help-10

[timid silo]

so now I have

3

that makes no sense what

[past sand]

If A = B then you can say A^2 = B^2

[timid silo]

what

[past sand]

^2 is "squared"

[timid silo]

where did 2 come from

this?

[past sand]

$A = B \implies A^2 = B^2$

[warm shale]

Nel

[past sand]

You need to keep sign in mind to get the correct solutions, but here you have a square root, so x must be positive anyway

Can you apply that to this?

[timid silo]

nope

[past sand]

Why not?

[timid silo]

bc I don’t know how to

that makes no sense

[past sand]

I don't know what makes no sense to you

[timid silo]

The whole thing

[past sand]

Can't really help in that case... read your course material

[timid silo]

I did

I can do the basic surds

not whatever is going on here

[past sand]

"whatever is going on here" is really not that complicated if you take it step by step

[timid silo]

I’m trying man

[past sand]

$\frac{\sqrt{x} \cdot \sqrt{18}}{\sqrt{3}} = 8 \sqrt{3}$

[warm shale]

Nel

[past sand]

Here, $A = \frac{\sqrt{x} \cdot \sqrt{18}}{\sqrt{3}}$ and $B = 8 \sqrt{3}$

[warm shale]

Nel

[past sand]

Do that

[timid silo]

Okay

Now what

[past sand]

Simplify it

[timid silo]

so I end up with

root x times 3 root 3 over root 3 = root 64 plus root 3?

[past sand]

...

[past sand]

That's not the same thing

[timid silo]

what

[past sand]

$A = \frac{\sqrt{x} \cdot \sqrt{18}}{\sqrt{3}}$

[warm shale]

Nel

[past sand]

What is $A^2$?

[warm shale]

Nel

[timid silo]

X times

18 divided by 3?

Equals 64 times 3?

[past sand]

I don't know where you saw a plus sign

[timid silo]

Ok

Then what sign is it

[past sand]

?

[timid silo]

What

[past sand]

Why would you edit instead of writing it again

Anyway... can you use actual signs, *, /, =, instead of words?

[timid silo]

nvm man

I think I got it

Can you end this help channel

Idk the command

[past sand]

.close

[obtuse pebble]

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[obtuse pebble]

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[hoary idol]

Need help to simplify this

[unreal musk]

May want to think about how ctg and tg are related

[hoary idol]

I wrote this

[unreal musk]

You could do it like that sure

[hoary idol]

Ok now what

[unreal musk]

Well you have “a fraction of fractions”, might want to common denominator stuff then work with that

[hoary idol]

So at the top would it be sina+sinb?

I don’t relly get it can you write it down what do i do

@unreal musk

[unreal musk]

Well not quite - remember to cross multiply, try and simplify
[
\frac{\sin(\alpha)}{\cos(\alpha)} + \frac{\sin(\beta)}{\cos(\beta)}
]

[warm shale]

@unreal musk

[unreal musk]

by itself

[hoary idol]

Do i find the common denominator ?

Got this

Is it right

@unreal musk

[unreal musk]

Looks good, then notice you have a fraction of fractions as per before, you know how you’d deal with (a/b) / (c/d) [well I hope so!]

[hoary idol]

not rilly😢

what do i do

[unreal musk]

Well a general fraction as per here becomes (ad)/(bc)

[hoary idol]

Like this?

@unreal musk

<@&286206848099549185>

[unreal musk]

Yep but you need a bit more

[hoary idol]

what?

@unreal musk

[unreal musk]

Like this

Then take a look at what you have, does anything stand out(?)

[hoary idol]

not relly

what do i do?

[timid silo]

Hey guys

What’s up

[hoary idol]

hi

[timid silo]

I need help for one really easy equation

[hoary idol]

open another channel

@unreal musk what do i do now?

[timid silo]

This ons seem easy

[unreal musk]

!occupied, get another channel

[obtuse pebble]

Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).

[hoary idol]

soo what do i do?

[unreal musk]

Take a look to see if anything cancels out

[hoary idol]

the thing in the purple

all of it right

?

and were left with tg ctg no?

[unreal musk]

Yep

[hoary idol]

i got one more problem like this but a bit diffrent can i try to do it and if i get something wrong can you help?

[unreal musk]

Sure

[obtuse pebble]

@hoary idol Has your question been resolved?

[hoary idol]

Need help with this

Did that but I think its wrong

I need to simplify it

@unreal musk

[unreal musk]

for reference
[
\sin^2(\alpha) - \sin^2(\beta) - \cos^2(\beta) - \cos^2(\alpha)
]

[warm shale]

@unreal musk

[hoary idol]

yes

[unreal musk]

I see you initially changed $\sin^2(\alpha) = 1 - \cos^2(\alpha)$ and I think you tried to change that $\sin^2(\beta) = 1 - \cos^2(\beta)$, right?

[warm shale]

@unreal musk

[hoary idol]

yes

[unreal musk]

Mind you, I wouldn't have changed the $\cos^2(\beta) = 1 - \sin^2(\beta)$, as it appears like you tried to do (and also I think forgetting some brackets too)

[warm shale]

@unreal musk

[unreal musk]

You could either just change one of them, so you have everything in terms of cos^2, which might make things easier - but, even better...

[unreal musk]

...if you factor this in a certain way, you'll notice that you get a very wonderful identity(!)

[hoary idol]

yea i think thats bad that i changed that

but what do i do?

[hoary idol]

factor how

[unreal musk]

Maybe try and get the beta terms together, and the alpha terms together

[hoary idol]

sin^2a-cos^2b-sin^2b-cos^2b

??

[unreal musk]

Yep, then add in some in one spot and something should shout out

[hoary idol]

add what i dont get it

[unreal musk]

Oh wait a moment, you should have this as
[
\sin^2(a) - \cos^2(a) - \sin^2(b) - \cos^2(b)
]
and factor the terms with $a$, and the ones with $b$ (saying more would make the answer too obvious!)

[warm shale]

@unreal musk

[unreal musk]

Maybe, alternatively, may I ask what trig identities you know?

[hoary idol]

sin cos tg ctg

[unreal musk]

identities, rather than functions

such as there's one where $\tan(a) = \frac{\sin(a)}{\cos(a)}$ (tan for tg)

[warm shale]

@unreal musk

[unreal musk]

Any others?

[hoary idol]

cts eaquels cos/sin

ctg eaquels 1/tg

tg eaquels 1 /ctg

1+ctg^2 eaquels 1/sina

1+tg^2 eaquels 1/cos

[unreal musk]

(squared, for the latter two, but fair enough )

Any other ones that have squares in them?

[hoary idol]

cos^2 eaquels 1-sin^2

[unreal musk]

That's the one, alternatively, sin^2 + cos^2 = 1

[hoary idol]

ye ik that

[unreal musk]

can you make the latter appear here?

[hoary idol]

no idk how to work this thng

is it -1-1?

[unreal musk]

Not quite, no almost(!)

let's try focusing on the $-\sin^2(b) - \cos^2(b)$ for now by itself

[warm shale]

@unreal musk

[hoary idol]

sin^2-cos^2 equals -1

[unreal musk]

One of the ideas you had was to change $\sin^2(b) = 1 - \cos^2(b)$, so if you did that, you'd have
[
-(1 - \cos^2(b)) - \cos^2(b)
]
maybe try simplfying that?

[warm shale]

@unreal musk

[unreal musk]

(and it doesn't)

[hoary idol]

0?

[unreal musk]

Not that either (there is another identity there, if you're familar with double angle identities)

[hoary idol]

what is it then?

[unreal musk]

Are you familar with double angle identities?

[hoary idol]

no

[unreal musk]

Well in that case, not too much you can really simplify that to, other than e.g. getting everything in terms of just sin^2 or cos^2

(there is the cos double angle identity, $\cos(2\alpha) = \cos^2(\alpha) - \sin^2(\alpha)$, but if you haven't learned that, don't worry about that)

[warm shale]

@unreal musk

[hoary idol]

ok so what do i do

with what ik

[unreal musk]

Best I guess is as per here tbh

[hoary idol]

if ik the duble angles what would i do?

[unreal musk]

You could change it to -cos(2a)

[hoary idol]

.close

[obtuse pebble]

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[sick grail]

Here I know I have 2 equations,

A=xy
P = x + 2y

[sick grail]

Do I plug in the 98 for my area equation?

And solve for one variable

[inland matrix]

Seems right, cuz the area is constant

[sick grail]

Okay let me see

I got 14ft for x and 7 for y

Can anyone double check that?

[hollow river]

You have to minimise the fence right?

Say one side is x and other is y

What will be the length of the fence?

Say x meters is covered by the wall

[past sand]

,w min x+2y, xy=98, x>0, y>0

[warm shale]

Results provided by WolframAlpha

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

Requested by nelfeal

[past sand]

That's right

[obtuse pebble]

@sick grail Has your question been resolved?

[sick grail]

sweet

thank you guys for the confirmation

[obtuse pebble]

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[warm shale]

Ransik

[lone dirge]

because $(1/2)^{x-3} = 2^{-(x-3)} = 2^{3-x}$

[warm shale]

Cain

[lone dirge]

yep

and you add

review power laws

[obtuse pebble]

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[coarse gate]

hi guys, chatgpt isnt being helpful with this and im not good at math lol. how would i accomplish this?
i need a set of 50 numbers with the following constraints:
-all numbers in the set should be integers
-the set must be exponential-ish in nature
-the first 5 numbers in the set should be 1
-the 50 numbers equal to 4416 when added together

[coarse gate]

gm eric

gm all

[meager aspen]

Oh

Shouldn't have done that

[brazen gorge]

"the first 5 numbers in the set should be 1" huh? a set is a collection of unique elements

[coarse gate]

sorry im making an rpg and i have 50levels of items. the first 5 levels of items are priced at 1 ... uwu

idk math at all

50 levels and each level costs money. all 50 levels must be 4416 moneys. first 5 levels is 1 each... etc etc xD

[brazen gorge]

okay so it's not a set, it's a sequence

[coarse gate]

oh i see

[obtuse pebble]

@coarse gate Has your question been resolved?

[coarse gate]

1
1
1
1
1
1
1
2
3
4
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
105
110
115
120
125
130
135
140
145
150
155
160
165
170
175
180
185
190
195
200

did it myself by hand

xD

thanks guys

/resolve

[obtuse pebble]

@coarse gate Has your question been resolved?

[obtuse pebble]

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[blazing mesa]

Geometry help: I am in 9th grade and we will have our final exam (called SOL in VB) very soon after christmas breaks.

My teacher gave me a Jefferson's lab to complete that I THINK relates to the SOL and gives us practice. (not the actual exam)

Anyways, there was so many things that I NEVER saw or just forgot, i am very stressed out and i don't want to fail

What i need to work on:
Reasoning, Lines, Transformations
Triangles, Polygons, Circles, Three dimensional figures (i just need to review and or learn)

(search up jefferson lab SOL and i think it's the first link that pops up, then just select geometry, 20 questions, additional options, and click on the first options that pops up, it starts with "All..."

No work provided

[blazing mesa]

if possible please provide me some articles or videos (if possible please videos) that talk about them

[tame narwhal]

this is for specific math questions. you can try asking in one of the discussion channels

or ask in #geometry-and-trigonometry

[obtuse pebble]

@blazing mesa Has your question been resolved?

[tardy epoch]

@blazing mesa listen to cwatson

.close

[obtuse pebble]

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[blazing mesa]

who that

oh nvm

[blazing mesa]

i already did

nobody is helping

[obtuse pebble]

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[muted delta]

What do you think the problem means by “ending position”? Does it mean the location of the box containing last line, or like the direction to that box? Please do not aid me in solving it.

[meager glade]

ending position is the last box the path has to visit

so in the example it ended at the top rightor top left

[muted delta]

k

[obtuse pebble]

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[abstract trench]

okay

so for number 1

in 2003

the population would be 105% of 2002

so u solve for 2002

and thats 105% of 2001

and solve for that

and for 2005

do 105% of 2003

and then do 105% of that

and u get ur answer

[slender scroll]

Write an equation first so you don't have to do it recursively

[abstract trench]

okay that too what they said

[slender scroll]

every year "t" after 2002 it gets multiplied by 1.05 so write an exponential function with that in mind

[abstract trench]

oh yeah

then you do that for number 2 as well

and for 3 its 92% of 42,000

38640

[spice steppe]

can any one come to teach over a call or something

i need them wit steps

[abstract trench]

which one

all?

[spice steppe]

ya

[abstract trench]

i wont do call cuz privacy but i can write it and send a pic of my work

[spice steppe]

ya sure dat works

[abstract trench]

okay do you want functional way for first one

or simplified way

cuz i kinda just did divided by 105% twice

but there's also the equation of like

y=a(r+1)^t or smtn

[spice steppe]

ya it is p[1+r/100]whole power n

[abstract trench]

hUh

i was never tought that one

[spice steppe]

the formula

can u solve and send a pic plz?

[abstract trench]

yeah

shoot i got this

2400000/49

heres the pic

for part 1

@spice steppe

dude

okay

i gtg anyways

but u apply this to everything else

sorry

bye

[obtuse pebble]

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[obtuse pebble]

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[mellow olive]

mind blank help

[mellow olive]

💀

[limber quartz]

make a right triangle.

[mellow olive]

yeah then am is 24

[slender scroll]

AO is 26 which is the radius

[mellow olive]

yeah

[slender scroll]

so XO is also 26

[mellow olive]

💀 mb im blind

thanks

.close

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[tough hull]

In how mays can Ireland be arranged

[tough hull]

Probability question

[uneven palm]

7!

[tough hull]

Now if every arrangement has to start with e how many is there

[rich plume]

Do you mean in how many ways letters of the word 'Ireland' can be arranged?

[tough hull]

Yes

[rich plume]

well you should be clear with your question

just in case

[tough hull]

Ok

[rich plume]

Okay so you basically have fixed 'e' on the first place

so you have 6 letters to fill and 6 spaces

This is same as the previous question if you think into it

[tough hull]

6! ?

[rich plume]

yes

[tough hull]

Ok

What if each letter is only used once

[low patio]

that's the current assumption

[leaden ginkgo]

each letter is only used once in 6!

[tough hull]

No they're not talking about e

But okk

.close

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[half lodge]

how can i go about solving this?

i dont understand what i should be looking for?

[obtuse pebble]

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[fallow pasture]

Help

[fallow pasture]

The answer should be 31,111 cubic yards

[obtuse pebble]

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[fallow pasture]

<@&286206848099549185>

[obtuse pebble]

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[wanton dune]

find rank of this matrix with respect to $ \lambda \in \matthbb{R}$

[wanton dune]

i tried finding kernel by row reducing transposed matrix

but the solution is by finding kernel of this matrix not trasnposed

why is that?

[fathom flicker]

well why did you transpose it to begin with

if I asked you to find the rank of

$$\begin{pmatrix} 1 & 0 & 0 \ 0 & \lambda & 0 \ 0 & 0 & 1 \end{pmatrix}$$

[warm shale]

Austin

[fathom flicker]

ignore that this is symmetric lol bad example

but

would u transpose it or just row reduce

I'd just row reduce

and yippee it's already reduced

[wanton dune]

but when finding image of kernel shoulnd i do it with transposed matrix?

[fathom flicker]

the rank is the image of the column space

so there is no need to find the image of the kernel

[wanton dune]

https://math.stackexchange.com/a/242282

as in this

[fathom flicker]

although you could do it that way, making use of rank-nullity theorem

[fathom flicker]

I'm not sure why you're bringing this up

the question you posted asks you to find the rank of the matrix

not the image

it wants the dimension of the column space

[wanton dune]

well isnt rank=dimImA?

[fathom flicker]

yes

you don't need to find the column space

just its dimension

[wanton dune]

hm alright

but dimension of what?

[fathom flicker]

so make use of rank-nullity

the dimension of the column space

is the rank

apply row operations to that matrix to reduce it

and find the null space in terms of lambda

then use rank nullity

to determine the rank

[wanton dune]

alright im gonna do that thanks for clarification

[fathom flicker]

np

[obtuse pebble]

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[obtuse pebble]

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[pallid mortar]

I was wondering why 1/n diverges?

I was thinking p series but then using that idea I couldn’t follow why (-1)^n/n converges

[royal basin]

$\sum \frac{(-1)^n}{n}$ is not a $p$-series at all

[warm shale]

Ann

[royal basin]

did you never see the proof for it?

[pallid mortar]

How do you know if something is a p series/ when to follow that/ when to now

And how does (-1)^n/n converge?

[royal basin]

a $p$-series is specifically a series of the form $\sum \frac{c}{n^p}$ where $c$ is a nonzero constant.

[warm shale]

Ann

[royal basin]

alternating series test.

also you didn't answer my question of "did you at any point see the proof that sum 1/n diverges"

was it ever explained to you in class WHY the harmonic series diverges? @pallid mortar y/n

[pallid mortar]

No to either. I haven’t seen proofs. Just was told that’s how it was

N

[robust raven]

A typical proof consists in assuming that the series converges and then showing that such an assumption leads to a contradiction.

hence the harmonic series is divergent

[royal basin]

really?

the "typical" proofs i've seen proceed differently

@pallid mortar ok so then would you like to know?

[robust raven]

let me shwo oyu

$\text{Let's assume that: }\sum_{n=1}^{\infty }\frac{1}{n}\text{ is convergent}\text{, then:}\\S_{2n}-S_{n}\longrightarrow 0\text{, but in same time:}\\S_{2n}-S_{n}=\frac{1}{n+1}+\frac{1}{n+2}+{\cdots }+\frac{1}{2n}>\frac{1}{2n}+\frac{1}{2n}+{\cdots }+=\frac{1}{2}>0\\\text{hence we got a contradiction, hence }\\\sum_{n=1}^{\infty }\frac{1}{n}\text{ is divergent}$

[warm shale]

Joanna Angel

[royal basin]

hm

yeah that works

a bit too technical for my taste though esp given op has stated she has never seen any proofs before

the proof i like better is the one where you group the terms from 1/(2^n + 1) to 1/2^(n+1) and show the sum of this group of terms is greater than 1/2, and in so doing show that your series is greater than an infinite series of 1/2's

[robust raven]

yes there is such a proof too 🙂

[royal basin]

holding off on presentation until OP gives unambiguous affirmative to me doing so.

[robust raven]

in my country, such a proof of divergence of harmonic series are only shown, when a student studies at the Math Faculty

and if a students studies, assume, chemistry, such proofs are not reuired, then theOP, needs only to know, whether it is convergent or divergent, so i asusme that concerns op here too

[royal basin]

<@&268886789983436800>

[obtuse pebble]

@pallid mortar Has your question been resolved?

[royal basin]

thats why im asking op whether she wants to see the proof. what's your point?

[robust raven]

all is ok 🙂 but , i thikn if op studies Maths, then op wud not ask us ab such simle things 🙂 so i assume OP studies soemthign that does not require such proofs, only that i mean 🙂

[scenic hill]

The original question was why the harmonic series diverges though

[royal basin]

was it

if so, my bad

[pallid mortar]

Am I allowed to solve this using ratio test? The prof used root test

[obtuse pebble]

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[pallid mortar]

.reopen

[obtuse pebble]

✅

[scenic hill]

Why? Your intention was to show them the proof of the harmonic series diverging wasn't it?

[scenic hill]

^

[royal basin]

no, but i could be said to have put up artificial roadblocks and a hoop for OP to jump through with my withholding of help

[scenic hill]

Afaik the best way to prove the p series test is using the integral test so one would have simultaneously explained why 1/n diverges and 1/n^p where p > 1 converges so it's a double whammy if you ask me

[robust raven]

integral test is ok but is not elementary

you have to agree with it

integral tests are shown fw lectures later when harmonic series is shwon

[pallid mortar]

I didn’t do the integral test? I did the ratio

[robust raven]

yes yes brownie, you ve got another case 🙂

but faster

is

Cauchy criterion in yoru case

check it

[obtuse pebble]

@pallid mortar Has your question been resolved?

[pallid mortar]

I’m still a little confused 🥲

Does it matter what way we solve for divergence/convergence

[robust raven]

no

but some ways are harder while other ones are easier

and also

soem methods = criterions do not react

on series

so it requires experience

which test

you have to use

for your case I would use this, look:

$R=\lim_{n \to \infty } \frac{1}{\sqrt[n]{\left| a_{n} \right|}}=\lim_{n \to \infty } \frac{1}{\sqrt[n]{n^{n}}}=\\=\lim_{n \to \infty } \frac{1}{n}=0$

[warm shale]

Joanna Angel

[robust raven]

hence, if radius of convergence is: R = 0, then the power series is convergent only in its center: x = 0

x = 0 is the only one point where your power series is convergent

[scenic hill]

Where did (n^n)^(1/n) come from?

[obtuse pebble]

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[azure herald]

is this following true:

[warm shale]

みつる

[kind hawk]

yes

[robust raven]

yes all ok so far

[azure herald]

js wanted ot make sure bc desmos said that it wasn't and i wasn'tsure

[royal basin]

show exact desmos input

oh wait

you didn't put parentheses around the (3n+4) did you

desmos considered that as $\paren{\sum_{n=1}^{10} 3n} + 4$

[warm shale]

Ann

[azure herald]

no i added parentheses

[royal basin]

ok then show exact desmos input

[azure herald]

i closed the page like an hour ago

[royal basin]

well then you will have to reproduce this

[robust raven]

$\sum_{n=1}^{k}\left( an+b \right)=\sum_{n=1}^k{}an+\sum_{n=1}^{k}b$

[warm shale]

Joanna Angel

[azure herald]

thanks i got that

[robust raven]

etc

[obtuse pebble]

@azure herald Has your question been resolved?

[obtuse pebble]

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[mental cipher]

Hey Everyone. When calculating an. Is it possible to multiply by 1.5 and integrate from [-2pi,0] to get the entire integral ?

[mental cipher]

Or would I have to do seperate integrals ?

Doing it this way would mean that I would do less work to get an ?

Sorry I meant from [-pi,0]* not [-2pi,0]

[timid silo]

You'd have to prove that multiplying by 1.5 gives you the entire integral

So it'd be easier to just do 2 integrals

However, if all you want is the answer, then you don't even need to integrate

[mental cipher]

But just by looking at the graph it seems possible no ?

[timid silo]

Yes, but it's not rigorous

[mental cipher]

just out of curiosity

[timid silo]

If you just want the answer then go ahead, use the formulas for the area of a triangle and rectangle and skip integration entirely

Though when you're not being rigorous because you don't need to you need to be extra careful

If you need to justify in any way however you have to do 2 integrals

[mental cipher]

Okay, I understand you now thank you

[pseudo swift]

wait, why would you multiply by 1.5 anyway ?

[timid silo]

The triangle part is pretty clearly half of the square part

[pseudo swift]

yeah ok

[timid silo]

Though that's a convoluted method admitedly

Like, if you're gonna try to simplify the calculations, at least do it fully

[pseudo swift]

it's just cause the bounds you stated are screwed then

1.5 * the integral from -pi to 0, this I get

the integral from -2pi to 0 is already "the entire thing"

[mental cipher]

Yeah sorry the [-2pi,0] was a mistake Im meant [-pi,0] my bad.

[timid silo]

Your PFP is so cursed

[mental cipher]

Oh my hahaha, I haven't even realised haha

So what im getting at is that its possible but just unnecessary.

[timid silo]

Calculating the area under the curve using geometry is useful if you just want the answer
Calculating the area under the curve using 2 integrals is useful if you need justification and/or to be rigorous
Calculating the area under the curve using your method is a weird mix of the 2 methods above which is unoptimal for both use cases

But yes, it's possible. It's not illegal or anything

[mental cipher]

Perfect that will do, thank you guys !

Ill do the 2 integrals since, Ill have to show the work for exams soon anyways

[timid silo]

Great. Don't forget to .close

[mental cipher]

/close

[timid silo]

bruh

[mental cipher]

.close

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[timid silo]

Unsure what to do here

[fossil crag]

which part?

[prisma smelt]

Change sec x to it's reciprocal, then cross multiply. You will get a double angle and use the appropriate formula. Inverse then divide. For part a)

[obtuse pebble]

@timid silo Has your question been resolved?

[obtuse pebble]

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[formal basin]

Why does the range not include [-1,0)?

[formal basin]

I figure I could figure this out by setting everything to a value in that interval but I think that would take too much time and I wouldnt really understand why it would act in such a way

[unreal musk]

Seeing it's a quadratic, you could argue - if $\frac{x^2}{3 - x^2} = k$ with $-1\leq k < 0$, then you'd need to solve $(1 + k)x^2 = 3k$

Actually, better, you don't even need the discriminant for that either

You'd have $x^2 = \frac{3k}{1 + k}$ if $k \neq -1$, and $ \frac{3k}{1 + k}$ is negative, so no solutions

[formal basin]

i see

is there a way for me to think about it that is kinda easier to wrap my head around

like this i can figure out but i need something to put in the notes yk

[unreal musk]

If k=1, then you'd be saying that 0 = 3, which isn't true

Oh wow I can't manipulate in my head today

[warm shale]

@unreal musk

@unreal musk

[unreal musk]

not sure I can think of an easier way to visualise it myself tbh

[past sand]

Why not just use the derivative?

[formal basin]

ibe never used derivative before in my life

[past sand]

Well here is a way of looking at it:

If 3-x^2 < 0:
    x^2 / (3-x^2) >= -1
=>            x^2 <= -3+x^2
=>              0 <= -3
=>              false
So x^2 / (3-x^2) < -1 by contradiction

If 3-x^2 > 0:
    x^2 / (3-x^2) < 0
=>            x^2 < 0
=>              false
So x^2 / (3-x^2) >= 0 by contradiction

Conclusion: x^2 / (3-x^2) < -1 or x^2 / (3-x^2) >= 0

[formal basin]

idk i just moved on from this one

lemme show you how i thought of it

that 3 / x^2 becomes so small its irrelevant

i used that to explain the asymptote

then all of the values beyond that between y=-1 and y=0 are explained away by some weird things with that -x^2 in the denominator

i didnt think hard about it its one subject thats probably not even gonna come up in the midterm since we spent like two days on it

oh well

.close

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[ruby elm]

I would like help with this one.

[ruby elm]

I got it.

.close

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[covert dagger]

yo can someone teach me how to multiply these umm vectors?

[obtuse pebble]

@covert dagger Has your question been resolved?

[covert dagger]

<@&286206848099549185>

[obtuse pebble]

@covert dagger Has your question been resolved?

[covert dagger]

<@&286206848099549185>

so apparently axa is zero and bxb is also zero cause theyre in the same direction

but idk how to multiply vectors

like if its about cross product i would have done it the determinant way

if someones answering, please ping me when you do

[glacial pulsar]

you didnt actually multiply the vectors, you just simplified the expression @covert dagger

axa is zero and bxb is zero because angle between them is zero

[covert dagger]

yeah but how do u do it 😭

[glacial pulsar]

there are two types of products in vectoes: 1. vector/cross product
2.scalar/dot product

[covert dagger]

in the question (a,b) is 45 degrees

[glacial pulsar]

the scalar/dot product gives the output as a number
and the cross product gives output as a vector

axb= absin(theta)*(normal vector)

a and b are magnitude of a and b vectors

theta is the angle between the vectors

and normal vector is a vector mutually perpendicular to both a and b vectors

area is a vector thats why we used cross product of the sides instead of dot product

[covert dagger]

but how did -6bxa turn into +6a x -a in this @glacial pulsar

[glacial pulsar]

in cross product, axb≠ bxa

axb=-bxa

this is because when you do cross products, there are 2 possibilities of the normal vector in opposite directions

[covert dagger]

ohhhh

[glacial pulsar]

so to determine the direction of axb we use right hand rule

[covert dagger]

cross product isnt commutative unlike dot product

[glacial pulsar]

yeha

do you know right hand rule?

[covert dagger]

no

[glacial pulsar]

ok let me explain

to find direction of axb, point all the fingers of your right palm in the direction of a and curl your fingers(except thumb) in the direction of b,
the direction in which the thumb is now pointing is the direction of axb

if we want direction of bxa, we will do the same process but we will point our fingers in the direction of b and curl in the direction of a

[covert dagger]

ohh

cool

thanks @glacial pulsar for explaining all of it really well

.close

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[kind meteor]

I cannot figure out the zero input response and zero state response for exactly in differential equation. I just need a very clear and simple explanation.

[kind meteor]

I want a tangible example. I mean that I need it explained by analogy with something.

[final thunder]

Zero state: this means the initial condition is set to be 0, and we examine the response of the system to some input. For example, if you have a room at the ambient temperature and turn on the heater (input). The room’s temperature will increase, starting from the neutral state; that’s the zero state response.

Zero input: Examine how the system evolves from some initial condition, without any external push (zero input). Suppose we have a heated room that suddenly loses its heat source. The way the room’s temperature gradually decreases without any new heat input is the zero input response.

[kind meteor]

That's very clear explanation. I got it for exactly. Thx

.close

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[junior iron]

F and g is 1:1 and covering functions f(2x+3)=3x+4 (gof)(x+7)=3x+5 g(-2) am i supposed to use the fact that f and g is covering and 1:1? I tried to do it by turning f(2x+3) into fx and found -10 but thats not right

[hushed pebble]

You did it the wrong way?

[junior iron]

I think so the answer i found wasnt in a b c d or e and i retried it 3 times

[hushed pebble]

wait what are they tho?

[junior iron]

-19 -13 -8 1 3

[hushed pebble]

ok

f(2x+3)=3x+4 then we can set y to 2x+3

to express f in terms of y

so x in terms of y is x=(y-3)/2

then u just resubstitute it back in

and you already know g(f(x+y))=3x+5 so

f(x+7)=-2

and g(-2) = -19

[junior iron]

Ah gotcha i see

Ty

[hushed pebble]

have a nice day!

[junior iron]

.close

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[timid silo]

inntegration of 1-x^2/ x^4+3x^2+1

[timid silo]

<@&286206848099549185> help please

[slim cove]

Please wait 15 minutes before pinging Helpers

[ashen dove]

Can u use brackets

[timid silo]

yeah sur

(1-x^2)/(x^4+3x^2+1)

[chilly scroll]

what have you tried

[timid silo]

x^2 as t

dt = dx?/2x

[unborn vigil]

Hi. Is there a task, or is it possible to do it yourself, in which you need to place different geometric shapes inside a limited space or inside another shape?

[chilly scroll]

nah that doesn't helps

these types you divide numerator and denom by x²

then look for creating 1/x²+a² form

[obtuse pebble]

@timid silo Has your question been resolved?

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[shell oar]

where is tanx not continous and why?

[shell oar]

would it be all real values except +- pi/2 ?

[forest sinew]

youre missing some

,w Plot[Tan[x],{x,-10,10}]

[shell oar]

oh is it all multiples of pi?

acc no

[warm shale]

Results provided by WolframAlpha

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Upgrade to WolframAlpha Pro!

Requested by janniku

[shell oar]

its all multiples of pi/2?

[forest sinew]

,w Tan(100 * pi / 2)

[shell oar]

um

[forest sinew]

just think, tan is sin/cos

[shell oar]

wait

sorry

[forest sinew]

where is cos 0

[shell oar]

oh shit

yh

one sec

[forest sinew]

thats where youll have a problem

so its not multiples of pi/2

well, its some of them

[shell oar]

so cosx = 0

when

x = pi/2

and 3pi/2

[forest sinew]

theres more

cosx has infinitely many zeros

[shell oar]

2pi?

[forest sinew]

huh?

[shell oar]

sorry

so pi/2

3pi/2

goes up ever pi

so 5pi/2

7pi/2

[forest sinew]

okay

where are they in general

[shell oar]

nth term of that sequence?

[shell oar]

this?

[forest sinew]

yea

,w Solve[Cos[x] == 0]

[warm shale]

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[forest sinew]

wonder if wa can do this

[shell oar]

so is this it?

thats it?

[forest sinew]

yea

W?

[shell oar]

idk about the W but

why the rhs?

[forest sinew]

why is there a W

also, convince yourself this is the same

[shell oar]

its defined as the largest open interval tan x is continous

[forest sinew]

so you looked up an interval but your problem is asking about points of discontinuity right

so theyre relatex but not the same

also these two answers are saying the same thing anyways

[shell oar]

yh

ok yh this makes sense

thanks a lot

[forest sinew]

np

[shell oar]

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[steady tartan]

For this question, I got a and b, but I can't figure out C. For my solution, I proved convergence with the alternating series test, as the series is positive and decreasing. What I don't understand is how they get the R, since two terms have an (n+1) exponent and one term has an (n) exponent, so I don't understand how they combine those?

[forest sinew]

they just pluck stuff off, you can do the same

like, if you have $\frac{a^{n+3}}{b^{n-2}}$ and you want $\qty(\frac ab )^n$

[warm shale]

jan Niku

[forest sinew]

just write $a^3 b^2 \qty( \frac ab )^n$

[warm shale]

jan Niku

[steady tartan]

Ohh ok

That makes sense

Thank you

.close

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[steady tartan]

Oh wait but how does that get them to 4/3?

[steady tartan]

When I do it I get -2(-2/3)^n

It looks like they're doing it twice?

Pulling out two 2's, two -1's, and a 1/3?

[steady tartan]

This is the original question, I now understand how they combine them but not why they get 4/3 as the coefficient

[obtuse pebble]

@steady tartan Has your question been resolved?

[steady tartan]

<@&286206848099549185>

[tardy epoch]

Show

[tardy epoch]

Your first term in your expansion should be 4/3 from the very first equation

[steady tartan]

Yeah, 2^2/3^1?

[tardy epoch]

And then the second term, is it-8/9?

[steady tartan]

Yeah? -1*2^3/3^2

[tardy epoch]

What is your first index

[tall flicker]

try putting n = 0 in the summation
if u did not have a 4/3 there, you'd have 1 as your 1st term which is not the case in the original question

[steady tartan]

Ohhh, ok

Does it have to be n=0?

Because initially they use n=1

Are geometric series always n=0 or something?

[steady tartan]

sorry to bother you, but could you help clarify this for me? I'm still not sure why I can't just keep using n=1

[tall flicker]

you could definitely start with n = 1, but for that you'd have to change the inside part to

4/3.(-2/3)^(n-1)

since, whenever we expand the summation, we always want 4/3 to be our first term, -8/9 second and so on

[forest sinew]

sorry, did you end up getting this

[steady tartan]

What I tried was -2(-2/3)^n, starting at n=1

[steady tartan]

Not yet 😅

[forest sinew]

refresh me whats the problem exactly

can you share the image

[steady tartan]

So I get -6/5 instead of 4/5

[forest sinew]

right so we start from $(-1)^{n+1} \frac{2^{n+1}}{3^n}}$

[warm shale]

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

[steady tartan]

Yeah

[forest sinew]

heres the goal

if an exponent has n in it

it cant have any other numbers

it can only be n

lets start with -1

how many extra -1 do we have

how do we fix the exponent

[steady tartan]

Pull out -1

[forest sinew]

okay, so lets remember to add it back later

[steady tartan]

And then do the same with 2, that's how I gte -2

As the coefficient

[forest sinew]

To add: $(-2)$

okay, now were at $\qty( \frac{-2}{3} )^n$

hmm you know what

lets go back

youre missing a key piece

this starts at n=1

but geometric starts at 0

you need to re-index this sequence

[steady tartan]

Every geometric?

[forest sinew]

its standard to start the geometric series at 0

[steady tartan]

Ok

[forest sinew]

you dont have to

the result just isnt as clean

and its a good exercise here

[steady tartan]

But if I don't I don't get the same answer?

[forest sinew]

the result of the series is what it is

$\sum _0 x^n = \frac{1}{1-x}$ right

[warm shale]

jan Niku

[forest sinew]

for |x|<1

[steady tartan]

Ok

Yeah

[forest sinew]

so what if you dont start at 0

what do you lose

[steady tartan]

I'm not sure

[forest sinew]

i mean, whats the loss of the series if we just skip the first term here

[steady tartan]

Oh

x^n-1?

[forest sinew]

whats the difference between that sum and $\sum _1 x^n$

[warm shale]

jan Niku

[steady tartan]

The x^0 term

[forest sinew]

which is what

[steady tartan]

which is 1

[forest sinew]

yea

so $\sum _0 x^n = x^0 + \sum _1 x^n = 1 + \sum _1 x^n$

[warm shale]

jan Niku

[forest sinew]

then $\sum _1 x^n = \frac{1}{1-x} - 1$

[warm shale]

jan Niku

[forest sinew]

not as nice

re-indexing is a good exercise but you could use this too

[steady tartan]

Ok I see

That makes sense

So to re-index I take out terms until the exponent is n-1 whcih would start at 0 and then just say that it is n?

[forest sinew]

to reindex you introduce a new index variable

your sequence starts at n=1

so m=n-1 seems reasonable

[forest sinew]

this way, m will start at 0, and we can use this

this means n=m+1 and you have some substitution to do

then a similar process of fixing the exponents

i believe that process will give you the results youre showing in the picture

[steady tartan]

Ok, thank you

[forest sinew]

[steady tartan]

I am incredibly not ready for this final 👍

Getting there I guess

[forest sinew]

hmm dont forget to take breaks and drink water

good luck

[steady tartan]

True, thank you

.close

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[solemn jungle]

For part c of the question, it is asking for the complex number formed by the middle point cubed. I found that point a and b is 4cis(pi/4) and 4cist(11pi/12) and tried to do find the trig form of midpoint but didnt get the right answer (4sqrt(2) - 4sqrt(2) i) and instead got 3sqrt2/4 + 3/2i

[timid silo]

Hi!

Are you trying to find the complex number formed by the midpoint cubed?

[solemn jungle]

yes

[tardy epoch]

Show your work

[timid silo]

Do you have the real and imaginary part of the midpoint?

[solemn jungle]

i got 1/2 cis(7pi/12) for midpoint but im not sure if thats right

[timid silo]

Dm me. I am not notified

[obtuse pebble]

@solemn jungle Has your question been resolved?

[obtuse pebble]

@solemn jungle Has your question been resolved?

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[surreal shell]

I need help approaching this problem

<@&286206848099549185>

[obtuse pebble]

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 badger]

Notice that $\left(\frac{x}{x + 1}\right)^{x} = \left(\frac{x + 1}{x}\right)^{-x}$

[warm shale]

jan Mikkel

[surreal shell]

I don't get it

Do i apply L'hopital?

brah

[obtuse pebble]

@surreal shell Has your question been resolved?

[surreal shell]

<@&286206848099549185>

[meager glade]

how cna you rewrite the fraction

in a way that is nicer

[smoky vigil]

do you know the famous limit $\lim_{x \to \infty}\left(1+\f{1}{x}\right)^{x}=e$

[warm shale]

🫎 A Certain User(Moosey) 🫎

[smoky vigil]

apply that here :)

once you get it into that form at least

[obtuse pebble]

@surreal shell Has your question been resolved?

[obtuse pebble]

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[wooden cipher]

. @late brook ask your question here

[late brook]

COnfused with factoring

[smoky vigil]

$(x+a)(x+b)=x^{2}+(a+b)x+ab$

[warm shale]

🫎 A Certain User(Moosey) 🫎

[obtuse pebble]

@late brook Has your question been resolved?

[obtuse pebble]

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[obtuse pebble]

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[strong sun]

Not sure how to setup this word problem so far I've gotten 7x+5y+2z=35

[scarlet gale]

There's also the weight equation.

[strong sun]

so it's just
7x+5y+2z=35
x+y+z=10

[scarlet gale]

Right.

[strong sun]

okay, i think i should be able to solve it now, thank you

.close

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[timid silo]

I need help with this question below, please describe it step by step if possible.

Boys length can be descried with the simple model f (x) = 78 ⋅ e^0,07x where f(x) is the length in centimeters and x is the boys age in years.

A) Use the model and decide how fast boys grow when they are exactly 6 years old.

B) Is the model also compatible with boys that are 18 years old?

[timid silo]

.close

[obtuse pebble]

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[dark idol]

Help

[low patio]

there's an infinite number of solutions, you provided 2

[wooden cipher]

You forget the n or k or whatever multiplier

[dark idol]

So what was I suppose to put

[low matrix]

2pi*k

[warm shale]

hypox

[low matrix]

it's k in this case

not n

[dark idol]

So which is it

And what’s the difference anyway

[low matrix]

no difference but it states to use k in the question

[dark idol]

👍

.close

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[worldly narwhal]

how do u do this

[worldly narwhal]

i dont understand the log because of that 7 in front

rn

[mighty torrent]

It will probably help you to make it into a word problem

[worldly narwhal]

ok

[mighty torrent]

so, log base 7 of 45 is the number to which 7 raised to the power of will equal 45

so, in the full equation, you are raising 7 to the power that, when 7 is raised to it, will equal 45

[warm shale]

Chai T. Rex

[scarlet gale]

Log and exponential undo each other.

[worldly narwhal]

OHHHHH

[scarlet gale]

So, if you have a log_7 of a 7 to the power of a, they cancel, and that power is the answer.

If you have a 7 to the power of log_7 of a, they cancel, and a is the answer again.

[devout yarrow]

Similarly, if u have pi to the power of log_pi of a, they cancel

[worldly narwhal]

is this in exponential form right now

like

wait

[scarlet gale]

Yes, 7 to something is an exponential.

[worldly narwhal]

i don't get it

[scarlet gale]

What part of it?

[worldly narwhal]

the answer

45

how do u get to it

i understand right now that 7 to the power of x gets you the log745

=x

[scarlet gale]

Oh, you use my second line here ^

[worldly narwhal]

its in exponential form

i still dont get it

[warm shale]

Chai T. Rex

[worldly narwhal]

yes but

[warm shale]

Chai T. Rex

[worldly narwhal]

yes

[scarlet gale]

The 7 to the power of and the log cancel out, leaving just the 45.

[worldly narwhal]

how.

[scarlet gale]

Because that's how logs work.

[worldly narwhal]

why does it go to log7

[mighty torrent]

because the functions are inverses

[worldly narwhal]

like i have 7^x=log7[45] on my paper

and now where do i go

that part i get

[scarlet gale]

Logs ask the question "What exponent do I need to get this number?".

[warm shale]

Chai T. Rex

[scarlet gale]

So, that exponent will give 45.

So, 7 to that exponent will give 45.

[worldly narwhal]

i have 7^x

what is x

[scarlet gale]

x is what power you're taking 7 to.

There's going to be a number that you take 7 to the power of that will give you 45.

Does that make sense?

[worldly narwhal]

yes

[scarlet gale]

No, that's not how it works.

[mighty torrent]

this is the wrong equation

[worldly narwhal]

oh ok

i thought i could rewrite it

as an exponential

to find x

[scarlet gale]

So, lets call the number that you take 7 to the power of to get 45 as q.

7^q = 45, right?

Because we picked the exact power that will give us 45 if we take 7 to that power.

Or you could do something different. I could find the exponent that could make 3^whatever = 86

[worldly narwhal]

ok

[scarlet gale]

There's going to be an exponent that gives exactly 86 there.

[worldly narwhal]

so we need to find cx

x

right

[scarlet gale]

No.

[worldly narwhal]

bc theyre equal

[scarlet gale]

No.

[worldly narwhal]

ok im listening

[scarlet gale]

There's going to be an exponent for 3 that gives 86.

[worldly narwhal]

right

3 to the power of x is equal to 86

[scarlet gale]

Like it'll be 4 point something.

So, like 3^4.1202 or whatever is equal to 86.

[worldly narwhal]

ok

[scarlet gale]

Does that make sense so far?

[worldly narwhal]

right

yes

[scarlet gale]

OK, so log_3(86) is that exact exponent.

Whatever log_3(86) is, if you fill it in here 3^that, you get 86.

Does that make sense?

[worldly narwhal]

okay

is that just another equivalent form of x

that log

[scarlet gale]

So, log is just a thing that says "Which exponent do I need?"