help-38

[shrewd ridge]

once 1 + twice 2 − once 2

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[zenith bane]

How to do 48

I got -3 but i dont know how to solve the other answer

[split chasm]

what's -3 supposed to be

oh

the works a bit messy

[zenith bane]

Answer

And i dont know how you get the other answer

[split chasm]

you didn't set up your equation properly

[zenith bane]

🤔

[split chasm]

since you don't care about the quotient, only the remainder,
remainder theorem would be less tedious

[zenith bane]

Emmmm

How should I solve it

[split chasm]

it looks like you ignored the equal to the square of the remainder when...

you just set the remainders equal to each other

[zenith bane]

Oh i miss that

What should i do

I’m kinda lost

[split chasm]

try write what the property says

[zenith bane]

So do the question again

[split chasm]

you have the remainders already

just set up that relation stated properly

[zenith bane]

So like i use the equation divide by x-1 and square the reminder?

[split chasm]

... = square of other remainder
instead of
= other remainder

[zenith bane]

Like that?

[split chasm]

poor wording

[zenith bane]

Sry is reflected

[split chasm]

,rotate

[zenith bane]

But i got the reminder of 1

Is that how it work

[split chasm]

no

[zenith bane]

Emmmmm

[split chasm]

please read what i'm saying

you already had your remainders

[zenith bane]

So square -3?

Sry i am kinda confused

[split chasm]

no

what's your remainder in terms of a when dividing by (x+2)

[zenith bane]

-3?

[split chasm]

no

[zenith bane]

Oh

Ok i think I should do it again, the question so now i do the same thing, but when i set the question

I square the x-1 reminder

[split chasm]

yes

that's what i've been saying

[zenith bane]

Sry🥲

[split chasm]

i actually don't see how you got
3 + a+ 1
for the remainder when dividing by (x-1)
but it is the correct remainder

since you don't care about the quotient, only the remainder,
remainder theorem would be less tedious
you can get the remainders simply by plugging in x=-2 , 1

[zenith bane]

I think I got it

Tyty

I got a other question

Question 49, my idea was synthetic division 2, and -1 figures out both reminder and set them equal to each other, and plus a other 12 to the “2” reminder side

Is my idea make sense or is it wrong

<@&286206848099549185>

[opal grove]

That was actually what I was thinking

Add 12 to -1, set equal

[zenith bane]

Oh shoot

[opal grove]

Then use the multi choice I think

[opal grove]

The remainder

[zenith bane]

Let me try it again

[opal grove]

Of -1

Not -1+12

[zenith bane]

I still got it wrong

Idk how

[opal grove]

2b+4a+3

Is the last part

[zenith bane]

That’s what I got

[opal grove]

-b+a-4

Add 12

[zenith bane]

Yeah

[opal grove]

-b+a+8

[opal grove]

Remainder for 2

[zenith bane]

What’s that

[opal grove]

Remainder for -1 +12

So I can set it equal to the x-2 example

[zenith bane]

How did u get that

[opal grove]

Last one was b+2a+4

Multiply by 2

Is 2b+4a+8 but subtract 5

(5 being constant

So

2b+4a+3 is remainder for x-2

[zenith bane]

8 need be times by 2

[opal grove]

Og

So

Oh

[zenith bane]

I do it wrong

Do my paper

[opal grove]

You suck lol (I saw the 2)

And thought

Hey that was accounted for

[zenith bane]

Bruh

😎

[opal grove]

It was not

(You don’t suck it’s just)

[zenith bane]

Don’t rely on Me lol

[opal grove]

2b+4a+16

[zenith bane]

-5

[opal grove]

So 2b+4a+11

-5+16

[zenith bane]

Yes

[opal grove]

-b+a+8=2b+4a+11

[zenith bane]

Yes

[opal grove]

(Sorry I’m on my bed rn so I’m just doing mental math)

[zenith bane]

Is fine lol

[opal grove]

A no work

B no work

C no work

Wait what

[zenith bane]

Yup

I think we are cooked

[opal grove]

Hold on

Sorry I’m like sleep deprived and doing it in my head

[zenith bane]

Is fine

[opal grove]

It’s C

Wait

[zenith bane]

How

Like what’s the step

[opal grove]

-b+a+8=2b+4a+11

-3b-3a=3

[zenith bane]

Ok

[opal grove]

-b-a=1

3=a

[zenith bane]

I mean what if we don’t have the multiple choice

[opal grove]

-4=b

[opal grove]

Um

Idk

[zenith bane]

Cuz on test I don’t think we do

[opal grove]

Some sort of algebraic manipulation

What

How

[zenith bane]

Idk our teacher just like that

[opal grove]

Okay

It’s possible

But very hard

(For this problem

-b-a=1

So this

[zenith bane]

I think we should plug like (-2,31) in to the equation.

[opal grove]

Technically it can be like -5=b and 4=a

[opal grove]

Yeah

[zenith bane]

But how

I got no idea

Cuz it give us this information

I think we should use it

[opal grove]

The only thing you can do

Is plug in -2

So

[zenith bane]

Basically we do it right

So we just need to plug in -2 to like get the answer

[opal grove]

Yeah

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[spring cradle]

the answer should be -100.8 but im getting -170.8. i cant find the error

[trim lichen]

2nd last line first term

28/12 * 6 should be 28/12 * 36

also tbh you have made your own life harder than it needs to be

[spring cradle]

oh i see

[spring cradle]

how so

is there an easier way of doing this

[trim lichen]

well yeah like you could have written the inner integral as $$(4-4x)\paren{\frac{7}{6}x - \frac{7}{10}x}$$ rather than have to drag 4 terms after yourself

[solid kiln]

Ann

[trim lichen]

and then this can get simplified a bunch algebraically

like $4(1-x) \cdot \frac{7}{30} (5x - 3x)$ or something

so that you end up with $\frac{14}{15} \int_0^6 2x(1-x) \dd{x}$ which is less heavy

[silver glade]

i think you missed an x

[trim lichen]

yes i did my bad

[solid kiln]

Ann

Ann

[trim lichen]

this actually makes it even simpler cause you can now pull a 2 out as well

[spring cradle]

oh i see

yeah that wouldve been easier for sure

ill keep that in mind. thank you

[trim lichen]

moral of the story is dont burden yourself with more arithmetic than necessary

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[split chasm]

wdym

[dusty sleet]

y = -√x

Is this what you want?

I don't think I understand your doubt

Also, lines are infinitely many, so what do you mean with linear graph?

[past cargo]

dk what your question is

but y = sqrt(x)

if you send x to x^3

then the line is above the "radical graph"

,w graph y = sqrt(x^3) and y = x for -1 < x < 2

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[past cargo]

that's a radical tho

[little wigeon]

radical graph but only one degree inside it

[past cargo]

then it's not possible

between 0 < x < 1

x^(1/2) > x

u can translate it yeah

but it's not going to be under the linear graph

the line can be tangent to it

or the "radical graph" can be totally to the right of the line and not intersect

something like

,w graph y = sqrt(x - 0.75) and y = x

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[past cargo]

well if you do the same transformation

then you can always define a new variable

v

v = x - c

then you have the same y = v and y = sqrt(v)

so the same situation

[past cargo]

as this

[little wigeon]

hm ok...

[past cargo]

sure

[little wigeon]

anyways tysm blesss

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[runic heath]

Can someone help me simplify the following

[orchid wagon]

observe that 11 + 2sqrt(30) can be turned into the form (a + b)^2

[runic heath]

Yeah

This ?

[orchid wagon]

yeah

[runic heath]

Then ?

I am lost

[orchid wagon]

we wish to turn 11 + 2sqrt(30) into the form a^2 + 2ab + b^2

[runic heath]

Ok

[orchid wagon]

i believe you are astute enough to know of course the 2ab term will be the 2sqrt(30)

[runic heath]

Look I have done it all

But I am still not sure

[trim lichen]

about what

[orchid wagon]

this mildly unsettles me

[trim lichen]

some of the equals signs at the start of your lines are misused

and that also

[runic heath]

I am not really sure about this

My friend was telling me to replace 11 with 6+5

[trim lichen]

well you try to match your a and b to the required values of their sum and product

[runic heath]

But why not other numbers like 7+4 or smth

[trim lichen]

7*4 isn't 30

[runic heath]

Oh ok

Then what about this ?

[orchid wagon]

they are similar

[runic heath]

can you elaborate?

[trim lichen]

is that an S or a five under the inner root?

[runic heath]

"5"

Abnormally heated ?

[orchid wagon]

well, turn it into the form a^2 - 2ab + b^2

to make the expression a bit clearer to you, you can move a part of the 8 into the square root; in particular, 8sqrt(5) = 2sqrt(80)

[runic heath]

so is it like the previous:
Where:
- a-b=21 ?

[orchid wagon]

no, the minus sign should not matter in your system of equations

it's essentially the same idea; find a, b such that ab = 80 and a + b = 21

[runic heath]

how'd you get ab to equate 80 ?

[orchid wagon]

it's a bit of a shortcut

[runic heath]

Could you tell me ?

[orchid wagon]

you want to find ab such that ab = sqrt(80), so a and b may be in square roots. but i recycled the notations by letting a and b denotes the number inside the square root instead of the roots themselves

[orchid wagon]

with this we have a simpler system of equations here, the terms for your (a + b)^2 shall be the square roots of the system's solutions

[runic heath]

but like how does ab= sqrt(80) ?

[orchid wagon]

well you want to turn 21 - 8sqrt(5) into the form a^2 - 2ab + b^2

[runic heath]

yeah

[orchid wagon]

so we have a^2 + b^2 - 2ab = 21 - 2sqrt(80)

if you match the coefficients, we have the following system of equations [
\begin{cases}
a^2 + b^2 = 21\
ab = \sqrt{80}
\end{cases}
]

[runic heath]

yeah

[orchid wagon]

but by doing what i said here, we have a simpler system of equations that you can solve

that is, a + b = 21 and ab = 80

find such a and b, then plug it back into (a + b)^2

note that the a and b in the system of equations and the a and b in (a + b)^2 are not the same thing

in particular, the latters are the square root variants of the former

[runic heath]

Yeah, I understand all of this.

My, doubt

is that how would you go about converting 21=a^2+b^2

like how would you solve for a, b

[orchid wagon]

yeah

[runic heath]

(Also sorry for wasting your time)

[runic heath]

how do you find numbers for the variable to equate the statement ?

[orchid wagon]

usually the numbers are simple enough for me to do it in my head

in the events they are not, there are methods that i think is not relevant to you at the moment

[runic heath]

Thanks

One final thing how did u learn this ?

[orchid wagon]

in school

[runic heath]

Anyways thanks for up your help

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[spice shuttle]

.reopen

.reopen

[glass radish]

how does this simplify?

[whole coral]

Multiply all the terms in the denominator by a

[glass radish]

ohh

i see now

thank you

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[halcyon harness]

hello

[halcyon harness]

is this formula correct for getting the variance of a sampled group data

[supple copper]

What’s the f for

[halcyon harness]

frequency

assuming we use midpoints

[supple copper]

Do you have more context

[halcyon harness]

for example we have a table of grouped data with each class and their midpoints and frequencies

trying to find the variance of the sample

n = number of data values
f = frequency
Xm = midpoint

i know that this version is for non grouped data

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[halcyon harness]

nvm i prooved it sorry

it is correct

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[frozen hedge]

Is it possible to l hopital this

Cause im doing it and it keeps stretching out

[plush vigil]

technically you can use lhopital but it's not the best way

try using cosA-cosB formula

[frozen hedge]

Thats what the solution says and it ends up doing a roundabout into taylor expansion

Both methods seem meh

[plush vigil]

taylor expansion?

i dont think you need that

after cosA-cosB you can use sinx/x result to simplify the limit

[frozen hedge]

Yeah...

[frozen hedge]

Dont think l hopital gonna work

💀

[plush vigil]

try cosA-cosB

[frozen hedge]

Aight

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[nimble bolt]

What did i do wrong here?..

[regal gorge]

Why did you square the 3x?

[nimble bolt]

Cuz i didnt know what to do with the encircled 2

[regal gorge]

Try rewriting it like this, and move the 1/2 to the front

[nimble bolt]

2 and 2 cancels out?

[regal gorge]

Yep

[nova dirge]

Try to make 1 as a base 3x log

And just square the sqrt54

Using the law of logarithms

[regal gorge]

There's many ways to do logs

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[nimble bolt]

This one alr?

[regal gorge]

Got it 👍

[nimble bolt]

Alr
Thanku 🙂

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[distant swallow]

Can anyone explain this to me? The questions is Prove 120 divides n^5 - 5n^3 + 4n for all values of n

[distant swallow]

Also is that pascals rule? I have no idea how they turned the polynomial into that form

[fleet bear]

$$ \binom{n+2}{5} = \frac{(n+2)!}{(n + 2 - 5)! 5!} $$

[solid kiln]

StrangeQuarkAL

[distant swallow]

ah right

so like (n+2) C 5?

[fleet bear]

Yeah

[distant swallow]

how did they arrive there?

to the 5!

[fleet bear]

$$ 5! \cdot \binom{n+2}{5} = \frac{(n+2)!}{(n + 2 - 5)!} $$ = (n + 2)(n +1)(n)(n-1)(n -2) $$

[solid kiln]

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

[distant swallow]

ohhh shit yeah

yeah yeah that makes sense

Ah but wouldn't it be .1/5! ?

Oh wait no mb I get it

[fleet bear]

Epic

[distant swallow]

thanks bro

[fleet bear]

No problem

[distant swallow]

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[verbal breach]

💀 what do i take median class as

[verbal breach]

i know it’s either option b or c

n/2 is 45

[zinc ginkgo]

https://www.wikihow.com/Find-the-Median-of-a-Histogram

[verbal breach]

but do i take median class above 45 or 45

[zinc ginkgo]

45 is between 40 and 50

[verbal breach]

but 45 is n/2

not median

o wait

but i see some answers say median class should be the cf which is greater or equal to n/2

[zinc ginkgo]

median index is (n+1)/2

[verbal breach]

dk what that is

[zinc ginkgo]

same as their step 3 here

oh your n=90

so you use the median is the average of the middle two numbers in step 4 ^

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[abstract junco]

What do I do

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[jaunty scarab]

you should put a bit more effort into the diagram, tbh. It's basically impossible to see anything there

anyways, your parabola has directrix x=-5, so it will be of the form x = ay^2 + by + c

you know that the equation holds for x=7 y=1, and for x=-2 y=13, since you're given two points

since both circles are tangent to the parabola, to each other, and that the parabola passes trough the center of the circles, you also know that the focus needs to be at the intersection of the circles

having the focus, you also know that the point midway between the focus and x=-5 will be the vertex of the parabola

which gives you a third point for the equation. Solve the system for a, b and c @abstract junco

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[quick sand]

how do i do A in this

[strong delta]

the waiting times have been given as a range instead of discrete values

so you'll take the mean of each class to find the estimate of the mean

[quick sand]

i dont know how to interpret it into a mean

[strong delta]

so from 0<x<=10
5 would be the mean right
so we just assume the value to be 5

for 10 to 20 we assume it to be 15

[quick sand]

Yeah I get that

[strong delta]

and we do the same thing for all the classes and carry out the standad way to find mean after that

it will be an estimate and it will not be accurate

[quick sand]

I don't know how to find the mean from that

[strong delta]

(mean of waiting time)*frequency for every class

add them up

divide them by sum of frequency

[quick sand]

so the cumulative frequency is 100

[strong delta]

yea

that is what u divide it by

[quick sand]

so the sum of waiting time / 100

[strong delta]

no

take the mean of all the waiting times

for example for 0<x<10 the mean is 5 and frequency is 2

so we can assume 2 people waited for 5 minutes

for the next one, 11 people waited for 15 minutes

and so on

so you find the total time all of these people waited by doing multiplication as follows:
2x5
11x15

and so on

for all the frequencies given

then add them up

[quick sand]

Then divide by 100

[strong delta]

yep

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[flint rock]

i need help with geometry. Theres a right angle trapezoid abcd with diagonals that are normal to each other so that ab and cd are parallel to each other. Prove that ac * bd >= 2* ab * cd

[hidden linden]

What have you tried so far?

[trim jolt]

@flint rock Has your question been resolved?

[flint rock]

ive written what the sides are equal to using Pythagoras theorem, but i have no idea what to do from there T-T

[hidden linden]

I'm trying to imagine this geometry. Is this the one?

[flint rock]

yup

[hidden linden]

And how exactly are the vertices placed?

Actually, that might be redundant.

[flint rock]

this is where im at rn

i named the point where the diagonals meet F

do you have any idea?

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@flint rock Has your question been resolved?

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[mint agate]

how do u do this?

[real rose]

It's asking for the zeroes. Where is $f(x)= 0$? (Hint: use the information given).

[solid kiln]

mathisfun

[mint agate]

yeah but i dotn know how to actually figure it out

is (-2,0) one

[real rose]

Yep

[mint agate]

is that it

[real rose]

No

[slender shard]

The problem is giving you two characteristics about the function. One is the interval with respect to x and the other is if along that interval, the curve is above the x-axis or below the x-axis.

[marble wharf]

imagine you tried to draw the graph. how might that look like

[inland forge]

They gave you the intervals of where it is positive and negative

We know negative is below the x-axis, and positive is above the x-axis, when the intervals are told to you, they are pretty much telling you when the function is below and when it is above

We would want to focus on the ends of those intervals, since that is where we can figure out it changes

The reason (-2,0) is a zero is because for the interval with -2 as a starting point and -2 as an ending point, one of them was below the x-axis (f(x)<0) and the other was above (f(x)>0), meaning for that to happen, it must have passed through the axis to switch between, intercepting the x-axis in the process

The intervals may look similar to how point are written, but just to clarify, the first number represents the starting point which is smaller, and the second number represents the end point which is written to be larger, and it means all the numbers in between those start and end locations on a number line

[trim jolt]

@mint agate Has your question been resolved?

[mint agate]

is it -2, 0, 2, and 5?

also

[real rose]

Yep.

[mint agate]

the point A (1,-2) lies on function f(x)= x^3 - 3x, determine if point A is a local max/min using rates of change

how do u use rate of change in this

i owuld just

use a table of values

or smth

[real rose]

Pretty much.

[trim jolt]

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[trim jolt]

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[strong canopy]

need help

[strong canopy]

how do i do this

[real rose]

Can you test each out?

[strong canopy]

I did and got the third one

but idk if thats right

[real rose]

Check it again.

Try it with, say, a quadratic.

[strong canopy]

ok

O

Is it the 2nd one

[real rose]

Yep.

[strong canopy]

thanks

.close

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[real herald]

Can someone explain how they go from 2->3 in this proof? Mostly why they need the gN(g^-1)^-1?

[boreal harbor]

define $g'\coloneq g^{-1}$. then $g^{-1}ng=g'ng'^{-1}$, which implies $g'ng'^{-1}\eqcolon n'\in N$ is an element of N. from there you can rewrite n as $n=gn'g^{-1}$ proving that for some arbitrary $n\in N$, $n\in gNg^{-1}$.

[solid kiln]

esca

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[boreal harbor]

theres really not much that can be explained unless you articulate what it is youre having trouble with

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[trim jolt]

@acoustic sphinx Has your question been resolved?

[acoustic sphinx]

.close

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[zealous badger]

Heyyoo, I want to create a term that can be used to calculate if an ISBN number is correct. I want to use ∑ (i=1→10) and multiply it by the individual digits of the ISBN-number. Is this possible with a normal calculator?

[trim lichen]

what do you mean by "normal" calculator

is this what you are looking at rn?

[zealous badger]

this can help

Thanks

.close

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[outer kiln]

What’s the difference between dS, dA, dr in vector calculus?

[outer kiln]

For example greens formula uses dr but divergence theorem uses dS, I don’t really get the difference

[cunning cliff]

Arent these different types of integrals

by any chance

[outer kiln]

greens formula is for curve integrals right

or line integrals I mean

And then divergence theorem is double integrals

I’m just a bit confused by the difference

[cunning cliff]

greens equates a line integral to a double integral

[outer kiln]

Yeah and then the double integral is integrated over dxdy

Or whatever variables

But then when we take double integrals in like stokes or divergence dS is used

[cunning cliff]

yes

You might want to ask in topic specific channel

since u have less of a question

and more clarification/understanding

#multivariable-calculus

[outer kiln]

Oh okay I’ll do that ty

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[fresh mesa]

in the first place, i do not understand what is a "projection" of a vector over another

and why it is a numeric value

[trim jolt]

@fresh mesa Has your question been resolved?

[fresh mesa]

<@&286206848099549185>

[elder linden]

do you know the formula for the projection of a vector along another?

[cunning cliff]

when you project x onto y

[fresh mesa]

i dont know

[cunning cliff]

you are finding the y-directional component of x

[fresh mesa]

so, which is the projection

[fresh mesa]

so it should be $|y|=|x|cos\theta$?

[cunning cliff]

p(x) is the projection in my drawing

[solid kiln]

therealtdp

[hidden linden]

Projection is along a vector.

[cunning cliff]

not |y| but your projection

[elder linden]

the projection of a vector A on a vector B is given by B.A/|B|

[cunning cliff]

|p|

but thats the right idea

and yes these are vectors

[fresh mesa]

hmmmm yes yes where p is the modulus of my projection

[fresh mesa]

yeah man i got it

thank you guys

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[obtuse talon]

hi

[obtuse talon]

i need help

what exacly did he do to turn the brackets into equations

[trim lichen]

you're almost certainly misusing the word "equation",

but what exactly are you talking about?

are you talking about how (x-1)^2 became x^2 - 2x + 1 and how (2x-3)^2 became 4x^2 - 12x + 9?

[obtuse talon]

LMAO yesnits an inewuality

[obtuse talon]

yes

[trim lichen]

right

do you know how to expand (a-b)^2 in general?

[obtuse talon]

ik its a dumb equstion

[obtuse talon]

nope, my question is what exacly he did so i can search a utube video and learn

bracket expansion?

[trim lichen]

the key words you're looking for are "distributive law" and, if you are so inclined, "FOIL"

sure, bracket expansion should bring something up.

[obtuse talon]

ty very much dear ann

[trim lichen]

if you know how to expand (a+b)(c+d) with the distributive law, you can re-derive the expansion for (a ± b)^2 on your own

[obtuse talon]

oh

i will learn all i can from indian men on utube

thx alot ann

[hidden linden]

This is binomial multiplication if you want to look it up.

[obtuse talon]

binmial multiplication

got it

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[woeful harbor]

can someone explain to me why ax + by + cz = d is the equation of a plane in 3D?

[dapper swift]

you have two independent variables and the third variable is dependent

if you let x, y be the independent variables

you have z = (d - ax - by)/c, so it depends entirely on what x, y are

2 free variables and 1 dependent variable, and of course it's a linear combination of x, y, z

so you know it's a plane and not a line (1 degree of freedom) or a single point (0 degrees of freedom)

degrees of freedom is also the number of parameters

[woeful harbor]

i’m not sure i understand how this implies the graph of ax + by + cz = d is a plane. take z = (d - ax^2 - by^2)/c. it has 2 free variables and one dependent one but it’s not a plane

[dapper swift]

well it's quadratic in x or y so can't be a plane

you can check these conditions for the subspace of a vector space

[dapper swift]

it breaks when it's quadratic, cause if you replace (x, y, z) with (kx, ky, kz), it simplifies to a different equation, so not a vector space

you get ak^2 x^2 and bk^2 y^2 which doesn't cancel properly with kz

[woeful harbor]

i’m a little confused. so how does z = (d - ax - by)/c mean it’s a plane? i.e. why is it that linear combination => plane?

[dapper swift]

should I talk about the normal vector definition of a plane then

[woeful harbor]

yes

[dapper swift]

let $n = (a, b, c)$ and $r = (x, y, z), r_0 = (x_0, y_0, z_0)$

[solid kiln]

south

[dapper swift]

so you fix the base of the normal at position vector r0
and then r = (x, y, z) is free to move around, giving you the locus of all such points such that n and (r - r0) are perpendicular

[woeful harbor]

okay, wait, so you have a normal vector at position r0 which defines the plane. what’s r?

and n?

[dapper swift]

n is a given normal vector

[dapper swift]

r is any vector, (x, y, z)

you want a condition on (x, y, z) such that (x, y, z) is on the given plane

[woeful harbor]

what is the condition?

[dapper swift]

so x, y, z are the only variables and the others are all constnats

[dapper swift]

the red box

follows from $n \cdot (r - r_0) = 0$

[solid kiln]

south

[woeful harbor]

okay, i get how this implies the red box. n * (r - r0) = 0 means n and r - r0 are perpendicular. what’s r - r0 though?

ah

ah, okay, i get it

[dapper swift]

yeah and if you're okay with the red box

$ax + by + cz = (ax_0 + by_0 + cz_0)$

[solid kiln]

south

[dapper swift]

we just define $ax_0 + by_0 + cz_0 = d$, since those are all constants

[solid kiln]

south

[dapper swift]

all constants on the LHS of the one just above

[woeful harbor]

okay, thanks, this makes sense

[dapper swift]

no worries!!

[woeful harbor]

.close

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[cyan narwhal]

hi what to i do here?

[zinc ginkgo]

the slope of the tangent line is given by the derivative

have you learned derivatives yet?

[cyan narwhal]

BRUH

ofc

oh yeah isnt first deriv the gradient

or is it 2nd

[zinc ginkgo]

2nd derivative does not give you the slope

1st derivative gives you the slope

[cyan narwhal]

ok cool

yeah i did that a year ago just forgot

thanks

i get that thewn i just sub 3 right?

[neon dirge]

what do you have for a & b

[cyan narwhal]

well i hav 15x^2

didnt you mean integrate?

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[cyan narwhal]

.close

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[river cosmos]

I'm so confused

Is it A(y) or A(x)?

[timid imp]

what are you trying to use

disc/washer or shell

[river cosmos]

Disc

We haven't done shell yet

[timid imp]

you can do it 2 ways

A(y) with x=y^15

or A(x) with y=x^(1/15)

[timid imp]

this one is especially valid since it's only the first quadrant that you're concerned with

[river cosmos]

And the limits should be 0 to 1 right if I'm doing x?

[timid imp]

think so

yea

it'd be 0 to 1 for both A(x) and A(y)

and you understand how to go from the A(y) to the A(x) right

you just solve x = y^15 for y

[river cosmos]

Yeah

Wait I did it but the answer's wrong

[empty orchid]

!show

[trim jolt]

Show your work, and if possible, explain where you are stuck.

[timid imp]

wait my bad you can't use A(y) if you want to use disk/washer

I dont think

you have to use A(x)

[river cosmos]

I did use A(x)

[timid imp]

the area you're trying to use should be 1-x^(1/15)

not just x^1/15

it's a washer

[river cosmos]

But there's no hole? Is there?

[timid imp]

so it'd be $\pi \int_0^1 1 - x^{\frac{2}{15}} dx$

[solid kiln]

NahhFam

[timid imp]

wym

[river cosmos]

Like to use the washer method there has to be a hole in the solid right? π (R2-r2)? There has to be like a big R right?

How did you get the 1 to be the big R?

[timid imp]

y = 1 would be the top function

and x^1/15 would be the bottom function

big R is 1 since it's ontop

like in order to find the area between those

it'd be 1-x^1/15

int of that

from 0 to 1

[river cosmos]

Ohhhhh

I see

[timid imp]

I dislike volume questions

[river cosmos]

Wait it's still wrong 😭

[timid imp]

shouldn't be

what's the answer you're given

and from where

[lament locust]

this looks fun

[river cosmos]

It's a question from lumen in my homework- I don't know the answer - it just says it's wrong 😭

[timid imp]

what did you input as the answer

[river cosmos]

0.3696 (the decimal answer correct to four decimal points)

[timid imp]

is it supposed to be put in as a decimal? that should be right

if not then 2pi/17 would be the way you input it

[lament locust]

this is what ive gotten so far

[timid imp]

it shouldn't all be squared

it should be R^2 - r^2

[lament locust]

thats the volume of revolution between 2 curves

[timid imp]

y=1 is technically the curve

the 2nd curve

[lament locust]

our radius is from 1

[timid imp]

this does give a more reasonable answer

[lament locust]

so that radius that we are actually revolving is 1-x^1/15

[timid imp]

I suck at volume lol

[lament locust]

so its (1-x^1/15)^2 as r^2

[river cosmos]

it's still wrong 😔

[lament locust]

then multiply by pi

[timid imp]

do what @lament locust says

[lament locust]

0.0231 is the answer i hope 😭

@river cosmos

[river cosmos]

Yes!!!

[timid imp]

sorry lol

[lament locust]

gg 🫡

[timid imp]

I thought I understood it my bad

[river cosmos]

No no it's okay thank you so much

[lament locust]

now to make sure this is proper

@river cosmos find the volume of revolution, of some general curve f(x), from a to b, around the line y = d

[river cosmos]

Is this correct?

[lament locust]

from a to b

x = a to b

but justify to me

why does this work

[d-f(x) ]^2

[river cosmos]

Because we're using the disc method and the radius is like the top function minus the bottom one (like if we take a random point)

[lament locust]

hm

that may be the case

[lament locust]

but what if the function is above the axis

then what is this radius

[river cosmos]

f(x) - d !

[lament locust]

yea

now that seems very different from d-f(x)

and it is

but there is something very nice about r and r^2 that we can do

[river cosmos]

ooo what is it?

[lament locust]

rewrite that f(x) - d

in terms of d - f(x)

[river cosmos]

-d + f(x) ?

[lament locust]

you might need to factor out a negative

[river cosmos]

-(d - f(x))

Ohhhhhh

[lament locust]

yea#

[lament locust]

now when we do r^2

(d-f(x))^2 is just [d-f(x)]^2

[lament locust]

what about this one

[river cosmos]

That's also just [d-f(x)]^2 because the negative cancels out right?

[lament locust]

yea

(-1)^2 = 1

so both of those radiuses, above and below our axis y = d

will give us the same general radius

lets try something else though

[river cosmos]

Ooo what?

[lament locust]

if we want to find the volume of revolution between two curves

there is some conditions involved

that being one must be greater then the other for all values in that interval range a to b

[river cosmos]

But if that's satisfied we can still use the radius thing? With the radius being f(x)-g(x)?

No wait that's a washer right?

[lament locust]

im not too sure what i washer is

but the disk method works very nice here

[lament locust]

not exactly since that method was used for "making y = d the axis of revolution"

but our axis of revolution

still remains as the x axis

[river cosmos]

Oh... but how do you calculate the radius around the x axis? Is it just f(x)?

But that can't be right?

Because then the g(x) doesn't come into the equation at all

[lament locust]

so in this diagram we are gonna assume that f(x) satisfies that condition that its gonna be greater than (in y value) then g(x)

the goal is to find the volume of revolution between these two curves yea

[river cosmos]

Yeah

Do you just calculate the volume of f(x) and g(x) separately and then subtract?

[lament locust]

im going to indicate this g(x) as little r for small radius, and f(x) as big R for big radius

[lament locust]

yea

what is the area of the pink region

[river cosmos]

pi R^2 - pi r^2

[lament locust]

ye

applying this to the disk method now

what should we get for that volume of revolution between the curves

[river cosmos]

?

[lament locust]

goated

now factorise the pi

and substitute R and r with the functions

[river cosmos]

Integral of π [f(x)^2 - g(x)^2]?

[lament locust]

yep

from a to b

but that will give

the volume of revolution between two curves, (given f(x) is greater than g(x) in that interval range a to b)

how cool is that

[river cosmos]

Very cool! Thank you so much! I actually understand this way better now - thank you for asking all the questions 😅

[lament locust]

🫡

make sure to always to ask

the most important question of all

what if ...

• why but what if is just cooler 🔥

[river cosmos]

Very true

.close

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[undone crown]

Package holidays to Costa Natura are priced per person.
The current brochure only gives departure dates up to December 17th, but holidays continue after that date.
The total price per person is made up from the flight price for the outbound flight, the flight price for the flight back, and their cost for the room.
The room price does not depend upon the number of people sharing the room, but may change from week to week.
The flight price changes frequently, but is the same in both directions on any date.
There are no discounts.
The prices in the brochure (shown below) are total prices, per person, given in dollars; they refer to holidays starting on the date shown.
Someone has put a coffee mug on my brochure, and I can't read some of the figures.

[trim jolt]

Please don't occupy multiple help channels.

[lapis plaza]

what is the question?

[zinc ginkgo]

!1c

[trim jolt]

Please stick to your channel.

[zinc ginkgo]

.close

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[empty tulip]

can someone help explain this

As I've done something wrong here

[austere cedar]

Why do you think it's wrong?

,w integral of 5x^2e^x

[solid kiln]

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Requested by kaynex

[austere cedar]

Wolfram even gives it into the same form

[empty tulip]

oh

okay

wrong questions

oops

.close

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[nimble bolt]

If any1 gets to this
Please dm me..

[wraith hinge]

Which part?

Also what exactly is your doubt?

[nimble bolt]

Just to confirm answers

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[gilded ruin]

i am confused about my professors lecture, need some help understanding something

[gilded ruin]

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

how can this function (3x^3-9x) be injective

[cunning cliff]

,w graph 3x^3 - 9x

[solid kiln]

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[cunning cliff]

the function is Z -> Z

[gilded ruin]

yes

[cunning cliff]

this function isn't injective if it was R -> R

but for Z -> Z it looks like it might be

[gilded ruin]

what about something like -1 and 2

both are y=6

[cunning cliff]

oh then it isn't

[gilded ruin]

thats what im saying !

[cunning cliff]

,calc (3(-1)^3 - 9(-1))

[solid kiln]

Result:

6

[cunning cliff]

,calc (3(2)^3 - 9(2))

[solid kiln]

Result:

6

[cunning cliff]

yes isn't not injective.

[gilded ruin]

is my prof just wrong ??

this was a like take home question he had all day to think about it

[cunning cliff]

yes, its not like they don't make mistakes

[gilded ruin]

gg

he mentioned later on

its surjective if m does not divide n

but i dont rly understand that part either

[cunning cliff]

what is m what is n

[gilded ruin]

m is 3 in this case and n is 9

[cunning cliff]

and what is the overall context

are they referring solely to functions of the form f(x) = mx^3 - nx ??

[gilded ruin]

mx^3-nx

ye

[cunning cliff]

well if m and n share a divisor that isn't 1

then clearly this function cannot be surjective

so if gcd(m, n) > 1

[gilded ruin]

why tho

[cunning cliff]

then the function isn't surjective

because f will always be divisible by the gcd

in particular, f(x) will never be 1

[cunning cliff]

====
As for this claim

I'm not sure it is true

and I don't think it is

[gilded ruin]

idk

im just rly confused xdd

[cunning cliff]

if m is very large compared to n

then this function grows in magnitude far too quickly

for it to possibly hit small y-values

you can write down a more concrete proof with some counterexample, but yeah

[cunning cliff]

so this sounds like bogus.

[gilded ruin]

hm

ty ig

i will look online more

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[near flax]

idk how to find the domain and range for the graph that looks as wonk as that

wonky*

[severe fjord]

Do we know how we find range and domain through graph?

[near flax]

yea i think? domain is x and range is y'

[real rose]

Domain is just the numbers that a function takes in. Range is just the numbers the function outputs.

So what numbers are taken in, and what are outputted?

[severe fjord]

Domain is values of x that are valid to be use in it, and range the corresponding output

[near flax]

uhh for question number 1 i think the domain is 5,- infinite?

(5,-infinite)

[real rose]

That is not correct notation. Make sure you order it correctly.

[near flax]

order?

[real rose]

Why do you say so?

[real rose]

Yes.

It should be $(-\infty, 5)$.

[solid kiln]

mathisfun

[real rose]

If that is what you mean.

[near flax]

yea

cuz like it goes to the left infinitely

[real rose]

Good work.

But, you're missing something.

Notice the closed dot?

[near flax]

oh wait yea

i forgot what that means

[real rose]

It means the point is included within the domain/range of the function.

[near flax]

above the 5 right?

[real rose]

Yes.

[real rose]

^

Do you know the notation for a number included within domain/range?

[near flax]

so is it (5,10) and (-infinity,5) ?

[near flax]

notation im not shure what that means :,c

(5,10) less than or equal to?(-infinity,5)

[real rose]

Unsure of the word "notation" itself, or don't know the notation?

[near flax]

i dont knw

:,D

both

[real rose]

Is this all for domain?

[near flax]

i just need to find domain and range for the first one

for domain i pout - infinity,5

put*

[real rose]

(With a closed bracket at the end.)

$(-\infty, 5]$

[solid kiln]

mathisfun

[real rose]

Represents that the value is included within the domain/range.

[near flax]

ohhh ok!

and then range is(-5, 9]

[real rose]

Eh, not really.

[near flax]

?

[real rose]

Ah.

Does the function have outputs at y=-5?

[near flax]

oh wait no

i don think so

[real rose]

(The answer is yes.)

(Look at the graph.)

[near flax]

i just remembered its y not x

:,D

[severe fjord]

"Is there any moment when graph goes to y = -5" @near flax