help-23

[safe radish]

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[kind willow]

how do i create my own function using the base function y=cos(x) and give the function an amplitude of 3 and a period of 90

[winter summit]

y=acos(b(x-c))+d

A is amplitude

[kind willow]

yeah

[winter summit]

B is 2pi/period

[kind willow]

so 2pie /90

[winter summit]

Which is pi/45

Wait a second

Is the 90 in radians or degrees

[kind willow]

degrees

[winter summit]

Alright

So 90 degrees is pi/2

So 2pi/(pi/2)

Which is 4

[kind willow]

so its y=3cos4(x)

[winter summit]

The amplitude is 3

Not 2

so it’s y=3cos(4x)

That is 90 is in degrees

[kind willow]

so for example

if i had to do a period of 720 degres

it would be 2pi/720

@winter summit

[winter summit]

You have to convert 720 degrees into radians first then use the 2pi/b

[kind willow]

how do i convert it into a radian

[winter summit]

Degrees * pi/180

[kind willow]

so 720 multiplied by pi/180

[winter summit]

Yes

[kind willow]

so thats 4pie

and i do 2pi/4pi

wich is 0.5pi

[winter summit]

0.5

pi’s cancel

[kind willow]

oh ok

thanks

.close

[safe radish]

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[hollow ridge]

Hello I don't understand how to get the answer

[winter summit]

sin(x+pi/2 or 45 degrees)=cos(x)

I think

That might not help

[winter summit]

cos(a-b)

Do you know how to expand this?

[hollow ridge]

cosacosb + sinasinb ?

[winter summit]

Ok

So it’s gonna be 3cos(x)cos(45)+sin(x)sin(45)=-1

Which would be cos(x)cos(45)+sin(x)sin(45)=-1/3

Understand?

[hollow ridge]

up till now yes

[winter summit]

Alright, do you know cos(45) and sin(45)

[hollow ridge]

yes, but don't know how to type

[winter summit]

Latex notation?

$\frac{\sqrt2}{2}$

[flat frigate]

% Openglobe %

[hollow ridge]

yes

[winter summit]

So it’s gonna be $\frac{\sqrt2}{2} cos(x) + \frac{\sqrt2}{2}sin(x)$ =$-\frac{1}{3}$

Idk why it’s not working

[flat frigate]

% Openglobe %

[winter summit]

@hollow ridge it’s working

[hollow ridge]

not sure how to find x

[winter summit]

We’re getting there

Can you find a GCF

[hollow ridge]

sorry what is GCF

[winter summit]

Greatest common factor

[hollow ridge]

this?

[winter summit]

Yes

Find GCF for the left side

[hollow ridge]

only this?

[winter summit]

Left side of equation

I have a failsafe plan if this doesn’t work

[hollow ridge]

to my way of thinking it is this only

[winter summit]

$\frac{\sqrt2}{2} (cos(x) + sin(x))$ =$-\frac{1}{3}$

[flat frigate]

% Openglobe %

[winter summit]

See it now?

[hollow ridge]

yep

[winter summit]

What’s next

[winter summit]

Actually @hollow ridge I have a better idea

sin(x-45)=-cos(x)

[winter summit]

Use this from the top

And cos(x-45)=sin(x)

So it would be 3sin(x)=-1

That should be much easier

3cos(x-45)=3sin(x)

[winter summit]

How’d you get the third line

Second line should be sin(x)=-1/3

[hollow ridge]

sin^-1(-1/9)

oops

i forgot the 3 is already divided

[winter summit]

Yes

Now take sin^-1(-1/3)

It will give the answer in radians so make sure you have it in degrees

[hollow ridge]

i don't know how to convert it

[winter summit]

You can use this as an exact answer

Because it’s irrational

x=sin^-1 (-1/3)

[hollow ridge]

that's what i got

[winter summit]

Do you need decimal form?

[hollow ridge]

cor to 1 dp

[winter summit]

I don’t understand 3rd line

[hollow ridge]

I copied how my textbook presented it

[winter summit]

Hmmm

[winter summit]

So second to third line was cross divide

[safe radish]

@hollow ridge Has your question been resolved?

[hollow ridge]

That's what I wrote, but it contradicts the answer in the textbook

[winter summit]

Have you been taught that identity

[safe radish]

@hollow ridge Has your question been resolved?

[safe radish]

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[normal hedge]

Can i have some tips on how i should continue

[molten heart]

http://advancedscience.org/2016/3/079-082.pdf maybe try with this

[unique bison]

It's natural to try u = tan theta or sec theta

Turns out ||the former|| works

[normal hedge]

the former?

[unique bison]

tan

Former means the first option and latter means the second one

[normal hedge]

i see

so let u tanx

?

[unique bison]

Seems like you already used u so use another variable

[normal hedge]

wouldnt tanx be sec^2

[unique bison]

No

[normal hedge]

wait waht

im sorry its like 2 am our teacher gave us a long assigmnet for 10 questions around 11 my brain aint working hahaha

[safe radish]

@normal hedge Has your question been resolved?

[safe radish]

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[tiny gorge]

Hello, can someone explain what equation i have to use? I don't know where to start

[plucky elk]

is this calculus?

[tiny gorge]

yeah

but i think i know what im confused about

i'm not sure what equation to differentiate or start with

because there is the volume equation and the area equation

[vast obsidian]

That's a good start

[tiny gorge]

give me a second i will try

[vast obsidian]

OK, ping me if you need. But I'm sure you don't 👍

[safe radish]

@tiny gorge Has your question been resolved?

[glass carbon]

firstly let set the dimensions of the aquarium

let's say x is the side length of a base and y is height

Now formulate equations for the volume and the exterior surface area in terms of x and y

[tiny gorge]

V = x^2 * y

A = x^2 + 4xy

[glass carbon]

yeah, great

now what you can say about the volume?

[tiny gorge]

i can set v = 32

and then isolate one of the variables

[glass carbon]

exactly, I'd recommend isolating y

[tiny gorge]

oh ok, but it works with both

[glass carbon]

yes it's your choice np

[tiny gorge]

32/x^2 = y

now i think i have to replace y in the area equation and find the derivative

A = x^2 + 128/x

x=4

[glass carbon]

yea, we have the minimum at x = 4

[tiny gorge]

and so then y = 2

[glass carbon]

gj

[tiny gorge]

damnnn

thank you very much!

.close

[safe radish]

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[trim condor]

which formula is the solution for this?

[plucky elk]

look at your list of trig identities

for each of the options

[trim condor]

addition formula for sine?

oh wait no

its addition formula for cosine right?

[plucky elk]

what is this formula

use t for theta

[trim condor]

first one

[plucky elk]

what happens when a=b

[trim condor]

uhm

im not too sure

[plucky elk]

write it out

[trim condor]

wait hold on

my teacher said its add for sine

i was right

[safe radish]

@trim condor Has your question been resolved?

[safe radish]

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[safe radish]

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[potent solstice]

What is this asking me to do

I obviously am not inserting any number

Am I just suppose to simplify it and make it equal to what it is in terms of a?

[glass carbon]

imagine 'a' is a number

now you can't plug x = a into because it gives you indeterminate form

0/0

try to get rid of that

[potent solstice]

How did you know it was zero over zero if there's no limit to put in

[glass carbon]

there is an expression under the limit

try plugging x = a then you'll see

in denominator x - a becomes a - a = 0, similar thing happens in numerator

[potent solstice]

You mean x ▶️ a?

[glass carbon]

x approaches to a, yes

generally calculating limits is about plugging that number into limit and checking if it gives the result

if not, then we need to apply some tricks to get the other form

[potent solstice]

So the x in this equation would become a

[glass carbon]

Maybe some easier example would make it clear

look at this one

$$\lim_{x \to 1} \frac{(x-1)^2}{x-1}$$

[flat frigate]

Modus

[glass carbon]

I guess you can do this one simply

[potent solstice]

Yup you plug in one and get the limit as x approaches 1

Or before you may want to cancel out the top and bottom

[glass carbon]

now you have 0/0, after cancellation it becomes just x - 1 and we can plug x = 1 to get the result

yes

in your query situation is similar but a bit more complicated

[potent solstice]

The only thing I'm doing there is taking the x values, changing them to a, then simplifying the equation?

Or do I first simplify the equation, then change the x values to a and solve for a

[potent solstice]

It's probably this

[glass carbon]

firstly you need to get rid of the indeterminate form, so yes - simplify firstly

then if it's possible plug x = a and just calculate the limit

[potent solstice]

Does this simplified form look solid?

[glass carbon]

wdym

[potent solstice]

Sorry took a sec to get that picture

[glass carbon]

what's that? by "simplify" I meant to cancel out the denominator

the idea is to plug x = a at the end, how we can do that having x - a in denominator? we can't divide by 0

[potent solstice]

Ah okay

[glass carbon]

hint: try to factorize numerator

[potent solstice]

My calculator doesn't factorize past what I have in the numerator, is that because I am trying to factorize the wrong thing

[glass carbon]

you have (x+a)^2 - 18x^2 + 14ax

after some operations it becomes

-17x^2 + 16ax + a^2

this is a right quadratic you need to factorize, notice that x = 'a' is probably a root

[potent solstice]

So I should try to factor this

I would try factoring by grouping

[glass carbon]

ok

[potent solstice]

@glass carbon Yup that was it, thank you for your help!

.close

[safe radish]

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[safe radish]

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[lucid charm]

Hello, I am not sure if this is the right place to ask this.
I am trying to program an script that gives me one equation that passes thru specified points.

Ex. I input 1,1,2,2,3,3,4,4,5,5 and the program should reply: f(x)=x
(That is a very simplified example, 99% of times the equation wont be linear)

Thanks in advance!

[lucid charm]

(Please, ping me when you answer ^^)

[jagged bluff]

@lucid charm There are many ways to answer this... You can just connect the points with straight line segments, for example. Or you can create a polynomial that passes through the given points, which will give a smooth curve, but may oscillate wildly (go up and down more than seems necessary). If you're doing something computer-graphics-y, look up composite Bezier curves, and splines -- these are ways of making smooth curves that pass through given points... They're used a lot in computer graphics.

If the polynomial thing sounds good to you, start by looking up polynomial interpolation, e.g. on Wikipedia or YouTube.

[lucid charm]

Okay....

The thing is that I have 15 years and this is hard to do for me

But I will try

Thank you so much!

[safe radish]

@lucid charm Has your question been resolved?

[safe radish]

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[slow fern]

What is the probability that delay in train arrival will be more than 11 minutes but less than 14 minutes?

height = 1/6

Area between 11 to 14 is required.

= Area(11-12) + Area(12-14)

How do I calculate Area between 11 - 12 ?

[quasi bison]

it's a trapezoid

[slow fern]

Base is 1 unit, height is (1/6) unit

Area of trapezoid is (a+b)/2 * h

b=1, h=1/6

what is a?

@quasi bison

[quasi bison]

i would consider the vertical sides of your trapezoid as your bases

so the height is 1, and the two bases are 1/6 and 1/12

[safe radish]

@slow fern Has your question been resolved?

[safe radish]

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[safe radish]

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[gusty roost]

Does anyone know the answer for 2,3,4,5 ?if so please teach me

[bold aurora]

Let’s do 2. The inside of the double integral is integrating along y from 0 to x. However the integrand is constant with respect to y. So we can move it out of the inner integral

[gusty roost]

U mean like this?

[gusty roost]

1/x**

[bold aurora]

Nope, I meant $\int_0^{\pi} \frac{\sin{x}}{x} \int_0^x dy dx$

[flat frigate]

Learath2

[bold aurora]

sinx doesn’t involve y either, so it’s also constant

[gusty roost]

Oh ok

[bold aurora]

Now the inner integral should be trivial to take, no?

[gusty roost]

Yes

[bold aurora]

Which leaves you with a single integral, which should also be trivial. So let me know when you are done with it and we can try another

[safe radish]

@gusty roost Has your question been resolved?

[safe radish]

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[neat kiln]

Draw the three things as described first

[lean otter]

I sent the wrong one srry

[safe radish]

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[safe radish]

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[safe radish]

@devout fossil Has your question been resolved?

[safe radish]

@devout fossil Has your question been resolved?

[plucky elk]

what don't you get?

[safe radish]

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[unkempt solstice]

!15min

[safe radish]

Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.

[safe radish]

@lean otter Has your question been resolved?

[safe radish]

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[stuck agate]

Is 3+2(2+3) same as 3+2 * (2+3) or is it 3+(2 * (2+3))

[marsh schooner]

It's 3+(2*...

[quasi bison]

these are all the same as written

however if you meant for the left addition to be done first in 3+2 * (2+3) then you did not communicate that properly

[stuck agate]

.close

[safe radish]

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[stuck agate]

.reopeb

.reopen

[safe radish]

✅

[stuck agate]

Umm

.close

[safe radish]

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[safe radish]

@lean otter Has your question been resolved?

[ivory trellis]

alr i'll try to help you

so what do you need help with

show me the different answers i guess

i'm not sure which one's right cuz idk about iqr or quartiles or whatever this stuff is

im not sure how i can help lol

[safe radish]

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[lean otter]

no thinking method: put all the numbers in and check.
proper method: move the sin(x) to the other side then factor the quadratic in sin(x) and find the roots

[fossil wraith]

wdym factor the quadratic

so i moved the sinx and got 2sin^2x - sinx = 0

[lean otter]

yup

now factor out a sin(x)

[fossil wraith]

sinx(2sinx -1)

now i set sinx = 0 and 2sinx-1 = 0

?

[lean otter]

yes

[fossil wraith]

holy

why is your brain so huge

let me work it out rq

[lean otter]

experience ._.

[fossil wraith]

so i got this basically

all the numbesr are the same as what i got

but shouldnt they be in ascending order?

[lean otter]

yup thats right

it would be nice if they were in ascending order, but not necessary

[fossil wraith]

alright

appreciate it

wait actually

would you mind helping me witht he next one?

[lean otter]

sure

[fossil wraith]

it looks similar but still no clue how to start

so i moved the csc to the left side

2sinx - cscx -1 = 0

and now idk what i can factor

[lean otter]

rewrite csc as 1/sin then clear denominators

[fossil wraith]

clear? as in make them all the same?

[lean otter]

yeah, get rid of it by multiplying everything by sin

[fossil wraith]

ok

i made them all the same then combined them

now i have (2sin^2x - sinx - 1) / sinx

[lean otter]

yup, now multiply both sides by sin(x) (and keep in mind sin(x) cannot be 0)

[fossil wraith]

wait

so i mulitply the sinx by the 0 on the other side just to get rid of it?

[lean otter]

yup

[fossil wraith]

ok

2sin^2x - sinx - 1 = 0

it looks like i could probably factor something

[lean otter]

yeah, you can factor it

[fossil wraith]

(2sinx + 1)(sinx - 1) = 0

is that the right factor?

[lean otter]

yup, thats right

[fossil wraith]

alright

i did all the rest

i got {π/6, 5π/6, 3π/2}

i gotta change it to degrees but

[lean otter]

hmm that doesnt look quite right

[fossil wraith]

oh

[fossil wraith]

so after i made this factor

[lean otter]

I think you messed up signs

[fossil wraith]

OH

i did

[lean otter]

sin(pi/6)=1/2, but you want -1/2 etc

[fossil wraith]

when i wrote it on my paper i flipped the signs in the factors

let me fix it

[lean otter]

hmm

[fossil wraith]

ok i got

{π/2, 7π/6, 11π/6}

[lean otter]

yup perfect, just need to make it degrees now

[fossil wraith]

{90, 210, 330)

ALRIGHT

THANKS BRO

[lean otter]

yw :)

[fossil wraith]

o7

.close

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[plain hedge]

is there a good way to solve this that I can memorize

[stoic dune]

You just want to integrate under it, right?

[plain hedge]

to find the volume with tripple integrals

I has to be with tripple integrals idk

[stoic dune]

You sure? This would be much easier as a double integral

But if you have to do it as a triple, this is a pretty simple one

[plain hedge]

oh ok

[stoic dune]

https://tutorial.math.lamar.edu/classes/calciii/tripleintegrals.aspx

[crimson field]

Find the x,y and z intercepts to be able to generate a rough sketch of the region.

Solve for x, y or z, solving for z will feel the most natural then you are between two surfaces z = 0 and z = f(x, y) above a certain region on the xy-plane.

[safe radish]

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[safe radish]

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[bold aurora]

I’m having a bit of trouble following this definition. $\gamma : I \subset \mathbb{R} \mapsto \mathbb{R}^m$ and $\gamma = (x_1, x_2,…, x_m)$

[flat frigate]

Learath2

[bold aurora]

I understand the second part, gamma is differentiable at t_0 iff all components are also differentiable at t_0

However the first one is a little baffling. \dot{\gamma} hasn’t been defined anywhere, any idea what it could be trying to signify?

[plucky elk]

it's just stupid notation for "derivative with respect to time"

[peak estuary]

it's using this as a definition for gammadot

[plucky elk]

https://en.wikipedia.org/wiki/Notation_for_differentiation#Newton's_notation

[peak estuary]

pretty common in e.g. physics to write dots if you differentiate with respect to time

[bold aurora]

Oh, so it’s saying \dot{\gamma} is the same thing as \gamma’ ?

[peak estuary]

yes

[bold aurora]

Okay, and the limit part. From what I gather it’s saying if this limit exists, then \gamma’(t_0) is equal to it and we say the curve is differentiable at t_0?

[peak estuary]

yes

the limit is just the definition of a derivative

same as in one dimension

$f'(x) = \lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$

[flat frigate]

Denascite

[plucky elk]

hence my use of "stupid"

[quasi bison]

it's the same definition except your function is now vector-valued

[bold aurora]

It’s almost as if we need a different version of =, this one confused me quite a bit as I wasn’t sure if this gammadot thing was being defined or being used as a part of the definition

[quasi bison]

it isn't stupid. it is convenient at times.

as a neat shorthand

[peak estuary]

some people use $a:= b$ or $a\overset{def}= b$ or something

[flat frigate]

Denascite

[quasi bison]

yeah := is a good one

[plucky elk]

okay i retract my statement since i'm not a physicist.

i like :=

[quasi bison]

i'm not a physicist either

[plucky elk]

also, there should be a symbol for contradiction

[quasi bison]

i still like to make physics-y analogies sometimes

like arclength of a curve

[quasi bison]

?! and !!! are things ive seen used

[bold aurora]

Ah, so that’s what the := is used for. TIL

[plucky elk]

Lol ?!

[glass carbon]

Newton vs Lagrange 💀

[quasi bison]

you mean leibniz.

[bold aurora]

Ok, I think I get it now. Thanks 🙃

.close

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[vagrant sand]

how can i solve this question?

[tame charm]

Hmm

[safe radish]

@vagrant sand Has your question been resolved?

[vagrant sand]

<@&286206848099549185>

[glass carbon]

Try to find a, b, c such that:
v = u1 * a + u2 * b + u3 * c

[unique bison]

You need to solve $\begin{pmatrix}\bf{u_1} & \bf{u_2} & \bf{u_3}\end{pmatrix} \bf{x} = \bf{v}$ for $\bf{x}$.

[flat frigate]

giannis_money

[unique bison]

Row reduce $\begin{array}{ccc|c}(\bf{u_1} & \bf{u_2} & \bf{u_3} & \bf{v}) \end{array}$.

[flat frigate]

giannis_money

[safe radish]

@vagrant sand Has your question been resolved?

[vagrant sand]

wb this?^

[safe radish]

@vagrant sand Has your question been resolved?

[unique bison]

literally the same thing, on a computer

[safe radish]

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[kind patio]

How would you solve

[final halo]

trying making it look like just sqrt(stuff)

[kind patio]

how would i do that

[dusty sable]

Factor by x^2 in the square root

[hearty egret]

$x=\sqrt(x^2)$ you can do that because x is positive

[flat frigate]

everg

[hearty egret]

than $\lim \sqrt{\dots}=\sqrt{\lim \dots}$ because squareroot is continuous

[flat frigate]

everg

[safe radish]

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[safe radish]

Please don't occupy multiple help channels.

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[idle parrot]

Hi, can ODE have parametric solution with. dy/dx as parameters?

[idle parrot]

I’m trying to solve this ODE, and ended up here, is this correct?
Here p= dy/dx

,w derivative of p/sqrt(1+p^2)

[flat frigate]

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Requested by Deep.#0001

[idle parrot]

,w derivative -1/sqrt(1+p^2)

[flat frigate]

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Upgrade to WolframAlpha Pro!

Requested by Deep.#0001

[idle parrot]

This is the integration part, apparently wolfram can’t solve that integral, so i used derivative for verification

[safe radish]

@idle parrot Has your question been resolved?

[idle parrot]

Here i used solvable for x method

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[idle parrot]

.close

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[frosty moth]

how do I find x given the instantaneous rate of change

I can do the opposite but I can't do this

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[trim condor]

how do i answer this

[lean otter]

Use this rule

$$P(A \cup B) = P(A)+P(B)-P(A \cap B)$$

[flat frigate]

Legolas

[trim condor]

0.16

thats what i got

[lean otter]

Nice

I also got that

[trim condor]

perfect

how would i solve this? if you can see it hahah

[lean otter]

I can't really lol

[trim condor]

okay just a sec

ill write it out

there we go

[lean otter]

does not drink tea means not in that tea bubble

so 80 people don't drink tea

[trim condor]

but the 40 people in the middle drink tea aswell no?

oh oops

yea

[trim condor]

you are right idk why i said this lol

ok so that means 80 out of the total 140 people dont drink tea

[lean otter]

wait

weren't there 150 people surveyed

[trim condor]

ah that is so weird

[lean otter]

so 10 people don't drink tea or coffee

[trim condor]

trick type of question

[lean otter]

so we have 80 + the extra 10 who don't drink either

[trim condor]

good eye though!

[lean otter]

so 90/150 right

[trim condor]

yea

that simplifies to 3/5

[lean otter]

yeah

is it the correct answer?

[trim condor]

im not sure my teacher didnt give us an answer key sadly

but i think its correct

[lean otter]

Ok

[trim condor]

thanks for the help

:))

[lean otter]

No problem

[trim condor]

.close

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[lean otter]

Hi

[winter pivot]

split the fraction and integrate term by term

[dire crypt]

Hi

[winter pivot]

or alternatively factor the denominator

[dire crypt]

um

[winter pivot]

um

what's the issue

[dire crypt]

how would i integrate x^3 / x^4+x^2+1

[winter pivot]

on second thought let's not do that

let's try factoring the denominator

[lean otter]

Yeah I'd say factor then do partial fraction

[dire crypt]

i did it after that what can i do

if i remember it correctly it should be x^2+x+1)(x^2-x+1

[lean otter]

do pfd

[dire crypt]

,wolf simplify (x^2+x+1)(x^2-x+1)

[flat frigate]

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[dire crypt]

whats pfd

[lean otter]

partial fraction decomposition

[dire crypt]

oh

for each of the 3 terms

oh

[lean otter]

wdym 3 terms

[dire crypt]

oh wait

ill solve it first

@lean otter x^3 / x^4+x^2+1, cant be integrated by partial fractions

cause degree of x^3 ( ie 3) is greater than that of each factors ( ie 2)

[lean otter]

i meant partial fraction on the entire original expression

[dire crypt]

yes but

[dire crypt]

^

[winter pivot]

you should have (x^3 + x + 1)/[(x^2+x+1)(x^2-x+1)]

[lean otter]

It can be

[dire crypt]

right

[dire crypt]

hm

okay ill try give me 5 mins

@lean otter @winter pivot this is what i got

can someone check if its right, pls?

[final halo]

differentiate it

[dire crypt]

br

[safe radish]

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[rigid kernel]

How do I do part B

[still charm]

3 - 5x = 2.995

@rigid kernel

[rigid kernel]

u get the x value but then

how do u find the approximation

[final halo]

use a

plug in your value of x

[rigid kernel]

why do we do 3 - 5x = 2.995

thoguh

[still charm]

Because that's what we're approximating

We want to approximate 2.995^k

So we can just find (3-5x)^k at x = 0.001

And then use the expansion

[rigid kernel]

what does the x value represent

[safe radish]

@rigid kernel Has your question been resolved?

[still charm]

x value represents where we center the approximation

[rigid kernel]

oh they both 4

exponent

we can do that

[rigid kernel]

so we would then do (3-5(0.001))^4

right?

[still charm]

Yes

[rigid kernel]

oh

ok tghanks

i uinderstand now

.clsoe

.close

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[stray socket]

\geq is >= btw

[still charm]

Looks to be an am gm problem

[flat frigate]

willump

[still charm]

Imho

[peak bramble]

Cauchy-Schwarz?

[unique bison]

Move 2b^2 to the rhs and treat that as a quadratic in b?

Comple the square

Right hand side of the inequality

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[lean otter]

hi

[lean otter]

this is the problem but I can't use any other methods

I have to use elimination method to solve it

someone helped me solve it but he used substitution method which was still 2 equations instead of 1

.close

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[proven pollen]

Hello

[proven pollen]

How do intergrals work?

I know they are opposite of derivatives, but are they the exact opposite?

?

[oak wraith]

have you look up textbooks about it?

[proven pollen]

I don't find great resoruces

[novel onyx]

Integrals are used to find the area under a curve while derivatives are for the slope of the line

[oak wraith]

3b1b, for intuition understanding.

[proven pollen]

Aren't integrals and Derivatives opposites?

[oak wraith]

and any calculus textbook can give you a proper definition

[fossil forum]

Yes the process to take the derivative and the integral are the same

But when you differentiate, you lose the constant

So when you integrate you denote the lack of constant with c

[proven pollen]

the derivative of a constant is 0

but what about integrals?

how do you get that constant?

[fossil forum]

Well you can’t know whether a constant existed before differentiation so you can’t get it. It’s denoted by a c, like I said

You integrate what you can and add + c to the end

[proven pollen]

so the constant is always unknown in integration?

[oak wraith]

they can be view as inverse operation of each other at some cases (fundamental theorem of calculus). but that is not the definition of integral

[fossil forum]

If you were to be given a coordinate on the line you could solve for c

[proven pollen]

oh

[proven pollen]

really?

[fossil forum]

But otherwise it remains unknown. When you have a definite integral you’ll cancel out your C’s to find the area

[novel onyx]

or you may be using a definite integral, which is the area under the curve from a to b, which you don't need a constant for

[proven pollen]

oh

it makes a bit sense

[proven pollen]

then what is the true definition of an integral?

[oak wraith]

area under a curve

[proven pollen]

ok

thanks

.close

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[lean otter]

Find the radius and centre of the given circle
2x²+2y²-4x+6y+3=0

[lean otter]

How did the centre became 1,-1.5?

What step confuses you

The last stepp

Centre is ...

The general form of a circle is

$(x-h)^2+(y-k)^2=r^2$, where (h,k) is the center, and r is the radius

[flat frigate]

Legolas

[lean otter]

Those are the corresponding values in the equation shown in the notebook

[lean otter]

I didn't get u since the h is positive and k is negative

On the centre

1,-1.5

?

Ohhhh

NVR mind

I understand now

Thanks

And for the radius?

How it became root

1.75

@lean otter

Because in the equation we have r^2

And 1.75 is the value of r^2 in the equation we have

So r^2=1.75, so r = sqrt (1.75)

Uh

[lean otter]

We have r^2 at the end of the equation, and 1.75 is in the place of r^2

So r is just the sqrt of 1.75

Oh i see

Thanks...

.close

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[stark tulip]

is this answer correct?

both of them

please tag me when your answering or reply to a text

[winter pivot]

what does the sign of the derivative mean?

[foggy zinc]

i think its asking when its a decreasing function

[stark tulip]

negative when the function is decreasing and positive when the function is increasing.

[winter pivot]

can a function be positive, but decreasing?

[stark tulip]

no

[winter pivot]

really?

[stark tulip]

wait no

it can

[winter pivot]

think for a bit

[stark tulip]

my bad

yes it can

[winter pivot]

give me an example

[stark tulip]

so if a point is a 5,5

and it decreases to 4,4

its still positive

[winter pivot]

yeah!

could you draw a function that looks like that?

[stark tulip]

Like just plot the dots?

[winter pivot]

sure

[stark tulip]

Okay

[winter pivot]

not quite

it would have to go down from left to right

[stark tulip]

Okay

4,4 and 6,3

[winter pivot]

yep

[stark tulip]

ok so

[vast obsidian]

Maybe drawing a function, which has a derivative, instead of two points would be helpful.

[stark tulip]

im not sure how

[vast obsidian]

You could just draw a decreasing line

[stark tulip]

Okay

[vast obsidian]

So no Weierstrass function please

[stark tulip]

wait so what bout this?

[vast obsidian]

Proof by counterexample. f(x) > 0, but f'(x) < 0

[stark tulip]

im confused, wdym by that

[vast obsidian]

This is an counterexample of what she caims

[stark tulip]

So shes wrong?

hello?

[vast obsidian]

You know I help other people, too?

[vast obsidian]

Obviously

[stark tulip]

Sorry

[stark tulip]

But why is she wrong

[vast obsidian]

You just made an counterexample? Your y value is bigger than 0 and your derivative is less than 0 as the function is decreasing

So you cannot say if the function is positive the derivative has to be positive as well

[stark tulip]

Oh i understand

and im sorry again for the rush, im just really stressed out

.close

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[stark tulip]

would it be (+inf, 1) U (2, +inf)

[plucky elk]

draw tiny slopes along the curve

if the tiny slopes are point down, then the derivative is negative

[stark tulip]

Tiny slopes?

but dont i need intervals

[stark tulip]

and (0.5, 2)

[plucky elk]

it's just for you to get intuition

[stark tulip]

Oh

well ik where it is its just that

idk how to write the actual intervals

[stark tulip]

woah

[plucky elk]

this is the right interval notation, just not the right answer

[stark tulip]

Oh what

could u help me with what I did wrong

[plucky elk]

draw the red tangent lines along your entire curve. maybe 10-15

[stark tulip]

i dont understand

[plucky elk]

do you know how to draw tangent lines?

they're line segments really

[stark tulip]

I don't

like idk how to

[safe radish]

@stark tulip Has your question been resolved?

[safe radish]

@stark tulip Has your question been resolved?

[stark tulip]

.close

[safe radish]

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[alpine adder]

Factor the denominator

Then see what value of a can get you a factor of x-2 on the numerator

[safe radish]

@stark tulip Has your question been resolved?

[safe radish]

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[stark tulip]

I factored but im not sure how to get a

[trim swan]

You factored the denominator you mean?

[stark tulip]

yes

[trim swan]

So

$\frac{3x^2+ax+a+3}{(x-2)(x+3)}$

[flat frigate]

tatpoj

[stark tulip]

Yes

wait

yes thats what i got

sorry i was just checking

[trim swan]

All good

And we want to be able to take the limit at x=2

Do you agree, at x=2, the denominator is equal to 0.

[stark tulip]

2 - 2 = 0

and then 2 + 3 would be 5

wait yes

0 x 5 so 0

[trim swan]

Exactly. x=2 causes the denominator to be 0 because of the (x-2) factor

[stark tulip]

yes

[trim swan]

That's why we can't just plug in x=2, the function is not defined there.

[stark tulip]

Right

[trim swan]

But if the numerator also has a factor of (x-2), then you can cancel it with the one in the denominator

So we need to find a value of a that allows the numerator to have a factor of (x-2)

[stark tulip]

Okay

How do I find that?

[trim swan]

You agreed with this

[stark tulip]

Yes

[trim swan]

So, when x=2, we want the numerator to also be equal to 0.

[stark tulip]

ohh

so for that

would i just

3(2)^2 + a(2) + a + 3

and see what

would make it 0?

[trim swan]

yep

[stark tulip]

okok

Im gonna try one sec

[trim swan]

Okay

[stark tulip]

so a is 5?

[trim swan]

I get -5

[stark tulip]

if its 3a + 15

and u divide by 3

wouldn't it be a = 5

[trim swan]

Where's your equal sign?

We want the numerator to be equal to 0

[stark tulip]

3a + 15 = 0

[trim swan]

Right

[stark tulip]

Oh wait

ur right

my bad

[trim swan]

All good

[stark tulip]

that helped out

[trim swan]

Good!

If you substitute a=-5, I think you'll find you can factor the numerator and cancel the factor of (x-2)

[stark tulip]

Thankyou so much

I can get the limit from here

[trim swan]

You're welcome, happy to help

[stark tulip]

Will u still be online for a while?

I will come back soon because I have some more questions I need help with

[trim swan]

If not I'll be back pretty soon

[stark tulip]

Okay thankyou

[safe radish]

@stark tulip Has your question been resolved?

[stark tulip]

just to make sure

[stark tulip]

the limit is undefined right?

[trim swan]

No, it's defined

After finding a=-5 what did you do?

[stark tulip]

oh..

[stark tulip]

1 sec

nvm i have no idea what i did

[trim swan]

Did you substitute a=-5 into the numerator?

[stark tulip]

yes

[trim swan]

So you have

$\frac{3x^2-5x-2}{(x-2)(x+3)}$

[flat frigate]

tatpoj

[trim swan]

Right?

[stark tulip]

i did

[trim swan]

Well, that will give you 0/0

Because we already knew the denominator would be equal to 0. And we picked a=-5 specifically so that the numerator would be equal to 0.

[stark tulip]

Oh right

[trim swan]

So don't plug in x=2

[stark tulip]

Okay so just a then

[trim swan]

Just look at it like this

[stark tulip]

okok

[trim swan]

And factor the numerator

[stark tulip]

alright

(3x + 1)(x-2)

[trim swan]

Awesome, so that gives us factor of x-2 to cancel out

[stark tulip]

Okay

then what do I do with it

[trim swan]

$\frac{(3x+1)(x-2)}{(x+3)(x-2)}$

[flat frigate]

tatpoj

[trim swan]

This is undefined at x=2

But everhwhere else, it's equivalent to

$\frac{3x+1}{x+3}$

[flat frigate]

tatpoj

[stark tulip]

oh so x - 2 cancel

[trim swan]

Right

But only because we're talking about a limit

At exactly x=2, it would still be undefined

[stark tulip]

right

[trim swan]

But extremely near x=2 (and everywhere else), it is the same as this

[stark tulip]

Okay got it

[trim swan]

So you can just plug in x=2 to that to get the limit

[stark tulip]

Ohh okay so ill try 1 sec

7/5

[trim swan]

Yep

[stark tulip]

Wow

it seems so much more easy to me when u explain it

vs my teacher

[trim swan]

Happy to help

[stark tulip]

Can I show u 3 other questions i need help with?

[trim swan]

Go ahead and DM me, but I might not be able to look at them for a little bit

[stark tulip]

I acc need to hand this in 1.5 hours

thats why

.close

[safe radish]

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[keen pilot]

so what do you need help on?

is it like everything or just want the first step?

@stark tulip

[stark tulip]

everything yeah

im not great at this unit

[keen pilot]

so lets start by evaluating the limit on the left side of function describing x = 0.5

that'll know what value we have to approach

and then from there we can create a system of linear equations

[stark tulip]

So what do i do

.close

[safe radish]

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[timber bison]

I need help with this proof im stuck on what to do next

[timber bison]

<@&286206848099549185>

[unkempt solstice]

!15min

[safe radish]

Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.

[timber bison]

[safe radish]

@timber bison Has your question been resolved?

[severe smelt]

@timber bison you'll have better luck here probably #advanced-analysis

[timber bison]

hopefully, i will try thanks

[safe radish]

@timber bison Has your question been resolved?

[safe radish]

@timber bison Has your question been resolved?

[safe radish]

@timber bison Has your question been resolved?

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[tropic wharf]

i need helpl please

[lean otter]

It’s magnitude is the mass times the change in velocity

[tropic wharf]

ye

[upbeat dragon]

change in velocity can be found using a triangle (Probably should have drawn it more like an isoceles triangle oops)

[tropic wharf]

oh

oh

okk thanks xD

can i ask why that side is the change in V

[upbeat dragon]

The line on top represents your velocity after it got hit, and the one below represents the one before it got hit. So the change there is from the tip of the initial velocity to the tip of the final velocity

[tropic wharf]

oh

oh understood, thank you sm

.close

[safe radish]

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[safe radish]

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[languid crystal]

What does the arrow to the right mean like this =>

[plucky island]

⟹ means "implies"

[iron tulip]

\[
  \implies
\]
means `implies': $A \implies B$ is the same as `not (A and not B)'

[flat frigate]

Lance

[plucky island]

X⇒Y . This means that if X is true, then Y is also true.

[iron tulip]

yeah identical explanations

[plucky island]

oo anime

[iron tulip]

[languid crystal]

But why cant we use 2+ 2 = 4

[iron tulip]

wdym

[languid crystal]

Well does 2 + 2 => 4 also work

[iron tulip]

oh

do you mean $\geqslant$?

[flat frigate]

Lance