help-23

[flat frigate]

Azyrashacorki

[soft lava]

I just don’t understand how it relates to the other problem cus this is how we did it in class but when I try to do it that way it did not work

[quiet plume]

Yes that's the easy way you can solve this.

You have $20e^{-0.0001216t}$ and you want to know when you'll have half of what you started with (20)$

[flat frigate]

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

[quiet plume]

So you can write $10 = 20e^{-0.0001216t}$ and solve for t.

[flat frigate]

Azyrashacorki

[quiet plume]

This will tell you the half life

What did you get when doing this then?

[soft lava]

I try to solve it but I get stuck here

[quiet plume]

Then you can divide both sides by -0.0001216

[soft lava]

570.02?

On the answer key it is 5700 years so idk what I did wrong

I double checked and I wrote down the original problem correctly

I’m just really confused I don’t know what I did wrong even with the answer key

[quiet plume]

You just lost a 0 transcribing

Its -0.0001216

Not -0.001216

[soft lava]

Ohhh

[quiet plume]

That should give you the right answer now

[safe radish]

@soft lava Has your question been resolved?

[safe radish]

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[honest idol]

Hello , I need help with this problem please

[upbeat veldt]

Okay, so this is a standard optimization question in Calculus.

[honest idol]

Yeah

[upbeat veldt]

What is your understanding of optimization. What do you think the first thing we should do here?

What are we trying to solve for?

[honest idol]

We need to find our objective function and our constraint function

[upbeat veldt]

Right, so we our trying to find the largest Volume we can obtain, so quantity is volume and we are maximizing it.

[honest idol]

Yea

w*l *h?

[upbeat veldt]

Volume = LengthWidthh

[upbeat veldt]

yes.

Now we need to rewrite it in terms of one variavble

[honest idol]

Wdym

[upbeat veldt]

So what relations do we know to rewrite it in terms of one variable?

[honest idol]

Shouldn’t it be f(w,h,l) = whl

[upbeat veldt]

We need to rewrite the volume in terms of one variable, so we can solve it and differentiate it.

[upbeat veldt]

it is, but we need to know rewrite this in one variable.

You can do that utilizing the other information you are given.

[honest idol]

Why tho?

In similar problems I didn’t have to rewrite anything

Ohhh btw

[upbeat veldt]

Because what would you do if we didn’t rewrite it in terms of one variavle

[honest idol]

I’m doing Lagrange multipliers

[orchid mirage]

I can help you with question 6

[upbeat veldt]

lol

[upbeat veldt]

Optimization.

[honest idol]

Yea

[upbeat veldt]

Okay so to rewrite it in terms of one variable.

[honest idol]

The problem is I have a test ina few hours and I didn’t learn it this way and I’m tryna build off of my previous knowledge so I don’t get too confused also because I’m lowkey running out of time

So

[upbeat veldt]

Well then you need to know this as this is essential

Do you want me to give you the answer

you may learn better that way

[honest idol]

I’m trying to do Lagrange multipliers

[quiet plume]

You already have the volume of the box as your objective function

Now you have to write down your constraint equation

[upbeat veldt]

Cost = 4sides + 1 top + 1 bottom
-> 4(lh)2 + (lw)2 + (lw)*3
15 = 8(lh) + 2lw + 3lw

[upbeat veldt]

Hmmm lagrange multipliers, I see its slightly different than optimization but same idea at end of day.

[honest idol]

The dollars are confusing

[quiet plume]

It’s really a constraint on the surface area you’re given

So you need to find the cost of the box in terms of l, w and h and then your constraint says this cost is fixed at 15$

[honest idol]

2(lw) + 3(lw) = 5?

Idek 😂

I mens

Mean*

[quiet plume]

Sketch out the box. You need the surface area of the top and sides along with the surface area of the bottom

I need to bike home I’ll be back in 20min

[honest idol]

For sure take your time man

[safe radish]

@honest idol Has your question been resolved?

[quiet plume]

Okay have you got a sketch?

[quiet plume]

Okay I feel it's hard to see what's going on with this. I was expecting more a 3d sketch

Now you need the surface area of the bottom, and the surface area of everything else

[honest idol]

so when it comes to contraint formula, we want a formula that the volume cant exceed right?

so how are we doing that here

[burnt notch]

Don't overthink

Do you need to maximise the volume? Then, well, you have to write the volume formula, using some suitable variables

[burnt notch]

So with these names for the main dimensions, can you write down the volume formula?

[honest idol]

V(h,l,w) = wlh

[burnt notch]

Good

Now., with your own words, what is the constraint for this problem?

[honest idol]

i have no idea tbh,

i didnt really give anything besides the volume formula

[quiet plume]

Say we disregard the fact that the cost must be 15$ for a second.
Can you write down what the area of the bottom side would be in terms of those variables ? What does this part cost then?
Can you write down what the surface area of the rest of the box is (without its bottom)? What does this part cost?

[honest idol]

Can you write down what the area of the bottom side would be in terms of those variables ? What does this part cost then?

l*w

Can you write down what the surface area of the rest of the box is (without its bottom)? What does this part cost?

2(w x h) + 2(l x h) + l x w

idk about the cost part tbh

[quiet plume]

l*w is correct for the bottom

But for the rest your box should also have a top.

[honest idol]

Oh there fixed

[quiet plume]

Okay.

They're telling you the bottom costs 3$ per unit area to make.

How much does the bottom cost?

[honest idol]

3 dollars?

[quiet plume]

So the bottom has area 1?

[honest idol]

Oh wait

Okay so how did we know cm^2 is our unit area

[quiet plume]

You can assume w, h, l are given in cms, it doesn't matter.

[honest idol]

Oh alright

[honest idol]

2(wh) + 2(lh) + lw = 3

[quiet plume]

If I tell you the area of the bottom is 4 cm^2, how much does the bottom cost to make if it costs 3$ per cm^2?

[honest idol]

12?

[quiet plume]

Right, so you multiply the area by the cost per unit area to get the cost.

[honest idol]

wait so its 2(wh) + 2(lh) + lw = 12

[quiet plume]

The whole point of the setup is to express the total cost of the box in terms of w,h and l. You have no equation yet since the constraint is specifically a constraint on the total cost of the box.

[honest idol]

true

[quiet plume]

So I'll ask again. If the bottom of the box has area w*l and you're given that the bottom costs 3$ per unit area to make, what is the cost of the bottom of the box?

[honest idol]

w*l = 3

[quiet plume]

No equations.

It's the same thing as here but with w*l instead of 12.

[honest idol]

i know youre probably asking a trivial question but its not clicking for me rn idek why

[quiet plume]

The thing is you answered it with a fixed number.

For a bottom with area 4 cm^2, you said that the cost was 4 * 3 = 12$

For a bottom with area w*l cm^2, it's not a different process.

[honest idol]

3wl?

[quiet plume]

Good. So the bottom costs 3wl to make.

Now you need to compute the cost of the rest of the box.

You computed its area earlier

And you know it costs 2$ per cm^2.

[honest idol]

so 2( 2(wh) + 2(lh) + lw )

[quiet plume]

Good. So the rest of the box costs 2(2wh + 2hl + wl) to make

What is the total cost of the box, then?

[honest idol]

2( 2(wh) + 2(lh) + lw ) + 3(wl)

= 4(wh) + 4(lh) + 2lw + 3wl

= $15

[quiet plume]

Yes, so you have your constraint given by 4wh + 4hl + 5wl = 15.

[honest idol]

ohhhhhhh

[honest idol]

the constraint equation is kinda confusing

is it not supposed to be expressed in the way im expressing it?

[burnt notch]

Why? 😅

[honest idol]

because of the = 15

[burnt notch]

How's it troubling you?

You can (and should) bring the 15 to the left hand side

So that you have g = 0, where g is your constraint equation

[honest idol]

ohhh

your constraint equation is always in the form g = c?

[burnt notch]

No 😭

Remember what Azy told you yesterday?

It is of the form "g(...) = 0"

[burnt notch]

Note that I also what I wrote here... @honest idol

[honest idol]

thats after differentiating it or what

[burnt notch]

No? I haven't even mentioned differentiation

[honest idol]

i mean

yeah i think im confusing the = 0 part but thats just f(x,y) = z

such that z = 0

so its not really different from any funciton in space

[burnt notch]

Wait wait

[honest idol]

well that was a bold claim

[quiet plume]

Whatever you get as your constraint, (expression)(x,y,z) = c, you can always rewrite it as g(x,y,z) = (expression)(x,y,z) - c = 0.

It's standard to do this.

[honest idol]

again, always this freakin part

[quiet plume]

Try (equation 1) - (equation 2).

[honest idol]

bruh what am i doing

[honest idol]

oh its a linear system 💀

i just realized

[quiet plume]

It's not

[honest idol]

howcome

i always get stuck on this part

[quiet plume]

You have variables multiplied together.

It's not a system of linear equations

[honest idol]

yeah youre right

the thing is every time we solve these its like we do something random every time

[quiet plume]

It's not random. You have to spot things that you can cancel.

[honest idol]

were always allowed to combine functions right

[quiet plume]

You can rearrange and substitute. You can eliminate, like this

You can do whatever you would do to solve equations in general, there just isn't a cookie cutter way to do it like for linear equations.

[honest idol]

@quiet plume in theory would this have worked

[quiet plume]

From here you could've multiplied through by lambda l - 4

This would give you a nice relation between w and l

[honest idol]

oh shoot

all this is cuz of the overthinking

test is in like 30 minutes so ill just lock on off of what i know

@quiet plume thank you very much for your help

i really appreciate it

@quiet plume one last thing tho you said we dont care about lamda in these problems right

[quiet plume]

We don't care about the particular value of lambda, no

It does have a meaning, but for the purposes of optimization it's not relevant.

[honest idol]

so you wanna avoid making lamda substitutions? or is it just what lamda is that we dont care abt

[quiet plume]

No, you can still use it.

All I mean is you don't have to find its value

Like you do for x,y,z.

[honest idol]

would you say using lamda over other varibales matters or does it just depend on the problem itself

[quiet plume]

It doesn't change anything.

You could solve for x,y and lambda and then get z from one of the equations

[honest idol]

i see

[safe radish]

@honest idol Has your question been resolved?

[safe radish]

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[lone shuttle]

Hey there, I'm trying to prove that if a topological space X is compact and Hausdorff, then X is normal (T4). The professors hint was to recall the proof for compact subspaces of Hausdorff spaces being closed and applying twice (once for each closed set), which i included below, but I really don't see how to do it. I'd really appreciate a further hint or something to guide me in the right direction :)

[lone shuttle]

(I know this is a fairly standard result in general topology and that proofs exist out there but I feel like I miss out on a big part of the learning process by just finding the proof online)

Let $K\subseteq X$ be compact, $X$ Hausdorff. Let $x \in X\setminus K$. $\forall y\in K$, by Hausdorffness, $\exists$ open $V_y \ni y, U_y \ni x$ s.t. $V_y \cap U_y = \emptyset$. Then ${V_y}{y\in K}$ is an open cover of $K$, hence $\exists$ finite subcover ${V{y_1}, ..., V_{y_k}}$. Then $U \coloneq \bigcap_{i=1}^k U_{y_i}$ is disjoint from $\bigcup_{i=1}^k V_{y_i} \supseteq K$, so $U\cap K =\emptyset$, i.e. $U \subseteq X\setminus K$, so $X\setminus K$ is open QED.

[flat frigate]

dan

[safe radish]

@lone shuttle Has your question been resolved?

[quiet plume]

Take any two disjoint closed subsets A,B of X.
They're both compact as closed subsets of compact spaces are compact.
Iirc the idea is you can parameterize open covers of B in terms of each point of A using Hausdorfness (like in the proof you sent kinda) which also gives you an associated open cover of A. You should be able to get something nice if you tinker with those a bit.

[lone shuttle]

thanks for your reply! I'm finding it pretty tricky to construct the necessary open cover but I think I get the general idea of how you need to do it now (by basically simultaneously constructing an open A and B cover from what I understand)

maybe I just need to approach the problem with a better rested brain, ive always found point space topology to be a difficult subject

[quiet plume]

Essentially, you should find a way to construct, for each element of A, an open neighbourhood of it and an open cover of B which doesn't intersect this open neighbourhood of A. Then you could string those along A and do more stuff to make sure everything is nice and disjoint.

But yes, rest helps a lot!

[lone shuttle]

tysm! in which field of math are you studying if i may ask?

just curious

[quiet plume]

Hmm I’m doing masters atm but mostly differential geometry / algebraic topology type stuff

[lone shuttle]

is there any diff geo textbooks you'd recommend? I'm taking a class on it this semester and my prof is so disorganized

[quiet plume]

I think the standard one is Do Carmo’s diff geometry of curves and surfaces

I think he explains relatively well and the first 3 or 4 chapters cover pretty much what one would see in a first course in differential geometry I reckon

There’s Ted Shifrin’s notes online as well which may provide outside perspective

[lone shuttle]

I think I understand now, is the general idea to basically fix an element $x \in A$, and then for each $y\in B$, you get (similar to the proof above) some $U_y \ni x, V_y \ni y$ disjoint. You then get an open cover of B by ${V_y}{y\in B}$, you use compactness for a finite subcover ${V{y_1}, ..., V_{y_k}}$, then you set $\mathcal C\coloneq {U_{y_i} : 1\le i\le k_y}$, and repeat this $\forall x\in A$ to get a cover $\mathcal C \coloneq {U \in \mathcal{C}_X : x\in A}$, and you can then play around with it pretty easily to show its disjoint from B?

[flat frigate]

dan

[quiet plume]

Yes, that’s pretty much the main point of the proof.

[lone shuttle]

alright perfect I think I can do that now! thank u so much I really appreciate your help :)

[quiet plume]

Of course

[lone shuttle]

I'll make sure to take a look at them, we use Do Carmo in our class but I find that I often find myself wondering why things are defined the way they are or why any of the results matter which I find that Do Carmo misses a bit off

thanks again for the advice!!

[quiet plume]

Yes that is true I found that as well reading it. Again Shifrin is pretty good and I think videos from, say, the bright side of mathematics, explain more obscure stuff nicely

Happy learning!

[lone shuttle]

.close

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[clever tinsel]

do you have an actual question?

[safe radish]

@rigid hemlock Has your question been resolved?

[safe radish]

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[subtle crown]

I need help with Excel, if anybody is familiar with the Goal Seek tool

[subtle crown]

B16 is my formula cell, F9 is my variable cell (y)

I don't know why it isn't solving for y, so if anyone has some insight, I'd appreciate it.

.close

[safe radish]

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[safe radish]

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[crude light]

Hi guys could someone help me please, I think this is a binomial question but i dont use the formula i use a table

[crude light]

is this the correct layout, as in did i split it correctly

[safe radish]

@crude light Has your question been resolved?

[crude light]

.close

[safe radish]

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[safe radish]

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Remember:
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[dapper grail]

https://claude.ai/share/62f8dac0-2d52-4470-8bc4-99bf3f9a3e02 Can someone explain me this Errichto iterative method for Chinese remainder thoerem? I haven't dropped the link of the video as some people may not like to watch a video just to explain a concept therefore they can read the claude explanation using this given link.

[buoyant shadow]

I can't open the link

i found the video, which part confused you?

[dapper grail]

Whole

Actually i am on starting, I am not able to understand the starting part

so i don't have any idea if i will be able to understand the rest or not

https://youtu.be/EHDEvFuYPRQ?si=BYPmHCVfJ-jkAtVY here is the video i am refering to

[vague phoenix]

o this not working

[vague phoenix]

you have 2 channels opened, please close one

!1c

[dapper grail]

Is it against rules

?

[vague phoenix]

ye

[dapper grail]

both have diffrent questions

well

[vague phoenix]

ask 1 at a time

[dapper grail]

ok

[vague phoenix]

type .close pls

[dapper grail]

done

[dapper grail]

Had linked wrong video, now corrected.

[buoyant shadow]

it's like you're at the bus stop, and you just look at buses, and there's 3 lines, one arrives every 4 minutes, then there's another that arrives every 6 minutes, and there's a 5 minute bus
and there's not enough space for all three to fit, so you really want to stay and see what they will do

3,5, 2 is when the next bus arrives

you jut trial and error to find how long to wait

until they all arrive at the same moment

[dapper grail]

Lemme read

[buoyant shadow]

it's not chinese remainder theorem, it's too different, but people don't have different names for these problems for some reasion

[dapper grail]

But that is finding us a value that will serve all equations of modular arithmetic, and that's what the purpose of Chinese remainder theorem is also

So it doesn't matter to me. If I am getting the answer

🙂

[dapper grail]

Ok

Alright

But how will I find that?

According to what i understand after watching some part of video

[buoyant shadow]

you solve it for subsets

first bus only, then first 2, then first three

[dapper grail]

Lemme explain

What I have understood

[dapper grail]

One by one you mean?

[buoyant shadow]

yeah

if there's 7 buses you do 7 sub steps, where you increase the step

it's bruteforce that gets faster and faster

wikipedia says you could optimize it if you change the order from big mod to small, that sounds right

[dapper grail]

Let,

x = a1 (mod m1)
x = a2 (mod m2)
x = a3 (mod m3)

Then,
x = m1 × k + a1

And also,
x = a2 × q + a2

Putting equation 1 into eq 2 -

m1 × k + a1 = a2 × q + a2

Hey wait, what am I doing? 💀

I think it is not what he is doing

In the video

Did I mixed up two idea? Maybe 🤔

Because I was studying diphontine equation as well

[nimble wyvern]

<@&268886789983436800>

[dapper grail]

Was gonna do it too

[buoyant shadow]

it would be clearer as python lol

from math import lcm
buses = (4,3),(6,5),(5,2)
time = 0
step = 1
for m, r in buses:
   while time % m != r:
      time += step
   step = lcm(step, m)

that's all

[dapper grail]

I know C/C++ but not able to understand this code 🤔

By the way phone is about to die

Will be coming back in half an hour.

[buoyant shadow]

mod = [4,6,5]
remainder = [3,5,2]
time = 0
step = 1
for(i=0, i < 3, i++) {
   while (time % mod[i] != remainder[i])
      time += step
   step = lcm(step, m)
}

more c less hiss

[dapper grail]

What is time and step variable?

[buoyant shadow]

integers?

[dapper grail]

Ah yes

Time = 0

And step = 1

What are these integers type variables

[silver current]

Is buses a list?

[buoyant shadow]

no, that makes a tuple

a tuple is a list without append method

[silver current]

I see

I know

[dapper grail]

What do u mean

[dapper grail]

What is this also

[buoyant shadow]

it's python things, sorry for spamming the channel

[dapper grail]

I think we shouldn't jump into coding before I understand general maths

😅

The method in that video is not that complex

I just need someone who can explain me what he is doing

Basically

I can explain the part I understand

To see if I am right or wrong

[safe radish]

@dapper grail Has your question been resolved?

[clever tinsel]

that's not the correct ping, just for your information. you pinged a user.

[dapper grail]

Oh I am dumb 🥀

<@&286206848099549185>

I hope it's right now

[clever tinsel]

that's the right ping.

[safe radish]

@dapper grail Has your question been resolved?

[dapper grail]

Why is there no single helper to help me

[safe radish]

@dapper grail Has your question been resolved?

[dapper grail]

Why is no one helping me

?

Have I asked a very bad question?

[zenith maple]

What do you want help with?

[dapper grail]

The question is above. Hidden in the Convo

Lemme tag

[dapper grail]

This

[dapper grail]

And this is the resources I am referring to

[zenith maple]

Okay do you understand how the modulo works at all

@dapper grail are you atleast familiar with Euclid's division lemma

[dapper grail]

No this

[dapper grail]

Modulo yes

[zenith maple]

Both are basically the same

So in the video the focus is to find the set of numbers that give certain reminders under the given modulos

[dapper grail]

Ok

I understand this much atleast

[zenith maple]

So you should be familiar with the idea that if a = b (mod c) a can be b + nc where n is any whole number

[dapper grail]

Ah basically this is the algebraical form of the equation of modular arithmetic

[zenith maple]

So for say two modular equations a = b (mod c) and a = p (mod q) we just need to find the first nc+b for which we get p under mod q

Then to get the full set we need to keep on adding the least number that is 0 under mod c and mod q

Which is the LCM

Now if more equations are there repeat the steps where we need to find the least n(LCM) + the first nc+b (that is p under mod q)

The guy in the video just focuses on how the remainders appear to get the needed value then to find the number corresponding to it he multiplies the LCM with the number of remainders till the needed one (since you have to add the LCM that many times to get the wanted number)

[zenith maple]

Or you could just write down a bunch of linear equations and solve for them how ever you want

[dapper grail]

Lemme read

It's like -

Let,

x = a1 (mod m1)
x = a2 (mod m2)
x = a3 (mod m3)

Then,
x = m1 × k + a1

And also,
x = a2 × q + a2

Putting equation 1 into eq 2 -

m1 × k + a1 = a2 × q + a2

---

Then what do we do further?

[dapper grail]

.close

[safe radish]

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[fathom dock]

Question 59

[fathom dock]

I figured out sin and cos, and i tried to use sin/cos to find tan but I can’t figure out where I went wrong

(Sqrt(6)-sqrt(2))/(sqrt(6)+sqrt(2))

[midnight cypress]

which exercise is this for

[fathom dock]

59

[midnight cypress]

Yeah you're right

as in your answer is right

[fathom dock]

Alr

So its just the solving part that’s wrong

[midnight cypress]

why do you think you're going wrong

[fathom dock]

Well the equation i got after solving for tan is wrong

[fathom dock]

After solving this

[midnight cypress]

ah

[fathom dock]

Multiplied the equation by the inverse of the denominator

[midnight cypress]

yeah you're making a mistake while rationalizing

[fathom dock]

I got (6-sqrt(12)-sqrt(12)+2)/4

[midnight cypress]

hmm

2-sqrt3

that's correct tho

[stoic torrent]

lmao

[fathom dock]

Does that simplify to 8-2sqrt(12)/4

[midnight cypress]

yes

[fathom dock]

Then 4-sqrt(12)/2

[midnight cypress]

yes

[fathom dock]

The answer in the key is 2-sqrt(3)

[midnight cypress]

you can simplify this further

[fathom dock]

Oh you can?

[midnight cypress]

sqrt(12)=sqrt(4*3)

do you see it

[fathom dock]

Ohhh

I see it

Ah

[midnight cypress]

yep

[fathom dock]

I forgot you could do that

Lmao

[midnight cypress]

happens with the best of us 🙏

[fathom dock]

Thanks

[midnight cypress]

np!

[fathom dock]

Number 36

This one i genuinely just dont know where to start

[midnight cypress]

this one you can do in a lot of ways

[fathom dock]

Or wait

Mb

Its 35

[midnight cypress]

oh okay

yeah this one too

let's begin with the rhs

[drowsy lake]

what is this??

[midnight cypress]

what's your first instinct seeing the big expression in cosx

[fathom dock]

Expand it

[midnight cypress]

no!

[fathom dock]

Oh

[midnight cypress]

your first instinct should be to take the terms common outside

[fathom dock]

Oh wait your going from right to left

I see

Thats smart

[midnight cypress]

ah

[midnight cypress]

either way works

[fathom dock]

Yeah ig

[midnight cypress]

after this do u see anything?

[fathom dock]

Yesyes

1-2cos^2x is cos

Cos2x

I think

[midnight cypress]

you're left with cos^2x(1-2cos^2x+cos^4x)sinx

the term in the bracket

any algebraic identity u remember?

[fathom dock]

This one

Cos 2x

[midnight cypress]

you're not wrong here, its -cos2x btw, but that's not helping us in getting closer to something that looks like the RHS

2cos^2x-1=cos2x

[midnight cypress]

hint: (a-b)^2

[fathom dock]

Its not coming back

[midnight cypress]

wdym

[fathom dock]

I dont remember the identity

[midnight cypress]

a^2-2ab+b^2

[fathom dock]

Ohh

I remember

[midnight cypress]

so u see how this is applied right

can you do this for me till here?

[fathom dock]

Wait hold on

[midnight cypress]

I think you'll be able to see the next step

[fathom dock]

I dont see how its applied

[midnight cypress]

1-2cos^2x+cos^4x

let a=1, b=cos^2x

do u see it now

[fathom dock]

Sorry its just not clicking

[midnight cypress]

1-2*1*cos^2x+(cos^2x)^2
Now?

[fathom dock]

Oh alr that makes sense

[midnight cypress]

so this is just

(1 - cos^2x)^2

[fathom dock]

OHH

I see it now

[fathom dock]

But I dont think I could have seen that without this

[midnight cypress]

you need to learn your algebraic identities fr

[fathom dock]

Yeah

[midnight cypress]

there's another way

you begin with sin^5xcos^2x

and then you notice how the RHS has a lone sinx

so you take that out of sin^5xcos^2x

sinx(sin^4x*cos^2x)

now you see that

sin^4x=(sin^2x)^2

and from here you could do it

it's basically the opposite of what we did

[fathom dock]

Im trippin

So you got sinx((sin^2x)^2 * cos^2x)

[midnight cypress]

yep

when u see sin^2x what identity come to mind

or u can see that in RHS

all the other terms are in cos

so how do u convert sin^2x in terms of cosx

[fathom dock]

Wait give me like 2 mins

Bavk

Back

[fathom dock]

I genuinely can’t think of how

[midnight cypress]

you've learned sin^2x+cos^2x=1

right

[fathom dock]

Yeah

[midnight cypress]

(use that)

(do u see it?)

[fathom dock]

If there was like a 1-sin^2x i would get it

[midnight cypress]

hmmmmmm

don't u think u can manipulate sin^2x+cos^2x=1 to get it that way?

this identity is true always

[fathom dock]

Well you subtract sin^2x to get it that way

[midnight cypress]

yeah

[fathom dock]

What do i do with that

[midnight cypress]

well you can also notice that you can subtract cos^2x instead

[fathom dock]

OHHHH

WAIT

I GET IT

sin^2x is 1-cos^2x

[midnight cypress]

correct

[fathom dock]

Idk why that took me so long

[midnight cypress]

it's okay

[fathom dock]

So then you got sinx((1-cos^2x)^2 * cos^2x)

1-2cos^2x+cos^4x

Times cos^2x is sinx(cos^2x-2cos^4x+cos^6x)

Yay

Thanks

[midnight cypress]

np!!

[safe radish]

@fathom dock Has your question been resolved?

[safe radish]

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[void locust]

Guys can someone confirm if this is correct

[void locust]

<@&286206848099549185>

[stoic torrent]

,w 300102 * pi /360

[flat frigate]

Results provided by WolframAlpha

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

Requested by u_seen_yaj

[stoic torrent]

K part a is correct

,w pi ( 100) / 6

[flat frigate]

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Upgrade to WolframAlpha Pro!

Requested by u_seen_yaj

[stoic torrent]

everythings correct!

Dont forget the units!!

[safe radish]

@void locust Has your question been resolved?

[void locust]

Ty

[safe radish]

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[stoic torrent]

Your wlcm!
Have a nice day!!

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[safe radish]

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as this action is irreversible, and this channel may abruptly lock.

[normal moss]

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[solar inlet]

how do i find the range

[mighty cairn]

Let f(x) is equal to y

Then find x in terms of y

[frozen veldt]

If x=0 then the expression gives 3

Beyond that, the function is strictly decreasing

however it will never equal 0 or be a negative number

but it will be arbitrarily close to 0

You can show that in more mathematical detail if you're interested

but that's the intuition

[solar inlet]

but like why do we include 0 in the function to get 3 cuz like the x values RE GREATER than 0 so should we not plug in 1

[frozen veldt]

That's an excellent question

The trouble is that 1 is already overshooting by a lot

because it would be better to start from 0.1, or 0.01, or 0.001, or 0.0001, or something like that

but if we play that game we'll always miss out on possible values

So it's a better idea to start from 0, then just exclude 3 from the range because x=0 is not actually allowed, and reason that anything slightly smaller than 3 must be doable

[solar inlet]

oh

[frozen veldt]

As an exercise you can try showing that there is an x for which f(x)=2.99 even though there is no x for which f(x) = 3

[solar inlet]

okay so what if it was x>=1

then what would the range be

[frozen veldt]

In that case you can plug in x=1 to get 1

and this time you don't have to exclude it from the range

[solar inlet]

oh

[frozen veldt]

then you can again reason that the function is strictly decreasing etc.

[solar inlet]

i see

tysm gng

[frozen veldt]

no worries ^^

[solar inlet]

wait

one last qs

@frozen veldt

[frozen veldt]

sure

[solar inlet]

what if it was

x>2

we plug in 2

right?

and exclude it from range

[frozen veldt]

that works

[solar inlet]

alr bet

[frozen veldt]

well done

[safe radish]

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[delicate radish]

i am back with the work

[delicate radish]

its saying the second coord for both version are incorrect

[mortal sandal]

,calc 1/sqrt(17)

[flat frigate]

Result:

0.24253562503633

[delicate radish]

il try and put all that in and see

[mortal sandal]

you can try using more digits, it could be that

[delicate radish]

thx ive never had that issue before where it wanted more than 3 past dcml

[mortal sandal]

strange

its off by about 0.2%

[delicate radish]

ts webwork

.close

[safe radish]

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[mortal sandal]

,calc 1/sqrt(5)

[flat frigate]

Result:

0.44721359549996

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[white pebble]

yo guys,
if i wanna get the inflection points of an function and the 3rd derivate is equal to 0, that doesnt necessarily means its no inflection point but what do i do then

[white pebble]

or what does it even means when its 0

[tardy mango]

If the third derivative is zero at a point where $f''(x)=0$, then the test is inconclusive. You'd have to keep checking higher order derivatives until you find the first one that is not zero.
\begin{itemize}
\item If the first non-zero derivative is of odd order, then you have an inflection point.
\item If the first non-zero derivative is of even order, then you do not have an inflection point.
\end{itemize}
(Note that this only works under the usual smoothness assumptions.) \

Honestly, you're better off just doing sign analysis on $f''$.

[flat frigate]

Civil Service Pigeon

[white pebble]

oh okay

thanks

[tardy mango]

[safe radish]

If you are done with this channel, please mark your problem as solved by typing .close

[white pebble]

.close

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[honest idol]

hello, i need help setting up the objection function and the constraint function

[plucky elk]

!show

[safe radish]

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

@honest idol Has your question been resolved?

[honest idol]

none

i dont know where to start

thats the issue

<@&286206848099549185>

[still stone]

do you know what a lagrangian is?

or using lagrange multipliers?

[honest idol]

yes i do

i understand how it works but not here

[safe radish]

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[plucky elk]

You messed up again and didn't respond to bot

[honest idol]

.reopen

[safe radish]

✅ Original question: #help-23 message

[plucky elk]

Anytime you're given a fixed number for Lagrange multiplier problems, that's probably your constraint. Write down two equations for volume and surface area first

[winter whale]

Waaaaarmaaaaaaan

[honest idol]

i dont know how to set up the volume due to the fact that it doesnt have a lid

idk whats gonna be different here

[honest idol]

yo

have we talked before

dude why am i lowkey known all over discord

[plucky elk]

Do you think if you take any face of a box the volume changes

[honest idol]

no?

i wouldnt say so

[plucky elk]

Whatever assumptions you made in 4.7 apply here too

[honest idol]

if then so whats the point of it having no lid i dont get it

[plucky elk]

Yes so what's the volume of a box

[plucky elk]

The surface area changes

[honest idol]

how are you tracking my sections 😭

[plucky elk]

Second equation you should find

[plucky elk]

I read the question

[honest idol]

oh

yeah

i thought you were lurking cu yestderday i was asking questions

[plucky elk]

"same problem from 4.7"

[plucky elk]

Do you need help doing this?

[still stone]

volume remains the same

like

uhh

imagine a cup

there is a certain volume of water that goes in it, will it matter how much water it can hold if it has a lid or not?

what matter are the dimensions

[honest idol]

yes please

[honest idol]

right

good point

[still stone]

now so the volume will be length x breadth x height = 4ft^2 like usual

[honest idol]

whats confusing to me is the idea of how this can show us the gradient of the level curve and that of the objective curve

like

[still stone]

?

[plucky elk]

That has nothing to do with volume and surface area

[still stone]

just focus on your functions first

[plucky elk]

Maybe you should learn to focus on basics first

[honest idol]

but isnt that the whole intuition behind lagrange multipliers

[still stone]

yes but our main goal is to find the functions to apply lagrange multipliers onto

[plucky elk]

No

Read the first two formulas and make sure you understand them before learning Lagrange multipliers

[plucky elk]

Only one part changes in the surface area for a box without a lid. Identify how it changes from the formula for TSA from the picture

[honest idol]

yes

[honest idol]

so

[plucky elk]

?

[honest idol]

i have no idea dude

like im actually confused

theyre saying they want the least amount of cardboard possible

[plucky elk]

Have you never learned these formulas before

[honest idol]

such that the entire volume = 4

[honest idol]

yes i understand the fomrulas in and of themselves

but not in terms of

what we need to optimize

and what the constraint is

[plucky elk]

So write the formula for the surface area then

[plucky elk]

.

[plucky elk]

That's not enough

[plucky elk]

Volume formula is given here

[plucky elk]

Look at the picture again and figure out what l, w, and h stand for

[honest idol]

length width and height

[honest idol]

wlh = 4

[plucky elk]

Right. Do the same for surface area

[plucky elk]

[plucky elk]

I told you here that there was one change from the given formula which you wrote down. Did you read that message

[plucky elk]

No

[plucky elk]

You're ignoring the "without a lid" part of the question

[honest idol]

i thought it didnt matter

[plucky elk]

.

[plucky elk]

Only the volume remains unchanged.

[honest idol]

ohhh yeah

so are we subtracting an area of one side?

which is the top

[plucky elk]

Right

[plucky elk]

This should be easy now

[plucky elk]

I mean you should simplify but yes that's right

[plucky elk]

Yea now and only now can you start thinking about calculus

[honest idol]

okay

[plucky elk]

If you don't know your basic volumes, surface areas, etc. you should not be thinking about calculus

[honest idol]

i do know

[plucky elk]

?????

[honest idol]

what

yeah thats one of the things that are confusing me

[plucky elk]

Yea that had nothing to do with volume or surface area of a box

[honest idol]

we want critical points restricted by a level curve of some other function

right

[plucky elk]

Do you know what those words mean

[honest idol]

such that the gradient vector of the level curve at some constant is a scalar multiple of the gradient of the objective function at that same point

[honest idol]

yes

but not how they look like here

[plucky elk]

What's the level curve in this question then

[honest idol]

i have no idea

thats the thing

but i know we need one

im gonna assume its the contraint fomrula since all our level curves are considered the constraint

but i dont understand why thats the case

[plucky elk]

Your choice of f and g are fine here. Do you know what equation you need next from the words you just said

[honest idol]

i dont know tbh

because to begin with i dont even know how the concept of critical points on g = c applies here

[plucky elk]

Maybe since you can't come up with an equation, you should just memorize the equation

$\grad f(x, y, z) = \lambda \grad g(x, y, z)$

[flat frigate]

pi_day

[honest idol]

i mean as much as i hate memorizing and not understanding, how do i "memorize" what my constraint is here

which one is the level curve

[plucky elk]

Do more problems and see what works

[honest idol]

but theres gotta be properties that renders one a constraint formula that we should be able to identify right

[plucky elk]

In reality, it doesn't matter which one is f and g. Solving

grad f = lambda * grad g

is the same as solving

grad g = (1/lambda) grad f

Since lambda doesn't matter, and only x, y, z, your answer should be the same

[plucky elk]

You should just memorize the blue box here
https://tutorial.math.lamar.edu/classes/calciii/lagrangemultipliers.aspx

And do the calculations in example 1 since you need to read a lot of examples. They will be nearly identical to your problem.

[safe radish]

@honest idol Has your question been resolved?

[honest idol]

Wait you kinda got a point

So I can scale whichever vector right

Like either one can be the scalar multiple

[plucky elk]

Did you calculate this yet

[plucky elk]

Or read example 1 yet

[honest idol]

I’ll continue tmrw in the morning or something

Thanks for your help tho

[honest idol]

Btw is this right

Like the vectors are parallel

So you can scale either one to get the other

Right

[plucky elk]

You keep focusing on the wrong things

Do one problem completely before moving on

[plucky elk]

.

And

[plucky elk]

.

[safe radish]

@honest idol Has your question been resolved?

[safe radish]

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[solemn glade]

any insight on how to solve since i don't find any option related to the outcome of the equation i worked

[pine wind]

what did you try?

[cobalt frigate]

!show

[safe radish]

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

[hasty wagon]

do you know that conventionally
(sin(x))² = sin²(x)

[vague phoenix]

yo biscuitt 👋

[solemn glade]

yes, then?

[hasty wagon]

and

[solemn glade]

1/sin^2(x)(1+tan(x))

[hasty wagon]

ah, sorry I didn't read well, lemme check

[hasty wagon]

i see now, so can you continue from the last line

[solemn glade]

no, that's why i'm asking

[near sky]

hm.. i would say it should have been $\sqrt{1- a^2} = |\cos x|$ but they do not seem to consider that

[flat frigate]

professor paradox

[hasty wagon]

like $\frac1{\sin x}+\frac1{\cos x}=\frac{?+?}{\sin x \cos x}$

[flat frigate]

Biscuity

[hasty wagon]

do you know what will be in the ?'s

[solemn glade]

sinx+cosx

[hasty wagon]

yea

now we have sin(x) + cos(x) as the numerator

we tryna divide the numerator by sin(x)

we can get 1 + cot(x)

all good till here?

[solemn glade]

yeah

[hasty wagon]

after that we just plug back in to get the answer

[solemn glade]

i think i get it

how to not get lost while doing htese kind of questions though

[hasty wagon]

usually for me, i will just first break all stuff into sines and cosines, and then compact the factors so that it can be in one term

finally check to see if which choice is good to transform into

e..g. i look at E, and see there's (sin x cos x) in the denominator

then I'll aim for that option E

Hope that helps!

[solemn glade]

thanks

!close

!stop

eh how to

[hasty wagon]

it's .close

[solemn glade]

.close

[safe radish]

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[zenith creek]

hello i had this question and i just wanted to know how it should be approached first and what the idea is here?

[upbeat swan]

do you know how to do polynomial long division?

[bitter fossil]

Guys what is 1 plus 1 equal

[upbeat swan]

um idk 2?

[vague phoenix]

Don't troll

[zenith creek]

sure

[upbeat swan]

well then do that

and see what you get

[bitter fossil]

Good boy correct

[upbeat swan]

thank you

[bitter fossil]

What is sin

[vague phoenix]

Dude don't troll

[upbeat swan]

its when you do a bad thing

[zenith creek]

thats the first step im guessing?

[whole hill]

Rebelling against God???

[upbeat swan]

yes

i mean

after long division its preety simple

[fathom jewel]

Try to do polynomial division

[zenith creek]

let me give it a try

[upbeat swan]

because long division allows you to identify the asymptote

[velvet lodge]

oh snap look who it is

asserting dominance

[upbeat swan]

yeah well the green ones have dominion over the help channels

[zenith creek]

my problem is the a

[upbeat swan]

just treat it like a nuber

number

[zenith creek]

hm

[velvet lodge]

are you looking for the answer

or an explanation

[upbeat swan]

instead of a

imagine that its a 2

or a 67

anything

and then just carry that 67 over

[zenith creek]

explanation

[velvet lodge]

ill let googoo explain then since hes already doing well

goodluck

[upbeat swan]

cheers

[vague phoenix]

Want mod ping again?

Didn't help and send random stuffs

[velvet lodge]

i was just talking and you came in and starting calling me a clown

[velvet lodge]

@upbeat swan did i not just offer to help?

[upbeat swan]

🤣

[zenith creek]

right now the asymptote i got is 2x+(a+2) so im guessing i should equalize the (a+2) and the -4?

[velvet lodge]

@vague phoenix lets just stop the beef alright

[upbeat swan]

yes

[velvet lodge]

and all be friends

❤️

[vague phoenix]

correct

[zenith creek]

and its equal to -6

[upbeat swan]

just set 2x+(a+2) = 2x-4

[covert shell]

Yoo

[upbeat swan]

yeah

[zenith creek]

the 2x cancels

[upbeat swan]

-6 should be right

yes

[zenith creek]

yeah thanks mate

[upbeat swan]

no worries

[upbeat swan]

awwwwww

cute

[upbeat swan]

sup

[zenith creek]

how do i unoccupy the channel?

[upbeat swan]

.close

[covert shell]

Am new here

[zenith creek]

.close

[upbeat swan]

cool

[safe radish]

Channel closed

Closed by @zenith creek

Use .reopen if this was a mistake.

[velvet lodge]

ofc

[upbeat swan]

do you have a question?

[velvet lodge]

@vague phoenix do you forgive me bro

[upbeat swan]

check #❓how-to-get-help

[velvet lodge]

you're a very smart guy

i apologise for interrupting you earlier

[covert shell]

I want you guys to asses me I like math but am bad at it

[velvet lodge]

how old are you

[patent citrus]

how did you end up getting a beef w him lol

hes chill

[covert shell]

Am junior at highschool

[upbeat swan]

i mean usually how it goes here is you ask a question and we guide you through it

[covert shell]

I don't have any question right now

[velvet lodge]

pretty much i asked a question about which maths to focus on in relation to quantitative finance, he told me i was asking "off topic questions" i tried explaining that it wasnt and it was directly related to maths, he had a go at me and called me a clown, then 3 other dudes actually came in and answered my questions

fun times

[patent citrus]

oh

lol

[velvet lodge]

i dont want beef with people in the server tho

so hopefully everything will be well and good

[velvet lodge]

im not american

just say the age 😭

[glass egret]

16/17

[velvet lodge]

ok

@covert shell what maths u doing atm

[covert shell]

U can tell how old I'm if I told you my grade

[velvet lodge]

no because the grade structure is different where i live

you would just be called yr 11 here

[covert shell]

Nothing

[velvet lodge]

in school

or u just

not doing maths

at all

[covert shell]

Oh am 18 yr old

[velvet lodge]

and you're in highschool

?

did you get held back

[covert shell]

No I started 1 yr later

[velvet lodge]

well

are you taking maths

in hs

?

[covert shell]

Ofc it's obligatory 😭

If there was a possibility I wouldn't learn it

[upbeat swan]

L

math is everything

[safe radish]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• 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.

[covert shell]

Can u convince me

[covert shell]

am open

[upbeat swan]

hm

this channel is for specific questions

if you wanna talk about it

we can go to the discussion channels

[covert shell]

Ok

[upbeat swan]

so if you dont have a question

you can type .close

[normal moss]

.close

[safe radish]

Channel closed

Closed by @normal moss

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• 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.

[safe radish]

Please don't occupy multiple help channels.

[upbeat swan]

interesting

[safe radish]

Channel closed

Channel closed due to the original message being deleted.
If you did not intend to do this, please open a new help channel,
as this action is irreversible, and this channel may abruptly lock.

[normal moss]

[safe radish]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• 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.

[terse burrow]

Hi. This might sound silly. I’m a secondary school maths teacher and I’m marking an assessment I gave to one of my classes. This was one of the questions. I attempted the question in two different ways and got two different answers. Method one is the correct way, but I don’t know where I have gone wrong in method 2.

[thorn dragon]

hmmm

[vague phoenix]

Strange, they looks both valid.

[terse burrow]

<@&286206848099549185>

[stuck basalt]

interesting

[vague phoenix]

I don't find an error for the second method.

[terse burrow]

This is the mark scheme for the question

The gaps between 12.7 and 3.2 should be +

[vague phoenix]

Ah I see why!

For the second way you found, if a = 14.8, then R = 54.8

This contradicts with the given information of the question, where PQ=20*sin(R)

Even though your second step is in fact correct, but if plug back into the original question information, it did not satisfy the given conditions.

[urban crown]

I'm finding inconsistencies in the triangle

[vague phoenix]

You can test for R such that $PQ=20sin(PRQ)$ such that $PQ=15.9$

[flat frigate]

Minλ

[vague phoenix]

Does this make senses @terse burrow ?

[terse burrow]

Yes

What inconsistencies do you see by the way?

[urban crown]

I don't think it's a possible triangle, which is why the answer for a cannot be consistent regardless (but I could be wrong)

[urban crown]

If we take the answer for a = 12.655 deg, the sine law is not working

When a = 12.655 deg, then <QSR = 180-90-a = 77.345
Hence, <PSR should = 180 - <QSR = 102.655
Using sine law on the upper (non right) triangle, sin(102.655) / 20 should equal sin(40) / 12.7 I believe?

But I'm getting 0.0488 ≠ 0.0506

[terse burrow]

I see that now. Thank you for your help.

.close

[safe radish]

Channel closed

Closed by @terse burrow

Use .reopen if this was a mistake.

[vague phoenix]

No problem, good luck in teaching!

[safe radish]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• 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.

[scarlet storm]

can anyone help me understand logarithms ?

[pine wind]

do you have a problem you were stuck on?

[split kayak]

Any particular question?

[scarlet storm]

nope

i just came into 11th and wanted to discover it as my first chapter

[vague phoenix]

Have you taken a look at your textbook?

[pine wind]

[scarlet storm]

yes yes

i have applied mathematics

i dont quite understand what logs are for ? are they for finding a power ?

[split kayak]

Well, just to start, you should know that the main purpose for logarithms to exist (as they were first created in ~1615) is that they have the main property of transforming multiplication into addition.

[scarlet storm]

OooooO

How so ?

[split kayak]

Since addition is a far easier operation, they created tables so you can find the logarithm of the numbers you want to multiply, and the number you get from it

This is because the logs satisfy the property

[scarlet storm]

like that a^m * a ^n = a^m+n ??

[split kayak]

log(a) + log(b) = log(ab)

[scarlet storm]

ohhh

that's pretty gooood

[split kayak]

So, for example

[scarlet storm]

uh huh

[split kayak]

Say we work in logarithm base 10 (we will see later what that means)
and you want to compute (easily)
134 * 211

Well, by property of logarithms we have that

[scarlet storm]

oh is that so, without multiplying ?

[split kayak]

log(134 * 211) = log(134) + log(211)

[scarlet storm]

damnnn

[split kayak]

you can then compute the sum of logs.

[scarlet storm]

ohhhhh so i would require to look into the table ?

[scarlet storm]

niceee

[split kayak]

log(134) + log(211)
2.1271... + 2.3243... = ~4.45138

[scarlet storm]

welp thanks anyways !!

uhhh wats that 😭

[split kayak]

mb, you can see above what those numbers are

its the numerical value for log 134 and log 211

[scarlet storm]

ow, so i would again need to look into the table to find the answer and then know what it is ??

[split kayak]

Then you look up what number corresponds to ~4.45138

[scarlet storm]

yeah, i get it now!

[split kayak]

and you would know that the multiplication = 28274

just by looking at 3 values on a table

[scarlet storm]

sweet!!

[stuck basalt]

here’s some intuition you should build

$\log_{x} y = z$
just means
$x^z = y$

so when you are given something like

solve $\log_{2} (4) = x$

that just means

“what number $x$ can i raise 2 to in order to get 4?

[scarlet storm]

thank you for your help man !! really appreciate it