help-38

[flint basin]

of the isotope?

[split chasm]

yes

[flint basin]

but why the background radiation?

[split chasm]

because no matter how much the isotope decays, the background will still contribute the same amount of counts to the detector

[flint basin]

ok so i shud add the background radiation to every calculation like this?

[split chasm]

total counts = [ counts from isotope that decays and will decrease] + [ counts from background that doesn't change]

[flint basin]

ok ty

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[sweet pagoda]

i understand how to solve this problem execpt i don't understand what they mean by roots in math

[runic dew]

The root of function is where y =0

[sweet pagoda]

oh

[runic dew]

So the root of y = x-1 is x=1

[cursive saffron]

Roots are basically the answers

[cursive saffron]

Yea

[sweet pagoda]

oh so i factor the gcf for the equation?

which is the root?

[cursive saffron]

Do you know how to split the middle term here?

[sweet pagoda]

yes i think its 3(-x^2+3x+18)

[cursive saffron]

Well that's the first step

Next you need to split 3x into 2 terms

Whose sum is 3x and product is -18

[sweet pagoda]

(blank)(blank)?

[cursive saffron]

Yea this will be the final answer

[sweet pagoda]

oh i can do that

[sweet pagoda]

for the root or the factored equation?

[prime crystal]

factored equation

[sweet pagoda]

k

[cursive saffron]

Yea

[prime crystal]

then you can equate the (blank) to 0 to get roots

[sweet pagoda]

thanks

i see now

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[little egret]

wat did I do wrong

ohh 2a

[astral ore]

hi bum lol

[little egret]

hiiii

my math test was bad

..

[astral ore]

,calc 46 * 7

[solid kiln]

Result:

322

[astral ore]

what did you fumble

[little egret]

vector word questions

why were they the hardest thing in my life

it was the plane and boat ones

one worth 8 marks one worth 12 marks

so if u got one thing at the start wrong u basically fumbled the whole question

they’re gonna be in my end of year exam too 😭

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[plain adder]

To find phase shift of a cosine graph do you see the horizontal distance from x = 0?

[verbal stream]

pretty much

[plain adder]

like the max?

so if the min is on x = 0 and the max is on x = 1 the phase shift is 1?

<@&286206848099549185>

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[pallid cedar]

i don't understand how you would come to this solution

[pallid cedar]

the answer i came to was:
-(1/4x - 2)^3

[spiral ocean]

So

Cubic function

x^3

[pallid cedar]

yeah

[spiral ocean]

Horizontally stretch by 1/4

So

Compress 4 times

Hence 4x

[pallid cedar]

wait why would you take the reciporcal

*reciprocal

[spiral ocean]

Stretch is to expand the thing right ?

[pallid cedar]

yeah

[spiral ocean]

What happens if u expand the thing by 0.25 its original size?

[pallid cedar]

ok so you would put 1/4 if they said horizontally compress by a factor of 4

[pallid cedar]

gets smaller

[spiral ocean]

So

It’s actually compression by factor of 4

[pallid cedar]

ok

[spiral ocean]

Then

[pallid cedar]

but why is it 4x-8

[spiral ocean]

-4x^3 to get vertical reflection

[pallid cedar]

you have to distribute that to the -2 as well?

[spiral ocean]

Substitute x-2 into x to move the thing by 2

[pallid cedar]

oh so c might not always be what it appears like

[spiral ocean]

Ja

[pallid cedar]

bc you have to distribute sometimes

ok that makes more sense

is there an easier way to memorize this compression and stretch stuff

[spiral ocean]

Yes

When multiply something > 1, what do u think will happen to the rate of change of the functions?

Or the slope in general

[pallid cedar]

goes up

[spiral ocean]

It goes faster

[pallid cedar]

yep

[spiral ocean]

So, the thing tends to be more vertical

Hence compression

[pallid cedar]

oh that makes more sense

[spiral ocean]

Apply the same logic for stretching

[spiral ocean]

👍

[pallid cedar]

that's why it's vertical stretch and it's the opposite for horizontal bc horizontal stretch would make it slower

ty this is actually really helpful

[spiral ocean]

Glad it helps 👍🥹

[pallid cedar]

ty for all the help

have a good rest of your day

.ckise

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[gentle sleet]

A drug company produces pills containing an active ingredient. The company is
concerned about the mean weight of this ingredient per pill, but it also requires that the variance (in
squared milligrams) be no more than 1.5. A random sample of twenty pills is selected, and the sample
variance is found to be 2.05 . What is the probability that a sample variance this high or higher would
be found if the population variance is in fact 1.5? Assume that the population distribituion is normal

[gentle sleet]

This is what I have done so far:

$\frac{(n-1)S^2}{\sigma^2}$, where $n = 20, S^2 = 2.05, \sigma^2 = 1.5$.

$\frac{19 \cdot 2.05}{1.5} = 25.9666 \approx 25.97$

We need to find $P(\chi_{19}^2 >= 25.97)$

Using the table it would be $0.1 < P(\chi_{19}^2 >= 25.97) < 0.9$ ?

[solid kiln]

Calc III Victim

[gentle sleet]

Idk how to find the exact value for $P(\chi_{19}^2 >= 25.97)$

[solid kiln]

Calc III Victim

[gentle sleet]

and (0.1, 0.9) seems like a crazy interval so idk if im doing this properly

[past comet]

Where does that come from?

[past comet]

Use software (for example in R type 1-pchisq(25.97, 19)) or look at a table

[gentle sleet]

This is a past midterm question im pretty sure they werent allowed to use R. but ye I was looking at the table and 25.97 seems to be between 0.1 and 0.9 for df 19

[past comet]

You can only approximate it then

[gentle sleet]

also why would it be 1 - pchisq for P(chi_19 > 25.97)? wont it be 1 - pchisq if it was P(chi_19 < 25.97)?

[gentle sleet]

so would 0.1 < P < 0.9 be good enough?

[past comet]

p…(q,…) finds P(…<=q) for any implemented distribution in R

[gentle sleet]

ah I see

[past comet]

You want P(…>=q) so the complement (since continious this works)

[past comet]

I would imagine they expect smth like linear interpolation then

But dk

[gentle sleet]

whats that? never heard of it before

[past comet]

A quick hand wavy estimate is from P(…)=0.9 to P(…)=0.95 it goes from 27 to 30. So linear interpolation the other way gives P(…)=0.85 would correspond to 24 ish. We want at 26 so like 0.88 is a reasonable approx

[past comet]

Basically you assume its linear in the interval to do approximation using change of near points

[past comet]

(The correct answer is 0.87 so fairly close)

[gentle sleet]

bettttt

tysm

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[stoic nimbus]

I’m new to implicit differentiation can someone explain why is this wrong

[spiral ocean]

Ok

It’s

x^2y’ + 2xy - 2ayy’ = 0

2xy = 2ayy’ - x^2y’

2xy(2ay - x^2) = y’

Also

U forgot the negative sign factoring

[stoic nimbus]

Which line is this?

[spiral ocean]

2 -> 3

Also, I’m not sure whether the answer u get is similar to mine

Cuz it is hard to rearrange

Nvm

Your answer = 2xy^2 / (ay^2 - x^2y + ay^2)

= 2xy^2 / (2ay^2 - x^2y)

= 2xy / (2ay - x^2)

[stoic nimbus]

Sorry I gtg will look at it ltr thx 🙏

[spiral ocean]

Damn

I found it

[stoic nimbus]

Oh

[earnest nymph]

a weird thing you did is divide by y, you dont need to divide the whole eq by y, youll get the answer easier

[spiral ocean]

U made two mistakes

But they correct each other

Miraculously 😭

In line 2 -> 3, u forgot to bring the negative after the differentiation out and also factored out dy/dx out without considering the sign

But they cancelled

So u have a correct answer

@stoic nimbus

[stoic nimbus]

Aight thx

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[knotty flume]

Can someone help me with q12?

[clear cloud]

Induction proof will work

[mild oxide]

Fr

But do u know sum of binomial coefficients from r=1 to n is 2^n -1

This is the hint

For real

I got it

How to explain him

Fr

@knotty flume

U know gp?

Gp

?

[knotty flume]

By gp u mean geometric progression?

[knotty flume]

Yes

[mild oxide]

See

Use gp here

Do u know sum of this 1+2+2^2+ for n terms

@knotty flume

[knotty flume]

Yes

[mild oxide]

Write the sum

[knotty flume]

(2^n - 1)

[mild oxide]

Now did u get the hint

[knotty flume]

Ohhhh

[mild oxide]

Yea I feel u got it

Fr

[knotty flume]

Thanks

[mild oxide]

Welcome now tell what u did

So I k u did correct :>

[knotty flume]

Am gm

[mild oxide]

Yes fr

We on top

Bro cooked fr 🔥

[knotty flume]

Nice

[mild oxide]

First person to literally understand the fastest

[clear cloud]

Teacher isint ready for this

[mild oxide]

Fr

Since 3 days bro is the fastest one to understand

[clear cloud]

Bros a fast learner

[mild oxide]

I got this idea of gp bcz below weighted means it is written fr 🔥

Bro is studying progression and series

[knotty flume]

Ye

[mild oxide]

Fr fr 🗣️🙊

[clear cloud]

Poor teacher doing induction while 2pages and you came with those 4 lines up

[mild oxide]

No

The teacher is not poor

Teacher has different ideas

Fr fr 🔥

[clear cloud]

Got u

[mild oxide]

Bro is goated fr 🗣️

Yaku the 🐐

[knotty flume]

Oppailol are u in hs?

[mild oxide]

hs?

[clear cloud]

Highschool

[mild oxide]

Ohoh

We in same hs in discord

Lol fr 🔥🗣️

We might not be in same hs but we in same zone fr 🔥🗣️

[ocean onyx]

Highschool

[knotty flume]

Ooh

[ocean onyx]

The good ol' days

[mild oxide]

Fr bro fr

[ocean onyx]

I wish I was back in high school

[mild oxide]

Fr bro

That time I used to think this is hard

Now after it I feel that was ez

[ocean onyx]

But it just gets harder

[mild oxide]

Fr 🔥

[ocean onyx]

Wait until you hear that you do triple integrals in college

[mild oxide]

So many languages but chose to speak fax 🗣️

[mild oxide]

I do aah

;-;

[clear cloud]

Final oral exam in hs was on it

[ocean onyx]

Triple integrals are fun tho

[mild oxide]

They fun but hectic sometimes

@clear cloud @knotty flume .close

[knotty flume]

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[mild oxide]

Welcome from the side of everyone who helped

Imma go now bye

@clear cloud @ocean onyx cya

[ocean onyx]

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[jade rampart]

how to solve thissss

[jade rampart]

should i expand it？

[cobalt cloak]

i dont think so

recall the range of sinx and cosx

which is from -1 to 1 inclusive

[jade rampart]

ohh thank you i get it

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[hazy vector]

I would like to prove the convergence of the sequence with the Cauchy criterion, how can I get rid of the plus one in my calculation? Or how should I have calculated differently?

[sudden mist]

In the last row, why estimate it like that?

If you have (1+n)/n^2

Then why not use that?

For example you could estimate 1+n like 2n for big n

If you want it neater

[hazy vector]

Ah yes 1 + n <= 2n is genius

[sudden mist]

Ur welcome aha

[hazy vector]

Thank you

!solved

[sudden mist]

.solved

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[fresh pendant]

why? how can visulize it?

[marble wharf]

what exactly do you mean with visualize

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[marble wharf]

the wikipedia article on radius of convergence has some examples of functions where it converges for some z and diverges for some other z with |z-a|=R

[fresh pendant]

I mean, like this:

Am i understanding it correctly?

[marble wharf]

no

z is in the domain

[fresh pendant]

ye so what should it be like

[marble wharf]

in the interval (a-R, a+R) it converges

at the points a-R and a+R it may or may not converges

[fresh pendant]

it's said in the pic but it's hard for me to visualize it correctly

[marble wharf]

I have no clue what you are thinking to get that picture

are you mixing it up with the epsilon-N definition of convergence of a sequence?

I mean thats related I suppose but not really helpful

[fresh pendant]

no, it's just inconvinient to plot it with discrete pionts so i used line

[marble wharf]

but what are you trying to plot

[fresh pendant]

Trying to understand

[marble wharf]

you have a series $\sum a_n z^n$, yes?

[solid kiln]

Denascite

[marble wharf]

and into the z you are plugging in different values

so for example you might to try to plug in $z=2$ which would give you $\sum a_n 2^n$

[solid kiln]

Denascite

[marble wharf]

or you might want to plug in $z=-1$ and get $\sum a_n (-1)^n$

[solid kiln]

Denascite

[marble wharf]

or $z=17$ which would give you $\sum a_n 17^n$

[solid kiln]

Denascite

[marble wharf]

all of these are different series

they might converge or they might not

the radius of convergence tells you for which z it converges

[fresh pendant]

can this sequence be an example

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@fresh pendant Has your question been resolved?

[nova spire]

well take the series $\sum (-1)^nz^n$ for example then

[solid kiln]

rafilou2003

[nova spire]

radius of convergence is 1

and for |z| < 1, it converges to 1/(1+z)

but if you take |z| = 1, in this example it's impossible for the series to converge

since |(-1)^nz^n| = 1 which doesn't converge to 0

so the series doesn't converge on any point of the boundary

another example : a_n = 1/n

becomes the series $\sum \frac{z^n}{n}$

[solid kiln]

rafilou2003

[nova spire]

radius of convergence is again 1

but here if |z| = 1, the series may or may not converge

if z = 1 then $\sum \frac{1}{n}$ diverges, harmonic series

[solid kiln]

rafilou2003

[nova spire]

if $z = -1$ then $\sum \frac{(-1)^n}{n}$ converges, alternating series

[solid kiln]

rafilou2003

[nova spire]

so it depends

@fresh pendant depending on a_n, the series may or may not converge depending on which point of the edge of the disk of convergence we're looking at

[fresh pendant]

ok so basically The radius of convergence R is the distance within which the series converges. For ∣z∣<R, the series converges, and for ∣z∣>R, the series diverges. On the boundary ∣z∣=R, the behavior of the series can vary—it may converge at some points and diverge at others?

[nova spire]

exactly

[nova spire]

note that what you wrote is for a power series centered at 0

$\sum a_n z^n$ specifically

[solid kiln]

rafilou2003

[nova spire]

for a series centered at a point $a$, it's gonna be $\sum a_n (z-a)^n$

[solid kiln]

rafilou2003

[nova spire]

and in this case the radius of convergence is "what is the MAX distance possible can you go from point a and still hope to converge"

because if you go further than R, meaning |z-a| > R, it diverges

meanwhile if |z-a| < R it always converges

so |z-a| = R are the cases we don't know about

[fresh pendant]

Very clear

Tysm

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[gaunt temple]

is f(x) = xsin(1/x) continuous on [0,1] ?

[gaunt temple]

f is not defined for x=0 so i am a bit confused by this question

[grim sparrow]

I don’t think a function can be continuous at a point it’s not even defined at

[gaunt temple]

im confused by point c

theorem 4.4.7 states that if a function is continuous on a compact set K then it is uniformly continuous on K

[grim sparrow]

(0,1), not [0,1]

[gaunt temple]

yes but the solution says that it is continuous over [0,1]

[marble wharf]

more precisely, it can be continuously extended to x=0

[hasty yew]

define f(0)=0

[gaunt temple]

what does this mean?

[hasty yew]

take the limit as x-->0

then define f(0)= that

[marble wharf]

there is a function g(x) such that g(x)=f(x) for all x in (0,1) and g is defined on [0,1] and g is continuous

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[gaunt temple]

so if f(x) = xsin(1/x) is continuous at (0,1] then i can extend the continuity over [0,1] by making a new function g(x) defined as g(x) = xsin(1/x) for x in (0,1] and g(x) = 0 for x=0 ?

and this can be done because 0 is an accumulation point, i.e. lim x-->0 f(x) = 0

conversely, a function like f(x) = 1/x can't be extended to be continuous at 0 because there's a vertical asymptote at x=0 and x is not an accumulation point

[gaunt temple]

so h(x) is not continuous at [0,1] because it is not defined at 0, but if we make a new function m defined as m(x) = xsin(1/x) if x is in (0,1] and m(x) = 0 if x = 0, then m is continuous

correct?

[marble wharf]

yes

but then we dont often care to explicitly mention all that stuff, so we just say that h itself is continuous

with the implicit understanding that we actually do this entire construction

[gaunt temple]

okay thanks very much 🙂

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[subtle ether]

Im rly confused here

the mark scheme says to work out the interquartile rang and whatever

and Ik how to do that ez

but idk why am I doing it

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[viscid veldt]

the question is

what I don’t understand is

why is this person

using the first derivative

[wraith hinge]

isnt the first derivative to find the gradient?

[viscid veldt]

yes

I think so

[vagrant marsh]

stationary point is when the first derivative is 0

[viscid veldt]

I see

ty

so essentially
first derivative

• related to gradient
  when dy/dx = 0, this is stationary
  second derivative
• related to nature of stationary point
  d2y/dx2 > 0, minimum
  d2y/dx2 < 0, maximum

[vagrant marsh]

second derivative is related to concavity

i.e. concave up or down

or inflection point

[worldly wing]

You can’t decide if a point is a stationary point or not just by using the second derivative

You need the n’th derivative test for that

[viscid veldt]

is this correct?

[viscid veldt]

let me check what this is

[viscid veldt]

ik

by those 3

I shortened it into “nature”

that’s the term my teacher uses

[viscid veldt]

this isn’t within my syllabus

but I see

ty

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[young minnow]

how do i find the equation of a reciprocal graph?

[young minnow]

v

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[wraith hinge]

Here, if x = 4, then thats 1/-1 which is = -1 right? so how is it the case as shown for x > 3 is 1 please?

[wraith hinge]

nvm i was tired, figured it out.

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[wraith hinge]

I know that the integral is going to be from 0 to 1 at the beginning

from there on I don't where to go from

[uncut aspen]

do you know how to find the volume of a revolution around the x-axis?

[wraith hinge]

yes

[uncut aspen]

so it’s basically the same, except you integrate in terms of y

so instead of dx, it’s dy

otherwise it’s the exact same

[wraith hinge]

So like this

[uncut aspen]

don’t forget to multiply by pi

and you forgot a parentheses at the end

before the squared

but other yea that’s good

[wraith hinge]

And the sqaure goes to everything making it 4tan^2(pi/3y)

My notes didn't include tan so that's why I'm having difficulty

[uncut aspen]

that’s what i thought

you can convert tanx into sinx/cosx

and then do u-sub

[wraith hinge]

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[wraith hinge]

hello so I was wondering if this formula would hold if instead of cos x it's cos(x + constant) and sin x is sin (x + constant)?

[nimble stone]

is it the same constant?

[wraith hinge]

Yes

[nimble stone]

then yes

[wraith hinge]

okay

I have another question

A question asked for the period of 1/2 cos(2t) + 1/4sin(t) + 3. Idk how to do this is there like some similar formula I can do to make the trig one function or something

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[winged solar]

1+2=3 1/3=a???

[winged solar]

my question and answer

I dont understand how they got -1 when solving for a

[warm nymph]

It's negative

See it increases as x gets closer to 3

And then starts decreasing

[winged solar]

is a always just a 1 or -1

[warm nymph]

Think about it like 2 -1/3 |x+3|

As x gets closer to -3 the function gets closer to 2

Because ur subtracting less and less

After x= -3

You start subtracting more and more

[winged solar]

so a is just displaing that>

?

[warm nymph]

So it increases and then decreases

If it was 1/3 | x+3 | +2 then jt would be the oposite

You would be adding less and less until x=-3

And then start adding more and more

[winged solar]

so i wouldnt solve for a I would just see if its positive or negative?

or is a the slope

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@winged solar Has your question been resolved?

[winged solar]

I dont understand whats wrong

[split chasm]

how did you get 2(x-1)

[winged solar]

slope is positive 2

nearest x point is (-1,0)

[split chasm]

and then?

[winged solar]

y = 0 so thats it

[split chasm]

wdym by that's it

you have your point and slope

what exactly did you apply

[winged solar]

I used my slope for a

[split chasm]

what formula are you applying to get the final equation

[winged solar]

and used my point for h and k

just the one it showed me when i got it wrong

[split chasm]

state the formula

[winged solar]

i think its a(x-h)+k

wait so it would be +h then

cuz its a negative

[split chasm]

it'd still be -h

that doesn't change

it's just that the value of -h isn't -1

[winged solar]

2(x+1)

[split chasm]

yes

[winged solar]

2(x-(-1))

it was correct

thank you

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[eager lynx]

Why is this not64x^4 because 16/4 is 4 but 256/4 is 64 no?

[tulip violet]

You're doing 256^(1/4) not 256/4

With values, you're taking that root

So the 4th root of 256

[eager lynx]

oh you're right

my bad

thank you

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[subtle ether]

for b I'm confused

how to know what OD is equal to?

[earnest nymph]

you can find what AD is equal to and therefore what OD is

cause its a parallelogram AD and BC should be equal

[subtle ether]

oh I thought BD and AC were equal

I drew the parrlalogram wrong then

[earnest nymph]

thats also right tho

[subtle ether]

oh crap did I just make algebra mistakes then

[earnest nymph]

wait, idk if im getting direction correct cause im just looking at this

[ocean mural]

Add directional vector BC to position vector A

[subtle ether]

also shouldn't there be 5 points

with the origin?

like how is it a parallelogram

[ocean mural]

ABCD form a parallelogram

[earnest nymph]

ABCD is the parallelogram

yea

[ocean mural]

We just know the position of A,B,C,D with respect to O origin

[subtle ether]

Where is the room for d?

[subtle ether]

Ohh

Thanks

I got BC/AD

not sure what to do now

oh nvm

kl

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[keen orchid]

Which point in a tetrahedron has the minimum total distance to the vertices of the tetrahedron?
i.e. P in tetrahedron ABCD with minimal AP+BP+CP+DP

[dusky gust]

you can find it easily,

show what you have tried

[keen orchid]

this is for disphenoids

[dusky gust]

one way to do it first find in a triangle, then extend it to the line perpendicular to it, then differentiate to find minima

and if you know thedefinition of centroid

you can do it quickly

[keen orchid]

Fermat point?

[dusky gust]

yeah

[keen orchid]

What if I find the Fermat points of each face and connect the perpendiculars

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[keen orchid]

<@&286206848099549185>

[keen orchid]

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[outer vector]

can someone please help me solve this (sorry if i am not supposed to be posting here its just confusing on where to post my questions)

[split chasm]

posting here is fine

what have you tried so far?

[outer vector]

i have tried changing to sin2x/cos2x

but im still ocnfused on what to do

and then i tried multiplying cos2x on both sides to get sin2x by itself

idk if thats right

[split chasm]

its fine, but unnecessary

if you know how to work with tan

do you know how to solve
sin(2x) = 0

[outer vector]

no

im confused on the 2

like '2'x

[split chasm]

do you know how to solve sin(x) = 0

[outer vector]

i solved sinx=0

yes

i jus looked at a unit circle

[mild oxide]

Yo i came

[outer vector]

and my answer was 0, 180 and 360

[split chasm]

ok.

so instead of getting
x = ...
directly

for sin(2x) = 0,
you'll have
2x = ...

then divide everything by 2 to get x

[outer vector]

i seee

what do i do with the tan tho since it equals 0 do i find any quadrants or something

ik what to do if its neg or pos but what if it = 0 what happens then

[split chasm]

tan(x) is 0 in the same place sin(x) is 0

[outer vector]

how do i solve that

like how i did before?

[split chasm]

yeh

[outer vector]

okay

[split chasm]

the work you did leads to

sin(2x) = 0

[mild oxide]

See inactive

[outer vector]

okay

[outer vector]

yes

[mild oxide]

Do u know when sin x is 0

In general

I mean sin(2x) is

[outer vector]

no

😓

how do i figure that out

im generally strugglign with any of these types of questions ending with = 0

and with '2'x but im starting to understand the 2

[split chasm]

$\sin(x) = 0\
\sin(\this) = 0$

[solid kiln]

ℝαμΩℕωⅤ

[split chasm]

list the solutions for "this"

they'd be the same as the solutions for the original x

[outer vector]

for sinx=0 i looked at a unit circle

so if i try solving without the unit circle

how do i do it

[split chasm]

know special angles, and periodicity of the trig functions

unit circle definitions of trig functions

sin(t) gives the y-coord of the point on the unit circle at angle t

[outer vector]

oh okay

im still stuck

so we're at sin2x = 0

[split chasm]

first list the angles at when
sin(this) = 0

'this' here just happens to be 2x

this next step very similar to what you'd do when solving sin(x) = 0

you've already identified that
sin(x) = 0
when x= stuff like 0,180,360
so sin(2x) will equal 0 when
2x = stuff like 0,180,360
(or their radians equivalent)

however note that since you're being asked for solutions of x in [0,2pi], you'd need to consider values where 2x is in [0,4pi] or [0,720°]
so you'd need a few more values

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[outer vector]

i seeeee

so 2 revolutions

if it = 0

do i consider using quadrants or no

like for anything > 0 i find quads but when it = 0

do i still use quads

[split chasm]

angles on the axis aren't considered to be part of quadrants

[outer vector]

okay

so we dont work with quadrants here

how do i work out the rest then

[split chasm]

consider periodicity and what you observed from looking at the unit circle

[outer vector]

do i go another time

like 2 revolutions

since its sin2x

[split chasm]

yes

[outer vector]

so pi, 2pi, 3pi, 4pi

[dusty sleet]

0 as well

[outer vector]

oh yeah

the answers on the worksheet r different tho

[dusty sleet]

Indeed, you haven't solved it yet

[outer vector]

oh sike

lmao

[dusty sleet]

Your answers are in terms of 2x so far

[outer vector]

oh yeahhhh

and so now we divide it all by 2????

omggg

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[next cipher]

i dont undertsand this

[lyric ether]

Do you understand what a reciprocal function is?

[mild oxide]

Do u understand what is a reciprocal?

[next cipher]

Its like a graph from what i remember

like this i think

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[vale abyss]

how did they solve for phi here?

[vale abyss]

oh I see

its the only value where they are equal

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[vale abyss]

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✅

[vale abyss]

How would you find phi in this case?

I got cot(phi) = sqrt(5)

but Im not sure how to solve for phi

with the basic unit circle

[vale abyss]

oh

nvm

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[radiant portal]

derivative of 2x^3 - 6x^2 > 0

[radiant portal]

2x^3 - 6x^2 > 0

6x^2 - 12x > 0

(6x)(x - 2) > 0

I know im supposed to get an answer x> something and x< something right?

[cerulean spruce]

I don’t know if you can differentiate within an inequality

[radiant portal]

This is the original question

[cerulean spruce]

Ohhhh

[radiant portal]

I want to find when its increasing, so I differentiate and set it > 0 right?

[dusty hound]

reduee it to a set intersection

[cerulean spruce]

^^

[radiant portal]

wym sry?

[dusty hound]

{f is concave down} n {f is increasing}
Both sets you know how to find

[radiant portal]

Yeye I think I can solve it, I was just wondering about this ^^ Like does dividing both sides by (x-2) count as dividing by a minus? to switch the sign direction

Like is it (6x) > 0 and (x - 2) > 0

or (6x) < 0 and (x - 2) > 0

[hallow cliff]

If x is greater than 2, will dividing by x-2 be like dividing by a plus or a minus?

[dusty hound]

the only correct and general thing you can say about handling inequalities is to be extra careful

[radiant portal]

I feel like its this one

But I thought I was supposed to get an x> and x<

Not x> and x>

[radiant portal]

plus

[hallow cliff]

Yeah, so you have to make two cases: when x>2, and x<2. Dividing by x-2 will be like dividing by a plus and a minus respectively

Then i hope you can see how to solve your inequality

[radiant portal]

I think so haha

Thanks guys!

❤️

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[queen lark]

I don't want the answer, just clarifications about what is asked. Question says to replace the ? by an explicit number so that the equalities are true for any x with denominator being different than 0.
Does the ? mean like a y constant or something ? Like, it should be only one number, it can't be different numbers ?

[dusty hound]

yes you're looking for 3 constants to replace the ?

[queen lark]

thank you !

So they can be different

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[dim kite]

hello

[dim kite]

im not really sure how do to this

i know how to determine if something is a subspace of another vector space

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[solemn wagon]

Hello, how does one come up with the formula for 1^2+2^2+...+n^2?

[viscid flower]

i saw a suggestion that was like

well, you might guess that, because the sum of the first n numbers is quadratic in n

that the sum of the first n squares may be cubic

so you guess it has some form like $an^3+bn^2+cn+d$

[solid kiln]

jan Niku

[solemn wagon]

yeah now its simple

[viscid flower]

and then solve for these coefficients

you have no reason to believe this will be true

[solemn wagon]

but the fact that it has to be a polynomial

[viscid flower]

its just a guess

[solemn wagon]

doesnt really click with me

[viscid flower]

do you know any calculus

i mean just the power rule provides some loose motivation for why you might expect this relationship

[cedar onyx]

caluculus

[solemn wagon]

i know basics of calc but not too indepth

if familiar with rules of derivatives

[viscid flower]

so you know for example like

take some linear function $x$

[solid kiln]

jan Niku

[viscid flower]

the area under this function $x$ from $x=0$ to $x=n$, say, is $\frac{n^2}{2}$

[solid kiln]

jan Niku

[viscid flower]

which we get by using FTC, power rule

maybe you can see where I'm going?

you can kind of think of the sum of the first n numbers as something like this

were adding up areas

theres a rectangle 1 wide, 1 high for n=1

then a rectangle 1 wide, 2 high for n=2

[solemn wagon]

ooh

[viscid flower]

all of these sit under the function f(x) = x

[solemn wagon]

yes

[viscid flower]

so our area of rectangles is gonna be less than this formula

and it is

sum of rectangles is $\frac{n(n-1)}{2}$

[solid kiln]

jan Niku

[viscid flower]

instead of $\frac{n(n)}{2}$

[solid kiln]

jan Niku

[viscid flower]

so, you might reason that if we instead took $f(x) = x^2 \dots$

[solid kiln]

jan Niku

[viscid flower]

hopefully that makes sense

[solemn wagon]

yes it does

[viscid flower]

you know what im actually wrong lol

but the same reasoning works out

[solemn wagon]

why

[viscid flower]

here's our actual picture

calculus gives us the area under the black line

[solemn wagon]

yeah

[viscid flower]

the sum 1+2+... gives us the area under the green rectangles

so we actually expect that formula to give us a bigger answer

and it does

i got the formula wrong

its actually $S_n = \frac{n(n+1)}{2}$ instead of $\frac{n(n)}{2}$ by calculus

[solid kiln]

jan Niku

[viscid flower]

sorry for the minor mistake

but the reasoning is the same

[solemn wagon]

no worries

I didnt catch it either cuz I wasnt focused on the details

[viscid flower]

heres the picture for n^2 vs x^2

you know the formula for area under the black curve, by just integrating

[solemn wagon]

yes similarly

[viscid flower]

idk im just tracking out the reasoning you seem to believe now anyways

[solemn wagon]

okay

ill do it myself

[viscid flower]

sure

its a system of 4 equations

but if you know any intro linear algebra

or feel comfortable solving systems

youll make quick work of it

[solemn wagon]

yeah its just computation

[viscid flower]

yea

you need 4 data points, but thats easy, you can just calculate them

yadda yadda

derive the formula, simplify

youll be there

[solemn wagon]

thank you!

[viscid flower]

np

[solemn wagon]

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[rancid ravine]

Hi

[rancid ravine]

This is my question

Work so far

I need to sketch a possible graph for y = f(x)

I also know that I have to find the antiderivative to do this

[nimble stone]

you have everything you need really
what do you think its general shape is?

[rancid ravine]

Well all i need to know is the antideriavtive of the function

y = 3x^2 -3

[nimble stone]

sure i suppose, though you dont need to

[rancid ravine]

Also is part a and b correct so far

Ok forgot the antiderivative part but how can I graph this

[nimble stone]

your table does not seem right

[rancid ravine]

Oh

What is wrong in it

[nimble stone]

everything after the first box

[rancid ravine]

Then what should i put in it

[nimble stone]

explain where you were getting your numbers and results from

[rancid ravine]

Well x = -3 is in the graph and y = 0

So i wrote zero

[nimble stone]

the graph is not 0 at x=-3

the y value there isnt even on the graph

[rancid ravine]

Well idk what the intervals are

[nimble stone]

have you really looked at the graph?

[rancid ravine]

Yes but i dont understand it

I am confused thats why i came here

[nimble stone]

what dont you get though

f'(x) is positive when its above the x axis, negative when below, and 0 when intersecting it

[rancid ravine]

Oh so is it like x < -3, -3 < x < 0, 0 < x < 1 and x > 1

What are the intervals thats all i am asking

[nimble stone]

x<-1, -1<x<1, x>1

[rancid ravine]

oh thats it

I thought it was 5 intervals

[nimble stone]

i wasnt counting the intersections

just slot them in

x=-1 and x=1

[rancid ravine]

ok

The rest is okay right

The second column

[nimble stone]

do you mean row?

[rancid ravine]

yes

[nimble stone]

if you fix the titles, then sure theyre fine

[rancid ravine]

Yes still what would the shape of the graph be

[nimble stone]

if f'(x) is a quadratic, what must f be

[rancid ravine]

Linear

[nimble stone]

youre going the wrong way

[rancid ravine]

Cubic

[nimble stone]

there you go

that would be its general shape

[rancid ravine]

Can you send a image of a cubic

Idk how to draw it

[nimble stone]

,w graph x^3

[solid kiln]

Results provided by WolframAlpha

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Requested by aldrnari_

[nimble stone]

you still need to use the info you have though to make it more accurate

[rancid ravine]

ok thank you

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[wraith hinge]

Hello I need help with multiple questions but this is one of them. Idk how to apply log laws to questions

[wraith hinge]

The correct working out is just this but idk how they got x^2/x^2

[split chasm]

exponent/radical laws

convert sqrt(x) to exponential form
or x^(3/2) to radical form depending on personal preference

also don't use x for multiplication especially when variable x is present

[wraith hinge]

I tried, wouldn’t that just be x^1/2 for sqrt x ?

And idk what radical form is

[wraith hinge]

Okk 👍

[split chasm]

where you see stuff like $\sqrt{}$, those are called radicals

[solid kiln]

ℝαμΩℕωⅤ

[wraith hinge]

How is 3/2 radical

Oh

But why would i do that

Wouldnt that make it harder to solve

[split chasm]

it's about the same

[wraith hinge]

3 sqrt x?

[split chasm]

no

just stick with what you did earlier for now

[wraith hinge]

Oh

What

[wraith hinge]

This isn’t my working out

This is the right working out but idk how they got that

Especially 2nd line

[split chasm]

after converting sqrt(x) to x^(1/2),
you'll have
x^(1/2) * x^(3/2)
on the numerator
can you simplify that?

[wraith hinge]

X^1/2

[split chasm]

no

[wraith hinge]

….

Oh

Im so stupid

💀💀💀💀💀

Oh ok

Wait

I have other questions if you have time

[wraith hinge]

I got this one, my brain was just malfunctioning

?

[split chasm]

what's your q

[wraith hinge]

It says express the following as logarithm statements with the index as the subject for 4^3/2 = 8

The index is 4^3/2 isn’t it? How do i make that the subject in a log statement?

Nevermind I got it, thank u

.close

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[atomic seal]

is this solution correct?

[atomic seal]

the first 2 lines are the question

pls ping

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[potent pewter]

Can someone explain me this

Ik if g(x)=0 it would be ND

[split chasm]

such that what

what's the problem

[potent pewter]

Like to solve p(x) /g(x)

[split chasm]

!original

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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.

[potent pewter]

We find zero of g(x)

[potent pewter]

I was just trying to understand it

So for example

[split chasm]

show the full thing

[potent pewter]

If g(x) is x-1

[split chasm]

its cut off

[potent pewter]

Ok

It isn't related but ok

[ocean siren]

!original

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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.

[potent pewter]

The half word is find

[split chasm]

ok. that's most definitely related

[potent pewter]

But I'm just asking

That

Like for example g(x) = x-1

It's zero is 1

So if we assume x value as 1

It makes g(x) =0

So I don't get it

[split chasm]

wdym

i don't see the issue

[potent pewter]

What does it mean by g(x) is not equal to 0

Does it mean that g(x) literally can't be 0

Or that whatever polynomial is there it can't be equal to 0

It says g(x) shouldn't be equal to 0

[split chasm]

means that g(x) isn't the constant 0

[potent pewter]

Oh

Ok

Thx

.close