help-39

[errant fable]

has some interesting properties

[errant fable]

i'll try to provide geometric solution when i'm free

[dapper kraken]

how are you even supposed to do this with trig

[errant fable]

law of sines

lots of it

there are some shortcuts with advanced trig

[dapper kraken]

no calculators :(

is it poss without calcs

[errant fable]

still shouyld be possible

if i were to ask you what sin^2(17)+cos^2(17) is

you wouldn't need a calculator to calculate sin17 and cos17

construct the circumcircle of the triangle ABD, and let where that circle intersects with BC be E

AD=DE=CE and BD=BE

chasing the angles, <EBD=20 and <BAC=100

[dapper kraken]

wtf er

hm i got ebd to be 22.5?

so EAD=EBD, ECD=EDC=2EBD=>BED=4EBD=>BEA=3EBD?

)3+3+2)EBD=8EBD=180=>EBD=22.5

[errant fable]

BD=BE BDE is isosceles

BA=/=BE

ABE isn't isosceles

[dapper kraken]

is it not?

AD=DE right

[errant fable]

yes

[dapper kraken]

by congruency BD=BD AD=DE ABD=EBD?

[errant fable]

nu

[dapper kraken]

why?

[errant fable]

B could be any point on the circle and AD=DE would still stay true

doesnt necessarily mean ABD=DBE

[dapper kraken]

but BD is the angle bisector of ABC?

[errant fable]

it is

[dapper kraken]

doesent thwt imply ABD=DBE

[errant fable]

let me draw it out

this is the situation

you're saying AB=AB and <CAB=alpha and DB=CB

so ABC=ABD by congruency

but you can clearly see that that's not the situation

if only angle alpha was between the known sides, the triangles would be congruent

but since it's not, there's this thing

[dapper kraken]

the the circumcircle of ABD wouldnt intersect BC

[errant fable]

SAS is not the same as SSA

[dapper kraken]

ok but how do yoi chase BAE to be 4x

[errant fable]

ABED is a cyclic quadrilateral

BDE=BAE=4x

[dapper kraken]

ADE is icoseles?

oh yea

ok that makes sense

how the hell do i get the insight to draw a circumcircle tho wth bruh

alight tyy

.close

[pearl pond]

Channel closed

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[errant fable]

try weird things

until they work out

the problem is not gonna be mad at you for failing

[pearl pond]

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[teal glen]

Can anyone give me a hint on how to start/ set up question 2 (grade 12 advanced functions)

[compact ridge]

it's just pattern recognition

[compact ridge]

you should know what $\frac{2 \tan u}{1 - \tan^2 u}$ simplifies to, if not search it up

[jolly parrot]

south

[compact ridge]

ah then you need to factor g(x)

more specifically, try dividing g(x) by x^2 - 7x - 8

also $\log_9 |x| = \frac{\log_3 |x|}{\log_3 9}= \frac{\log_3 |x|}{2}$

[jolly parrot]

south

[teal glen]

Oh sorry I forgot to clarify I meant the part about algebraically determining when f(x)g(x) is less than or equal to 0

I already did that part

[compact ridge]

the restrictions are g(x) not equal to 0, tan(pi/16 * x) not equal to -1 or 1

ahhhhh

[teal glen]

So sorry lol

I totally forgot to add that

<@&286206848099549185>

[pearl pond]

@teal glen Has your question been resolved?

[compact ridge]

a few ways but either f(x) and g(x) are both positive

or f(x) and g(x) are both negative

or one of them is zero, of course

hopefully you know how to curve sketch

there's this way of thinking about polynomial and rational inequalities, called the wavy curve method

for a polynomial the signs alternate between + and -

for a linear rational function like f(x), you just have to know the shape of 1/x

then the sign of 1125x/(x + 1) will be the same as x/(x + 1) = 1 - 1/(x + 1)

so like it occupies the top left and bottom right 'quadrants' (where the vertical line dividing the quadrants is the asymptote x = -1, and the horizontal asymptote is y = 1)

cause you are multiplying a horizontally shifted version of 1/x by -1, and you should know what the graph of 1/x looks like

[pearl pond]

@teal glen Has your question been resolved?

[pearl pond]

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[sharp smelt]

Determine if $\sum_{i=1}^{\infty} \frac{4+3^n}{2^n}$ converges or diverges.
\
We use the comparison test here $\sum_{i=1}^{\infty} \frac{3^n}{2^n} \leq \sum_{i=1}^{\infty} \frac{4+3^n}{2^n}$. $\sum_{i=1}^{\infty} \frac{3^n}{2^n}$ diverges. Thus, so does $\sum_{i=1}^{\infty} \frac{4+3^n}{2^n}$.

[jolly parrot]

math rocks

[vast berry]

can u just break it into 4/2^n + (3/2)^n

(3/2)^n is arithmetic series that diverges

[sharp smelt]

yeah, I can

But is what I did wrong?

.close

[pearl pond]

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[sharp smelt]

Determine if $\sum_{i=1}^{\infty} \frac{1}{\sqrt{n^2+1}}$ converges or diverges
\
I was thinking of comparing it with $\sum_{i=1}^{\infty} \frac{1}{\sqrt{n^2+2n+1}} = \frac{1}{\sqrt{(n+1)^2}}$
\
The series $\sum_{i=a}^{\infty} \frac{1}{n+1}$ diverges, thus so does$\sum_{i=1}^{\infty} \frac{1}{\sqrt{n^2+1}}$ .

[jolly parrot]

math rocks

[solid pier]

that works

[sharp smelt]

Thanks so much!

one more question, if you don't mind

same question, but the series is $\sum_{i=1}^{\infty} \sin(\frac{1}{n})$

[jolly parrot]

math rocks

[sharp smelt]

Not sure what to compare it to

I suspect it doesn't converge

[solid pier]

one might think about the common joke that engineers say sin(x)=x

[sharp smelt]

yeah, I did think of that, but comparing it to 1/n feels wrong

[solid pier]

you’d be right, and one way to do that is come up with something linear that’s smaller than sin(x) around 0

[sharp smelt]

you mean around infty?

[solid pier]

no

[sharp smelt]

as we appraoch infty

[inland ivy]

as n approaches infinity, 1/n approaches 0

[sharp smelt]

yeah

so we compare it with 1/n

[inland ivy]

no

1/n is an upper bound for sin(1/n)

you need a lower bound

[sharp smelt]

what do you have in mind for a lower bound

does 1/x^2 work

Looks like it does

[inland ivy]

Do you mean 1/n^2

[sharp smelt]

yes

[inland ivy]

That is indeed a lower bound

But it converges

To prove divergence, you need a lower bound which diverges

[sharp smelt]

yeah, ture

1/n -6/n^2

does that work?

[pearl pond]

@sharp smelt Has your question been resolved?

[tropic saddle]

think simpler

change one of the symbols of 1/n

[sharp smelt]

-1/n

hmm, no

I mean -1/n is a lower bound, but it's negative, so it doesn't help much, does it

[tropic saddle]

no

it only tells you the series is at least -infty

[novel apex]

try using the fact that sin is a concave function on [0,pi/2]

[sharp smelt]

hmm?

[tropic saddle]

this is an approximation of sinx with a line which was nearly helpful. concave is also about lines. try connecting those two

[sharp smelt]

concavity means f"(x)<0

right

[tropic saddle]

no

it means that a line between two points on the graph is below the graph

(or on the graph)

[sharp smelt]

hmm, okay

Okay, I'm lost

[tropic saddle]

lost how

where

[sharp smelt]

Like how does concavity help here

I can't think of anyway to construct a secant line here

[tropic saddle]

you have two functions, sinx and x. x is a line and much nicer than sinx, but sadly its above the graph

so its not helpful as a lower bound

but can you find a line below the graph of sinx?

[sharp smelt]

x-0.5

[tropic saddle]

preferably still positive near zero otherwise we have the same issue as before

[sharp smelt]

I can't seem to think of any such function

[tropic saddle]

you know that sinx is concave

so you literally only need to connect two points on the graph

[sharp smelt]

yeah, I get that

[tropic saddle]

then whats stopping you from doing that

[sharp smelt]

I'm confused, doesn't the function have to be a lower bound for sin(x) everywhere

[tropic saddle]

only where you want to compare it to sin(1/n) for big n

aka near zero

[sharp smelt]

Then $y- \frac{1}{\sqrt{2}} = \frac{2\sqrt{2}}{\pi} (x- \frac{\pi}{4})$ works, I suppose?

[jolly parrot]

math rocks

[tropic saddle]

again, should be positive near zero

wait

why are you deciding to write so complicated stuff

[sharp smelt]

I chose (0,0) and (\pi/4, 1/sqrt{2}) as my points

[tropic saddle]

good choice

but why did you write the line in such a complicated fashion

y=cx for some c

[sharp smelt]

oops

right

yeah

$y = \frac{2 \sqrt{2}}{\pi} x$

[jolly parrot]

math rocks

[tropic saddle]

anyway, the constant barely matters

the point is, now y=cx is below the graph

yes?

[sharp smelt]

yes

[tropic saddle]

so sin(x) > cx for all x near zero

[sharp smelt]

yes

[tropic saddle]

so then x=1/n gives?

[sharp smelt]

$sin(1/n)< \frac{2\sqrt{2}}{\pi n}$

[jolly parrot]

math rocks

[sharp smelt]

oops

no

$sin(1/n) > \frac{2\sqrt{2}}{\pi n}$

[jolly parrot]

math rocks

[tropic saddle]

or again, sticking with just writing c, sin(1/n) > c/n

can you now compare the series?

[sharp smelt]

yeah

c/n diverges

so sin(1/n) does

too

Thanks

.close

[pearl pond]

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[pearl pond]

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[stoic harness]

Hey!

[brave sluice]

what if you use two intervals

[stoic harness]

The question is which 2

There are like 10 combinations

[brave sluice]

f(5)-f(3)+f(3)-f(0)=1

[stoic harness]

So [0,3] and [3,5]

?

[brave sluice]

yeah

[stoic harness]

I can find 3f’(ξ1) with that and propably by extension 2f’(ξ2)

But I am not looking for those 2

[brave sluice]

you mean you found zeta1 such that... ?

f'(zeta1)=?

[stoic harness]

Yeah yeah

[brave sluice]

equals what

[stoic harness]

(f(3) - f(0))/ 3

[brave sluice]

ok so what is 3f'(zeta1) + 2f'(zeta2)?

[stoic harness]

With that I can find 3f’(zeta1) but I am looking for 2f’(zeta1)

[brave sluice]

then do the other interval first

instead

[stoic harness]

Yeah then I will find 2f’(zeta2)

But I am not looking for it

[brave sluice]

good

you are

you should be

[stoic harness]

It’s 2f’(zeta2)

Not 3

[brave sluice]

^

the equation says 3f'(zeta2)

[stoic harness]

Made a typo

[stoic harness]

.

Wait I confused myself

Give me a minute

[brave sluice]

ok

[stoic harness]

Ok yeah , I am looking for 2f’(ξ1) and 3f’(ξ2) but using those intervals I am finding 3f’(ξ1) and 2f’(ξ2)

[brave sluice]

you can name the zeta whatever you want

you can name the first one zeta2 instead of zeta1

or use [3,5] and [0,3]
rather than [0,3] and [3,5]

[stoic harness]

Yeah got it! Ok ok thanks for the help

.closeel

.close

[pearl pond]

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[sharp sonnet]

is this right

[sharp sonnet]

chatgpt cant correct my answers anymore 😔

nope 💀

[autumn fossil]

,w (6m^4 - 9m^3 - 7m^2 + 16m - 26) / (2m+3)

[jolly parrot]

Results provided by WolframAlpha

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

Requested by methisalwaysright

[autumn fossil]

looks like a sign error

[sharp sonnet]

darn 😔

[stoic harness]

Sorry for interrupting guys , axe by that you are saying basically that 2f’(ξ1) + 3f’(ξ2) = 3f’(ξ1) + 2f’(ξ2) , no?

[brave sluice]

not exactly

[sharp sonnet]

ok thanks yall idk how i got the sign wrong but ill figure it out

.close

[pearl pond]

Channel closed

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[stoic harness]

Shouldn’t this be the case in order to proceed

[brave sluice]

back up before you proceed

[stoic harness]

[0,2] and [2,5] work as well

[brave sluice]

yeah just use those

[stoic harness]

Are you familiar with some formulas?

[brave sluice]

such as?

[stoic harness]

This gonna take a moment to write

Of the form : k(1)f’(ξ1) + k(2)f’(ξ2) = λ then we split the intervals [a,b] to [a,c] and [c,b] in which c = a +k*d in which ( d= (b-a)/ (k(1) + k(2))

.reopen

[brave sluice]

you'll need a new room

[stoic harness]

Ok I am opening

[brave sluice]

can't reopen this one since it wasn't assigned to you when it was closed

[pearl pond]

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[quick star]

some definitions use whole numbers interchangeably with integers?

[quick star]

😭 i've never seen this^

have u guys seen it?

[open rivet]

!original

[pearl pond]

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.

[quick star]

wdym original???

my question is as is posted lol

[open rivet]

like, whats the question?

[quick star]

^

whole numbers = integers definition

[open rivet]

whole numbers are sometimes defined as only the non-negative numbers

whole numbers = N
integers = Z

[quick star]

yeah but apparently there's a definition which has that whole numbers could also mean integers. Who exactly uses this definition and why is it a thing? 😭

[quick star]

this one i know

[cunning comet]

can you show a "whole number = integer" definition somewhere? or is it possible that this "whole number = integer" was used by a native german speaker? if someone would use a word-by-word translation from a german phrase (expressing integers) the result woud be "whole number".

[plush bramble]

Show where you got this information from

[quick star]

wikipedia

[plush bramble]

Send link

[quick star]

yeah okay wait

https://en.wikipedia.org/wiki/Natural_number

weird right? 😭

[plush bramble]

Yea that reference in [3] is not a math reference

People in science abuse math words all the time

[quick star]

so that's an engineering definition?

it's just wrong right?

or is it bold of me to say "it's wrong"

i guess it's right for their context

[open rivet]

its just whatever is useful at that time

[quick star]

but in genearl it's wrong

[quick star]

okay but at this point then everything is ambiguous

[open rivet]

some people use 0,1,2,3,.... as natural numbers, some use 1,2,3,.....

[quick star]

Lol

yeah but that's known

integers = whole numbers

i've never seen

[open rivet]

i guess in some field(s) its common

[quick star]

those fields not being math?

[open rivet]

probably

engineers abuse math a lot

pi=e=3

[cunning comet]

well, you have one statement ("sometimes ...") in wikipedia with one (!) reference to a non-math book. Is that really an issue? What i am asking: have you ever seen this interpretation yourself when reading math-books or math-ressources?

[quick star]

i've not

i was asking if u guys have

[open rivet]

i have never seen that whole numbers = integers

but i can imagine a scenario where someone would say it

[cunning comet]

i have and i have explained in which situation this can appear.

[quick star]

okay but so if a standardized exam

says "whole numbers"

i can contest them by saying "whole numbers can be integers too"

😭

I just need to know whether this definition has any bearing in a mathematical field

i'm not an engineer (yet)

[open rivet]

i dont think you have to worry about it in an exam

if its ambiguous theyll clarify, and if they dont you can always ask during the exam

unless you are at some uni/college where its just students in an exam hall with no supervision

[quick star]

no u can't ask during an exam

first of all your teacher isn't present during an exam

secondly, it'll be thought of as cheating

and yeah this isn't cheating but u get the point

anything related ot the content of the exam is straight cheating

[cunning comet]

you always need to know whats behind the used words. some defines naturals with zero, some without. i ha this in lectures for the same mastter.

[quick star]

but this part is common knowledge i guess

the integers = whole numbers part isn't

😭

[cunning comet]

so if you have an exam (of your teacher) you should use what the teacher used.

[cunning comet]

different meanings (with/without zero) should be "common knowledge" - thats a contradiction.

[open rivet]

lol what

how can asking about ambiguity in an exam be considered cheating?

[quick star]

i asked whether i should write 22 or 22nd when they wrote "one word" on an exam and they said they can't help lol

anyway so that should probably give context

also this is a standardized exam

if it's a school exam then okay you're right

Appealing over a question that shows up on a standardized exam is a tall order so it's better to just avoid it

[quick star]

😭

i mean natural numbers could or could not include 0

i think many people have seen that

i doubt many have seen natural = whole number definitio tho, right?

[sharp sonnet]

why is 0 not natural

is it unnatural to have 0 friends?

[cunning comet]

my argument is: if "natural numbers" appears in an exam, you need to know if its with od without zero.
and the same with "whole numbers": if it appears and your teacher used it in an "equals integer"-way, you cant argue thats wrong.

[quick star]

if they specify then no problem

if they say "whole number including 0"

they usually do that to combat the "natural numbers include 0 or not" notion

then what am i supposed to do with that?

is Whole numbers = natural numbers with 0 or whole numbers = integer

my only source is a wikipedia sentence 😭

that's why i was asking if this other definition is even common

just because one book makes their own definition doesn't mean it's conventional right?

that seems to be one engineering book

[cunning comet]

if you never have seen "whole number" = "integer" then i wont think about it. thats all i can say. i guess your making a mountain out of a molehill

you have seen one statement ("sometines ...") with one reference to a non math book.

[quick star]

okay so if i see whole numbers including 0 on an exam

do i assume naturals or integers?

that's my overarching question

[plush bramble]

You just ask for clarification

[quick star]

😭 i can't

it's a standardized exam

i would have to email the government i guess

lol

[plush bramble]

You follow whatever standards the field defines

If it's a math exam, follow math convention

If it's engineering use theirs

[quick star]

which is?

yep it's a math exam

not an engineering one

what convention do you follow?

i think i'll just follow that

[open rivet]

the math convention is that they dont use "whole numbers"

[quick star]

as a mathematician (i'm guessing u are)

[quick star]

lol i guess

[plush bramble]

https://mathworld.wolfram.com/WholeNumber.html

[quick star]

okay fair then this is just ambiguous to the T

[plush bramble]

Don't rely on non math resources for math convention/notation

[quick star]

😭 i mean wolfram link u sent also says that right?

"Some authors also interpret "whole number" to mean "a number having fractional part of zero," making the whole numbers equivalent to the integers."

[open rivet]

just use integers, non-negative integers, positive integers, then you will never have problems (i think)

[plush bramble]

Just read the first sentence

[quick star]

Yeah fair enough

I'll email them if this ever shows up

Since there's no general consensus

thanks guys

[cunning comet]

do you have materials for the preparation for this "standardized exam"?

[quick star]

no definition booklet

[cunning comet]

no materials?

examples for practice?

[quick star]

oh i haven't searched for those

like haven't encountered any ambiguity relating to this

yet

in practice

but it's okay i just wanted to know if there was a clear cut ambiguity

and not just one 😭 sentence on wikipedia

[quick star]

i guess this works for that

anyway yeah okay thanks everyone

.close

[pearl pond]

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[humble tinsel]

How is this limit rewritten?

[bitter herald]

they performed a limit substitution

Basically, if x -> infinity, 1/x -> ?

[humble tinsel]

Never heard of that before

[bitter herald]

it is nothing fancy

you have probably done it without really explicitly noticing, too

[humble tinsel]

Strange

[ember spear]

let 1/x = t

[humble tinsel]

Yes

[pearl pond]

@humble tinsel Has your question been resolved?

[pearl pond]

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[pearl pond]

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[plush bramble]

Do you have a question

[pearl pond]

@storm tinsel Has your question been resolved?

[open rivet]

whats the question?

[pearl pond]

@storm tinsel Has your question been resolved?

[plush bramble]

.close

[pearl pond]

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[frail dune]

Are these two right?

[tardy reef]

You messed up pythagoras' in 1st

x^2 + x^2 = (sqrt3)^2

or 2x^2 = 3

[frail dune]

But isn’t it square root 3

[timber wagon]

wtf

pythagora?

take sinus

[tardy reef]

pythagoras' is a^2 + b^2 = c^2, not c

[frail dune]

Oh

[timber wagon]

sin(45°)*(sqrt3)=x

dunno where you get pythagora

[frail dune]

So if a hypotenuse was 4 in a isosceles right triangle

The two sides would be 2(square root 2)?

[timber wagon]

$sin(45°)*(sqrt3)=x$

[feral leaf]

Because it's a right triangle so you can apply pythagorean theorem

[timber wagon]

it's asking to find x

[feral leaf]

And you can use pythagorean theorem

[timber wagon]

we only know one side

[feral leaf]

You're assuming that they know trig

[frail dune]

So for the first problem it’s square root 3/2?

[tardy reef]

its isosceles

[timber wagon]

ahhh

[frail dune]

Okay thanks yall

[feral leaf]

It's an isosceles triangle

[timber wagon]

i saw root3/2

though it were trigo

[frail dune]

.close

[pearl pond]

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Closed by @frail dune

Use .reopen if this was a mistake.

[tardy reef]

actually thats not what you want

your problem asks for rationalizing the denominator

[frail dune]

Simplify yea

[tardy reef]

so you are 1 step away

[frail dune]

Square root 6 over 2

[tardy reef]

yea

[frail dune]

Thanks

[timber wagon]

2x^2=3

x=sqrt 3/2

yeah that's right

[feral leaf]

Not exactly the answer that it wants fyi

[timber wagon]

Square root 6 over 2

wanted to confirm that

sorry for misinterpretation

[pearl pond]

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[stoic imp]

A relatively prime date is a date for which the number of the month and the number of the day are relatively prime. For example, June $17$ is a relatively prime date because the greatest common factor of $6$ and $17$ is $1$. How many relatively prime dates are in the month with the fewest relatively prime dates?

[jolly parrot]

938c2cc0dcc05f2b68c4287040cfcf71

[pearl pond]

@stoic imp Has your question been resolved?

[timber wagon]

gcf AGAIN

ah nah that's not too bad

so a month is composed with 28, 29, 30 or 31 days

lets put d for day

January: 31, February: 28 (29 in leap years), March: 31, April: 30, May: 31, June: 30, July: 31, August: 31, September: 30, October: 31, November: 30, December: 31.

and m for month

gcf$(m,d)=1$ for each day d in the month m

[jolly parrot]

ibratoch

[pearl pond]

@stoic imp Has your question been resolved?

[vestal tapir]

12 famously has more divisors than 10, so it's maybe december

weird question

[brave sluice]

december and june seem like good candidates

[stoic imp]

31 days for {Jan Mar May Jul Aug Oct Dec}
30 days for {Apr Jun Sep Nov}
28 days for { Feb }

at least in 2025

[stoic imp]

wdym

[brave sluice]

for february you can cross off all the even days

for march every third day

for june you can cross the even days and every third day

for october you can cross off the even days and every fifth day, but that's not quite as good as june

[stoic imp]

ye

what about December

[brave sluice]

december is like june

[stoic imp]

but better?

[brave sluice]

i think it's worse because the 31st is a coprime day

[stoic imp]

ok so June it is

[stoic imp]

10? relatively primes

[vestal tapir]

oh yeah it's not that weird

of course it's either june or december

[stoic imp]

yeah

but 31 is prime

[brave sluice]

seems so

[stoic imp]

ok ty

.solved

[pearl pond]

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• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
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[sharp smelt]

For what value of $p$ does the series $\sum_{n=2}^{\infty} \frac{1}{n^p \ln(n)}$ converge . We compare this with $\frac{1}{n^p}$, to determine $p > 1$

[jolly parrot]

math rocks

[rough forge]

smells like integral test + p test

[sharp smelt]

what;s p test

[rough forge]

Essentially your last sentence

sum/integral of 1/n^p converges if p > 1

[sharp smelt]

So what I wrote works?

[rough forge]

Should work actually, if you intended to use comparison test

[sharp smelt]

thnaks

[brave sluice]

but how do you know if it converges for p=1 or not

[jolly parrot]

𝔸dωn𝓲²s

[rough forge]

Also interesting one

[sharp smelt]

This?, yeah, it is

We can do an integral test here

[rough forge]

with integral test you could show that

[sharp smelt]

integral test

.close

[pearl pond]

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[pearl pond]

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[sacred pasture]

How do I express cos240⁰ in terms of acute angle

[glacial kite]

cos240⁰ = cos(-120⁰ ) =cos(120⁰ )=-sin(30⁰ )

[sacred pasture]

I have this

[light helm]

is incorrect

[sacred pasture]

But my calculator doesn't give me the same answer for cos240⁰ and -cos30⁰

[light helm]

if you're using 270°, it'd no longer be cos

look up supplementary trig identities

[sacred pasture]

sure

[light helm]

and/or if you're using the unit circle for reference,

if still using cos, you'd want that blue angle

which is red - green

[pearl pond]

@sacred pasture Has your question been resolved?

[pearl pond]

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[mental solstice]

Question 17:

Let i be a unit vector pointing east and j be a unit vector pointing north. Ilama walks 6 km in the direction north 30° west, then 5 km east.

The position vector , relative to her starting point, is given by:

[regal finch]

j is north?

[mental solstice]

oops sorry

i is unit vector pointing east and j is unit vector pointing north

[regal finch]

Yep

What dont you understand?

[mental solstice]

how to do the problem lmao

[regal finch]

The problem is asking to create the position vector by adding the distances it travels

so we know that in the first step its travelling north 30 deg west

[mental solstice]

yeah

[regal finch]

for 6km

it might be easy to understand if you have a drawing of it

[mental solstice]

I think I get it now

I just have to split it up into components

I got A as the answer

[regal finch]

Thats not what i got

[mental solstice]

ah damn

[regal finch]

But i think its just because the wording for the first part is a bit tricky

[mental solstice]

what did you get as the answer

[regal finch]

when they say north 30 degrees west

it would be mean from the north point

then taking an angle of 30 degrees west

[mental solstice]

alright thanks

.close

[pearl pond]

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[dapper kraken]

i checked and got a strategy for all the greens, red jusr doesrnt work

im not sure about the ? yellows tho

ok nvm

.close

[pearl pond]

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[pearl pond]

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[mighty meteor]

the translation in english for this is: find the antiderivative/ primitive of f(x)

[pearl pond]

@mighty meteor Has your question been resolved?

[silk vine]

Interesting integral

Have you tried any substitutions yet?

[mighty meteor]

ok so

i have

i tried to write x^3 as u

but that did me no good

as for trig substitutions, i got no clue

nothing looks like it could work in terms of trig subs

[silk vine]

Hmm

No trig substitutions use x^3

[mighty meteor]

exactly

thats kidna what im saying

[silk vine]

Let me get some paper

[mighty meteor]

for sure

[silk vine]

My first though it's multiply top and bottom by the deonominater

[mighty meteor]

hmm

[silk vine]

oh actually

[mighty meteor]

why?

[silk vine]

hold that thought

[mighty meteor]

oki

[silk vine]

if we get the bottom to be sqrt(1-x^2)

then the substitution is easy

[mighty meteor]

for sure

[silk vine]

and we can get that by doing u=x^3/2

[mighty meteor]

i was thinking of writing 1-x^3 as

[silk vine]

i havent finished but im pretty sure that substitution works

[mighty meteor]

right, wierd subs once again

ok let me try

[urban briar]

try u = x^1.5

[mighty meteor]

ok

[urban briar]

that will give us du = x^0.5

which cancels with the top part

and we get 1 - x^2 in the denominator

[mighty meteor]

mhm

[urban briar]

sine sub for that

and we should get

[mighty meteor]

ill write it on paper and show you

[urban briar]

arcsin(u)

where u = 1.5

[silk vine]

this works btw

[urban briar]

im pretty sure its e

I believe the answer is e

[silk vine]

it is most certainly not

[urban briar]

ok

let me try doing it for real this time

wait

ferocious

are u sure the answer isn't e

option e I mean

[silk vine]

ohhhh

[urban briar]

not e as if in 2.7182818284590

[silk vine]

my bad

yeah its that

[urban briar]

thats what i said

[mighty meteor]

no it isnt

[silk vine]

thats on me i thought you meant a constant

[silk vine]

really?

[urban briar]

bruh

[mighty meteor]

yah

its d

[urban briar]

really

d?

ok

lets see

[silk vine]

ill take the derivative of this

[mighty meteor]

@silk vine do you have any other idea for this exercise?

[silk vine]

its totally e)

[mighty meteor]

other than the sub he just said?

[silk vine]

the substitution works

We said the same substitution

[mighty meteor]

oh wait my bad

i was looking at the wrong solution

yea its e

[silk vine]

balling

[mighty meteor]

ight ill try the sub now too and see where it takes me

[silk vine]

I was worried i completely messed up for a second

good luck

[urban briar]

i disagree

@mighty meteor I double checked

its definitely e

I'm 100% sure

[silk vine]

it was found to be e

[urban briar]

bruh

wait

[silk vine]

they read the wrong solution

[urban briar]

u already found out its e

[mighty meteor]

alright so lets see

[silk vine]

remember the x is cubed

[mighty meteor]

oh damn

alright

[silk vine]

on the last line

[mighty meteor]

alright thats good

[silk vine]

did that work?

[mighty meteor]

yeo

yep

alrighty

should i make a new channel on here

or do i send the following question here?

[silk vine]

i think you can just send the following one here

[mighty meteor]

ah ok sure

one sec

here

so

[silk vine]

can you drop a translation rq

[mighty meteor]

TRANSLATION: Let f:R->R and the let the following be: (and all those things). Which of these admits a primitive/antiderivative?

so basically, you gotta figure out which of those can be integrated

theres only one that can be integrated

[silk vine]

oh cool

[mighty meteor]

ok so

first things first

[silk vine]

what are the brackets around cos in 5

[mighty meteor]

i got no clue

[silk vine]

if its normal brackets then like

isnt it thatone

[mighty meteor]

probably meaning that x is between [-1, 1]?

[mighty meteor]

huh?

why would it be that one?

[silk vine]

like cos can be integrated

I suspect i dont fully understand the exercise

[mighty meteor]

no no

wait

so

just how f: R ->R

i suspect that x: (who knows) -> [cosx]

and since -1<= cosx<=1

then maybe it means that x -> [-1, 1]?

idk either honestly

im trying to figure it out

[silk vine]

im pretty sure its not 1,3 or 4

[iron stream]

Yo

[silk vine]

actually it could be 1

[iron stream]

It means greatest integer function

[silk vine]

idk they all look integrable to me

[mighty meteor]

first things first

whats Darboux's property?

[silk vine]

never heard of it tbh

[mighty meteor]

damn

its used here i think

here

[silk vine]

so is it like

tryna find out which of these functions are continuous

[mighty meteor]

yes

precisely

but see

theres so many things here

OH WAIT

yea XOR was right

[cosx] mean the integer part of cox

because we also have {sinx} which is the decimal part

so that means

since -1<=cos x <= 1

that the integer part of it would be -1, 0 or 1

[silk vine]

ohhh

its the floor function

[mighty meteor]

so x-> {-1, 0, 1}

for the 5th degree tho

for the others i got no clue

so why darboux's property is useful here i think

you can check if a function is continuous just based on the interval

[silk vine]

hmm

isnt the decimal part of sin the same as sin in the [0,1] range

[mighty meteor]

also i figured that for example, in the 5th case, what would mean is that f: [cosx] -> R

[silk vine]

except the drop off at sin(x)=1

[mighty meteor]

so thats what those mean

[mighty meteor]

(0,1)

and yea i think

unless they can be negative too

can decimal part be negative?

[silk vine]

i dont think so

its defined as

\

[iron stream]

Hey hii

Sry i had to go ! Had a work !

[mighty meteor]

ok

[iron stream]

The anti derivative would exist if and only if

The function is continuous over it's domain

That is R IN THIS case

And so it's max(1,x²)

[iron stream]

This is basically IVT, they chose a cool name !

[silk vine]

is not fractional part of sin also not continuous?

[iron stream]

No

[iron stream]

Fractional part won't be at 0

For 0- it will be 1- 0-

[mighty meteor]

you are right

it isnt

[iron stream]

So it gets discontinuous

[silk vine]

oh I see

[mighty meteor]

yea because if sinx= -0.25, then that would be {sinx}=0.25

which makes it discontinuous

ok so

we got 5 and 6 out of the way

[silk vine]

3 and 4 are discontinous

[mighty meteor]

why

whats sgn

does it just mean sign?

[iron stream]

Signum

[silk vine]

actually 4 may be continous

[iron stream]

,w y =sgn(x) graph

[jolly parrot]

Results provided by WolframAlpha

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

Requested by xor_11

[mighty meteor]

so what does sgn do

care to explain?

[iron stream]

Signum function

[silk vine]

its the sign of x

so if x = -200, then sgnx = -1

[iron stream]

Sgn (x) = 1 for x>0
-1 for x<0

[mighty meteor]

oh wait

[iron stream]

0 for x = 0

[mighty meteor]

isnt the answer 5?

because actually if x-> {-1, 0, 1} then its continuous

[silk vine]

isnt 5 super discontinous

[mighty meteor]

so its either -1, or 1?

[silk vine]

cause it jumps from -1 to 0 to 1

[silk vine]

its 0 at x = 0

[mighty meteor]

aha ko

ok

[iron stream]

Recall it's Greatest integer fn

[mighty meteor]

ok so

we are left with 1 or 2, right?

[silk vine]

think so

2 looks discontinuous as well

[mighty meteor]

id say so too

because that would just be -{x} right?

[silk vine]

yeah

[mighty meteor]

ok

so now lets check for 1

[silk vine]

1 looks pretty good for x isnt in (-1/1)

but then it stays at 1 so i guess its continuous

but not differentiable

[mighty meteor]

whats the difference between between being differentiable, and being able to be integrated

[silk vine]

some stuff thats integrable can't be differentiated

cause anything continuous is integrable

but theres continous functions which arent differentiable

[mighty meteor]

oh yea right

i know about that

but wait

[silk vine]

go on

[mighty meteor]

wouldnt we get this?

[silk vine]

yup

[mighty meteor]

well

isnt that discontinous at -1?

[silk vine]

no

cause it wouldnt be -1

itd be (-1)^2

[iron stream]

x² is a parabola

Now at 1 it will be straight line instead of that down curve

Wait lemme use wolfram

,w graph max[x²,1]

[silk vine]

woah

[mighty meteor]

well yea thats the point

it wouldnt be -1

[silk vine]

the bots here are crazy

[jolly parrot]

Results provided by WolframAlpha

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

Requested by xor_11

[iron stream]

This curves weird !

[silk vine]

so it wouldne be discontinous there

[mighty meteor]

how so

ill show you

[silk vine]

sure

[mighty meteor]

oh wait

i wrote it down and you are right

wow

ok so its 1

[silk vine]

really weird question

what kinda text book is this

its cool

[mighty meteor]

i told you, its the university entrance exam

its the hardest university in my country

[silk vine]

best of luck

[mighty meteor]

so its got some really wierd exercise

[silk vine]

are you going for maths

[mighty meteor]

thx

[mighty meteor]

the uni has the major in computer science

but they require you to know this level of maths to go there

[iron stream]

Which one brother ?

[mighty meteor]

how would you know?

you're from India

[silk vine]

I have mates in comp sci who still dont do this math

[mighty meteor]

yea well idk

[silk vine]

advanced integration

[mighty meteor]

oki

next exercise

[silk vine]

lets see it

[mighty meteor]

ight

so

obvious little thing to notice

we can write 2(x-1) -1

and then break it into two integrals

but then, thats still hard to solve

[silk vine]

hmm

theres a nicer thing to spot in this question

but its a tricky spot tbh

[mighty meteor]

hmmmm

tricky thing to spot u say

[silk vine]

yeah

[mighty meteor]

you mean to say x-1=u?

thats not tricky tho

what could be tricky

[silk vine]

theres a really nice substitution which clears the whole thing

[mighty meteor]

imma say something crazy

what if

(x-1)/(x-2)= u?

[silk vine]

uhh

not the one I had in mind but I think that could work as well

[mighty meteor]

oh wait

[mighty meteor]

i dont

maybe

(x-1)(x-2)=u?

could that do something?

[silk vine]

maybe

[mighty meteor]

what could u have spotted?

[silk vine]

try this

[mighty meteor]

sure

[iron stream]

Try x = z+1.5

[mighty meteor]

dude

idk where u find these subs

[iron stream]

Trust me

[mighty meteor]

it might work

[iron stream]

It would

[mighty meteor]

but its pointless if i cant come up with it

[iron stream]

It gives you 2z/(z²-0.5²)²

[mighty meteor]

yea ok

[mighty meteor]

it actually doessomething really cool

[iron stream]

Which can be easily integrated because d(z²) = 2z

So integral would be - 1/(z²-0.5)

[mighty meteor]

because then we would get (2x- 3) dx = du

[mighty meteor]

yea ok dude

[silk vine]

yupp

makes it super easy

[mighty meteor]

aha oki i got it

thx

[silk vine]

nws

if youre wondering how to spot something like that

it was mostly trying stuff out

[mighty meteor]

mhm i see

[silk vine]

but I wanted a sub for the denominator

and wanted the stuff on the top to cancel

and ive seen questions like this befpre

[mighty meteor]

makes sense

oki, thx

nexttt oneee

right so

[iron stream]

(1-sinx) multiply numerator deno

[mighty meteor]

here we can substitute tgx/2

[mighty meteor]

hmm

[iron stream]

Oh ok they want that answer

[iron stream]

This is a way but from the form of answer

We will switch strategy

[silk vine]

wallis substitution right?

[mighty meteor]

whos?

[iron stream]

sinx = cos(pi/2-x)

[mighty meteor]

ohhhhhhhhhhh

yea

[iron stream]

1+ cosx = 2cos²x/2

And integration

Sec²x is

tan x

[mighty meteor]

no need

ure always overcomplicating things

[mighty meteor]

after you know this

u juste use the sub with tg x/2

that should do it

ill try

@silk vine if you've got another idea

im listening

[silk vine]

id do the substitution you suggested

tan(x/2)

[mighty meteor]

yup

makes sense

but we should do sinx= co(pi/2 - x) right?

cuz otherwise i think we wouldnt get what we want

but let me see

[silk vine]

this also works really nicely

although im not too sure where it comes from

[mighty meteor]

its the half angle formula

[silk vine]

oh i see

i read it wrong

[mighty meteor]

just like how cos2x= 2cos^2x -1

[mighty meteor]

yea

[silk vine]

ill investigate this

[mighty meteor]

i did

here

idk if i can do some more magic with that

[silk vine]

isnt it 1+sinx

[mighty meteor]

oh yea

well, then just change the - to a + everywhere

i dont think mult else changes

[silk vine]

yeah

[mighty meteor]

oh, i forgot a 2 as well

there should be a times 2 everywhere

because of the tgx/2 sub

[silk vine]

yup

swapping from this to a form they want is a bit trivky though

[mighty meteor]

true

i did it

i could show you if you are curious

but i think its correct

[silk vine]

im pretty sure its c

[mighty meteor]

nvm, its not in the answers, mine answer isnt

[silk vine]

did you do tan addition formula

[mighty meteor]

ill show you i guess

but i wrote so ugly