help-42

[shy nova]

should i go with sum of sines

.solved

[calm coral]

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[zealous rose]

how am i supposed to do this..? just guess and check? the second photo is what the teacher said in class and kinda explains it ig

[zealous rose]

never mind i think ive gotten it

.close

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[cunning reef]

Consider hexagon ABCDEF that has all equal sides, there is a circle O such that O passes point A,B, is tangent to DE, and passes through BC at point P.
If CP = 9, find the area of ABCDEF

[cunning reef]

yeah i have no idea what to even do

this is what i drew

mahoraga helppp

<@&286206848099549185>

[summer gate]

You mean eight divine dial general thingy

Uhh

Lemme try

Uhhhh

Nothing comes to mind

😦

Sorry

[cunning reef]

💔

[twin forum]

Do we have any radii or anything or should we just make an equation

Actually nvm

[cunning reef]

nope

[twin forum]

I didn't see cp =9

[cunning reef]

oh

[cunning reef]

can you do it?

[twin forum]

No

😭

[cunning reef]

sade

<@&286206848099549185>

[calm coral]

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[fierce fog]

Are my diagrams correct?

[calm coral]

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[fierce fog]

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[halcyon stump]

So I have this assignment about fourier series. I said that f(t) = s(t) where f(t) is continous. In other words that the fourier series congerges to all points where f is continous and where f has discontinuity, we need to take the limit from left-handed side and right handed side divided by 2

[halcyon stump]

My professor is asking me what theorom is that

I just explained that my function satisfies all 4 dirichlet conditions, hence why it means that the fourier series converges to f where it is continous

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[hollow frigate]

Why are they squaring the lengths of PM and MT and then square rooting the addition of them>

I thought they are using vectors method

so its just PM - MT

so root 40 - 3?

PM is root 40 btw

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[wise shell]

Any suggestions on how I could solve this? I have no idea where to start

[calm coral]

@wise shell Has your question been resolved?

[wise shell]

<@&286206848099549185>

[wise shell]

[calm coral]

@wise shell Has your question been resolved?

[calm coral]

@wise shell Has your question been resolved?

[wise shell]

<@&286206848099549185> does any have any idea?

I still can’t figure it out

[arctic echo]

Nope

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[teal junco]

I got a question

[teal junco]

if there are 35 people in a room, what is the minimum number of ppl thatll have the same month of bd?

im thinking its 24

Let set A = 12 people
Then set A each people will have different bd months.

Set B = another 12 people.
All 12 are guranteed to match a month from SET A. Lets suppose they all have same BD so it minimizes it. SO it would be 12 + 1 = 13
Then set C = 11 people. So 11+13 = 24

<@&286206848099549185> who got me wit help

in some dire need of help rn

[leaden thunder]

!15m

[calm coral]

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.

[teal junco]

whatt

how is it 15

[teal junco]

sorry daddy

[runic jewel]

Assume each month has 3 people, except December with 2. Would that mean the answer is 3?

[teal junco]

no

its 24

first 12 ppl cant have same bd let say

13th person is gurnateed to have same month that makes 2 ppl sharing same month

14th person is also gurnateed to share same bd as person 13 OR if he shares it sum1 random from the 12 itll be 2 extra SAME month

[calm coral]

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[clear delta]

yep that looks good!

[remote mural]

mercii

[clear delta]

de rien!

[remote mural]

.close

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[blissful field]

F(x) = Sqrt(4x^2 - 12x + 9), whats the domain of f(x),

[blissful field]

So i set whats under the root to >= 0 bc yk- its a square root

And- i got (x-(3/4) )^2

= 0

And thus x >= 3/4

Which is wrong… why

[drifting seal]

how did you get this

[blissful field]

Simplification of whats under the root?

[drifting seal]

show your work brother

you can’t just say "i simplified"

[tepid wedge]

It's a quadratic, you need both sides of the parabola.

[blissful field]

Sheeesh i gotta write it all again in english

Alright

Gimmie a min

[drifting seal]

i’m assuming you divided by 4 first

to get

[blissful field]

I just realized that i shouldnt have done that

Well- ig lemme continue

Wait

Whats wrong with dividing all by 4

[drifting seal]

you can

[blissful field]

Alr..

[drifting seal]

$\sqrt{4(x^2 - 3x + \frac{9}{4})}$

[potent lotus]

knief

[drifting seal]

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

[potent lotus]

knief

[drifting seal]

that’s your mistake

[blissful field]

Whats wrong

[drifting seal]

you did (x-3/4)^2

[tepid wedge]

$$4x^2-12x+9 = (2x-3)^2$$ The quadratic is a square, i.e. always positive, so the domain is R.

[potent lotus]

LooseEthics

[blissful field]

Soooo

Alright i understand

But whats wrong with my simplification

[drifting seal]

brother

💀💀

i told you

[drifting seal]

and this is just wrong

[fleet verge]

3/2 is not 3/4

Because 2 is different from 4

[blissful field]

Wait so- this is always positive

[drifting seal]

since it’s (x-3/4)^2 inside the square root (x-3/4) can be anything

positive or negative doesn’t matter

[leaden thunder]

$\sqrt{x^2} = |x|$

[potent lotus]

riemann

[blissful field]

Making me feel dumb 😭

[drifting seal]

Euler was a bitch

[leaden thunder]

damn euler for making everyone feeling dumb

[blissful field]

Yea i got beef with him

[fleet verge]

Sorry, i just wanted to point out the very important fact that 2 is not always 4

[blissful field]

Oh so- i shouldve gotten it eitherway 😭 but i still didnt

[drifting seal]

indeed

[blissful field]

Yeah my method is lacking- is there a good way to calculate domains

[drifting seal]

seems like you already knew

[drifting seal]

^

[blissful field]

I also wanna know how this person simplified dis

[fleet verge]

Inside the square root you complete the square

[drifting seal]

me?

[fleet verge]

He said person

[blissful field]

No wait i forgot to reply

[drifting seal]

you’re right, i don’t qualify

[blissful field]

Can yall chill 😭

[drifting seal]

it’s pretty obvious tbf

the 4 and 9 should be giveaways

[blissful field]

🤨

[drifting seal]

but if you want you could factor by grouping to get that

[blissful field]

Grouping yes

Tell me more

Abt grouping

[drifting seal]

two numbers that multiply to 36 but add to -12

because 9*4 = 36

so the two numbers are -6 and -6

[blissful field]

Mhm

[drifting seal]

4x^2 - 6x - 6x + 9 = 2x(2x-3) - 3(2x-3) = (2x-3)(2x-3)

[blissful field]

.

Alright

Thank you

[drifting seal]

you’re welcome 🚗

[blissful field]

Snell is also a bish btw,

[drifting seal]

damn

[blissful field]

.close

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[blissful field]

Yeah he copied the concept and rule of ibn sahl and named it as his “snell’s rule”

[calm coral]

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[near mortar]

So like can I use the form on the left instead of the form on the right?

[balmy cape]

they specify different domains and ranges in this case, but both forms are valid in general.

[near mortar]

Ok thx

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[muted tundra]

might be a dumb question, but when I use the (p → q) = (!p or q) can i justify it by saying "Definition of conditional"?

[runic jewel]

Depends on the intended reader. In general that’s possible though. May be different if it’s for your professor you’re writing it for

[muted tundra]

they used it in one of their proofs once

so do you think ill be ok

[runic jewel]

If your prof said that it’s the definition of the conditional then you can most likely feel free to use that definition yourself, yes

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[remote mural]

hi was wondering if this is correct. if not can i get some help?

[graceful dust]

omg i love your aesthetic

[remote mural]

[graceful dust]

yep, looks good so far

[remote mural]

okie dokie that’s all thank you

.close

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[ionic igloo]

is this correct? or is my decimal placement correct

[ionic igloo]

<@&286206848099549185>

[fierce garden]

Я

Wyn

[ionic igloo]

is this correct

[fierce garden]

I’ll see

[ionic igloo]

okeii

[fierce garden]

Hmm

[ionic igloo]

is this the right formula

I = p x r x t

[fierce garden]

To calculate the total interest paid on a loan, you can use the formula:

Total Interest = Principal x Rate x Time

Where:

• Principal = 57,000 pesos
• Rate = 4.5% per year (0.045 as a decimal)
• Time = 34 months

First, you need to convert the time from months to years. Since 34 months is approximately 2.83 years (34 months / 12 months per year), you can now calculate the total interest:

Total Interest = 57,000 pesos x 0.045 x 2.83 years

Total Interest = 57,000 pesos x 0.045 x 2.83

Total Interest = 57,000 pesos x 0.12735

Total Interest ≈ 7,257.45 pesos

Therefore, the total interest paid would be approximately 7,257.45 pesos.

If you need additional clarification or information, please let me know.

[ionic igloo]

thank youuu

[fierce garden]

No

No problem

[ionic igloo]

.close

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[ionic igloo]

.reopen

[calm coral]

✅

[ionic igloo]

hi how does it become 7257.45

[fierce garden]

Uhh

I made an error in my calculations. Let's correct it:

Total Interest = Principal x Rate x Time
Total Interest = 57,000 pesos x 0.045 x 2.83
Total Interest = 57,000 pesos x 0.045 x (34/12) years
Total Interest = 57,000 pesos x 0.045 x 2.833333...
Total Interest = 57,000 pesos x 0.045 x 2.83
Total Interest = 57,000 pesos x 0.12735
Total Interest = 7,249.50 pesos

Therefore, the correct total interest paid would be 7,249.50 pesos.

Thank you for pointing that out. If you have any more questions or need further assistance, feel free to ask.

[ionic igloo]

thank you

.close

[calm coral]

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[fierce garden]

I’m a 11th grader 🙂 I’ve

[ionic igloo]

.close

.reopen

[calm coral]

✅

[ionic igloo]

.close

[calm coral]

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[calm coral]

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[tough star]

why are these two different graphs

[drifting seal]

2(x+1) = 2x+2

not 4x+2

[tough star]

shoot

yeah

mb

[remote mural]

this feels so much like chatgpt 🙏

[drifting seal]

it is

"Thank you for pointing that out. If you have any more questions or need further assistance, feel free to ask."

[remote mural]

you can use AI?

[drifting seal]

that’s the respond you get after correcting chatgpt

[drifting seal]

nope

!nogpt

[calm coral]

Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).

[remote mural]

so what should we do ig?

[remote mural]

!nogpt

[calm coral]

Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).

[drifting seal]

you can ping mods

i’m not sure how serious it is though

[remote mural]

ok

thanks g

[drifting seal]

you got it g

[remote mural]

where do i find the commands

like !no gpt

wait mbmb

ill make a new help channel

mb

[remote mural]

!nosols

[calm coral]

As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.

@tough star Has your question been resolved?

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[vast swallow]

so i was doing a problem in integration and i ended up with the solution u see there, but the answer provided to me is log (2-t)/(1-t) + c instead , whats wrong with my step?

in the solution they have used integration with partial fractions method

[mental tendon]

it should be -1/1-t + 1/2-t

[vast swallow]

oh right 💀 thank you

.close

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[timber abyss]

What is f(4) equal to?

That’s a good sub question to start with

[still coral]

.close

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[crimson zephyr]

didn't want to retype everything, but am i cooking or am i cooked?

[crimson zephyr]

🙏

[swift laurel]

selecting (e) would mean (a)-(d) are true, but one of them is false anyway

[crimson zephyr]

well heres the thing

if one of a-d are false

then e is false

because e means "all of the above are true"

so theres two answers! (aha? or uh uh?)

[crimson zephyr]

^?

[swift laurel]

it's a weirdly worded question but the intent is clear

[crimson zephyr]

yea

well, yea

my brain got caught up in a technical loophole

[crimson zephyr]

thanks anyway

[calm coral]

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[fading blaze]

Hello, I'd like help to figure out the values of C and T in an equilibrium problem. What I have so far are the two equations: Tcos40-Csin20-8=0 and Tsin40-Ccos20-3=0. I think i have to solve for C and T simultaneously using those two equations but I don't know how.

[fading blaze]

the first equation is for the summation of forces x and the second is the summation of forces y

<@&286206848099549185>

elp

may i ask how u got d forces

[calm coral]

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[keen parrot]

Can someone explain what this means?

[keen parrot]

I saw it on TikTok and it made me very curious.

And why would you use the first
method if the second one is obviously faster and what seems easier?

[honest swallow]

Well the first method is using a formula

[mental tendon]

when the quadratic looks not factorable

i guess

[sullen dock]

Because you will find an equation where second one is not that easy to do

[honest swallow]

Which works for all equations but is long

The 2nd gets the job done easily but you require high intuition to apply it

[calm coral]

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[tall moon]

find the value of $$\sqrt{1+\sqrt{4+\sqrt{4^2+\sqrt{4^3+\sqrt{\dots}}}}}$$

[potent lotus]

Skill_Issue

[tall moon]

status 1

[modern peak]

huh, interesting

where'd you get this from

[tall moon]

my teacher

[modern peak]

o

[tall moon]

not sure which olympiad (or if it even came from one) this is from

[limpid pilot]

me

[tall moon]

true

[modern peak]

?

[tall moon]

your name is simmilar to my teachers name

[modern peak]

notice you can factor out a two

[tall moon]

how

[potent lotus]

ren

[tall moon]

what

[modern peak]

using this recurrence relation

[modern peak]

square it, subtract one

and dividing by 2 is the same as multiplying everything inside the root by 1/4

which reduces each power by 1

[tall moon]

no

[modern peak]

yes

[tall moon]

i tried that and maybe im stupid but thst doesent work

lemme try again

[modern peak]

fairly certain it does

[limpid pilot]

context

[modern peak]

oh i see what the issue is

[potent lotus]

ren

[modern peak]

because the 1/4 in the first root changes to a 1/16 in the next-

wait no hang on

am i wrong again?

lemme try this

[modern peak]

yeah that's what i'm getting

?

[tall moon]

nvm

me dumdum

[modern peak]

interesting

[tall moon]

i remember my teacher giving a hint but it was really wierd

it was like $x+1=\sqrt{(x+1)^2}$

[potent lotus]

Skill_Issue

[modern peak]

u sure that was the hint

that's just x+1 = abs(x+1)

[tall moon]

i think so

yeah exactly

i was confused on that

[modern peak]

yeah not much man

[desert mauve]

A small(rather irrelevant) observation can be to see the GP under the radical.

[modern peak]

we're trying but can't seem to make any connections. any thoughts?

also love the pfp

[desert mauve]

Trying on my end too

[modern peak]

until just 4^3, i'm getting really weird stuff

see if you can spot a pattern

$$x_0 = 1$$
$$x_1 = \sqrt 3$$
$$x_2 = \sqrt{1+2^{1.5}}$$
$$x_3 = \sqrt{1+\sqrt{2^2 + 2^{1.5}\sqrt 3}}$$

[potent lotus]

ren

[jolly pilot]

It looks like it converges to 2

[modern peak]

probably

just wanna find a justification

[jolly pilot]

yea

[modern peak]

if we could only find a recurrence relation

[limpid pilot]

wait yeah

[potent lotus]

ren

[modern peak]

horrible ik

[tall moon]

wtf

[modern peak]

did you find smth

maybe factor it using diff of squares?

[tall moon]

add a background please ^w^

kinda hard to see

[modern peak]

how did u get this?

[limpid pilot]

$x+1=\sqrt{(x+1)^2} \
$= \sqrt{1+ \sqrt{(n^2+2n^2)^2}} \
$= \sqrt{1+ \sqrt{4n^3 + \sqrt{(n^4+4n^2)^2}}}$

[potent lotus]

Astar777
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[tall moon]

what?

[modern peak]

????

[limpid pilot]

keep doing this

[tall moon]

how did x change to n wtf

[limpid pilot]

oh shit

yeah thats all supposed to be x

[modern peak]

explain

[limpid pilot]

i was working with n, my bad

[tall moon]

lemme see

[jolly pilot]

At high enough n, the 2^n th root of 4^n is gonna be 0. So, I just deal with a very high n

[modern peak]

??????

0?

[jolly pilot]

Yea, its somewhat wrong

[modern peak]

dude-

[jolly pilot]

(4^n)^(1/2^n)

[modern peak]

$\lim_{n\to\infty}4^{n\frac{1}{2^n}} \neq 0$

[jolly pilot]

n succesive square roots

[potent lotus]

ren

[modern peak]

that is completely different

[jolly pilot]

Not 0, I meant 1 😔

[modern peak]

ah

[tall moon]

$$\sqrt{x^2+2x+1}$
$$\sqrt{1+\sqrt{(x^2+2x)^2}}$
$$\sqrt{1+\sqrt{x^4+4x^3+4x}}$
$$\sqrt{1+\sqrt{4x^3+\sqrt{(x^2+4x)^2}}}$

er latex

[modern peak]

forgot the $'s

[jolly pilot]

close your $s

[tall moon]

ugh how did astar do it

[jolly pilot]

in each line

[limpid pilot]

for new line its \\

[modern peak]

*claps*

[tall moon]

what

how the fuck is there a $ there

[potent lotus]

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

[tall moon]

dude

[jolly pilot]

thats why I use paint

[limpid pilot]

$\sqrt{x^2+2x+1} \\ 
$\sqrt{1+\sqrt{(x^2+2x)^2}} \\ 
$\sqrt{1+\sqrt{x^4+4x^3+4x}} \\ 
$\sqrt{1+\sqrt{4x^3+\sqrt{(x^2+4x)^2}}}$

[tall moon]

er what does this even do

[limpid pilot]

keep continuing if u want to

but its

1 then 4n^3 then 16n^4 then 64n^12 then 256n^28 ...

plug n=1

u get ur original question

[tall moon]

OH

wtf thats genius

but why is the first term so wierd knao

[modern peak]

4^0

[tall moon]

?

[limpid pilot]

what first term

[modern peak]

1 = 4^0??

[tall moon]

this

it looks wierd for some reason idk

oh wdll idk

alr ty

.close

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[calm coral]

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[hearty otter]

.close

[calm coral]

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[sonic mason]

How to find the log value here?

[leaden thunder]

,calc log(10, 10)

[potent lotus]

Result:

1

[leaden thunder]

Use calculator

[limpid pilot]

calculator isnt allowed

in the exam

[leaden thunder]

Then leave exact value

[sonic mason]

How to find the value then

I'm far from approximate one

[limpid pilot]

yeah u dont need exact value, just approximated

[sonic mason]

0.7 is log 5

So .74 should be log 5. Something

Antilog

@limpid pilot

[gray smelt]

looks like it

[sonic mason]

However the answer says this idk

Oh maybe because it's the -

[limpid pilot]

yes

it goes in denominator

1/value

that results in 0.18

[sonic mason]

😭what goes in denominator

1/.74 gives 1.35

[limpid pilot]

10^(-0.74) is 1/10^(0.74)

[sonic mason]

Okay then how to take the antilog

[limpid pilot]

idk

[gray smelt]

inverse fucntions

[sonic mason]

We haven't learnt this part in math but it's in chemistry (the negative antilog part)

[gray smelt]

yeah they don't teach anti log in jee

check the wikipedia page

also it's not in the syllabus

generally they will provide values for log

[sonic mason]

Do such questions come in the chemistry jee part tho?

[gray smelt]

I don't think you need to use anti log

[sonic mason]

Then?

[gray smelt]

eliminating log using property a^log_a(x)

not much

or remember certain log values

[sonic mason]

Yess okayy

Thanks

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[graceful crane]

What subjects are in https://en.wikipedia.org/wiki/Planimetrics ?

[graceful crane]

or Planimetry

[calm coral]

@graceful crane Has your question been resolved?

[calm coral]

@graceful crane Has your question been resolved?

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[junior zodiac]

i dont understand step 3

[leaden thunder]

this?

[junior zodiac]

yes

@leaden thunder

[leaden thunder]

$m$ is fixed so you can just take $\eps ' = \eps \sqrt{m}$ and you have the $K' \in \R$ such that for all $n > K'$ implies $|u_n - m| < \eps'$

[potent lotus]

riemann

[junior zodiac]

and how do we take epsilon = epsilon sqrt m

[leaden thunder]

m is fixed

[junior zodiac]

what does that mean

[leaden thunder]

if you're given an $\eps' > 0$, then you can set $\eps = \frac{\eps'}{\sqrt{m}}$ so that $|u_n - m| < \eps' = \eps * \sqrt{m}$

[potent lotus]

riemann

[calm coral]

@junior zodiac Has your question been resolved?

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[wintry echo]

$|z^2-1|^2=1$

[potent lotus]

crazytime

[wintry echo]

how is it solved?

[fervent obsidian]

Sorry, rather what can you say of a number with a module equal to one?

How can you write them?

[wintry echo]

Mmm

[fervent obsidian]

Hint: the exponential is your friend

[wintry echo]

$z^2-1=1?$

[potent lotus]

crazytime

[lapis breach]

hehe ±

[wintry echo]

Anyway

[obtuse briar]

this is not enough

[wintry echo]

$z$ is a complex number

[potent lotus]

crazytime

[neat cloak]

Put z = x + iy

It's not modulus when we talk abt complex numbers

It's magnitude

[wintry echo]

$|(x+iy)^2-1|^2=1$

[potent lotus]

crazytime

[wintry echo]

I'll start from here

[obtuse briar]

I'd use this hint instead @wintry echo if it's not clear what it means I can elaborate a bit

$|e^{i \theta}|=1$

[potent lotus]

Merosity

[obtuse briar]

using that you can say $z^2-1 = e^{i \theta}$. Make sure that makes sense

[potent lotus]

Merosity

[wintry echo]

I dont know

[obtuse briar]

have you seen e^{i theta} before or is that totally new to you

[wintry echo]

Totally new

[calm coral]

@wintry echo Has your question been resolved?

[obtuse briar]

Hmm, that makes it tougher, do you have any notes or reading from class that covers this?

[calm coral]

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[eager crown]

I tried thinking about something like the ways of picking 1 hat between all the possible kinds of hats but i have no idea about what how to think about h-1 choose r-1

[graceful dust]

right. i think the idea is to add up the cases where you purchase exactly r kinds of hats for each r in {1,2,…,k}

so basically you want to prove that if you order h hats of r different types, and you buy at least 1 of each, there are h-1 choose r-1 ways to buy

and the k choose r is there because when you have k kinda of hats there are k choose r ways to select exactly r types to buy

[graceful dust]

you can think about this is the number of solutions to x_1 + x_2 + … x_r = h where each x_i is at least 1

you want to show that’s h-1 choose r-1

as a hint try to think of an equivalent problem where the constraint is that x_i is at least 0, and the equation is a little different

[calm coral]

@eager crown Has your question been resolved?

[eager crown]

So the point is, the total number of ways is, the number of ways i can order it when i buy 1 kind of hats + the number of ways when i buy 2 kinds of hats + ...
And i need to buy at least 1 of each kind of hat if not i would be repeating the previous cases, thats it?

[graceful dust]

yep

[eager crown]

Oh ok thanks so i understood

Thanks!!!!

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[graceful dust]

np

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[floral rampart]

Derivatives, related rates, intro calc.

The question is three parts:

A: A batter stands 2 feet from home plate as a pitch is thrown w/ a velocity of 130 ft/s. At what rate does the batter's angle of gaze need to change to follow the ball as it crosses home plate?

• I think I got this right. Just some related rates w/ tangent and I got about 65 radians per second.

B: Given a maximum rotational rate of theta prime equals -3 radians per second, find d such that theta prime equals negative 3. (How close to the plate can a player track the ball)>

• This is where I grew confused, considering setting up arcctan(d/2) = theta and differentiating and solving for d gave me a sqrt of a negative number.

C: I got even more confused here!

Suppose pitching speed, d', is different. Then theta' is different and the value of d for which theta' is -3 will change. Find d as a function of d' for d' rangining from 30 ft/s to 140 ft/s.

• I don't quite know where to start. I set up a tangent relationship and isolated d, but it was the sqrt of (2d')/theta' -4. My function is also in terms of theta prime 😦

[calm coral]

@floral rampart Has your question been resolved?

[floral rampart]

<@&286206848099549185>

if the pics are too blurry ask and I can type out said part in full

[calm coral]

@floral rampart Has your question been resolved?

[calm coral]

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[floral rampart]

Derivatives, related rates, intro calc.

The question is three parts:

A: A batter stands 2 feet from home plate as a pitch is thrown w/ a velocity of 130 ft/s. At what rate does the batter's angle of gaze need to change to follow the ball as it crosses home plate?

I think I got this right. Just some related rates w/ tangent and I got about 65 radians per second.

B: Given a maximum rotational rate of theta prime equals -3 radians per second, find d such that theta prime equals negative 3. (How close to the plate can a player track the ball)>

This is where I grew confused, considering setting up arcctan(d/2) = theta and differentiating and solving for d gave me a sqrt of a negative number.

C: I got even more confused here!

Suppose pitching speed, d', is different. Then theta' is different and the value of d for which theta' is -3 will change. Find d as a function of d' for d' rangining from 30 ft/s to 140 ft/s.

I don't quite know where to start. I set up a tangent relationship and isolated d, but it was the sqrt of (2d')/theta' -4. My function is also in terms of theta prime 😦

[floral rampart]

anyone?

😦

[floral rampart]

^

[calm coral]

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[calm coral]

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[cedar ether]

I am stuck at showing the triangle inequality for the $L^2$ norm when we are considering complex-valued functions. The CS inequality states that $|(f,g)|\leq|f||g|$ and I'd like to apply it below, but I'm unable to because we have $(f,g)$ and not $|(f,g)|$ and the former is possibly complex-valued; I'm not sure if the CS inequality applies to something complex-valued: $$\int (f+g)(\overline{f+g})=\int|f|^2+\int |g|^2+\int f\overline{g}+\int g\overline{f}=|f|_2^2+|g|_2^2+2(f,g).$$

[potent lotus]

psie

[mortal orbit]

CS inequality does apply to complex inner products, but what you wrote under is incorrect

it's 2 Re((f,g))

[cedar ether]

oh ok, thank you. Why is that?

[mortal orbit]

this looks like $z+\bar{z}$

[potent lotus]

rafilou is not not born in 2003

[mortal orbit]

which is...

not 2z, but rather...

[cedar ether]

ah, ok, indeed $$\int f\overline{g}+\int g\overline{f}=\overline{\int g\overline{f}}+\int g\overline{f}=\overline{z}+z,$$ which is $2\operatorname{Re}((g,f))$.

[potent lotus]

psie

[cedar ether]

thank you

[mortal orbit]

yes exactly

you'll notice Re((f,g)) = Re((g,f))

because (f,g) and (g,f) are conjugates

[cedar ether]

yeah

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[sinful lion]

can someone help or explain how to get an intersection that isnt z=sqrt3

[sinful lion]

for 1b.

[calm coral]

@sinful lion Has your question been resolved?

[calm coral]

@sinful lion Has your question been resolved?

[calm coral]

@sinful lion Has your question been resolved?

[calm coral]

@sinful lion Has your question been resolved?

[severe bobcat]

Have you tried x² + 3y² = 12 - 3x² - y²?

This equation corresponds to "z must be the same for both surfaces at points of intersection"

[sinful lion]

thats how i got z=sqrt3

looking at the graph again z=sqrt3 is a point where the two surfaces intersect, but it doesnt give me exactly what i wanted

[severe bobcat]

Weird, because those equations clearly have infinitely many solutions

[severe bobcat]

That's equivalent to
x² + y² = 3

So a circle with radius sqrt(3)

Geometrically it means that the set of intersection points casts a circular shadow on xy plane

[calm coral]

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[severe bobcat]

.reopen

[calm coral]

✅

[severe bobcat]

Also don't forget about the "x must be the same for both surfaces at points of intersection" part

[sinful lion]

nvm im dumb z=3 had intersection points

but not z=sqrt3

[severe bobcat]

Okay, anyways, the task is to find a curve

Currently we know that the projection of that curve on the xy plane looks like x² + y² = 3

So we're looking for a curve that looks like this:
$$(\sqrt{3}\cos{t}, \sqrt{3}\sin{t}, z(t))$$

[potent lotus]

EQUENOS

[severe bobcat]

To find z(t), let's get another equation using this ^ logic

[sinful lion]

wdym

do i isolate x on both surfaces and reduce

[remote mural]

if u think about it you want the intersection of both curves, which is an ellipse, but also assigning an specific height depending of where the point of the ellipse is located

so something like (x,y,z) -> (ellipse both x and y, z(ellipse))

what is the standard equation of the ellipse given by x^2+((3))y^2 = 12 - 3x^2 - y^2 ? (here i meant x^2+y^2)

[sinful lion]

is it not this?

[remote mural]

in ur first image u sent z = x^2 + 3y^2 but i will assume u meant z = x^2 + y^2 since thats what you typed in geogebra

[sinful lion]

pretty sure this was in reference to the problem

one of the surfaces is z=x^2+3y^2

[remote mural]

ok

same logic, (x,y,z) -> (circle both x and y, z(circle))

what is the standard equation of the circle given by x^2+3y^2 = 12 - 3x^2 - y^2 ?

[sinful lion]

x^2+y^2=3

maybe im tired but i feel like im going in circles

no pun intended

i dont see how a circle is relevant here

[remote mural]

i didnt read u already sent the parametrization of the circle

[sinful lion]

is the intersection not an ellipse

[remote mural]

sorry the 3rd image you sent here confused me

anyways. what you get for z if you subtitute x = sqrt3 cost and y = sqrt3 sint?

[sinful lion]

6

[remote mural]

show steps

[sinful lion]

wait for which equation

[remote mural]

both should work, the first one is easier to manage tho

[sinful lion]

oh

12

sqrt3^2+3(sqrt3)^2=12

[remote mural]

?

z = (√3cost)^2 + 3(√3sint)^2

[sinful lion]

oh didnt see the cost or sint

[remote mural]

the parametrization u did for the circle was x = √3 cost and y = √3 sint

[sinful lion]

yeah thats mb

z=3cos^2(t)+9sin^2(t)

if im not mistaken

[remote mural]

yes, so what is ur final answer

[sinful lion]

<sqrt3 cos(t), sqrt3 sin(t), 3cos^2(t)+9sin^2(t)>

?

[remote mural]

yes, u can try to type that in geogebra 3d tho

[calm coral]

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[calm coral]

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[muted cargo]

ive been trying to do this question but i cant seem to simplify the numerator to the equal the right hand side

could anyone see if I'm doing something wrong?

[urban relic]

the RHS shouldnt be (k+1)(3k+4)/2, since now n=k+1, you should have the second parenthesis also include that no?

(k+1)(3(k+1)+1)/2

ah u already included that

my bad

hmmmm i mean the results are different but i havent spotted where u went wrong yet

[calm coral]

@muted cargo Has your question been resolved?

[sullen dock]

k(3k-1)/2 -(k+1)+(2k+1)+2(k+1)```

[calm coral]

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[marsh blade]

is it wrong if i do this:

[frank basalt]

Yes it is

[marsh blade]

why

[frank basalt]

That ' ( 'represents that the number set is starting from 7

You cannot just pull it back

Like if you put it at 0

It will be (0, infinity)

[marsh blade]

ok another quesion do i need to make the lines with patterns

like these

or can i just mark the 7

do you understand me or i need to elaborate further

[frank basalt]

You mean the ' ( ' ?

Or ' | '

Whatever it is you have to make it

You cannot just mark the 7

[marsh blade]

the |

[frank basalt]

Yes you have to

[marsh blade]

like in need the patterns of | to make sense

like i need to show it goes by 7

can i do 1 2 then 7

[frank basalt]

Number line should be drawn from a negative number to positive

You cannot just start from 1

[marsh blade]

like this

[frank basalt]

Put atleast 1 negative number

Like -1

On the left

[modern peak]

not necessarily

[marsh blade]

ok but is the positive one correct

cuz i just put the 7 anywhere

[modern peak]

yes

[frank basalt]

[marsh blade]

k thx

thx @frank basalt

[frank basalt]

Never did that though

But alright

[marsh blade]

btw when do i do this:

like arrows from both parts

when its to negative infinity?

we always put arrow on the positive part

[frank basalt]

It's on both sides

A number line is from -infinity to +infinity

When the interval asks us to show infinity you can just include the line as well in the respective direction to the infinity

As shown in the first number posted by you

[calm coral]

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[calm coral]

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[green cosmos]

I’m really slow so I’m having trouble understanding this formula

[green cosmos]

n —> infinity represents the infinite amount of rectangles

f(x) = is the height of i

But what is i?

I know delta x is the length

[eternal shard]

i is just an index, to numerate basically your "different heights"

as you can see in the gif you will have many rectangles but with different heights

so in order to distinguish them, we add an index i

i will give an example

[silver sonnet]

now do a lebesgue version of this integral

[green cosmos]

So are the subscripts index i

[eternal shard]

yes

i counts starting from 1

integre wise

1,2,3,4

[green cosmos]

I think I get it now

Thank you

[eternal shard]

depending on how you chose Delta x, you will have more or less "sections".
x_i is your i-th section, and you would need the according i-th height f(x_i) to calculate the rectangle at that section

[calm coral]

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[fickle flume]

Hello! I need help with checking if two functions are inverses of eachother.

[fickle flume]

I will show work

I know that if

and

are both true at the same time, they are eachothers' inverse function

I am given two functions

Range is [-1, 1]

Domain is [0, 1]

I decide to choose an x to insert into both functions

I insert F(sqrt2/2) into f first

Then I inserted f(sqrt2/2) into F

[fickle flume]

But this part is not correct I think

And I think I am confused about the + 2pi

is it inside of the arcsin or outside of the arcsin?

My intuition tells me they should be eachothers' inverse function

but sin^-1(sin(sqrt2/2) + 2pi isnt sqrt2/2

TL:DR I am struggling to understand where I have gone wrong/if I have gone wrong

[fickle flume]

Should I just conclude that they aren't eachothers' inverse function?

I feel like I have done my calculations correctly

But at the same time

2pi just means we end up at the same spot

but x is different technically

I guess I also forgot the +2kpi

<@&286206848099549185>

Also, the domain of sin(x) on the interval [3pi/2, 5pi/2] is [0, 1] right?

Range is [-1, 1]

!15m

[calm coral]

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.

[fickle flume]

Please ping me if someone ends up offering me help, I am trying to look up answers to my question online in the meantime.

[onyx shuttle]

I believe you have to prove it for all values of x that are in the domain
That is, you can't "choose a specific x" and show it works for it

[onyx shuttle]

I would guess it is outside

I am also not sure whhat the 2pi does, but I believe they are not inverses of each other

[onyx shuttle]

because as you said, x + 2pi is not x :)

[fickle flume]

Oh hey

Damn I have no clue how I would prove it for all x within the domain

I will try to do it for all x instead of sqrt2/2

They ended up being eachothers' inverse now that I did it for all x:es

I must have done something wrong

earlier

Thank you so much for your help

Now I am satisifed with my work

[calm coral]

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[simple musk]

the tangent line of the graph of f at the point $(3,f(3))$ is $y = -2x + 8$. $\$ Calculate the equation of the tangent line to the graph of $g(x) = xe^{f(x) - 2}$ for $x = 3$

[potent lotus]

938c2cc0dcc05f2b68c4287040cfcf71

[calm coral]

@simple musk Has your question been resolved?

[remote mural]

Are you still stuck

Ping me if so and I’ll explain

[simple musk]

still struggling

[remote mural]

Okay so

Differentiate g(x) using product rule

Also you know that f’(3) = -2 since the gradient of the tangent is equal to the gradient of the curve at that point

you can set up an equation for f(3)

y - f(3) =-2(x-3)
which is the same as
y = -2x + 8 so you can solve for f(3)

You can also then calculate g(3)=3e^(f(3)-2)

[remote mural]

And you can also find g’(3) using this result to find the gradient of the tangent to g at x=3

And the just use the straight line equation to find the equation of tangent

Which will be

y - g(3) =g’(3)(x-3)

Does that make sense

[simple musk]

it does not make any sense at all

you just jumped into solving without explaining why or when

[eternal shard]

They explained it really really well and in detail, I don't know if you're serious or trolling, but you should be more specific to what it is that didn't make sense, or explain your own thoughts

[simple musk]

why are they differentiating g(x) right away

[eternal shard]

that's the very first thing you do when you wanna determine the tangent line

[simple musk]

also how are they finding the derivative of f? or is that not needed?

[eternal shard]

because for the tangent line you need to determine the slope m

which is also known as the rate of change, at that particular point of f

and for that you use the derivative

here the slope m in x = 3 is equal to g'(3)

[eternal shard]

the derivative of f at x = 3 is given

read the first sentence

y = -2x+8 is the tangent line of f at x = 3 and from that you can derive the slope m = f'(3) which is -2

[simple musk]

the tangent line of f at x = 3 is given

oh

okay

I see what u mean now

[eternal shard]

f(3) can also be deduced using y = -2x+8

[simple musk]

f(3) = 8

no

[eternal shard]

How?

[simple musk]

y - f(3) =f’(3)(x-3)

y = f'(3)(x-3) + f(3)

-2x + 8 = f'(3)(x-3) + f(3)

mmm

[eternal shard]

it's pretty simple

the tangent line also goes through (3, f(3))

it's literally tangent to f at x = 3

so what you really need to do is plug in x=3 into y = -2x+8

y = -2(3)+8

y = -6+8

y = 2

aka f(3) = 2

[simple musk]

yes

[eternal shard]

that's why

it has to be

[simple musk]

that is very hard to see unless u said it yourself

like that f(3) = 2

[eternal shard]

very hard to see what?

[simple musk]

like plugging in x = 3 into the tangent line of f at x = 3

[eternal shard]

because it is the tangent line at x = 3 of f

so their intersection is (3,f(3))

so you can calculate f(3) using the tangent line

you dont need to know f for that

[simple musk]

yes but is very advanced what u said

[eternal shard]

what is very advanced

[simple musk]

I pressume it gets better with practice

we dont need f to find f(3) because the tangent line of f passes through (3,f(3))

but is very hard to see for myself that

[simple musk]

I guess this makes things less complicated but is a hard concept to grasp

[eternal shard]

it even says, the tangent line of the graph of f at the point (3, f(3))

that's screaming to you how to find f(3)

[simple musk]

yeah and now we can differentiate g(x)

[eternal shard]

wait

can you explain what it is that is complicated?

[simple musk]

tangent line passes through (3,f(3)) if u inject x = 3 into the tanget line equation I didnt know you will get f(3), like f is not even needed we just need the tangent line at x = 3

[eternal shard]

that could also say a robot

you really need to understand

not ingrain this in your head

what's a tangent line in your understanding?

[simple musk]

an approximation of a function

[eternal shard]

AHA

very well articulated

so around x_0 it's a good approximation of f

now what else, the tangent line intersects the function f at one point

else it wouldn't be tangent

[simple musk]

true

[eternal shard]

and that point is the point where you "lay" or "place" the tangent on f

here it is (3, f(3))

that's where you place the tangent line on f

so that must mean they have the same coordinates

at x = 3 you get f(3) whether you used f(x) or the tangent line of f(x)

[simple musk]

perfect yes

[eternal shard]

take a look at the formula

y = f'(a)(x-a) + f(a)

f'(a) = m slope we know that

y = m(x-a) + f(a)

now what do we do with (x-a) and + f(a)

we move the tangent line to the point (a, f(a))

y = -2x+8 = -2x+6+2 = -2(x-3) + 2

m = f'(3) = -2
f(3) = 2

[simple musk]

yeah that is very cool

[eternal shard]

So what you can ingrain is the tangent line of f at some point also has that point

it has to

that's how we defined it

you need to understand this so bad

it has to click basically

[simple musk]

that the tangent line approximation has same output as f when evaluated at x = 3?

[eternal shard]

yes but only because it is the tangent line at x = 3

if you would plug in say x = 2 or x = 4

you wouldn't get f(2) or f(4) but a rough approximation

of these values

[simple musk]

yes

[eternal shard]

ok good i feel it starts makign sense a bit

SO

you know

f'(3) = -2
f(3) = 2

you will need these calculating g'(3)

[simple musk]

y = g'(3)(x-3) + g(3)

[eternal shard]

yes

and g(3)

Can you already calculate g(3)?

[potent lotus]

bacc (unhelpful)

938c2cc0dcc05f2b68c4287040cfcf71

[eternal shard]

no

what about the x

[simple musk]

oh

g(3) = 3

because f(3) = 2

[eternal shard]

here

x it must be 3

not x

[simple musk]

,, g(3) = 3e^{f(3)-2} \quad \textbf{ my baad}

[potent lotus]

938c2cc0dcc05f2b68c4287040cfcf71

[eternal shard]

yes!

f(3) = ?

[simple musk]

2

[eternal shard]

ook you already said

[eternal shard]

correct

g(3) = 3

50 % done

y = g'(3)(x-3) + 3

now g'(3) you need first of all g'(x)

that is why the other helper jumped to it straight

[simple musk]

xd

[potent lotus]

bacc (unhelpful)

[eternal shard]

what rule

[simple musk]

product

[eternal shard]

very good

[simple musk]

• chain rule

[eternal shard]

try it

[eternal shard]

very very good

[simple musk]

xd

,, g'(x) = (x)' \cdot e^{f(x)-2} + x \cdot \frac{d}{dx} \left(e^{f(x) - 2}\right)

[eternal shard]

f(3)?

f(x)!!!

[simple musk]

myy baaad

[potent lotus]

938c2cc0dcc05f2b68c4287040cfcf71

[eternal shard]

good.

derivative of x

[simple musk]

,, g'(x) = e^{f(x) - 2} + x \cdot \frac{d}{dx} \left(e^{f(x) -2}\right)

[potent lotus]

938c2cc0dcc05f2b68c4287040cfcf71

[eternal shard]

good.

I assume