help-39

[silver sequoia]

Lim x -> 0 sin x/x = 1 ?

[acoustic path]

yes this is pretty much the idea

[silver sequoia]

Ohhh

[faint scaffold]

Sinx =x when infinitesimal

[acoustic path]

you might have figured it out but the way to connect this to your problem is since you now have limit x -> infinity of sin(1/x) / (1/x) = 1

[faint scaffold]

Well except for the sin, would be easier

[acoustic path]

... no

that is enough of a reason not to use it

[faint scaffold]

After Taylor expansion, it would be able to to be used

[silver sequoia]

I did this but i dont understand whats next

[acoustic path]

ah that's good enough

yeah just simplify that expression

[faint scaffold]

Becomes lim x to infinity (x^3+x^2)/(1+x^3)

[acoustic path]

can you just let the guy do the problem

[silver sequoia]

Its okay

We can learn together..

[silver sequoia]

Like this?

[acoustic path]

yes very good

[silver sequoia]

Thanks!

Im gonna do other question now

.close

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[pulsar stump]

why 1/cosx -> 1?

[sharp smelt]

What?

[acoustic path]

cos(0) = 1

[pulsar stump]

• cos x is continous?

cos 0 alone isnt enough

[acoustic path]

the limit of cos(x) as x approaches 0 is 1*

[pulsar stump]

alright ty

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[prime bramble]

,, \lim_{1 \to 0} \frac{\sin(1)}{1} = 1

[jolly parrot]

higher!

[pearl pond]

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[brisk steeple]

consider ax-by = 1

[brisk steeple]

gcd (a,b) = 1

(a,b) -> Z

Can any value of 'a' and 'b' have x_1 and y_1 such that the result is 1 (as in the equation)? (x,y) ->Z

for example 29x-99y = 1

[twin fox]

Umm

Its a diophantine equation

[brisk steeple]

yes

[brisk steeple]

first solution is x =41 and y = 12

Not so easy to get

Is it possible for every such equation?

[twin fox]

Use the algorithm?

[brisk steeple]

which?

[twin fox]

Yes

Ahh what was its name

This algorithm has been derived from euclid divison lemma only

[quiet tendon]

extended euclidean algorithm

[twin fox]

To find the value of x and y where coefficients are too big

[twin fox]

Kinda ig?

[quiet tendon]

https://cp-algorithms.com/algebra/linear-diophantine-equation.html

[brisk steeple]

thanks

[twin fox]

Great

[brisk steeple]

🙂

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[shy ocean]

Hello. Could someone please explain to me what is happening in this step?

[random ermine]

d[(x+1)^2 + 4] = 2(x+1)dx

[shy ocean]

hmm

let me think for a bit

okay I got it

Thanks

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[pulsar stump]

why the lim = 1?

[pulsar stump]

it's 0/0

[raven girder]

L'hopital is fun

[upper drum]

log(1+t) can be written as t, if t is ending to 0

[upper drum]

^

[snow sail]

you dont need lhopital

[pulsar stump]

didnt learn yet

[raven girder]

Also equivalent infinitesimals are fun

[upper drum]

expansion?

[snow sail]

yes

[pulsar stump]

taylor? no

[raven girder]

Care has nothing to do with mathematics. If it is correct it is okay

[snow sail]

$\log(1+t) = \sum _{n=1}^\infty (-1)^{n+1} \frac{t^n}{n}$

[jolly parrot]

jan Niku

[pulsar stump]

requiring rigirous proof

[snow sail]

does not

[feral sedge]

This server despises lhopital to an unbelievable extent

[snow sail]

its not despising it, its not always taught at the point a person is presented with some limit

and assuming @pulsar stump doesnt want to use it or theyd have used it and got the answer rapidly

[snow sail]

the derivation is very fast

[pulsar stump]

i cant read cursive

[tender siren]

you got them notes

[snow sail]

u just need fubini

[pulsar stump]

it's not in the syllabus this semester as well

[upper drum]

uh

[pulsar stump]

didnt learn to f`

[snow sail]

f`?

[upper drum]

wha

[pulsar stump]

defreniate

[snow sail]

you actually dont need to

the derivation uses integration not differentiation

it uses fubini but unless you want a proof its not necessary to worry

[pulsar stump]

didnt learn it

[snow sail]

so you are doing limits without lhopital differentiation or integration

have you just begun calculus 1?

[pulsar stump]

yeah, using t substition

[pulsar stump]

mid course, next 2 are defrentations

[random ermine]

are you doing epsilon delta stuff

cuz that's analysis

[frosty ravine]

One cool approach is to let the limit be L and then you have
$$e^L = \lim_{t \to 0} (1 + t)^{\frac{1}{t}}$$

[pulsar stump]

yeah of course

[jolly parrot]

Pluton

[pulsar stump]

alright yeah i see it now

ty

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[rough forge]

so true

[pearl pond]

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[lime river]

If [.] denotes greatest integer function then
Lim [ x²/sinxtanx] is equal to :
x->0

[lime river]

I'm confused between 0 and 1

[pearl pond]

@lime river Has your question been resolved?

[tulip mason]

I mean as the greatest integer function is surrounding the function inside its probably fine to take it outside of the limit (although we'll look at justifying it later.)

After that the limit should be a product of two well-known limits...

[lime river]

How can we take the gif out

[tulip mason]

If it was surrounding a single part then we couldn't buts it's surrounding everything right?

[lime river]

It's surrounding x²/sinxtanx

[tulip mason]

By continuity you can take it outside, we'd then need to consider the conditions on that continuity

[lime river]

I don't think you can if we can do that take the function sinx/x
Limx ->0 [sinx/x] is 0 but [limx ->0 sinx/x] is 1

[tulip mason]

We're expecting the function to approach 1 as x goes to 0 (ik this is not proof but we can construct the proof once we know how the limit works)

[lime river]

Bruh you just plotted it how can we deduce this

[pearl pond]

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[tulip mason]

Nevertheless using some "bad" maths we know that as $x\rightarrow 0$:
[\frac{x}{\sin(x)}=1]
[\frac{x}{\tan(x)}=1]

Then as these limits are finite you can deduce their product

[lime river]

I tried deducing it by using the Taylor series expansion for tan and sin

[tulip mason]

.reopen

[pearl pond]

✅

[lime river]

Lim x ->0 yes okay

[jolly parrot]

Max

[tulip mason]

So we know that without the greatest integer function it will approach 1

[lime river]

Yea

[tulip mason]

But we need to understand how it approaches 1:

There are 3 cases:

It approaches 1 from above, in which case the greatest integer function would output 2

It approaches 1 from below in which case the greatest integer function would output 1

It approaches 1 from above and below

Now we know sinx and tanx are odd so sinx tanx is even and x^2 is even, so the whole function must be even. So it's one of the first two cases. I'll let you deduce which one it must be from that

[lime river]

Yes okay

[tulip mason]

Btw, we will need to apply the continuity argument, we know the greatest integer function is continuous in any interval:

(n,n-1), for integer n, so if we can prove that between (-k,k) that the expression x^2/(sinx tanx) is between two integers then we can just take the greatest integer function outside of the limit

[lime river]

No wait x²/sin²x would be a little greater than one and cosx would be a little less than one for x -> 0

[pearl pond]

@lime river Has your question been resolved?

[pearl pond]

@lime river Has your question been resolved?

[pearl pond]

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[slate ravine]

I need help, I am trying to get a log function into polar form, but I am having trouble isolating "r"

[slate ravine]

This is what I have so far

Can someone help me get further

I don't know if I am in the right direction

[pearl pond]

@slate ravine Has your question been resolved?

[pearl pond]

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[deep obsidian]

what does this mean

[gritty orchid]

for every i between 1 and n-1 calculate the sum of i(i+1)

[unborn abyss]

$\blue1(\blue1+1) + \green2(\green2+1) + \red3(\red3+1) + \pink4(\pink4+1) + \cdots + \orange{(n-1)}(\orange{(n-1)}+1)$

[jolly parrot]

hayley 🥥 🌴

[gritty orchid]

nicely done!

[deep obsidian]

I don't get the n-1 part

[unborn abyss]

if n = 10, for example, then it goes all the way up to 9(9+1)

[deep obsidian]

what if it's n+1

[peak wren]

The n is a number of your choice

Then it would go to 11

[deep obsidian]

what if it's just N

[peak wren]

Then its any number

n in math just means a number

A number of your choice or given to you by an exercise

I dont know what that symbol means in english but whats above that symbol shows to how far it goes

[unborn abyss]

remember how in geometry you had formulas for things like the area of a circle or a square and they had symbols in them

,tex .plane geom

[jolly parrot]

hayley 🥥 🌴

[unborn abyss]

this is kind of like that, we'll figure out what n is later

what is the question asking you to do with that sum?

[deep obsidian]

I just want to know what the sum symbol means

this is induction

[unborn abyss]

well i demonstrated it above

[deep obsidian]

yeah but I was still a bit confused but I think I got it now

[unborn abyss]

see how the colorful number starts at 1, and goes all the way up to n-1

and each time we do $\bsq(\bsq+1)$

[jolly parrot]

hayley 🥥 🌴

[peak wren]

Hayley can we private talk to each other i have a really serious question

[unborn abyss]

um... sure i guess? dm me

[peak wren]

Not about a math problem 😭

[deep obsidian]

idk what you guys are gonna do but

I'll attempt the question

[rough forge]

determination

[deep obsidian]

have fun

[slender viper]

wow that latex

[unborn abyss]

thank you i like my color macros

[pearl pond]

@deep obsidian Has your question been resolved?

[deep obsidian]

I thought I was cooking but I think it burned

[rough forge]

related to this q?

[deep obsidian]

yeah I burned something

I'll reattempt it

[jolly parrot]

𝔸dωn𝓲²s

[rough forge]

isn't there gauss and sum of square formula?

[deep obsidian]

hold up forget this for a moment

what does ! mean

[rough forge]

I suggest to look Little Gauß formula and Sum of squares formula

faculty

,, i! = 1 \cdot 2 \cdot 3 \cdot 4 \cdot ... \cdot (i-1) \cdot i

[jolly parrot]

𝔸dωn𝓲²s

[rough forge]

all numbers from 1 to i are being multiplied

[deep obsidian]

oh alright

[pearl pond]

@deep obsidian Has your question been resolved?

[deep obsidian]

what is -k multiplied by k

my brain isn't braining

[rough forge]

-k²

[deep obsidian]

wow it finally did not burn

thanks guys for explaning the sumation thing

also is that symbol not called sigma or did I confuse it with something else

[light helm]

its a capital sigma

[deep obsidian]

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[snow stratus]

hi

[snow stratus]

HOw do we do 3/10 + 1/6

[iron basin]

find the LCM of the denominators

what's the LCM of 10 and 6?

[pearl pond]

@snow stratus Has your question been resolved?

[snow stratus]

60

[north talon]

incorrect

there is a number smaller than that which is a common multiple of 10 and 6

[iron basin]

hmm, can you show how you got this number?

[snow stratus]

oh shit, its actually 2

[iron basin]

ok, so do you know what LCM means?

[snow stratus]

lowest common multiplyer

[iron basin]

yes
so, for example, if you have 3 and 5, their LCM would be 15

since 15 is a multiple of 3 and a multiple of 5

so obviously, to be a multiple, the number must be greater than or equal to both the numbers

2 is the HCF or the GCD of 6 and 10

[pearl pond]

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[potent valley]

(ab^-4) / (a^-2 • b)

Can someone help me simplify this

[pseudo oxide]

hi ur back

yea same deal

[potent valley]

(a/b⁴) / (b/a²)

Is this right?

[pseudo oxide]

yes

then the a^2 goes to the top and the b^4 to the bottom

so a^3 / b^5

[potent valley]

So that's the answer?

[pseudo oxide]

yes

[pearl pond]

If you are done with this channel, please mark your problem as solved by typing .close

[potent valley]

Thanks

.close

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[pulsar stump]

if D is continous nowhere, then f should either be continuous in all x or nowhere to acoording to my understanding. it shouldnt be continous in x0 only it makes no sense

[pearl pond]

Please don't occupy multiple help channels.

[pearl pond]

@pulsar stump Has your question been resolved?

[feral sedge]

f is continuous at x_0

[pearl pond]

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[weak cipher]

I am expecting s(k+1) as a stament would be
1 + 3 + 5 + 7 + ... + (2(k+1)-1) = (k+1)^2

however the closest option to this is
1 + 3 + 5 + 7 + ... + (2k+1) = (k+1)^2

[weak cipher]

Can someone explain to me why my expected statement is incorrect

[midnight haven]

Hello can you recommend a program to write a book of mathematic because the software i found are very complex and i want a note of my class but very nice

[sly dagger]

what looks wrong?

[cosmic charm]

have you tried simplifying 2(k+1)-1?

[tender siren]

this channel is occupied, try another one

[midnight haven]

ok sorry

[weak cipher]

Cant see how to simplify that

[vernal grove]

2k + 2 - 1

2k + 1

[weak cipher]

oh ahahah

thanks guys dumb moment

.close

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[rose sapphire]

hi! is there a shorter way to solve this problem rather than manually taking the roots and substituting in the equation stated? thanks

[thin sigil]

Try manipulating the first expression into the form of k * (x - a) * (x - b) * (x - c)

Just a thought

oh hold on this is a quadratic lol

so k * (x - a) * (x - b)

then

I mean you can always just multiply and simplify and solve, but there's likely some trick here

[rose sapphire]

I think it can also be solved by getting the sum and product of the roots of the quadratic and then expanding the denominators and substituting a * b and ab

Is this correct?

[thin sigil]

Try it

[tropic saddle]

it feels liek there should be something nicer. I mean if you plug a and b into the poly you get all those expressions like a+1, a-2 etc

[rose sapphire]

Yep, works and takes a shorter time than manually computing. Since the expression expands to 1/[ab + 2(a+b) + 1] + ...

Thanks!

[tropic saddle]

without the 2 but yes

[rose sapphire]

oh made a mistake, thank you

.close

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[midnight haven]

math help

[midnight haven]

how do i find the algebraic form

what are the steps to do

[next dove]

You have to count the number of dots involved to make each square

and then observe the pattern they follow

[midnight haven]

is there any formula

or steps

[next dove]

you have to create the formula yourself

[midnight haven]

how do i

[next dove]

by observation

maybe hit and trial

[midnight haven]

ill try

[north talon]

(notice the amount of dots: 4, 9, 16, 25,... look similar to something)

[midnight haven]

yea they are squares

[north talon]

correct

[midnight haven]

ooo

got it

thankyou

👍

[north talon]

nw

[midnight haven]

where can i find more questions like this lmao

[north talon]

maybe try search sequences in math

[pearl pond]

@midnight haven Has your question been resolved?

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[lethal ruin]

What is the sigma notation for 6, 12, 24, 48, 96, 192...?

[north talon]

6 * 2^(k-1)

general term

@lethal ruin

[lethal ruin]

thanks

what about 6, 18, 42, 90, 186...? (If you add all previous, like 6+12 = 18, 18+24 = 42 etc)

[north talon]

you find the difference between those terms

(you get 12, 24, 48, .. etc)

i think i remember the formula holdon

@lethal ruin sorry to ping you a bunch

[lethal ruin]

np

you're right it works

[north talon]

havent used this in a while haha

[north talon]

its supposed to be 2^n+1

[pearl pond]

@lethal ruin Has your question been resolved?

[pearl pond]

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[quick stratus]

can someone explain how a^2 disappear

[prime bramble]

by assumption, we have b^2 + c^2 - a^2 = -bc

look at the second line

they just substituted -bc for that sum

[quick stratus]

o

thank you

.close

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[prime bramble]

[pearl pond]

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[cobalt needle]

I have been 'creating' an integral which I want to be done by IBP. The first trick is to multiply and divide by 4cos²x and multiple them as you can see how I have done. But the issue is I want to find a function (?) with cosx in denominator whose derivative cancels out with ln(2xcos²x-10sinxcosx). Help me out🙂

[cobalt needle]

Or like the integral of{( 2* something)/(2cosx) * ln(2xcos²x-10sinxcosx)} results in elementary functions not like power series or infinite product.

[pearl pond]

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[ivory breach]

can anyone tell me how to determine which symbol to use
answer is supposed to be x<-5 , x > - 5/3

[cursive wraith]

If you wanna determine when a product is positive, either both terms are positive or both are negative

positive*positive or negative*negative

this leads to, in the particular case of a trinomial : the sign is always the sign of the leading coefficient, except between the roots

[cobalt needle]

. reopen

[ivory breach]

i dont get how that helps find the symbol sorry 😅

[cursive wraith]

?

positive means > 0

negative means < 0

[ivory breach]

-5/3 isnt positive tho

but its >

[cursive wraith]

I already told you

If a*b > 0

then you either have

BOTH a > 0 and b > 0

OR

BOTH a < 0 and b < 0

so

(3x+5)(x+5) > 0

either (3x+5) > 0 and (x+5) > 0

or 3x+5 < 0 and x+5 < 0

[ivory breach]

ahhh ok

how do ik which one to choose tho between these two

or does cambridge not care

[cursive wraith]

you don't choose

both are possible

explore both options

and see where it leads you

first option should lead you to x > -5/3

and second option leads you to x < -5

so x > -5/3 or x < -5

[ivory breach]

ok thx

.close

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[tacit arrow]

Is the answer to b 163 or 330 or 350?

and c is it 500 or 2100 schools

and for d is it

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<@&286206848099549185>

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[pearl pond]

@tacit arrow Has your question been resolved?

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@tacit arrow Has your question been resolved?

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@tacit arrow Has your question been resolved?

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[fading quarry]

i haveno idea where to start

[junior chasm]

Your main goal in these types of problems is to find similar triangles, angles etc and exploit those to find the relevant lengths. Maybe start by labelling things on the triangles?

[fading quarry]

i just labeled the giving lengths idk where to go from there

[junior chasm]

There's a couple pieces of information that stand out, first is that the triangle PQR is isosceles and that the interior circles are tangent to the other edges. Maybe those could prove useful. Also notice that the length YV is euqal to the length SQ minus SY and VQ, maybe you could form some triangles to find those lengths from the ones given?

[fading quarry]

but im trying to form triangles but i dont have enough information to solve for the side that i need

like idk how to use the two given lengths

[junior chasm]

Likely that'll happen towards the end

Remember the line from the center of a circle to the edge is perpendicular to the tangent running through that point

[fading quarry]

ik that

i just dont know how to solve for the neccessary sides with the two given sides

[junior chasm]

What's your diagram look like so far?

[fading quarry]

im just drawing right angled triangles to create tangents

but idk how to even use 15 and 21 to solve for any sid

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[brittle onyx]

Is this right?

.rccw

[dapper kraken]

,rccw

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<@&286206848099549185>

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[fast shuttle]

Hey, in the range [1,100000]
how many numbers have the digit sum of 37?

[fast shuttle]

the question deducts to numbers between [19,999-99,999]

since 19,999 is the lowest possible number that has the sum of 37

[flat plover]

you have 3 commas

[fast shuttle]

how do i answer this combinatorically?

d1+d2+d3+d4+d5 = 37, when 0<=d1,d2,d3,d4,d5<10

i ran a brute force program that gave me the final solution, but i am having a hard time coming up with a solution

[tropic saddle]

you can google "stars and bars with upper bounds" to find a few solutions to that problem on stackexchange

the tldr is: its not that easy

[fast shuttle]

stars and bars and inclusion exclusion

i`ll have to make a case for each d

with 5 variables, it seems quite tedious, and i am sure there is another way

[flat plover]

this seems hard

[oblique river]

there's another restriction that if any digit repeats it has to be >=5

[fast shuttle]

not really, helpful, but i thought of that

[flat plover]

there are way too many combos

to resort to that

[fast shuttle]

the lowest possible combination is definitly 19,999

the one after it is 29998

[flat plover]

there is 28999

its too hard to count like that

[fast shuttle]

there are 5 possibilites with 2xxxxx

4*

[flat plover]

ye but try with 7xxxx

there are way too many

to do it that way

[fast shuttle]

i mean, i can reduce the question for each first digit

[flat plover]

38899, 37999

[fast shuttle]

so for 3,
its d1+d2+d3+d4 = 34

but i am still unsure

[quartz void]

מאריק לשלוח לך?

[fast shuttle]

שלח כפרה

[quartz void]

עשיתי בפונקציות יוצרות

[fast shuttle]

חחחחחחח הלוואי והייתי יודע מה זה אומר

[quartz void]

שמע זה הנושא הכי סבבה לדעתי חחחחחח

לא עברת על זה עדיין?

[fast shuttle]

בדיוק עובר

לא ידעתי שצריך את זה בשאלה חחחח

[pearl pond]

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[prime bane]

Guys question b is impossible

[prime bane]

We dont have any angle to use trigonometry

Nor do we have the lengths to us Pythagorean theorem

[tardy reef]

If I understand the problem right, the OY and OX are the same wire, so their length has to be equal (by common assumption of inextensibility in such problems)

[prime bane]

Oh that might be true

But it's not specified

[tardy reef]

it does mention that the ball swings with line straight

[prime bane]

Hmm

[tardy reef]

If this is not a physics question, I would say go ahead with that assumption

[prime bane]

Ok but how is ZY 15 then?

[prime bane]

Nah it's a normal math question

[tardy reef]

You have just shifted the ball by a horizontal distance of 15 cm like a simple pendulum

[prime bane]

Ok thanks ig

[tardy reef]

but it is tied with the string at the point O so it has to rise in height

[prime bane]

Ok

.close

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[prime bane]

. reopen

.reopen

[pearl pond]

✅

[prime bane]

I think it's wrong

If I use Pythagorean theorem

It's going to be Square root of 0 which is 0

And hence not valid

[dapper kraken]

thats wrong

[tardy reef]

On which triangle are you using the theorem? It is applicable only for the triangle OZY and ZYX

[prime bane]

Zyx

[prime bane]

Then how do I do this

[tardy reef]

you do not know any angles of the triangle ZYX except for the 90 degree one

[prime bane]

Yah

[tardy reef]

ZX = OX - OZ

and you get OZ from trigonometry

OX is wire length which is known

[prime bane]

I don't think so

Oz is from trigo

[tardy reef]

ZY / OZ = tan 40

[prime bane]

But ox cannot be calculated

[tardy reef]

OX = OY because they are same wire

[prime bane]

Ya I am talking about ox

[prime bane]

How

[tardy reef]

.

[prime bane]

Ohh

.close

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[north talon]

"Find the angle GMN"

[north talon]

i'm stuck midway

here's where im at

i have no idea where to go next

<@&286206848099549185>

[wary ibex]

pls tell the question

[north talon]

"find the angle GMN"

[wary ibex]

oh the previous one , sorry I should have seen it

[north talon]

nvm

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[midnight haven]

Never differentiated a vector, could someone please explain this.

aren't we supposed to differentiate i cap?

and j cap

[limber glacier]

no i cap is like a constant

[tardy reef]

yes, but they are constant so their derivatives are 0, so they can be neglected

[fluid oar]

that is actually happening here, you just dont notice it

[midnight haven]

why are we taking it as a constant ?

[limber glacier]

u r just taking a vector i cap, and scaling it with a quantity that varies with time

[midnight haven]

direction is same and magnitude is 1 ?

[fluid oar]

it will be clearer once you move onto the polar coordinates, in them, the derivative of the unit vectors is not equal to the unit vector itself

[limber glacier]

if u want, u can think in terms of the limit definition of a derivative

[tardy reef]

It is about this specific co-ordinate system, so it is constant by definition

[midnight haven]

oh

[limber glacier]

Let $\textbf{s}=x(t)\hat{\textbf{i}}+y(t)\hat{\textbf{j}}$. The derivative of $\textbf{s}$ is $\dfrac{d\textbf{s}}{dt}=\lim_{h\to0}\dfrac{x(t+h)\hat{\textbf{i}}+y(t+h)\hat{\textbf{j}} - x(t)\hat{\textbf{i}}+y(t)\hat{\textbf{j}}}{h}$

[jolly parrot]

SilverSoldier

[limber glacier]

yeah ig this is the same thing i was trying to show

[limber glacier]

well from here u can break this up into two fractions. frmo one u factor out the i hat, and the other u factor out j hat and u shud get the result

[midnight haven]

how ?

[midnight haven]

i will try it now

[limber glacier]

the unit vectors are not really functions

they are just constants

[fluid oar]

they are functions that happen to be constant

[midnight haven]

its kind of hard to understand the fact that we are differentiating a quantity which has direction

[limber glacier]

well yes ig.

[limber glacier]

to differentiate kind of means just to look at the difference between two very cloesly spaced things

it doesnt matter if they have direction or not

as long as u can find what the difference is

[midnight haven]

Okay

[midnight haven]

Could you also help me with this one too :p

Not able to solve it

[limber glacier]

u can take the i hat terms together first

what does that give u?

[midnight haven]

x(t+h)-x(t) / h i hat

[limber glacier]

yes, i hat multiplying that

yes

[midnight haven]

oh

isn't this d(x)/dt i hat

[limber glacier]

$\left(\lim_{h\to 0}\dfrac{x(t+h)-x(t)}{h}\right)\hat{\textbf{i}}$

[jolly parrot]

SilverSoldier

[limber glacier]

yes

[midnight haven]

nice

[limber glacier]

that limit is dx/dt

and u do the same with the j hat terms

[midnight haven]

which is v_x

[limber glacier]

yes

[midnight haven]

okay

thanks a lot, everyone

[pearl pond]

@midnight haven Has your question been resolved?

[pearl pond]

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[rancid mirage]

$$ \int^{\frac{\pi}{2}_{0} \frac{(\cos(x))^2}{1+\sin(x)} $$

[jolly parrot]

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

[sharp smelt]

what?

[rancid mirage]

I need help with this integral

[sharp smelt]

Is it $ \int^{\frac{\pi}{2}_{0}} \frac{(\cos(x))^2}{1+\sin(x)} $

[rancid mirage]

Though It seems I can't format it correctly

[sharp smelt]

Let me try to fix it

$\int_0^{\frac{\pi}{2}} \frac{cos^2(x)dx}{1+sin(x)}$

uh

[jolly parrot]

🏳🌈f(why am i here )= idk

[sharp smelt]

this?

[rancid mirage]

Yes

[sharp smelt]

OK, what's $cos^2(x)$ in terms of $sin^2(x)$?

[jolly parrot]

🏳🌈f(why am i here )= idk

[rancid mirage]

1-(sinx)^2

[sharp smelt]

Cool!

Factorise that further!

[rancid mirage]

This what I have now

[sharp smelt]

uh

[rancid mirage]

Oops error

[sharp smelt]

$\int_0^{\frac{\pi}{2}} \frac{(1-sin(x))(1+sin(x))}{1+sin(x)}$

[jolly parrot]

🏳🌈f(why am i here )= idk

[sharp smelt]

Yes?

[rancid mirage]

I never would've thought about that though I understand what you did

[sharp smelt]

Sorry.

Should have given a hint instead.

[rancid mirage]

This is very convenient because now it simplifies the integrand into something easy to integrate

[sharp smelt]

Yeah!

OK, have to go now, sorry.

Have fun integrating !

[rancid mirage]

Well reason I asked for help is because is turns out I need to solve this integral to solve an integral part of the main integral

[sharp smelt]

Post the entire question here, someone will help!

[rancid mirage]

Though kind of feel guilty because this isn't a homework question it just practice to get better at integrals.

[chrome river]

@rancid mirage do you still need help? feel free to post the full integral

no need to feel guilty!

[rancid mirage]

I might need help later though I haven't attempted the second integral yet but I can post the main integral in the mean time.

This what I’m trying to do, making some good progress

[chrome river]

oh cool cool

gl then!

[pearl pond]

@rancid mirage Has your question been resolved?

[strong glade]

i've already done this

kinda crazy

[pearl pond]

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[fallow aspen]

I am going through this book for self-study. I was wondering how w=4R^2-x^2 was a tip on how to differentiate √4R^2 -x^2

[fallow aspen]

I have referred to the designated chapter it references, but I am not sure

[sharp vigil]

it's a way to make the use of the chain rule more clear

[fallow aspen]

Oh I see, this is just an instance of it you mean?

[sharp vigil]

yes, we can write that [ \odv yx = \odv yw \odv wx ] and solve for each individually

[jolly parrot]

cloud

[fallow aspen]

I see, thats all I needed help with. Thanks!

.close

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[brisk steeple]

In the same circle.

Side 'c' is already drawn.

-as the chord

[acoustic path]

depends on the diameter of the circle

[brisk steeple]

Diameter is the semiperimeter

(a+b+c)/2

c>a, b

(a+b+c)/2>c

as a+b>c

[fickle schooner]

I think considering the upper bound on a and b (which is c) should be enough

So you get an equilateral triangle of side length c. If that's in the circle then all other triangles which a, b < c can be drawn within the circle

[brisk steeple]

I have a question.

If c is the largest side, then c must be more than 60 deg right?

I mean the angle opposite to 'c'.

[acoustic path]

yes

[brisk steeple]

ok

I took and example can calculated the value of angles and found that it can't be constructed with that chord lenght a triangle on either sides. I have to think more.

Aim: prove that any triangle can be perfectlyinscribed in a circle.

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[fickle schooner]

that problem sounds equivalent to proving any triangle has a circumcircle

[brisk steeple]

yes

o

ofc

circumcircle has some properties that I don't wanna use.

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[silk ivy]

Hello, can anyone help me with the following problem:
Let's look at numbers 49, 4489, 444889, ..., which are obtained if the number 48 is entered between the digits of the number 49, and then continue to enter the number 48. Prove that all of the obtained numbers are perfect squares.

(This problem was translated from Serbian, so sorry if the translation is goofy.)

[silk ivy]

I got to here and got stuck :/

[glass salmon]

try to find a closed form

[acoustic path]

@brisk steeple

they wrote it down right here

[brisk steeple]

Ok

[silk ivy]

i tried to prove it with induction

but it ain't working

[glass salmon]

let f(n) be n+1 4's in a row, n 8s, and a 7. then, the n+1 4's will be 4(10^(2n+1)+10^(2n)+10^(2n-1) + ... + 10^(n+1)) which simplifies by geometric sum

and u can do the same with the n 8s and then add 7

and then simplify to the desired form

[glass salmon]

i think i messed the indices up but u get the point

[silk ivy]

genius

thanks man

[glass salmon]

np

[silk ivy]

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[silver nest]

Is it possible to solve this by "inference by resolution"?

[silver nest]

or is truth table the only way to proof

?

[junior chasm]

No, you can just "translate" the entailments to see if they make logical sense

I'm guessing entails means the same as implies

[pine jay]

for simple cases like these, proof by contradiction is usually the smoothest way imo

sometimes though theyre basic enough that you can sort of see that it must be true i guess

[silver nest]

how can I proof by contradiction?

[pine jay]

let me run you through an example if thats fine?

so what's neat also is that if an entailment happens to be false, you get a counterexample for free when attempting a proof by contradiction

[pine jay]

alright, im taking that checkmark as a yes.

let's i guess start with the first one.

We want to show
"Sentence 6 entails Sentence 2"

let's maybe first abstract some of the predicates in each sentence as a start

so for Sentence 6 we have that "Ron is in the library or Hermione is in the library"

What would be suitable Boolean variables in this case to introduce, in order to translate the sentence logically?

[silver nest]

Sentence 6: (R V H), sentence 2: H

[pine jay]

Sure

so let's attempt this by contradiction

Assume the hypothesis is true while the conclusion false

I.e H must be false**

while (R v H) must be true

right?

[silver nest]

correct

[pine jay]

Now since H is false, we have that (R v H) is only true if R is true right?

[silver nest]

yes

[pine jay]

Cool, in such case we've found a counterexample!

i.e we've found a scenario where the entailment is false!

[silver nest]

wait, i got lost. what's the counterexample

[pine jay]

If R is true and H is false.

Then the entailment is false

makes sense?

[silver nest]

so then the entailment is false?

[pine jay]

Yeah

Let me ask this, when is an entailment false?

[silver nest]

when the right part of the entailment symbol is true and the left part is false

[pine jay]

yup!

and check now what happens with ours when R is true while H is false!

[pine jay]

Uh wait

the other way

I.e True entails False

is false

if youre not sure about this, i would recommend checking with ur textbook or notes/etc

or a simple google search i suppose

[silver nest]

I think I got it. H V R is true while H is false

then no entailment

[pine jay]

Yeah

[silver nest]

thanks

[pine jay]

And the reason this happened is by construction of our attempted proof rememeber?

However

if the entailment happened to be true

in our proof, we would end up with a contradiction

in our case that didnt happen

so we constructed a counterexample that way

(for free!)

and this is why i think this way of going about these are really smooth

both in terms of showing or disproving

[silver nest]

yeah

thank you very much

you are very helpful

[pine jay]

Im glad!

[silver nest]

.close

[pearl pond]

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[pearl pond]

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[silent flower]

Can someone help me figure out how to find side CD

[unborn abyss]

you can't

without more information

[silent flower]

What

[latent quail]

0 approach

[silent flower]

Would finding the angles help?

It says AD is parallel to BC

[unborn abyss]

ah well now that changes things

[silent flower]

It does?

[unborn abyss]

yes, finding the angles would be a reasonable way to do this

[silent flower]

Cos law is SSS right

[unborn abyss]

yea

[silent flower]

Which angle should I find first?

Does it matter

[unborn abyss]

doesn't matter, find as many as you can

[silent flower]

Angle B is 44.415

So 44 degrees

Angle a is 78

[unborn abyss]

(our target is angle DBC)

[silent flower]

Angle D is 58

[silent flower]

So the other B angle

I’ll call it G

[unborn abyss]

errr okay, i'd rather you call it DBC

but okay

[silent flower]

Okay I’ll call it DBC

Could I use the Z pattern?

[unborn abyss]

yeah that sounds good

[silent flower]

I was thinking maybe bc they are parallel

DBC is 58 degrees?

SAS is either sine or cosine law

I think it’s cosine

[unborn abyss]

generally cosine

[silent flower]

Yea I’m doing cosine

Got the answer

5.94

Thank you hayley 😄

Ima close this now

Bye thanks again

.close

[pearl pond]

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• Show your work, and if possible, explain where you are stuck.
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[dark flint]

Im not really sure how to approach this question D:

[junior chasm]

It's asking for the largest and smallest prices for which coupon A is as good as or better than B or C

Does that make sense?

@dark flint

[dark flint]

Does Coupon A mean taking 25% of the price that is higher than 100$ then subtracting it from the original price?

[junior chasm]

Coupon A is taking 15% off the price

I think you're talking about coupon C

[dark flint]

Oh yeye

Coupon C

[junior chasm]

So how coupon C works (I think): Say the item costs $120, this is $20 above $100. So the amount you save is 25% of $20, so $5. This is the value that would be subtracted to give the final price of $115.

[dark flint]

Ahhh

but then

how would you continue with the problem :(((

[junior chasm]

Maybe set up some equations for the savings of any given coupon given the price as a variable, say P?

Then you can set up some inequalities

[dark flint]

Ooo

[junior chasm]

Is that a comfortable method for you?

[dark flint]

Uh

ill try cuz thats what this practise is aiming to do i think

[junior chasm]

I see

[dark flint]

but im not sure if ill do it right :(((

[junior chasm]

Well I can help you check your work!

[dark flint]

:DDD

D::

idk how to set x and y as the largest and greatest as an equation 😔

[junior chasm]

I think it's a lot easier if you just look at your savings, for example A would be 15/100 * x and B would just be 30

C would be a bit simpler too (and don't forget your parentheses!)

[dark flint]

oh mane D::

[junior chasm]

But this will work too if it makes more sense to you!

[dark flint]

:DD

how would u go on from here 😭

how would u find y?

[junior chasm]

First I think your expression for C is wrong? Like you've subtracted the original price from the savings and I think that's backwards

[dark flint]

D:

[junior chasm]

Again it may be simpler to only look at the savings

Does that make sense?

[dark flint]

wait but doesnt that represent the savings?

a,b and c are like how much u save

for a number greater than 100

what do it represent 😭

[junior chasm]

Well look at the expression you wrote for coupon B: you subtracted $30 from the price, which would give you the final price after applying the coupon. I'd suggest writing expressions for the value of the coupon in each case (what you would save over base price) instead of the final price

[dark flint]

oooooooo

wait so then would this be the correct inequality?

[dark flint]

ooh mane

[junior chasm]

Well your C price is still wrong, I think it should be x-25/100(x-100), you can plug in values yourself to see if that makes sense

[dark flint]

😭

[junior chasm]

Next you should have two inequalities, one comparing the price comparing the A and B coupons and one for the A and C coupons, then determine the lowest and highest values that are in both intervals if that makes sense

[dark flint]

waitiatititwiat

does

x-(25(x-100)/100) work as c?

[junior chasm]

I would think so

[dark flint]

:DD

alr then thx u so much :DD

ill finish it and then see if i have any questions left over >:)))

[junior chasm]

Sounds good!

Glad I could help

[dark flint]

wait when multiplying from both sides of an inequality does the sign change?

[junior chasm]

If you're multiplying by a negative value I think

May wanna look that up it's been a while since I've had to do that

[dark flint]

D::

mane idk what to do 😭

@junior chasm

it flips the equation

so the inequality where a>=b

is like x<=150/17

😔

what i do wrong 🙏 😭

[junior chasm]

I can't really tell without seeing your work

[dark flint]

mane theres like a typhoon outside mb if the replies be bugging 😔

[junior chasm]

Oh you don't flip there since 100/85 is positive!

[dark flint]

😭

mane 😭 😭 😭

is this working correct? 😭

am so unconfident

[junior chasm]

Not quite, on the second line you need to multiply the -100 by the 25

Or third line

[dark flint]

MANE

DUDE

😭

THE FUMBLING

MY BAD

[junior chasm]

Nah that's fine!

[dark flint]

mane the multiple choice only has integer solutions

what happen 😔

mane im so dum dum 😔

got the answer tysm :DDD

[junior chasm]

No problem!

[dark flint]

:DDD

.close

[pearl pond]

Channel closed

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[vapid bough]

Why does the order of operation matter when we are integrating, ie, why is $\int_{a}^{b} f(x) = -\int_{b}^{a} f(x)$

[jolly parrot]

PianoDolphin

[vapid bough]

I mean if integral is just the area under the curve, then are would be the same whether you measure it from right to left or left to right

also what does the later even mean, area from smaller point a to a bigger point b makes sence

[west sapphire]

it's just a convention

[vapid bough]

we are assuming a<b

[inland ivy]

integrating from the left to the right gives us positive area, while integrating from the right to the left gives us negative area

[midnight haven]

The area has an orientation

[west sapphire]

the convention is used because then you have theorems like $\int_a^b + \int_b^c = \int_a^c$ holds for all $a,b,c$ regardless of how they are ordered

[jolly parrot]

Bungo

[vapid bough]

oh ok

[inland ivy]

you can think of area like a vector

[vapid bough]

I have always just thought of area a how many 1 unit squares you could fit in that shape

[vapid bough]

also if we follow the limit definition of integrals we can just plug values to get this identity

but I was not able to get that

[severe quarry]

That's not the oriented area though

Otherwise e.g. $\int_{0}^{2 \pi} \sin(x) \dd x$ should not be $0$.

[jolly parrot]

Kepe

[vapid bough]

we have $\lim_{n \to \infty} \sum_{r = 0}^{n} f(a+r\frac{(b-a)}{n}) \times \frac{b-a}{n}$

right?

so if we assume that b > a

and then we plug in that value

this is for the upper bound

[jolly parrot]

PianoDolphin

[vapid bough]

no one saw anything

so if we calculate this for b to a

then we will get a negative value for b-a/n

but also it will become f(a + r(b-a)/n)

so f(a-r(b-a)/n)

ie

we have $\lim_{n \to \infty} \sum_{r = 0}^{n} f(a-r\frac{(b-a)}{n}) \times -\frac{b-a}{n}$

[jolly parrot]

PianoDolphin

[severe quarry]

but also it will become f(a + r(b-a)/n)
so f(a-r(b-a)/n)

[vapid bough]

yeah'

[vapid bough]

but how is this equal to negative this

[severe quarry]

Ok yes, where -(b - a) > 0 now

[vapid bough]

yeah so how is this identity true, it would have been true if all we had was -(b-a)/n cause then we could take the - sign out and then write everything the same as if we were going from a to b

But however we also have f(a-r(b-a)/n) in this

and when we were going from a to b we had f(a+r(b-a)/n)

they are supposed to be the same thing apprently, I don't see how tho

[vapid bough]

why do we have that modulo over there

|b-a|

if we are assuming b>a then b-a is always > 0

[jolly parrot]

Kepe

Kepe

[severe quarry]

Otherwise it's just this

[vapid bough]

wait

say a < b here

[vapid bough]

so in this case will this identity still be true

measuring from b to a the area will be negative right

but why tho

I am getting $- \lim_{n \to \infty} \sum_{r = 0}^{n} f(a-rh) \times h$

[jolly parrot]

PianoDolphin

[vapid bough]

as you can see the stuff inside function is f is different

unless f(a - rh) will be equal to f(a + rh) in this context

so is that true?

but if a < b what does it even mean to go a-rh, this would mean that we are substracting values from a but we are going in the wrong direction

say this is the function for which we want the value

[pearl pond]

@vapid bough Has your question been resolved?

[vapid bough]

<@&286206848099549185>

[pearl pond]

@vapid bough Has your question been resolved?

[pearl pond]

@vapid bough Has your question been resolved?

[pearl pond]

Channel closed

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[livid bolt]

can I cancel like dis or no

[next dove]

definitely no☠️

[livid bolt]

this is so sad

im cooked for my test next week

wait

omgg i have an idea wait

[next dove]

if there is any plus or minus sign, you can't unless that thing is common in all of the terms separated by those plus/minus

[sly egret]

I got that there are 180 3-digit numbers, but i can't understand the parts e and f

[next dove]

!occupied

[pearl pond]

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).

[tender siren]

you NEED TO SIMPLIFY EVERY +/- SIGN ON NUMERATORS BEFORE DOING ANYTHING

you need products for that

imagine brackets are there

yeah

its better

[next dove]

why did it took that long to write only that
I was literally waiting here for something interesting

[tender siren]

I just came here, lol

[tender siren]

can you simplify more?

[livid bolt]

omg i removed a root

wait