ln(a)/ln(x)

imagine if x = 1

[unreal marsh]

Oh

But you described it in terms of ln

Wouldn’t that still happen

Even if you used ln

[hidden plaza]

log is base e so it’s fine

[unreal marsh]

But why do they use base e

Don’t you have to use base x

Since the base isn’t e

[vestal forge]

it doesn't matter what base it is, you can take any base you want as long as it is defined

[unreal marsh]

But you would get a different answer?

Since you are now asking

[vestal forge]

here you could also take log base x too, since x^x is defined on [0, infty), however, this would exclude the case x=0, but that is not really our problem

[vestal forge]

no

[unreal marsh]

yes

log base 10 for 100 = 2

2

If you used base e

You would get something else

[vestal forge]

im sorry, but i dont understand what you are trying to say

log_10(100) is not equal to code

neither is "code" a numerical value iirc

[unreal marsh]

It’s my phone I’m sorry

[hidden plaza]

Not relevant we’re talking about applying log to both sides of an equation

[unreal marsh]

There

I edited it

[hidden plaza]

You’re talking about 2 diff quantities

[unreal marsh]

OHHH I’m so dumb

[vestal forge]

yes, log_10(100)=2 but like pure said, that's not really related here

[unreal marsh]

Yes th Anel you

So you could use any base really

Why do we use base e though

[vestal forge]

because sci liked to

[unreal marsh]

Sci!