help-26

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[drifting swift]

$\sum_{x=1}^{10} x$

[thorny flame]

Ann

[atomic ermine]

"imagine that you write the numbers from 1 to 10 on one note each and put them in a box, and then take out a note randomly to see the number. what is the sample space?

thank you

is the above summation correct?

i mean could i write that instead of {1,2,3,4,5,6,7,8,9,10}, or would that notation mean the sum of all those numbers? how would you go about writing this series

[mellow path]

the notation with the sigma means the sum. it is very different to writing {1,2,3,4,5,6,7,8,9,10},

[atomic ermine]

ok i figured

[mellow path]

{1,2,3,4,5,6,7,8,9,10} is a set of numbers (and yes, it is the correct sample space)

but $\sum_{x=1}^{10}x$ is a real number (it's not even a set)

[atomic ermine]

is there a shorter way of writing the sample space?

[thorny flame]

LayneTheAndroid

[mellow path]

${i\in\mathbb{N}: 1\leq{}i\leq{10}}$

[thorny flame]

LayneTheAndroid

[atomic ermine]

oh i see

[mellow path]

not a lot shorter tbh but definitely if the sample space was up to 10000 or something

[atomic ermine]

yeah that's what i was worried about haha

[clear lodge]

or if you use N*, you can remove the "1<="

[atomic ermine]

maybe i should use such notation to impress the teacher and get better grades 🤔

[clear lodge]

if it's a small sample space they usually want you to write the "brute force" method of listing all of them

[neon iron]

$\mathbb{N}^* _{\leq 10}$

[thorny flame]

IllIIIllIlIIl

[neon iron]

but usually high school, undergrad doesn't consider 0 in N

[clear lodge]

that really depends on the convention used

[atomic ermine]

what is the star called?

[acoustic tangle]

Undergrad =/= high school

[acoustic tangle]

It means that the 0 is excluded I think

[atomic ermine]

i can't find any info about it

it just says it means multiplication

[acoustic tangle]

What question are we dealing with exactly btw?

[atomic ermine]

how to write a sample space like {1,2,3,4,5,6,7,8,9,10} efficiently

@neon iron could you please explain blackboard bold N* =<10 briefly?

[acoustic tangle]

Just ${ i \in \bZ: 0 \le i \le 10 }$?

[thorny flame]

A Lonely Bean

[acoustic tangle]

Or maybe even $[0, 10] \cap \bZ$

[thorny flame]

A Lonely Bean

[neon iron]

naturals less or equal to 10

if you consider 0 in N, then you need to use the star

otherwise you can use only <= 10

[acoustic tangle]

Why not just avoid all of the ambiguity about whether or not 0 is excluded and use Z?

[neon iron]

where is the ambiguity?

[atomic ermine]

how does the star show that 0 isn't included?

[neon iron]

definition bro

[acoustic tangle]

It's not a universal thing, some consider 0 to be in N and others disagree

Because definitions may differ

[atomic ermine]

is it a country thing? like how some countries like mine write decimals with comma instead of point

[neon iron]

nah bruh

[acoustic tangle]

In my country 0 is never a natural number

[neon iron]

nah either

advanced studies use 0 in naturals

most os them

[acoustic tangle]

?? You know what country I live in?

[neon iron]

no, but it doesn't matter

[atomic ermine]

does the asterisk have a special name when it plays the role of "no 0"?

the no zero asterisk?

[acoustic tangle]

Anyway, I think that using $\bZ_{>0}$ and $\bZ_{\ge0}$ is a better choice than dealing with the controversial topic

[neon iron]

bro just use it, stop making questions about notation

[thorny flame]

A Lonely Bean

[neon iron]

notation is a personal thing

in general

[clear lodge]

your solution would be incorrect, since 0 is not within the sample space.

[atomic ermine]

so blackboard bold Z or N 0<i=<10

[acoustic tangle]

Uh kind of in the context of ring theory as far as I'm aware

[acoustic tangle]

Oh I think 0 is there, my bad

Then I meant to say ${ i \in \bZ: 1 \le i \le 10 }$ or $[1, 10] \cap \bZ$

[clear lodge]

there's two sets of natural numbers very commonly used. One includes 0, one doesnt.
You call one N, and the other N*
which one you call each is a convention

[neon iron]

[1,10] in Z 💀

[thorny flame]

A Lonely Bean

[acoustic tangle]

Fixed

[neon iron]

[1,10] isn't an interval of reals numbers?

[atomic ermine]

ok i see now

[neon iron]

alright

[acoustic tangle]

It is

[clear lodge]

$[1, 10]_\bZ$

[thorny flame]

LordFelix

[acoustic tangle]

That's a thing?

[atomic ermine]

this is the most elegant

[acoustic tangle]

Interesting, I have never seen that

[clear lodge]

i mean, we were aiming for "shortest way to write it" werent we?

N instead of Z would also be valid

$[1, 10]_\bN$

[thorny flame]

LordFelix

[atomic ermine]

Thank you, that's exactly what I was looking for

.close

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[ebon zenith]

at the moment i have R5 = -cos(z)/6! *x^6 where 0<z<x

or x<z<0

if i were to choose z=x=pi i get pi^6/6! is that right?

ok i got it now

.close

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[stoic storm]

how do i solve this quadratic equation (2x-3)^2 = 18 using the umm square root method? im kinda at a lost on what to do firstt

[radiant tapir]

so the first thing I would do is square root both sides. You'll be left with two equations by that point, then you can solve for x separately in both.

$\sqrt{(2x-3)^2} = \sqrt{18}$

[thorny flame]

MellowDramaLlama

[topaz sinew]

@stoic storm Has your question been resolved?

[stoic storm]

oki oki then whats after that? for 18 i need to find other perfect squares then multiply it until i get 18 right?

[stuck forum]

That's not really too important at this point

Now you need to recognize that you have the absolute value function

[hardy anvil]

first :
the equation becomes: abs(2x-3)=sqrt(18)
which means there're now 2 solutions for x:
2x-3=sqrt(18) v -(2x-3) = sqrt(18)

solve the last 2 equations to get x

[stoic storm]

ohhh i see iseee

i get it now, thankzz

.close

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[pearl bane]

why is $\sqrt{3}$ the base?

[thorny flame]

张元英

[foggy bronze]

he took tan 60 not 30

or from Tan 30 u will get same

[pearl bane]

but it doesn't explain "1"

[foggy bronze]

See uk tan 30?

[pearl bane]

what?

[foggy bronze]

value of tan30

[pearl bane]

why did he set up the right triangle that way

[foggy bronze]

according to you what should be the triangle tell

[pearl bane]

?

it's for solving a trig equation

[foggy bronze]

bruhhhh

what exactly is your doubt

[pearl bane]

why did he set up the right triangle that way

[foggy bronze]

Tan 30 = 1/root 3 = opposite / adjacent

[pearl bane]

bro

that's not my question

[foggy bronze]

according to you how should the triangle be placed ?

[pearl bane]

i don't know thats why i want to know why $\sqrt{3}$ is the base and the hypotenuse is 2

[thorny flame]

张元英

[foggy bronze]

hypo aint 2

oh sorry hypo is 2

opposite is 1

See considering tanx

[valid stag]

tan = opposite / ajacent

[pearl bane]

https://tenor.com/view/crying-happy-but-sad-face-gif-19422780

[valid stag]

and which angle is the x

[pearl bane]

its you my friend

[valid stag]

yep

[foggy bronze]

See Tanx = O/A and X=30 Thus O/A = 1/root 3

[pearl bane]

fam, my question is, why did he set up the right triangle that way

[foggy bronze]

...

[pearl bane]

like why is the base not 2?

[foggy bronze]

how it can be 2

[valid stag]

what?

[foggy bronze]

coz adjacent is root 3

[valid stag]

tan(x)=opposite of x / ajacent of x

[pearl bane]

dude

helper is a helper

[valid stag]

ok so elaborate your problem

what don't you understand

[foggy bronze]

im outside

[pearl bane]

ait

like $Tan(x)=\frac{\sqrt{3}}{2}$

[thorny flame]

张元英

[foggy bronze]

how root 3/2?

[valid stag]

nope tan(x) = 1/√3

[pearl bane]

this is a trig equation

[valid stag]

orelse that's cos(x)

[pearl bane]

https://tenor.com/view/crying-happy-but-sad-face-gif-19422780

dude

[foggy bronze]

See the ratio You taken is adjacent / hypotenous which if os Cos

[pearl bane]

NO!!!

[foggy bronze]

yesss

[pearl bane]

listen to me first

[foggy bronze]

go ahead

[pearl bane]

the question is

$Tan(x)=\frac{\sqrt{3}}{2}$

[thorny flame]

张元英

[valid stag]

and?

[pearl bane]

and then solve x

[valid stag]

x=atan(that crap)

x=arctan(that crap)

x=tan^-1(that crap)

[foggy bronze]

x=tan^-1(root3/2)

[valid stag]

if tan(x) = y then arctan(y) = x

tan finds ratio between legs and arctan finds angle from ratio

[pearl bane]

what do you mean by legs?

[foggy bronze]

it will get more complicated for her/him ig

[valid stag]

two sides ajacent to the right angle

in a right-angled triangle

hypotenuse = opposite of right angle

legs = ajacents of right angle

[foggy bronze]

refer this

[valid stag]

(yeah I'm not a good teacher)

[foggy bronze]

i didnt mean that

but acc to his/her understanding

[valid stag]

IK

[pearl bane]

$tan(\frac{2\sqrt{3}}{2}})=x$

[foggy bronze]

2 root 3????

[thorny flame]

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

[valid stag]

?

[pearl bane]

don't you flip it?

[valid stag]

you solving this right?

[pearl bane]

yep

[valid stag]

x=tan^-1(√3/2)

or arctan(√3/2)

[pearl bane]

then flip it?

[valid stag]

no

x=tan^-1(√3/2)

that's it

[pearl bane]

so what is tan(root3/2)

[valid stag]

tan^-1(√3/2)

[valid stag]

use calculator

[pearl bane]

but can you flip

[foggy bronze]

if u flip it it will be in terms of cot

[pearl bane]

so it will not be tan?

[foggy bronze]

dont flip it then you will get it in terms of tan only

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• Show your work, and if possible, explain where you are stuck.
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[dull yoke]

is this tan tan theta = 8.2/9?

[versed cairn]

Arent u going to solve these questions ur self?

This is the 3rd question u asked of the same topic , u should prob have enough knowledge on how to solve

[dull yoke]

mb thought this was the math help group

.close

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[versed cairn]

It is

But it's help group not solve ur questions group

Also always draw a diagram in these kind of questions and make sure u label it

[topaz sinew]

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[runic imp]

To my understanding for A it would P=k*s^4

so then to start out B it would be 96=k*20^4

[spare smelt]

how many sides does a cube have

(hint: it's not 4)

or rather what are the dimensions of a cube is a more accurate way to phrase it

[runic imp]

oh 3

[spare smelt]

yep

P = k s^3

[runic imp]

idk why i thought 4

[spare smelt]

then the rest of your process looks good

[runic imp]

but A is right other then the cube part

[spare smelt]

yes

does it make sense as to why we say "the cube of s" to refer to s^3

[runic imp]

yes

[spare smelt]

why? just want to make sure you understand that

[runic imp]

the three dimesnions of a cube is 3

while a square is 2

with length and width

while cube has also hieght'

also how do i start the B question

[spare smelt]

The volume of a cube with side length s is s^3, yes

For part B you just have to plug in the numbers

You know what P and s are. Find k

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[nova fiber]

<@&286206848099549185>

[drifting swift]

!15m

[topaz sinew]

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.

[drifting swift]

also you have to post a question first

(but DON'T delete that first message)

[nova fiber]

-e^-ln1/2

[nova fiber]

oh ok mb

[drifting swift]

missing some parentheses here?

[nova fiber]

uhh

nah thats it

-e raised to the power -ln1/2

[drifting swift]

did you mean $-\frac{e^{\ln(1)}}{2}$, $-e^{-\frac{\ln(1)}{2}}$ or $-e^{-\ln(1/2)}$?

[thorny flame]

Ann

[nova fiber]

the 2nd one

[drifting swift]

ok

do you know how logarithms work in general?

[nova fiber]

wait srry

the 3rd one

the last one

[drifting swift]

ok, but still.

do you know how logarithms work in general?

[nova fiber]

yh i do

[drifting swift]

ok

do you know log laws?

[nova fiber]

yh

[drifting swift]

ok, so then what's troubling you her

[nova fiber]

I know how to solve these kind of questions

but this one is just a bit tricky thats all

i would easily understand if you could just solve it showing step by step procedures

its not like im new to this

Its natural logs

the negative sign is confusing me a bit

hello?

@drifting swift

s

[drifting swift]

was away, sorry.

[drifting swift]

we don't do that here.

do you have a list of log laws handy?

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@nova fiber Has your question been resolved?

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Remember:
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• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
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• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[upbeat shoal]

bro i dont understand any of this someone help please

[upbeat shoal]

@cinder rock

i think that was your name

i need someone to help@me@out

<@&286206848099549185>

[cinder rock]

kinda very much against the rules but hello

do you know what the word gradient means

(in terms of x and y)

[upbeat shoal]

hes

yes

[upbeat shoal]

oh

mb

isnt it like

y1+y2/x1+x2

?

$\ y1+y2/x1+x2$

wwit no

it was negative

$\ y1-y2/x1-x2$

[thorny flame]

eyas

[upbeat shoal]

$\frac {y1+y2}{x1+x2}$

[thorny flame]

eyas

[upbeat shoal]

wait no

$\frac {y2-y1}{x2-x1}$

[thorny flame]

eyas

[upbeat shoal]

yeah like that

$\g= \frac {y2-y1}{x2-x1}$

[thorny flame]

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

[upbeat shoal]

$\ G= \frac {y2-y1}{x2-x1}$

[thorny flame]

eyas

[cinder rock]

exactamundo

for questions like 7-9 its sometimes easier to think of it as "for each square right, i go G squares up"

ie if the gradient is 2 then for each square to the right you go two upwards

[upbeat shoal]

wha

[cinder rock]

nvm

which bit of the questions are you struggling with

[upbeat shoal]

3

[violet fractal]

what kinda math is this

[upbeat shoal]

fr brk

bro

the past 2 days have been hell

[violet fractal]

no i'm askin like what math class is this lol

[upbeat shoal]

oh

i have no idea

[violet fractal]

👀 ?

ain't this your math?

[upbeat shoal]

yes

its just math class

its just called math

[violet fractal]

how u not know what math class you in LOL

[upbeat shoal]

geometry

i think

[violet fractal]

every math has a title. you got a syllabus or anything?

[upbeat shoal]

uh

whats a syllabus again

[violet fractal]

you in grade school?

[upbeat shoal]

uh

what does that mean

bro its 2 am

[long stirrup]

what's a grade school

[upbeat shoal]

i dont know anything

[violet fractal]

lol I'm tryin to gauge

what help I can provide to ya

is this a college course?

[upbeat shoal]

question 3

[upbeat shoal]

nah

[violet fractal]

yeh I've already looked at the problems, I was wondering why the word gradient was used here.

y I was askin what class ur in

[upbeat shoal]

im in

9th grade

[violet fractal]

got it OK

So do you know what slope is of a line?

[upbeat shoal]

nope

[violet fractal]

a slope, mathematically, is described as the rise/run. It tells you how much your line is going up as you go from left to right

So for example, in your problem 3

A to B has a really high slope (because it's almost vertical) where C to D has a lower slope

It's like climbing a mountain

the larger the slope, or gradient, the more work you're gonna have to do to get up it

does that make sense?

[upbeat shoal]

man

i give up

[violet fractal]

why

[upbeat shoal]

because i am sleep deprived

and i sidnt understand a single word

you just said

[violet fractal]

yeh you probably just need to rest. is this due anytime soon?

[upbeat shoal]

tomorrow

but its one question

it wont affect anythint

so im good

[violet fractal]

let's just try one more time

we'll do this real quick

[upbeat shoal]

alr

[violet fractal]

So look at your lines

[upbeat shoal]

okk

[violet fractal]

Which one of those lines goes up the fastest?

[upbeat shoal]

BA

or AB

[violet fractal]

you got it. that's the one with the highest gradient

that's all there is to it

[upbeat shoal]

wait what

no calculations

or anytjing?

[violet fractal]

nope

[upbeat shoal]

fr?

[violet fractal]

yup that's all they wanted

[cinder rock]

you might have to if the lines are similar in gradient but in this case it's really obvious

[upbeat shoal]

forget anything i said

i love math

[violet fractal]

🤣

there you go. now go rest in peace

[upbeat shoal]

i sure will

goodnight bro

.close

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[civic stone]

Question 8

[topaz sinew]

@civic stone Has your question been resolved?

[civic stone]

Someone pls?

[wide fog]

Eww statistics

Well try counting how many entries you have that are between 3 and 4

Then 4 and 5...

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[civic stone]

Alr thanks

[topaz sinew]

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[muted solstice]

HELP

[lucid pond]

[muted solstice]

what happened

[fossil coyote]

Which line

[muted solstice]

hy did it switch from greater than or equal to less tahn or equal?

why*

[fossil coyote]

Oh

[pastel juniper]

they factored out -1

[fossil coyote]

Because when you multiply both sides by a negative number, you have to flip the signs

[muted solstice]

ohhh that's something new that i learned

THANK YOU GUYS

[fossil coyote]

2 < 3 but -2 > -3

as an illustration

[muted solstice]

that's brilliant

why did i not think of that

THANK U BOTH OF U

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[cyan creek]

in this solution, they use e^-ipi but why not just e^ipi

[topaz sinew]

Please don't occupy multiple help channels.

[cyan creek]

oops

[distant nymph]

Close your other help channel.

[cyan creek]

is it simply just a matter of choice and it doesn't mtter which one you use?

[distant nymph]

Kind of. I think it's not wrong if you just leave it in the first form.
However, since r represents magnitude of the complex number, i.e. |z|, it's often preferred to write r as positive. That's what i think.
It doesn't seem wrong to write as the first one either.

[cyan creek]

so i can use e^ipi instead?

[distant nymph]

Well, i think so. Unless, your question somehow specifies some particular form.

[cyan creek]

alright ty\

[distant nymph]

Close it if you are done.

[cyan creek]

Could I keep it open since I’m expecting to ask more questions

Or should I close and open new one

[ashen eagle]

You can keep it open.

[cyan creek]

not sure about this one. I could probably doing it by changing it to another form but would like to keep in expoennetial form

[distant nymph]

How can you write i in the form of $e^{i\theta}$?

[cyan creek]

oh

e^pi/2 i

ah ty

[thorny flame]

Enemagneto

[distant nymph]

Finally.

[cyan creek]

,w solve e^-pi

[thorny flame]

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[cyan creek]

,w solve e^-(pi/2)

[thorny flame]

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[cyan creek]

could i make this e^-pi/2 i?

although when i do that, i get -i when the asnwer is i

<@&286206848099549185>

[clear lodge]

you've written it wrong in discord.
It would be e^[-(i*pi)/2]

[thorny flame]

annyeong

[cyan creek]

what

isnt it just dividing the power by 2

[clear lodge]

yes, but the way you wrote it, i would be in the denominator with the 2

[cyan creek]

oh right

anyways

using tha ti get -i but the answer is i

is it that it has multiple answers or am i just wrong

[pastel juniper]

it has multiple answers ig

,w sqrt(e^(-i pi))

[thorny flame]

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[pastel juniper]

yeah 2 roots

[cyan creek]

although is i more correct in some way?

[thorny flame]

Dyssrupt

[pastel juniper]

unnecessary lol

[pastel juniper]

i dont think so, but confirm this with other helpers too.

havent revised complex in a long time

[cyan creek]

the answer is i in the textbook, and the bot said i first which makes me quesiton

alright

umm halllppp

lol

dont really want to ping already done so ..

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[agile ledge]

going through solutions and i dont understand where they got this line from

[agile ledge]

oh i realised

gradient function is left side

and 3 is gradient

[neon iron]

y12 differentiation, throwback

but yes, they differentiated y = xrootx + k, and know that it must be equal to 3 at some x

[agile ledge]

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[trail flame]

!help find the domain and function

[trail flame]

! help

[topaz sinew]

Please read #❓how-to-get-help

[trail flame]

!help

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Please read #❓how-to-get-help

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@trail flame Has your question been resolved?

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[steep tiger]

$2-x^2 \geq 0$

[thorny flame]

Hyperlix

[steep tiger]

when solving this, when do I know to flip the inequality sign?

[clear jewel]

whenever you are multiplying with a negative(-) you have to flip the symbol

[steep tiger]

like, there $x \geq -2$

[thorny flame]

Hyperlix

[alpine obsidian]

You flip it when you’re multiplying or dividing by a negative

[steep tiger]

I meant the $x \geq -2$ part

[thorny flame]

Hyperlix

[steep tiger]

I squared both sides but I have to flip one of the signs

how do I know which one I have to flip?

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[ocean forum]

When applying the triangle equation, does it apply to a half triangle?

[ocean forum]

E.g. ur given a half triangle, would the bh / 2 formula apply to it?

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[neon iron]

Am I doing this wrong?

[neon iron]

.close

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[keen raptor]

consider the question of whether 2x^2 is O(x^2)

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[keen raptor]

np

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[distant lintel]

May someone help me with this problem?

Would it not just plug these into the general formula to get this:
(x−(−6))2+(y−(−4))2+(z−4)2=42
(x−(−6))2+(y−(−4))2+(z−4)2=42

Simplifying, we get:
(x+6)2+(y+4)2+(z−4)2=16
(x+6)2+(y+4)2+(z−4)2=16

The equation is already normalized so that the coefficient of x2x2 is 11, as the question asks. Therefore, the equation of the sphere is:
(x+6)2+(y+4)2+(z−4)2=16
(x+6)2+(y+4)2+(z−4)2=16?

[unique crypt]

that looks correct

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[sharp laurel]

I’ve a sheet of similar problems. I’m not sure where to start. It’s about limits from graphs with transformations.

[sharp laurel]

.rotate

,rotate

[distant lintel]

Are you familiar with graphing cross sections?

[sharp laurel]

For the first one, I thought about separating the functions. I got -3. No idea if that’s right. For the one with -5 from the left, I got 8 with same method. Separating the functions

[sharp laurel]

Not sure what that is. Maybe?

[topaz sinew]

@sharp laurel Has your question been resolved?

[sharp laurel]

<@&286206848099549185>

[sharp laurel]

<@&286206848099549185> I’m really confused. Any help I’d appreciated

[sonic escarp]

i am not sure what the (your) qeustion is. for example $\lim_{x\rightarrow 3}\sqrt{g(x)}$. if you look into the graph of g(x) you see g(3) = 4 and g(x) looks continuous at x = 3, so $\lim_{x\rightarrow 3}\sqrt{g(x)}=2$

[thorny flame]

ThM

[sharp laurel]

I also got 2 for that one. For the first 3 I had: -2, -3, 2, idk, 7, 8. Not sure if any are right.

For the fourth one, do I look where x=5 and the x simply move 2 to the right?=1?

[sonic escarp]

i do not understand what you mean with this For the first 3 I had: -2, -3, 2, idk, 7, 8. you have 6 solutions for 3 excercises?

[sharp laurel]

Yeah sorry. That was all of them except the last one

[sonic escarp]

but didnt we have 2 as solution for the second one?

[sharp laurel]

Sorry again. I cut off the first two in the pic. So yes, we agreed on the second one

It should be 2, idk, 7, 8, idk. And I’m not even sure about the ones I answered

[sonic escarp]

???? lim sqrt(g(x)) is the second one .in the picture you posted ...

[sharp laurel]

Yes. Sorry I’ve been confusing you. I said that one was 2. The one after I didn’t know how to do

[sonic escarp]

$lim_{x \rightarrow 5}h(x-2)$: so we need h(x) at 3 and in the near of 3. as the graph shows h(x) is not continuous at x = 3. waht would h(3) be if h(x) would be continous at x = 3?

[thorny flame]

ThM

[sharp laurel]

3?

[sonic escarp]

yes

[sharp laurel]

What’s the next step

[sonic escarp]

which next step?

[sharp laurel]

Is that the answer?

[sonic escarp]

yes, it is. do you understand what you should do?

[sharp laurel]

Did you plug 5 in for x, then subtract 2 ti get 3?

[sonic escarp]

i take x = 5, subtract 2 as we need h(x-2), so i get that we need h(3). then i looked into the graph what happens to h(x) in the near of x = 3.

[sharp laurel]

Oh. I didn’t realize it was that simple. Can I do this for all of them?

[sonic escarp]

yes of course.

[sharp laurel]

Oh. That makes everything easy then. Okay. Thank you. So for the last one, is it 4?

Plug in 1 into g of x. Get -1. Then plug in -1 into f and get 4?

[sonic escarp]

yes, it is. but you have to be careful. you need that g(x) is continuous at x = 1. you could try lim f(g(x)) if x goes to 2.

[sharp laurel]

Okay. So just watch out for normal limit stuff. Thank you so much, I think I’ve got it now!

.close

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[fierce lake]

Hi! I have the following function as seen in the screenshot here:
Im trying to solve the following:
I have to find out when the graph is growing at the most per hour. So how would i do this? Do i need to find f'(x) and then f''(x)?

[craggy haven]

you just want to know when f is growing the fastest so you want the max of f '

[fierce lake]

Yeah i want to know when it's growing the fastest

[craggy haven]

if calculating that requires calculating f '' then sure

[fierce lake]

Is doing f''(x)=0 the correct thing to do then?

[tall wolf]

yes

[fierce lake]

Alright, thanks a lot Hayley and Arctrices ^^

[craggy haven]

make sure to check the boundary points too

[fierce lake]

This is what i did

And then since x = 8.27 is negative on the graph the correct value would be x = 20.36

If i understand it correct?

[tall wolf]

yeah

since x=8.27 is decreasing the fastest

[fierce lake]

Yep, alright thanks both of you so much 😄

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[versed salmon]

I need a polynomial function that looks like the rough sketch I drew in blue which connects the purple curve to the black , please.

[versed cairn]

I wanted to claim channel

[versed salmon]

Like this

im stuck

<@&286206848099549185>

anyone?

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@versed salmon Has your question been resolved?

[versed salmon]

No

[versed salmon]

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[next fjord]

What does a thin E mean in math

[next fjord]

Like example)
a, b E R

[odd pagoda]

"in"

"element of"

[next fjord]

You sure?

[odd pagoda]

yes

[next fjord]

Thx

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[stone spindle]

Hi

[stone spindle]

On the 2.19 i don’t understand how he go from (1+1/n)^n to the otherside of the equality

,rotate

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[stone spindle]

<@&286206848099549185>

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[undone flicker]

For this i got |sin thita| =1/6

[ionic oar]

Which question

28?

[undone flicker]

Yes 28

[drifting swift]

,rccw

[topaz sinew]

@undone flicker Has your question been resolved?

[undone flicker]

very strange

no one is responding, no problem I will need to choose other servers.

<@&286206848099549185> even the solution is availble

but still I am eager to learn the new thing from here

[nimble zenith]

Let $\mathbf{2u \times 3v}$ be a unit vector.

$\left| \mathbf{2u \times 3v} \right| = 1$.

[thorny flame]

sigmascribe

[undone flicker]

see

[nimble zenith]

[
6 \left| \mathbf{u} \right| \left| \mathbf{v} \right| \sin \theta = 1 \Rightarrow \sin \theta = \frac{1}{6}
]

Hence, there is exactly one value of $\theta$ for which $\mathbf{2u \times 3v}$ is a unit vector.

[thorny flame]

sigmascribe

[undone flicker]

why?

and it should be mod

[nimble zenith]

Since, θ is an acute angle, then there is exactly one value of θ for which (2u x 3v) is a unit vector.

[undone flicker]

acute okay then fine

i thought this condition is not okay with our new vector

[nimble zenith]

$$[ 6 \left| \mathbf{u} \right| \left| \mathbf{v} \right| \sin \theta = 1 \Rightarrow \sin \theta = \frac{1}{6} ]$$

[undone flicker]

$$

[thorny flame]

sigmascribe

[undone flicker]

I have told you I have got this step by these steps and my concern was just to know about thita

now it is cleared

.close

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[molten marlin]

im confused on how the answer is -1/2. i solved by graphing the function but i want to know how to solve it without graphing

[frail dove]

Hint: since we know that f is continuous on (-π,π), f(0)=limit as x→0 of f(x)

[sweet shard]

did you try multiplying top and bottom by cos(x) + 1

[molten marlin]

no i didnt. why do i have to multiply cos(x)+1?

also i dont understand the hint

[sweet shard]

top and bottom meaning numerator and denominator

it lets you do the hint

[frail dove]

It follows from the definition of continuity

[molten marlin]

how do i multiply them?

am i supposed to get cosx^2-1 for the numerator

?

[sweet shard]

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

[thorny flame]

riemann

[sweet shard]

oh yes this is the numerator yes

[molten marlin]

is it cos^2x or cosx^2? or is that the same thing

[sweet shard]

$\cos(x^2) \neq cos^2(x)$

[thorny flame]

riemann

[sweet shard]

the first one is

,w plot cos(x^2)

[thorny flame]

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[sweet shard]

the second one is

,w plot cos^2(x)

[molten marlin]

and the denominator is xsinx*cosx+xsinx right?

[thorny flame]

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[molten marlin]

so its cosx^2 right?

[sweet shard]

no

[molten marlin]

cos^2x?

[sweet shard]

$\cos^2(x) = (cos(x))^2$

[thorny flame]

riemann

[molten marlin]

ah yea that makes sense

[sweet shard]

this is right

but for this problem it's better to not distribute

you'll see why when you use a trig identity later

[molten marlin]

so do i leave it like this or do i change it

is this better for the denominator

i forgot the 1 at the end but yea

[sweet shard]

this is better

[sweet shard]

use a trig identity to re-write the numerator

[molten marlin]

this one?

wait no this one right

[sweet shard]

yea this one lets you cancel a power of sine in the denominator

[molten marlin]

ok so i have this now

and then i cancel and get this right?

[sweet shard]

did you learn this

[molten marlin]

no

[sweet shard]

https://www.youtube.com/watch?app=desktop&v=4Ik0M0hbqv0&ab_channel=WrathofMath

err shoot

wrong image

[molten marlin]

i havent learned that either

[sweet shard]

welp that's the easiest way to do the problem so i don't know what else you could use

[molten marlin]

what about the hint the other helper gave?

was this all for the hint

[sweet shard]

yes

[molten marlin]

well i might as well learn what you said rn

how does it work

[sweet shard]

watch

[molten marlin]

i watched it but idk how i would use it in this problem

[sweet shard]

apply it here

[sweet shard]

and try to understand the hint

[molten marlin]

ohhhh and then cos(0) is 1 plus the 1 would give me -1/2

that hint has me so confused😭

@sweet shard is there no simpler way of solving this problem

[sweet shard]

maybe!

[sweet shard]

if you're doing these types of problems the hint shouldn't be too difficult

if it is hard, you should watch some khan academy videos on limits

watch as many of those as it takes for you to be comfortable with this problem
https://www.khanacademy.org/math/calculus-1/cs1-limits-and-continuity

[molten marlin]

ok thank you

@sweet shard also how did you immediately know to multiply by cosx+1?

[sweet shard]

Common trick you learn to cancel something in the numerator and denominator for limits

You'll learn more of them

[molten marlin]

alr ty

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[vivid wagon]

why is c the right answer

[vivid wagon]

because technically its asking for the derivative of the function at 3

which would (with direct substitution) result in indeterminant form

and then what

how do you approach it from there

[wary tulip]

isn’t the value in question the derivative of the function at 3?

[vivid wagon]

oh yeah im sry

I meant to put in 3

[wary tulip]

you don’t have to do anything like that

[vivid wagon]

then how do you do it

[wary tulip]

what does the line tangent to the graph at x = 3 look like?

[vivid wagon]

a negative linear with a slope of -2/3???

[wary tulip]

yea c sounds wrong

who said it was c?

[vivid wagon]

the answer key 😭

[wary tulip]

😭

should be b

[vivid wagon]

thats what I was thinking too

but my teacher wouldn't explain it 😭

[wary tulip]

not slay

[vivid wagon]

ikk

well thank you

[wary tulip]

lol np ^_^

[vivid wagon]

🙂

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[cerulean solar]

find the domain

[slate pebble]

well mr eleven have you tried anything yourself

[cerulean solar]

yea..

[neon iron]

domain may means input, while the range is the output

[cerulean solar]

yea

[neon iron]

so in this case, is x axis is the domain? or y axis?

[cerulean solar]

x axis

..?

[neon iron]

I guess you have all the information to know the answer, idk how to simplify it more, sorry

[cerulean solar]

bru

i don’t know the answer like is it

(2,0)???

somebody..?

obviously i wouldn’t be on here bothering folks if i knew the answer or what to do

<@&286206848099549185>

[frigid sedge]

what happens to the x values as you zoom out? Ignoring the y axis entirely for now

[cerulean solar]

gets smaller?

[limpid violet]

is each square a single number? (1,2,3)?

[cerulean solar]

yea

1-5

[limpid violet]

(-infinity, 1) [1,3)u(3,infinity)

thats as far as i got

not sure about the single dot on the bottom though

have this double checked by someone though, still learning

[cerulean solar]

ok preciate i

u

[limpid violet]

I edited the 4, with a 3

[cerulean solar]

ok so nun of those were rt but it’s cool. thanks for trying

[limpid violet]

sorry bro I tried, the answers they might be looking for would be written differently

[cerulean solar]

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[full coral]

one last question on this same one

where does the 1/2 come from after we integrate 1-cos(2x)

[knotty ledge]

you have two options, learn about u subsitutions if you dont already as thats what was done
or just differentiate 1/2sin(2x) and see that it is indeed the antiderivative of cos(2x)

[full coral]

yeah i know u substitutions

[knotty ledge]

okay then you can just do u = 2x

[full coral]

even then

how is 1/2sin the antiderivative of cos(2x)

[knotty ledge]

do the u substitution

[full coral]

ugh

and let me guess

we have to do this because we can't technically integrate cos(2x) right

yeah it comes out

[knotty ledge]

unless you already know something that differentiates to it, then no we dont know its integral directly

[full coral]

so i shouldn't even attempt to use wolfram at all for these

is basically what the deal is then

[knotty ledge]

?

why does this mean you shouldn't use wolfram

,w int cos(2x)

[thorny flame]

Results provided by WolframAlpha

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

Requested by icaird

[full coral]

that is not what it gave me

it gave me the top two

[knotty ledge]

yeah well you could have just scrolled down a bit

[full coral]

i did

[knotty ledge]

or know that sin(2x) = 2sin(x)cos(x)

[full coral]

yep i got that as well

[knotty ledge]

then i dont see the issue

[full coral]

the issue is plugging those in don't return the given answer

that was what i first tried before coming on here

it didn't give me that

[knotty ledge]

what exactly did you plug in

[full coral]

i honestly wasn't about to plug pi/6 into both of those and then have that blow out into another half an equation

[knotty ledge]

it would've worked out the same

[full coral]

i ended up with pi/9

which definitely isn't it

[knotty ledge]

i mean that would be on you not wolfram

[full coral]

2pi/9(pi/6 - sin(pi/6)cos(pi/6) ) - 2pi/9(pi/4 - sin(pi/4)cos(pi/4)

that's just a bit much to grind through

personally

so moral is

unless its exact, i have to substitute or notation manipulation to get the exact identity or integral before i can actually process

[knotty ledge]

more like, if you dont know how to integrate something, try and use an identity to get it into a form you do know how to integrate

might not always be possible, but no point staring at an integral you dont know how to do

[full coral]

yeah for sure

and if there arent any identities

substitute to try and cancel something, right

[knotty ledge]

there are tons and tons of integration techniques so

you just try as many as you know

but yeah identity --> looking for a sub --> look for IBP is usually a good order to check things

[full coral]

awesome

thanks again man

really big time

[knotty ledge]

no worries:)

[full coral]

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[pseudo stream]

Can someone make sure this graph looks right to me? to me I think it does - I just wanna be sure

[topaz sinew]

@pseudo stream Has your question been resolved?

[pseudo stream]

<@&286206848099549185>

[frail dove]

Yeah it looks right

[pseudo stream]

thank you!

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[twilit robin]

Can someone explain how 1/x+5/x(x-5) = x-5+5/x(x-5)?

[empty sail]

Combining fractions using a common denominator

[twilit robin]

Ohh wait

I thought of that

but I just did it wrong

I thought x * x(x-5)

but its (x-5) * 1/x

wait hmm

is that right?

[empty sail]

No you multiply 1/x * (x - 5)/(x - 5)

[twilit robin]

Yeah yeah yeah my bad

Okay yeah that makes sense

ty

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[neon iron]

hey!! i've been struggling with these stupid things for many frustrating hours, and am really close to giving up lmao.
i was hoping to find help with a specific conversion process? the practice questions my teacher provided are clear-cut and most of them make sense. i can do regular conversion, but the problem is she wants it done in a specific way, one that i've never seen done and never learned how to do (provided on the picture with relatively bad handwriting). i get the gist of it, but something in my head isn't letting me put the answers together... any tips?

*edit, it's a part of my chemistry class if that provides any more detailing

[craggy haven]

the idea here is that you're multiplying by 1 although the paper messed it up slightly

[craggy haven]

so if you have 0.00592 L and you want to convert it to mL

and you know that 1000 mL = 1L

then you set up your fraction so that the L cancels out

so like

$0.00592\cancel{\text{ L}} \cdot \frac{1000\text{ mL}}{1\cancel{\text{ L}}}$

[thorny flame]

hayley!

[craggy haven]

and you're left with mL that hasn't been canceled

[neon iron]

so in this case, the 0.00592 would be multiplied by 1000

[craggy haven]

yeah

you often have a long chain of these

[neon iron]

mmmmmmm okay

[craggy haven]

here's an example i had earlier

the idea is that you can check each fraction individually and be like yep uh huh those are equal, and you can make sure all the units cancel appropriately, and then just multiply the tops and multiply the bottoms

[neon iron]

AH okay i see

weeping rn thank you

[craggy haven]

aww don't be sad i'm sorry this was confusing, it took me a while to get used to it as well

and the writing wasn't helpful since it didn't show multiplication but that weird table thing

[neon iron]

yeah, the class i'm taking is more of a catch up fast or fall behind, and there's often tables or diagrams that are explained too fast

powering through 💪

thank you!!!

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[random charm]

like how t is the angle between a line drawn to the point cost, sint and the horizontal. What is t in the case for a hyperbola?

[clear lodge]

it would be the length from the origin on the curve to the current point (this is not as precisely explained as it should)

[random charm]

uh could u help with a diagram?

[clear lodge]

I can try to explain it better, but i can guarantee it will still not be exact enough.
Let t start at 0. The point will be (1, 0)
As t increases, you'll be moving along the top right arc of the hyperbola.
If it decreases, you'll be moving along the bottom right arc of the hyperbola.

you can graph it in geogebra or equivalent to get an idea, but you'll need someone with better understanding of these curves to get a more proper definition

[random charm]

ahhh

i think i get it

thanks tho

[river condor]

Hyperbolic trig is specifically generated to satisfy cosh²x-sinh²x=1 for all x. You can observe this in their definitions:
Cosh(x)=½(e^x+e^(-x))
Sinh(x)=½(e^x-e^(-x))

They're not really about angles, here. The argument is determining what goes in the exponential functions. If you square them and subtract them you'll notice that everything cancels out to 1. Further, cosh is even and sinh is odd, just like their circular counterparts

They're named as they are because of how similarly they act (in relation to each other) to regular trig

[random charm]

oh ok

so Cosh(x)=½(e^x+e^(-x))
Sinh(x)=½(e^x-e^(-x)) came first

and then we started deriving expressions?

[river condor]

I can't speak about "came first" exactly, but I am pretty sure those definitions were made to satisfy cosh²-sinh²=1.

I.e. they wanted that property, and those definitions ended up behaving nicely.

[random charm]

oh okay, and sorry if this is silly, where did we get cosh^2 h - sin^2h = 1 from?

[river condor]

That I don't know, but probably somebody going "you know how cos²+sin²=1? What if we subtracted instead of added?"
A lot of math identities and facts are from mathematicians just asking silly "what if" questions out of sheer curiosity.

[random charm]

oh ok thanks so much

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[neon iron]

how to convert 5.37(23) to a fraction?

[drifting swift]

do you know in general how to convert repeating decimals into fractions?

[desert pulsar]

Could you elaborate on this

[neon iron]

nope

first time doing it

[drifting swift]

right ok

... might want to start with some easier ones first if you're able

try something like 0.(4) or 0.(3)

[neon iron]

my teacher gave us homework to convert the decimals with repeating part to a fraction(i'm from russia btw)

I got no idea why she used that notation

[desert pulsar]

Oh so 2 and 3 are repeating

[neon iron]

yep 23 is a repeating part 5.37232323...

[desert pulsar]

Okay I understand

[neon iron]

so I need to get a fraction

[desert pulsar]

Well

The way I learned this I presume would be very different

Essentially

You’d want to isolate the repeating portion of the number