help-33

[topaz tide]

wondering if its correct

[latent coral]

your expression P(t) is correct

[topaz tide]

My online notes only show examples of them finding expressions where the base was e. Is there a way to find an expression using e?

[latent coral]

2 is the same as e^(ln(2))

technically 500(2)^(t/15) is a totally accurate "exponential model"

if they don't tell you it has to include e then it might be unnecessary, but you can rewrite P(t) as 500 * (e^ln(2))^(t/15) = 500e^(t * ln(2) / 15)

,w 500(2)^(87/15)

[elfin berry]

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[topaz tide]

@latent coral great thanks for the help! clarified it for me 😄

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[marsh citrus]

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[timid sedge]

I need help with this:

[still temple]

integration

[lusty heart]

more specifically when integrating to find distance, make sure to use absolute value since distance cant be negative

[woven plinth]

absolute value of the part where the integral function is negative

[timid sedge]

This is helpful but class was canceled at my uni due to a massive power outage so I was never taught how to actually do this.

So I dont even know the first step or how to do it

[woven plinth]

are you familiar with power rule?

[timid sedge]

yup

and its inverse

[woven plinth]

yep

so remember that you integrate velocity to find distance/displacement

[timid sedge]

Do I integrate the v(t) equation?

[woven plinth]

yeah

[timid sedge]

aahhh

ok

So its t^3/3 - t^2 - 8t

Then what

[woven plinth]

then you plug in your bounds

2 and 7

[timid sedge]

F(b)-F(a)

right?

[woven plinth]

yep

[timid sedge]

sweet thank you so much.

[woven plinth]

well yea for displacement at least

[timid sedge]

Got it

Then for distance?

[woven plinth]

^

so you might need to split the bounds up depending on whether the function goes negative or not

for example: if the integral function is negative on (4,7) you would take ( f(4) - f(2) ) + | f(7) - f(4) |

does that make sense

[timid sedge]

Not really sorry

Like do I integrate the abolute value of v(t)?

im lost

[woven plinth]

ok wait let me graph it

[timid sedge]

may not respond for a bit but thank you so much I have to run an errand

[timid sedge]

if you could dm me a detailed explanation or walkthrough of how to do the second part that would be so helpful

[marsh citrus]

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[marsh citrus]

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[dense hemlock]

am i tweaking

[harsh falcon]

it depends on t, not x

and for the second part, look at what it actually wants

for example, look at the units

they don't want the amount of bacteria

[dense hemlock]

OH

[harsh falcon]

yeah i misread that as well at first

[dense hemlock]

letsgooooo

thanks

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[cunning elbow]

Isolate sin(theta) using algebra, then use arcsin (presumably a calculator if you're allowed to) to solve for theta

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@tall sun Has your question been resolved?

[tall sun]

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[marsh citrus]

@edgy sleet Has your question been resolved?

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[rocky nexus]

What does the d and y axis label mean

[marsh citrus]

@rocky nexus Has your question been resolved?

[rocky nexus]

<@&286206848099549185>

[winter sedge]

No idea about these notations, do you need any help in understanding how we can plot it?

[rocky nexus]

i can ploit it

but it wont let me

if i dont get these first

[pliant inlet]

x: 5/6 pi

if then it seems like a |x| function

[rocky nexus]

whats the

y

[rocky nexus]

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[mortal lintel]

Calc 1:
I am dealing with using the chain formula, but I am stuck on this one
It's got a lot going on and I get the idea, but its confusing me.
Specifically with finding the derivative of 4^cos(x^6)-1.
Chat GPT says I have to use ln(4) but I have no idea why.

[shadow venture]

So if we want to differentiate 4^x how do we do it?

[mortal lintel]

xa^x-1

[shadow venture]

Ah no not quite

So we are differentiating with respect to x

Do you know that the differential of e^x is e^x ?

[mortal lintel]

oh right I was thinking backwards like x^a

[shadow venture]

Yeah, common mistake, I do it all the time

[mortal lintel]

right yeah e^x just = e^x I understand that

[shadow venture]

So if we want to do a^x, well we don’t really know what it is off the top of our heads

So we want to write a in terms of e

So a = e ( ln (a))

[mortal lintel]

oh interesting

[shadow venture]

Which seems a bit weird until you do a^x = e( ln(a) * x )

So then you just differentiate e( ln(a) *x ) which is by chain rule, ln(a) * e( ln(a) *x )

Hope this helps 🙂

[mortal lintel]

😵‍💫

[shadow venture]

So it’s e^u * du / dx

Where u = ln (a) * x

You’ll have to work through this example yourself to get a proper understanding

Than hopefully your question will make sense

[mortal lintel]

Yeah I appreciate it. Just super confusing

[shadow venture]

For sure

[marsh citrus]

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[marsh citrus]

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[rocky moat]

Number 4 is confusing me

This is the lines graphed

Idk what it is asking me or what a lattice point is

How am I supposed to divide this up into 7 regions?

[brave marsh]

Lattice points are points (a,b) where both a and b are integers.

[rocky moat]

Can they be anywhere?

[brave marsh]

Yes, but here they are only concerned with those inside the triangle

[rocky moat]

Wait

Hold up I think I found the seven regions

[brave marsh]

Yes, it's just how the lines delimit the triangle.

The triangle itself isn't split in 7 regions, it is one of htem

[rocky moat]

Nvm

[rocky moat]

Huh??

Wdym delimit?

[rocky moat]

OHH

So how am I supposed to calculate the lattic points?

Am I able to calculate it or do I have to use the graph?

[smoky plover]

What are the intersection points?

The coordinates, just wondering

[rocky moat]

They aren’t integers

I’ll get it for u

(1.011,3.078)

The red and blue intersection ^

Blue and green: (12.283, 3.945)

Red and green: (2.405, 12.835)

[smoky plover]

Hmm I thought you might be able to shift the plane and cleverly apply picks theorem but the numbers don’t work

I gtg but it’s probably a tricky application of picks

[rocky moat]

Yea one of my classmates asked the prof and he said you can’t use it

[smoky plover]

Bruh

[rocky moat]

Ikr 😭😭

[smoky plover]

Ig just casework

Idk a better way

[rocky moat]

I never heard of that theroem but whatever ☹️

Is there some type of theorem to calculate lattice points or no?

[brave marsh]

One way is to see that he triangle is bounded, so you can draw a square around it and count the lattice points in there that are withing the triangle's bounds (think of it as inequalities)

[rocky moat]

Bounds?

Lemme draw a square and see if I get what ur saying

[brave marsh]

Like. Points inside the triangle respect a certain set of inequalities.

[rocky moat]

?

[brave marsh]

Yeah well you don't need the square to be exactly around it.
The idea is that if you draw a square with vertices on lattice points, then say a square with side c will have exactly c^2 lattice points inside it.

[rocky moat]

Ohh

[brave marsh]

And since the square is finite you can check the lattice points yourself

[rocky moat]

So a square with all it vertices on lattice points and use picks theorem?

Idk what picks theorem is but I’ll figure out how to use it

I assume picks theorem is to find lattice points?

[brave marsh]

A square or a rectangle would work as well

[rocky moat]

Interesting

[brave marsh]

Picks theorem gives you the area of a polygon, but its vertices must be lattice points.

It won't help too much.

[rocky moat]

So how am I going to find lattice points?

[brave marsh]

What I mean is if there are c^2 lattice points in there, well c^2 won't be that big so you can check them

[rocky moat]

C²?

What is c?

[brave marsh]

The sidelength of the square you draw

[rocky moat]

Ohh okay

[brave marsh]

But I suppose the square needs to be quite big here, so it might be too many points to check

[rocky moat]

I’ll try to figure it out now

Thank you!

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[fresh copper]

Is it increasing, stationary, decreasing and why

[final skiff]

!status

[marsh citrus]

What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above

[fresh copper]

I dont know where to begin.

1*

[final skiff]

If the derivative of a function is positive do you know what that means for the original function

[fresh copper]

not too sure

that means

the slope is positive?

of the tangent

i think

[final skiff]

Yes

[fresh copper]

so its incr4easing

OH

its increasing

[final skiff]

Yes

[fresh copper]

OK

so wait

the question is asking

what is the bevavior of the 1st derivative if the 2nd one is greater than zero

its increasing because its over 1 means it positive

correct?

ok i think thats right, thank you @final skiff

[final skiff]

Idrk know what you mean by “its over 1”

But since the derivative is a function itself, then the second derivative is the derivative of the 1st derivative so we apply the same logic as before.

More mathematically we can let g(x)=f’(x). Then g’(x)=f’’(x).

[final skiff]

Then use your answer to this

[fresh copper]

ohhh ok

i meant positive

not over 1 idk why i said that

[final skiff]

Yeah that is correct

[fresh copper]

thank you

have a great day unless i see u again

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[near meteor]

Could anyone check my working so far, and help me with where to go next with this implicit differentiation problem
Image

[marsh citrus]

@near meteor Has your question been resolved?

[sinful shuttle]

i dont think anythigns wrong here

[near meteor]

Cool I’m just struggling a bit with algebra and what the next step would be

Trying to write it in terms of y’ on the left

Do you think you could tell me what to do next

[marsh citrus]

@near meteor Has your question been resolved?

[marsh citrus]

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[hard cave]

Could someone check my answers/help with the ones I haven’t done^^

[marsh citrus]

Please don't occupy multiple help channels.

[hard cave]

Oh I closed the other

[hard cave]

^^*

[marsh citrus]

@hard cave Has your question been resolved?

[warped bane]

hello!

[white brook]

They have a rule about assisting with tests/exams. Not sure if that extends to quizzes, or I'd have attempted to assist.

[hard cave]

Oh it’s not a test or an exam

It’s a test my knowledge/just a homework

I attempted the bottom ones

[hard cave]

hii!

[marsh citrus]

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[raw quest]

im not rly sure how to go w this

i changed it to like

$$2n!/3!(2n-3)! * 3!(n-3)!/n! = 11$$

[elfin berry]

✯✡ρєввℓєѕ✡✯

[raw quest]

but then idk what to do next

[marsh citrus]

@raw quest Has your question been resolved?

[late geode]

ttry simplfying the factorisals

[raw quest]

im not sure how

[late geode]

do you know how factorials work

and/or the definition of factorial

[raw quest]

yes

but i dont know how to

simplify

2n!/n!

[late geode]

you don't

[raw quest]

cos i know it will technically

oh

[late geode]

you're focussing on the wrong component to simplify

[raw quest]

then what do i do next

rly

OHHHH

OHHHH

okok

i think i got it now

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[vestal rock]

If I am trying to find the sine of angle theta

I have to do opposite over hypotenuse, correct?

However, my work source says the answer is 0.909

11.9/28.7= 0.414

not 0.909

[latent coral]

the opposite side to the angle theta is 26.1

so sin(theta) should be 26.1 / 28.7

[vestal rock]

omg what a dumb mistake

makes sense now, ty again

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[lethal bridge]

how do i find this area

[keen stump]

imagine unrolling that rectangle

what are the side lengths corresponding to when you reroll it

[lethal bridge]

2rh

[keen stump]

no

thats the top part

[lethal bridge]

idk then

[keen stump]

use the circumference

[lethal bridge]

i got it

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[vestal rock]

what happens to the graph of a trigonometric function if it has a positive phase shift?

[vestal rock]

positive phase shift indicates movement to the left, but does anything else happen?

[devout mauve]

what kind of thing do you want to happen

[sour timber]

only change is moving to the left

[latent coral]

nah just horizontal translation

[sour timber]

nothing else

because x values change

[vestal rock]

horizontal translation,,, only to the left?

[vestal rock]

nothing specific, I just should know what will

[latent coral]

depends on whether the phase shift is positive or negative

if i have a trig function like sin(x), then i consider sin(x + c)

when c > 0, this translates sin(x) to the left. when c < 0, this is a horizontal translation right

[vestal rock]

I see

[vestal rock]

and for positive change it will always move to the left?

[latent coral]

for positive change?

as long as the number being added to the x is positive, yes it's a shift left

[vestal rock]

alr ty

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[turbid heath]

Help with all starting with d

[turbid heath]

i was thinking for d) 2sinx = cos 2x

Cos 2x is Cos ^2x - Sin ^2x ( i dont know how?

And Cos ^2 x is 1- sin^2 x

So we get 2 sin x = 1- 2sin ^2 x

[marsh citrus]

@turbid heath Has your question been resolved?

[blissful olive]

cos2x=cos^2x-sin^2x

An identity

@turbid heath

$2sin^2x+2sinx-1=0$

[elfin berry]

L'empereur

[marsh citrus]

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[blissful olive]

That's a quadractic in sinx

[marsh citrus]

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[atomic sparrow]

Is this correct

[lavish onyx]

you can use symbolab to see if you've got the right answer

[atomic sparrow]

Whats symbolab

I dont trust ai

Ive already messed up too much cuz of trusting ai

[lavish onyx]

its a website, no ai biz

lemme show you real quick
nvm

[atomic sparrow]

Can you help here for now?

I am not willing to go on a website

[spice frost]

its correct

[atomic sparrow]

Tysm

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[atomic sparrow]

But the highlighted 3 didnt got multuplied

[spice frost]

wait

i was just tryna figure out

why it wasnt right LMAO

i didnt catch that

\

gimme ur new answer

my bad :(

[atomic sparrow]

I cant believe an Indian maths guy betrayed me

[spice frost]

im not indian 💀

[atomic sparrow]

The video i was watching was Indian

[spice frost]

oh LMAO

alr alr

yo my bad tho i trolled

☠️ ☠️

[atomic sparrow]

I was tryna learn rationalisation

[atomic sparrow]

Why subtract them? Its =

[spice frost]

well yes

but if u subtract them and it equals 0, then its the same thing

[atomic sparrow]

Why it isnt the same 😭

Welp how do I learn rationalisation of such type of questions

[spice frost]

u were doing it right

lemme just see why its not the same

hol on

hm u did do it right tho

i cant find any error

[atomic sparrow]

But i didn't multiply that highlighted 3

[spice frost]

nah u did

2sqrt3 * 3sqrt3 = 18

u did that correctly

[atomic sparrow]

3^2 =9 then *2

Thats what i did

[spice frost]

yup

oh its wrong bc

its not sqrt5, its sqrt15

[atomic sparrow]

Why would the 3 be squared

[spice frost]

you transcribed it wrong here

damn that was hard to catch

from 3√15 to 3√5

then its right

[atomic sparrow]

My point is how do I do 2 √3(2 √2 - 3 √3)

$2 √3(2 √2-3 √3)$

[spice frost]

distribute it

2√3 * 2√2 = 4√6

2√3 * 3√3 = 2 * 3 * (√3)^2 = 2 * 3 * 3 = 18

so 4√6 - 18

[atomic sparrow]

.reopen

[marsh citrus]

✅

[atomic sparrow]

Oh i got it

Can you give me a question of same kind and i try to solve it to see if i understand please? @spice frost

[spice frost]

try simplifying (2√2-3√5)/(4√2-√7)

[atomic sparrow]

kk I'll mention you once am done

[spice frost]

bet

[spice frost]

two of those terms in the end r incorrect

[stoic slate]

Your second step is wrong

(a+b)(a-b) is not (a-b)^2

[spice frost]

mhm

but bro still got the right number lmao

on the denominator

[marsh citrus]

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[marsh citrus]

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[atomic sparrow]

Can you show the correct way completely as am not able to understand

[atomic sparrow]

Or can someone else do it with explaining each step

[iron meadow]

$(a+b)(a-b)=a^2-b^2$

[elfin berry]

∫oosh (lemonsaurus appreciator)

[iron meadow]

$a^2-b^2\neq(a-b)^2$

[elfin berry]

∫oosh (lemonsaurus appreciator)

[atomic sparrow]

I mean the whole question

[atomic sparrow]

This

[marsh citrus]

@atomic sparrow Has your question been resolved?

[elfin berry]

faiyrose

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[tall sun]

is more information needed?

[vital oracle]

no that should be good enough

[tall sun]

aight

thanks!

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[fickle jungle]

how does one find the integral without finding the function

[fickle jungle]

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[fickle shell]

The white is the question the gree is the solution

I dont really follow their steps

u = x^2 - 3

du = 2x

[glacial hedge]

that appears to be the same thing to me, basically

[fickle shell]

Its the next bit

[fickle shell]

to get x I need to rearrange this to get sqrt(u+3)

Then subbing everything in I get sqrt(u+3) (u)^4 (2du)

Which is very messy and probably wrong

how did they get (1/2) (x^2-3)^4 (2x dx)

[glacial hedge]

u-sub when you don't do weird things

[fickle shell]

Yes I know I did something wrong and they did it right... what was I supposed to do

[glacial hedge]

no

[glacial hedge]

x²-3 is your u so that will be u^4
Then you have dx, want du
du/dx = 2x
So dx = du / 2x
Hence it's integral of x u^4 du / 2x
= u^4/2

And if you had x terms left over, then convert them to u

[fickle shell]

Okkkk I think I get it

Thank you!

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[peak spire]

I'd like to know what was done in 2-(3-x) to get to this

[mild perch]

Typo

Fixes itself on the next line

The first x is a 3

[peak spire]

Oh, it is actually just that

Ty

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[mild perch]

Np

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[humble sage]

i am completly stuck on how to go about solving this

the answer is 54/11 how would i solve this quuestion to get that answer

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[winter osprey]

Is this correct ?

[latent coral]

yes, you can also double check them by plugging them back in for x on the LHS and seeing if you get 0

[winter osprey]

One more

[latent coral]

yes

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[tall sun]

How do i do this without a calculator?

[tall sun]

how do i solve this without a calculator?

[dry prawn]

do you know what csc is in general?

[tall sun]

uh

[tall sun]

cosecant

[dry prawn]

right but it's related to another trig function that you've hopefully practiced calculating before

[tall sun]

oh

probably have not

i started trig about 2/3 days ago now

[dry prawn]

so you don't know how cosecant is defined?

[tall sun]

mmno

i can

look it up

[dry prawn]

i'm curious how it was introduced to you in the first place, then

[tall sun]

barely is how

independent study leads to a lot of dumb questios

ok so

co·​se·​cant (ˌ)kō-ˈsē-ˌkant. -kənt. 1. : a trigonometric function that for an acute angle is the ratio between the hypotenuse of a right triangle of which the angle is considered part and the leg opposite the angle.

that seems right

[dry prawn]

sure

is there another trig function that involves the hypotenuse and this "opposite" side?

[tall sun]

sine?

[dry prawn]

sine

sine and cosecant are related; your task is to figure out precisely how

[tall sun]

sorry, went for food

cosecant is the opposite right?

reciprocal

[dry prawn]

reciprocal is a better term, yes

so csc(330) = 1/sin(330)

[limber hearth]

Now you have to know how to find sin(330)

[marsh citrus]

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[winged ivy]

im so confused on how the first circled step turned into the second circled step

[winged ivy]

im not so good at math but they transposed the 5/6r to the other side right?

how did it become 19/6 r??

[final skiff]

they subtracted 5r/6 from both sides

[winged ivy]

OH

WAIT

im stupid

sorry

thanks lmao

[final skiff]

np

[marsh citrus]

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[tropic hill]

Am I misunderstanding the idea of a null hypothesis? I'm doing a review assignment and I want to make sure I understand everything before I move on

[trim hornet]

so your answer is incorrect?

[tropic hill]

I want to know if it is before I submit it

I want to be sure

The assignment can't be resubmitted

if i'm wrong, i'd like an explanation of what i did wrong, please

[trim hornet]

So what does the claim "fewer than 16 in ten thousand male fireflies are unable to produce light" look like, using an inequality with p?

[tropic hill]

"fewer than 16 in ten thousand" looks like p<16/10000, which is p<0.0016

which is the null hypothesis

and p>=0.0016 is the alternative

hence why i answered the way i did

[trim hornet]

What led you to believe that this is the null hypothesis

[tropic hill]

Isn't the null hypothesis usually the initial claim being made?

[trim hornet]

So, the null hypothesis is the initial position stating no relationship/effect, that a study seeks to either reject or will fail to reject. In our case, the entomologist is claiming that fewer than 16 in ten thousand (or 0.16%) male fireflies are unable to produce light due to a genetic mutation. So, the entomologist is claiming that some relationship does in fact exist, and is testing this against the null hypothesis

[tropic hill]

What technical statement would be saying that there is no relationship at all? What you said means that the alternate hypothesis has to be p < 0.0016, but the null hypotheses are all completely equal statements or inequalities without inclusion

would it then be c, because that's the only one with alternate = p < 0.0016?

[trim hornet]

Null hypotheses don't exist until a hypothesis is made. As soon as a hypothesis is made, such as some proportion of fireflies not producing light due to a mutation, the null hypothesis comes into existence asserting the opposite of that

[tropic hill]

The opposite of that would be h0: p>=0.0016

but the issue is that there's no option that allows it

only p=0.0016, or changing the alternate hypothesis to be inclusive

[trim hornet]

So it would be C, and the reason why the null hypothesis isn't p >= 0.0016 but is p = 0.0016 is for a few reasons, mostly simplicity and to be conservative

[tropic hill]

alright, understood. can we go on?

It's left-tailed, and the critical value for a=0.05 is -1.645

Correct, right?

[trim hornet]

Actually, often the null hypothesis in a class is sometimes always said to have an equals sign. YMMV though

[tropic hill]

I see

[tropic hill]

is my reasoning here logical?

[trim hornet]

How did you arrive at that

[tropic hill]

The critical value when alpha is 0.05 is usually 1.645 in a standard normal calculation, and due to it being left-tailed, the answer is negative.

[trim hornet]

Looks fine, the answer should be right

Sorry I was just working on my hw and was actually trying to get help myself on a diff topic

[tropic hill]

this should be a basic plugging of numbers into a formula

[tropic hill]

also, no worries

thank you for helping in the first place

if you wanna leave and do your own work, that's fine

[trim hornet]

Yeah I got the same thing

[tropic hill]

null hypothesis has been rejected, which is the opposite of the hypothesis

right?

[trim hornet]

I'm not sure if it would be A or D just because of the phrasing of it

[tropic hill]

alot of the questions are like that

[trim hornet]

I'm leaning a bit towards A since it follows a sort of double negative pattern in the same way that the question is asked, and D makes a stronger statement about what the null hypothesis is, when it's not explicitly stated

So the alternate hypothesis is "the cereal packets weigh >14oz on average", and the null hypothesis is the rejection of that claim

[tropic hill]

since h0 and h1 are mutually exclusive, wouldn't they be more or less the same?

the null hypothesis is that it's less than 14 oz

rejecting the null hypothesis would be rejecting that

[trim hornet]

So yes, but your professor (and others) seem to follow a format where the null hypothesis is that the mean weight is equal to 14oz (based on the answer from earlier being $H_0 : p = 0.0016$

[elfin berry]

Menelik

[tropic hill]

ah, fair

this is why stats confuses me, alotta room for interpretation

calc feels more peaceful

ordered chaos

instead of chaotic order

[trim hornet]

In this case, referring to the null hypothesis as "the rejection of the claim that the mean weight is at least 14oz" is always true, whereas referring to it as "the claim that the mean weight is less than 14oz" is a bit more roundabout, and can be interpreted as an alternate hypothesis itself

[tropic hill]

well, that's a headache and a half

[trim hornet]

And then, "reject the null hypothesis" is sort of a double negative, which the first option follows better since it says "there is not sufficient evidence to warrant rejection of the claim that ____"

[tropic hill]

i have to head off soon, pretty important test to take tomorrow

[trim hornet]

Yeah, especially since it uses words, and we kind of go into stats already having ideas of how to organize sentences and the meanings of specific words

Which you have to get used to

vs. calc is like "i've never seen $\int$ before so I believe it means whatever you tell me it means"

[elfin berry]

Menelik

[trim hornet]

Good luck w that, sorry if you had more questions you were trying to get through

[tropic hill]

You don't need to apologize, friend. you've done enough and i really appreciate you for it

thank you.

damn, no worries though.

[trim hornet]

Oh my god I am so sorry

So I was yapping

[tropic hill]

You don't have to be!

It's confusing and a headache

That's all of stats

You still helped me understand things i wouldn't have otherwise

i appreciate the help, lad

thank you for the help, @trim hornet 💙

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[steady nymph]

can someone please help me understand this? i’m completely lost right now

please help teach me how to solve this

[still light]

You want to find all $n$ such that $6n > 42$

[elfin berry]

Ari

[marsh citrus]

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[thorny heron]

Anyone how to do this qn.

[thorny heron]

😭 No one present

Hello

How you did 3 rd step

Oh Tysm.

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[tough cradle]

can anyone help me learn how to graph in desmos? i've been trying to mess around with it for the past hour but have had 0 success.

[weak fog]

Here's the answer

[velvet cedar]

type the function out verbatim

[tough cradle]

do you mean like f(x) = -7882x + 7470 into the calc

[velvet cedar]

Yep

[tough cradle]

https://files.catbox.moe/juqgi2.png
this is what i currently have and apparently (1,-412) isn't correct

or the others in the table

and i don't know how to proceed from here

feels like i keep going back and forth

[velvet cedar]

(1, -412) is definitely a point on that line

Also the reason you can't really see the line is because the gradient is too large for the scale

Go to settings and extend the viewing range of the x axis

[tough cradle]

what would be a good number?

https://files.catbox.moe/7l7c18.png

currently it's just this

[velvet cedar]

-12000 < x < 12000 for starters

[tough cradle]

done, just leave the y-axis alone or is there a way to figure out the right number?

[velvet cedar]

Oh wait, other way round. Change the y-axis

The range of x-axis should be between 0.8 and 1

While the y-axis should have the large range: try -1200 < y < 1200

@tough cradle

[tough cradle]

https://files.catbox.moe/f3o8u3.png

sorry forgot to fix the y-axis

so the x interval is just (1, -412)?

[marsh citrus]

@tough cradle Has your question been resolved?

[velvet cedar]

Interval? Don't you mean x-intercept?

[tough cradle]

my question is asking for the intervals

https://i.imgur.com/eOGTooo.png

and i really have no idea how to get this as i've never done something like this before

i've been re-reading the book and watching the example but i feel stumped

https://i.imgur.com/FXd5cGE.png

for reference, this is the only example i'm given

[marsh citrus]

@tough cradle Has your question been resolved?

[marsh citrus]

@tough cradle Has your question been resolved?

[marsh citrus]

@tough cradle Has your question been resolved?

[velvet cedar]

the intervals would presumably be the viewing range

[marsh citrus]

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[lethal bridge]

not sure how to start, my only thought is finding the stationary points or something but not sure

[still temple]

yea u should start by finding the stationary points

what do you think you can do next?

[lethal bridge]

no way

yajat

[lethal bridge]

i cri

[still temple]

https://tenor.com/view/cat-dance-gif-25818512

[lethal bridge]

help

[still temple]

ok so to ensure that the cubic equation has three distinct real roots its derivative should have two distinct real roots that basically means that the functions changing its direction twice

what are the critical points?

[lethal bridge]

sooo

$y'=3x^{2}+2ax$

[elfin berry]

water beam

[lethal bridge]

$\ x\left(3x+2a\right)=0$

[elfin berry]

water beam

[lethal bridge]

$x=0,\ -\frac{2a}{3}$

[elfin berry]

water beam

[lethal bridge]

these

[still temple]

right

so now

for the cubic polynomial to have three distinct real roots the function should change direction at both critical points meaning one should be a local maximum and the other a local minimum so you should evaluate the second derivative at the critical points

let me know whatever you evaluate

[lethal bridge]

$y"=6x+2a$

[elfin berry]

water beam

[lethal bridge]

$y"\left(0\right)=2a$

[elfin berry]

water beam

[lethal bridge]

$y"\left(-\frac{2a}{3}\right)=-2a$

[elfin berry]

water beam

[lethal bridge]

done

i dont know what A is so i cant really tell

[still temple]

we can make cases

case one is when a>0 and case two when a<0

[lethal bridge]

okay

[still temple]

but it cannot be zero as that would eliminate the quadratic term and reduce the cubic equation to a form that doesn’t provide the necessary three distinct roots

[lethal bridge]

yes

[still temple]

yea so if a>0
y''(0)=2a>0 (local minimum)
y''(-2a/3)=-2a<0 (local maximum)

and if a<0
y''(0)=2a<0 (local maximum)
y''(-2a/3)=-2a>0 (local minimum)

so you can clearly see a>0 and a<0 both works

[lethal bridge]

yes

[still temple]

so that's it

[lethal bridge]

it says determine values of a

so what do i write as final answer?

[still temple]

is there like a slot where you can write whatever you want to?

[lethal bridge]

this is what solution provides

but i dont rlly understand

[still temple]

💀

[lethal bridge]

lol

[still temple]

i mean my theory isnt wrong either

[lethal bridge]

💀

[still temple]

bruh how can they say that

we're literally getting 2a and -2a

how tf are they not inflection points

😭

do you wanna watch something funny

[lethal bridge]

yajat

u failed me

ur supposed to be number 1

jee adv

😭

[still temple]

💀

bruh my thing sounds so right

[lethal bridge]

https://tenor.com/view/family-feud-survey-says-strike-buzzer-wrong-answer-gif-16312959

[still temple]

REACT TO THAT FUNNY VIDEO

[lethal bridge]

i did

it wasnt that funny

[still temple]

stfu

[lethal bridge]

wow

u dont get to say that after getting my question wrong yajat

https://tenor.com/view/thanos-infinity-gauntlet-snap-finger-snap-gif-12502580

[still temple]

NAH BRUH tell me what part of my explanation sounds wrong

[lethal bridge]

bro it dont matter if ur explanation is wrong or right what matters is that u didnt find the value of A or get the question right

😭

[still temple]

ok man

maybe im not good enough...

[lethal bridge]

but u know what

if it helps

i thought u were cooking

🧑‍🍳

[marsh citrus]

@lethal bridge Has your question been resolved?

[marsh citrus]

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[hollow snow]

No matter what i do i reach a dead end

[fathom ridge]

substitute cot and then IBP

[lucid zenith]

Won’t you need the exponential integral for this

has that been taught to you?

[hollow snow]

You mean integrating e to the power of anything? Im not really that good at that

[lucid zenith]

No

$\int\frac{e^x}{x}\dd{x}=\text{Ei}(x)+C$

[elfin berry]

kheerii

[lucid zenith]

You’re right in saying that the function is non-integrable

Without the use of non-elementary functions

[lucid zenith]

This is an example of one

[hollow snow]

I should be able to solve it with u-substitution, atleast this is what the professor says

Idk if he also included IBP tho

[lucid zenith]

They’re wrong

,w integral of e^(-cotx)*sec^2(x) dx

[elfin berry]

Results provided by WolframAlpha

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

Requested by shikhargupta1207

[hollow snow]

I’d not be surprised

[lucid zenith]

See, the answer has a non-elementary function in it

Maybe they meant $\int e^{-\cot x}\csc^2(x)\dd{x}$

[elfin berry]

kheerii

[lucid zenith]

Try doing this

[hollow snow]

Yeah here i can do -cot x = u where dx = du/-csc^2
Then make it integral of e^u -du, and the result would be -e^u

[lucid zenith]

Almost

You added an extra - sign somewhere along there

[hollow snow]

It should just be e^u?

[lucid zenith]

Remember that $\frac{d}{\dd{x}}(\cot x)=-\csc^2(x)$, not $\csc^2(x)$

[elfin berry]

kheerii

[lucid zenith]

Yes

[hollow snow]

Oh yeah yeah

Thanks, appreciate the help man

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[marsh citrus]

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[shy hill]

how to find the value of inverse trignometric function value if the input is negative, doesnt include in principle value

[marsh citrus]

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[ebon depot]

How many integer solutions does the equation $x_1+x_2+...+x_7 = 24$ have that satisfy $x1+...+x4 = 10$ , $x_5+x_6+x_7=14$ and $x_i \ge 0 for i = 1,2...,7$ ?

[elfin berry]

Merineth

[ebon depot]

my solution was $\binom{30}{24} - \binom{13}{10} - \binom{16}{14}$

[elfin berry]

Merineth

[ebon depot]

But the book says it's $\binom{13}{10}\binom{16}{14}$

[elfin berry]

Merineth

[ebon depot]

Why is that the case?

My assumption was that i take the total sum of possible solutions and then remove the possible solutions for the conditions?

[spark otter]

if you have this

then you have this

so you can choose (x1,...,x4) independently from (x5,...,x7)

so how many ways to have x_1+x_2+x_3+x_4 = 10 and all >=0?

it's 13C10

how many ways to have x_5 + x_6 + x_7 = 14 and all >= 0?

16C14

[spark otter]

These 2 conditions

[spark otter]

make this condition useless

[ebon depot]

seems odd to include that condition if it isn't used?

[spark otter]

well that's the point, it's your job to find out which conditions are useless

though let's say this condition wasn't included for example

well you can still find back the same number of ways

by saying:

x_1 + x_2 + x_3 + x_4 = 10

x_1 + .... + x_7 = 24

so x_5 + x_6 + x_7 = 24 - 10 = 14

[ebon depot]

Why doesn't my solution work tho?

[spark otter]

well I don't understand how you even got that

[ebon depot]

If i take the total amount of solutions from the equation $x_1 + ... + x_7 =24$

[elfin berry]

Merineth

[ebon depot]

Right? That would be 30C24

[spark otter]

sure

[ebon depot]

and then i want to reduce tthat total with my two conditions

[spark otter]

but then you have to get rid of the solutions to ... = 24 such that ... is NOT 10 and ... is NOT 14

[ebon depot]

oooh

right my conditions were wrong

it should still work tho if i adjust that?

[spark otter]

I don't know how you would adjust that

[ebon depot]

$x_1+x_2+x_3+x_4 = 10$ Well we know that this conditon has to be met?

[elfin berry]

Merineth

[ebon depot]

Nah idk

I dont get it att all

$\binom{13}{10}$ and $\binom{16}{14}$

[elfin berry]

Merineth

[ebon depot]

I get that the sum of these two are the total combinations for my two conditions

I don't see how multiplying these two together results in the answer i want?

why do we multiply and not add?

[spark otter]

you don't ADD here

ADDING is for this situation:

"You have 5 ways to get an A+ if you didn't go to class.
OTHERWISE, you have 11 ways to get an A+ if you DID go to class.
In total, you have 11+5 = 16 ways to get an A+"

Multiplying is for this situation :
"You have a 2-digit passcode.
You have 10 choices for the first number.
AND, you have 10 choices for the second number.
In total, you have 10*10 = 100 choices for your passcode"

[velvet cedar]

I've been following along with rafilou's explanaton and how I've understood is:

For every solution to x_1 + x_2 + ... x_4 = 10, you can pair with each solution to x_5 + x_6 + x_7 = 14 to sum up to 24

[spark otter]

• Adding is for INCOMPATIBLE CHOICES, "A or B". (meaning both can't happen at the same time)
• Multiplying is for INDEPENDENT CHOICES, "A and B". (meaning one doesn't affect the other)

[ebon depot]

Hmm i guess i'll just have to accept that it's multiplying

[spark otter]

@ebon depot try to understand what is being done in those 2 scenarios

and understand what is the difference, meaning why we should add/multiply instead of doing the other

[ebon depot]

Yeah i see what you mean by the passcode example but i can't seem to apply it to my problem

But it is what it is

I got it nearly correct

[spark otter]

well, what you choose for your first digit doesn't affect what you choose for your second digit

[spark otter]

here, no matter what you choose for x1,x2,x3,x4

it won't change the fact that x_5,x_6 and x_7 need to be picked such that x_5+x_6+x_7 = 14

so what you choose for x1,x2,x3,x4 doesn't affect what you choose for x5,x6,x7

since you have 13C10 ways to choose x1,...,x4

and 16C14 ways to choose x5...x7

you have 13C10*16C14 ways to choose x1,....,x7

[ebon depot]

\o/ i dont know what to say haha i sincerely dont get it but it's alright, i'll just try and remember when i want the total amount of combinations such as the passcode i'll just multiply them together

thanks for helping tho 🫶

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[wind kettle]

Is this correct? I tried different apps and they all say it’s 1 except one said it doesn’t exist. It looks like it’s not approaching anything from the left side so that means the limit doesn’t exist, right?.

[quiet anvil]

I would quibble about the final ln(L) = 0 = e^0 = 1, but surely you meant ln(L) = 0 => L = e^0 = 1.

I don't see a mistake. We are approaching pi/2 from the left side. So the fact that is only defined on the left side is fine

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[potent reef]

a canon is at an angle 45 deg to the x,y- axis. A canon ball is fired at a velocity of 600 m/s. A man is standing 300 km away from the canon. at what height will he see the ball above him? Also, a man is standing 60 km away from the man and canon. The weight of the ball is 5000 kg. The weight of the man is 143 kg. There is a nail 1 km away from man_2. Find how inside the nail will pierce his skin. Consider the deceleration of the nail with the man to be 5000 m/s.

[potent reef]

the breadth of man_2 is 20 cm. If the nail passes half of his body, he shouts. Tell how many times he will shout, i.e., how many times he passes half the breadth of his body.o

if g = 1 m/s**2

[marsh citrus]

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[still temple]

If 𝒇(𝒛) = 𝒖 + 𝒊𝒗 is an analytic function ,find 𝒇(𝒛) in terms of 𝒛 𝒊𝒇 𝒖 − 𝒗 =𝒆 𝒚−𝒄𝒐𝒔𝒙+𝒔𝒊𝒏𝒙 𝒄𝒐𝒔𝒉𝒚−𝒄𝒐𝒔𝒙 𝒘𝒉𝒆𝒏 𝒇(𝝅𝟐) =𝟑−𝒊𝟐

[marsh citrus]

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[still temple]

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[iron onyx]

I need help with this calculus problem.
x= (v^2sin(2θ))/32 where 45 ≤ θ ≤ 135
and I need to find the equation for dx/dt

it also says that dθ/dt is constant, which I think is important

[iron onyx]

this is the actual problem I'm looking at

I think I only have one more chance to try answering it before the question locks up and gets stuck as incorrect

[desert dirge]

note v is a constant

[iron onyx]

something about my chain rule has to be off, yeah

if v is a constant, then should the derivative function remove it completely?

or just not derive it?

like pull it out front along with 1/32 before deriving what's actually left

Does this look right?

[azure temple]

yes but theta being a function of t is messing with my gut checking

[iron onyx]

So t is the independent variable, v is the velocity at which water is moving away from (0,0), theta is the angle it's going in, and x is how far from (0,0) the water lands

While v is constant, the water is moving away from (0,0) at a constant speed. The rate that theta changes at is constant, and then the ultimate dependant variable then is x, how far away the water lands, or dx/dt, at what rate that landing point is changing relative to the center (0,0)

[marsh citrus]

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[iron onyx]

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[modest trellis]

The first picture is a definition of Lebesgue integral written in RCA by rudin, and the second picture is a definition of Bochner integral written in Integrals with values in Banach spaces and locally convex spaces by Piotr Mikusinski. (https://arxiv.org/pdf/1403.5209)
Since the Bochner integral is Lebesgue integral defined on Banach spaces, I thought that these two integrations should be constructed in almost the same way.
But as we can see from the picture, the Lebesgue integral is constructed using supremum, while the Bochner integral is not.
Is it just difference of notation and are they using actually the same method? Or is there a reason that Bochner integration cannot be constructed using supremum, just like Lebesgue integral in RCA?
I want to rewrite a definition of Bochner integral like (2.1) in the first picture.

[marsh citrus]

@modest trellis Has your question been resolved?

[modest trellis]

<@&286206848099549185>

i uploaded the same question yesterday, but it closed while i was sleeping 😦

[marsh citrus]

@modest trellis Has your question been resolved?

[modest trellis]

<@&286206848099549185>

[long osprey]

try in #advanced-analysis maybe

oh you're already there

nvm

[modest trellis]

yeah i think i should there lol

.close

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[marsh citrus]

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[iron onyx]

Can anyone help me solve this? I really don't know how to do this and I'm struggling

[clever spade]

related rates problem?

you most likely want to find an expression for the area of the rectangle and then use calculus to find the maximum value of that area

[iron onyx]

the unit is called optimization problems

I think what I'm supposed to do is make the derivative formula for the rectangle and then find where it equals 0 and... do something with that. I don't know how to use the formula for the circle to help me either. It has to be there for a reason

[clever spade]

that sounds good

remember that when you take the derivative it gives the instantaneous gradient

so at f'(x) = 0 your gradient equals 0

which will be either a minimum, maximum or stationary point of inflection (you can find which using the second derivative)

[iron onyx]

This is all I have so far

[clever spade]

so your thinking process should be

1. How can I find the area of the rectangle given an expression for y = some function of x
2. How can I maximise this? As our area will form a graph we can take the derivative, find stationary points and then your maximum point
3. At maximum point substitute in to find the maximum area

At least that's my thought process

[clever spade]

I think you might have the area expression wrong

since it's the area of a rectangle we will most likely use the form width*height of rectangle

[iron onyx]

it should be 2xy, shouldn't it?

[clever spade]

and both sides should be the same as the shape is symmetrical

I'm looking at this and your width of rectangle seems to be 2x

[clever spade]

when going to the point x,y you can see at x = 4 width is [-4,4] = a length of 8 so the length will be 2x

your height would be y as given on your circle but as you can see y on the circle is equal to sqrt(25-x^2)

this should give you an expression with just x

A=2x*sqrt(25-x^2)

[iron onyx]

oh my god

you're right, the x substitute for y is already given at the start

I was just wracking my brain over how I was gonna find it and how that circle formula was supposed to help me

[clever spade]

the key thing about optimisation problems is that you know the height and width are dependent upon each other

so you just need to think how each part of your area will change and how to calculate them 👍

[iron onyx]

right, and the thing that was confusing me about that earlier when I tried to derive the area was because I felt like deriving x shouldn't just give me a dx/dx. Like I feel. like there should be a rate of change for x to correlate to dy/dx

but I think this fixes that whole problem

I think

[clever spade]

you should be able to take the derivative of this using product and chain rule!

[iron onyx]

right, so that'll be, 2dx*sqrt(25+x^2) + x/sqrt(25-x^2), right?

no wait, gotta add in the derivative of what's inside the square root too

2dx*sqrt(25+x^2) + 2x^2dx/sqrt(25-x^2)

[clever spade]

I assume you mean 25-x^2 for the first?

also I don't think you should have dx left over because d/dx(2x) = 2

[clever spade]

and I think it should be 2/srt(25-x^2) for your second term? Let me check

[iron onyx]

Right, ok, so d/dx of x should still cancel itself out here

[clever spade]

here is the answer if you are curious

I would encourage you to go through the working out ofc

note that the first part of the product rule was multiplied by sqrt(25-x^2)/sqrt(25-x^2)

[iron onyx]

I see

Yeah I definitely want to show my work, I need to understand what I'm doing

I think I've seen that sort of thing before, multiplying the top and bottom by the square root to get it into the denominator. I don't understand why you would do that though

[clever spade]

it's just a simplification thing

the reason for it in this case is to fit to the denominator of the second part of the product rule which is already sqrt(25-x^2)

since it's in one fraction it is simplified further in this case

also conveniently this also removes the square root of the left expression which gives you a non-root for the denominator

[iron onyx]

I see

I'm gonna need to do more work I think to fully understand this

For now though, idk. I should actually go tend to dog. She's getting restless

[marsh citrus]

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[valid hinge]

|sinx|=sinx+2cosx find solutions between [0,2π]

[valid hinge]

for positive

2cosx=0

[static quarry]

you can get rid of the abs vals by considering separately the cases where sinx >= 0 and sinx < 0

[valid hinge]

which is 0 at π/2,3π/2

2 solution for sinx now -sinx

2 sinx=2cosx

sinx-cosx=0

[lucid zenith]

Could just directly say that tanx=1

[valid hinge]

π/4

[late geode]

messed up signs

[valid hinge]

5π/4

[lucid zenith]

But is this a valid solution?

[valid hinge]

Yeah it gives 0 i guess

1/√2 -1/√2 =0

[lucid zenith]

Wait, you messed up here

[valid hinge]

2sinx-2cosx=0

[late geode]

how are you getting that

[lucid zenith]

$-\sin x=\sin x+2\cos x$

[elfin berry]

kheerii

[valid hinge]

Yes from here

sinx-cosx=0