help-10

[gleaming gazelle]

I might just be very stupid.

yeah?

[elfin cosmos]

that is why 8√10 + 60 is the final answer

[gleaming gazelle]

Oh wait i kind of finally get it

i guess

but its so complicated

[elfin cosmos]

it is okay

just do more problems

you will get it

[fossil pendant]

🤨

[gleaming gazelle]

i got that but theres so many more i dont understand

but atleast

[fossil pendant]

well you're at the very start of algebra, thats where most students struggle... but once you perfect everything, math will be very easy! atleast until like college or smth

[obtuse pebble]

@gleaming gazelle Has your question been resolved?

[obtuse pebble]

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[wary badger]

so it was comparison test im assuming

[last ore]

The previous part was

Would I let B_n be 1/n^2

Idk how to really make these into formal proofs

Since

I only have done A level proofs like this which are nowhere near rigorous

It would just be cancelling

How would I go about this kind of proof

[last ore]

Is that what it’s called?

[wary badger]

limit comparison test yea

but did you try working it out with your choice of b_n

what did you get for lim a_n/b_n

[last ore]

not tried working it out but

Lemme try first

[last ore]

I’d get 1 for bn = 1/n^2

[wary badger]

yep

[wary badger]

so what does the conclusion of the theorem here tell you

[last ore]

That an and bn converge?

[wary badger]

well

we know b_n converges

1/n^2 converges is something you should already know

[last ore]

It would only converge if and only if a_n converges

[wary badger]

🤔

sure but we know it converges

if we don't know that either of them converge then its useless

b_n converging implies a_n converges as well

[last ore]

Hm

they both converge

[wary badger]

yes

[last ore]

Idk what u mean by conclusion

[wary badger]

🤔

we found a b_n > 0 such that the lim a_n/b_n = l \neq 0 so the conclusion of the theorem here tells us that b_n converges since we already know a_n converges

[last ore]

Oh I see

So what use does this test rly prove, the fact that the sequence we thought of converges?

[wary badger]

it offers a simple way to show a series converges?

not sure what you mean

[last ore]

In this case, b_n?

[wary badger]

🤔

we're showing a_n converges by comparing it to a known convergent series that we speculate behaves similar to a_n

[last ore]

Ahhh it’s to show a_n converges

Alright

For second one do I just let b_n be 1/n ?

[wary badger]

yep

[last ore]

Ah alright

Tysm

[wary badger]

you're welcome

[last ore]

Sorry I just had one last a

Q

Would this be sufficient?

Or should I show the division by n^3

[wary badger]

yea i would say so

you can do that if you'd like

to be more precise

[last ore]

.close

[obtuse pebble]

Channel closed

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[wary badger]

also you should write $\sum b_n$ converges so $\sum a_n$ converges

[warm shale]

knief

[wary badger]

i was just referring to it as b_n and a_n in text to avoid using latex

[last ore]

Oh yea fairs

Will do

[obtuse pebble]

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[slim ibex]

I really dont understand part b

[slim ibex]

image loading

[royal basin]

did you mean part b or part d

you pasted the MS for part d

[slim ibex]

part d oops

[royal basin]

ok so if you put x=0.01 into the original expression you get 1.01/sqrt(0.98) unsimplified, yes?

[slim ibex]

you see the last line of the MS, idk what is going on

[royal basin]

do you understand the second M mark

[slim ibex]

kinda yea, but id prefer if you can explain it

[royal basin]

$\frac{1.01}{\sqrt{0.98}}$ simplifies down to $\frac{101\sqrt{2}}{140}$ when you rationalize and remove all fractions-within-fractions

[warm shale]

Ann

[slim ibex]

i see

wait part d is just making sqrt2 the subject

[royal basin]

and then $\frac{101}{140} \cdot \sqrt{2} \approx 1.02025$ --- from here you multiply both sides by $\frac{140}{101}$ in order to get $\sqrt{2}$ on its own

[warm shale]

Ann

[royal basin]

yes exactly

[slim ibex]

tbh the way the question is worded is kinda weird to me

[royal basin]

it's a weird source of approximations for sqrt(2), tbh.

[obtuse pebble]

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[broken otter]

it just says $\vec{CA}$ with the coordinates of A (2,5) and C (8,6)

[warm shale]

opal

[broken otter]

how do i know which one is x2 and x1 y2 and y1

A is x2 and y2?

[trim portal]

Hmm, can u send the exact problem statement?

Are u supposed to find the vector CA?

[fossil pendant]

CA = A - C,
(2 , 5 ) - (8 , 6)

[broken otter]

theres no other information the rest is of the problem is 6 other vectors that i alredy solved

[fossil pendant]

if you wanna understand where CA = A - C came from, I can elaborate.

[broken otter]

components

[broken otter]

wait

what is C - A then

AC?

[fossil pendant]

yeah

that's just multipyling both sdies by -1

[broken otter]

oops

[trim portal]

-AB is BA

[broken otter]

C - A = -AC

[broken otter]

oh yeah

[fossil pendant]

well, with examples like AB = B - A, notice how the latter letter of the vector's name is the one that's written first

[broken otter]

so why do we do A - C and not C - A how do u know

[fossil pendant]

so smth like K - D should be equivalent to DK

[broken otter]

gotcha

[fossil pendant]

if the question is asking for CA

then that is A - C

if you do C - A

[broken otter]

ohh

[fossil pendant]

you are getting AC

[broken otter]

its not -AC?

[fossil pendant]

what is?

[broken otter]

C - A

[fossil pendant]

C - A = CA = -AC

[broken otter]

oh

alright i see

thanks

[fossil pendant]

this AB = B - A shtick also expalins why AB = -BA

AB = B - A
BA = A - B

the difference between 9-3 and 3-9 is the -

BA = A - B
BA = - (B - A)
BA = -AB

[broken otter]

i gotta note this somewhere lol

anyways appreciate it

[fossil pendant]

np ofcourse

[broken otter]

.close

[obtuse pebble]

Channel closed

Closed by @broken otter

Use .reopen if this was a mistake.

[fossil pendant]

there's one last thing I could say

[broken otter]

yeah?

[fossil pendant]

let me draw

if you-

[broken otter]

gotcha

[fossil pendant]

want to

[broken otter]

.roeopen

[fossil pendant]

continue

[broken otter]

.reopen

[fossil pendant]

yes

[obtuse pebble]

✅ Original question: #help-10 message

[fossil pendant]

if you have this rod, where A marks 3cm of it and B marks 6 Cm of it, what is distance AB equal to (in cms)

[broken otter]

B-A

[fossil pendant]

exactly!

AB = B-A even in lengths! (which work for vectors Ofcourse)

now you can close 🤩

unless you have other concerns

[broken otter]

but what if A was 6 and B 3

[fossil pendant]

well that means B came before A

[broken otter]

oh BA then right

[fossil pendant]

yes.

[fossil pendant]

i was thinking of this image (sorry)

[broken otter]

i see

[broken otter]

np

alright well again ty

[fossil pendant]

🤩 dw you can close now

[broken otter]

.close

[obtuse pebble]

Channel closed

Closed by @broken otter

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[broken otter]

.reopen

[obtuse pebble]

✅ Original question: #help-10 message

[broken otter]

@fossil pendant got 1 lat question

[fossil pendant]

ofcourse!

go ahead

[broken otter]

why do we do A-C to find the components for CA and not A + C

[fossil pendant]

what is adding 6 and 3 here gonna contribute to evaluating AB?

[broken otter]

oh i see

but to find the components we subtract then right

[fossil pendant]

what do you mean?

give me an example

[broken otter]

like to find the components of CA

and we had the coordinates A (5,2) C (8,6)

[fossil pendant]

well yeah! if you are given the coordinates of point A and C then you can get CA by substracting C from A (A-C)

[fossil pendant]

we know CA= A - C

and you got both point A and C

do you know how to substract** points** 🤩

...

def use your calculator

😭

[broken otter]

wait i got the coordinates wrong

[fossil pendant]

hm?

[broken otter]

it was A (2,5) C (8,6)

[fossil pendant]

mhm

[broken otter]

(2-8 + 5-6)

(-6,-1)

[fossil pendant]

+?!

[broken otter]

oh mb

[fossil pendant]

huh

im talking about the + here

[broken otter]

the A is positive no?

[fossil pendant]

hm

well yes but thats not how we denote it

[broken otter]

cause 5 is in A and A is positive

like C would be -(8,6)

[fossil pendant]

A - C is
(2, 5) - (8 , 6)

which is translated to
( 2-8** , **5-7 )

[broken otter]

ohh

gotcha

[fossil pendant]

your + here suggests (-7) which... is not a point

atleast in 2D

[broken otter]

gotcha gotcha alright

so components are (-6,-1`)

[fossil pendant]

preciesly

idk how to spell that

[broken otter]

so everytime i want to find the components i gotta subrtract

what does it do if i addition?

[fossil pendant]

try to not memorize "the way you do it" but rather "this cou8ld be helpful in some situation and its good information to know" so that hte solving process becomes more intuitive

[broken otter]

yeah true

[fossil pendant]

hm...
A + B = A - (-B)

calling point -B as U
it gives us the vector UA

not sure what you wanna do with that...

[broken otter]

nothing nvm

ig it doesnt do anything lol

alr sorry again ty

[fossil pendant]

I mean- If you are given A+B

in some situation

It could lead to... another thing that could be useful?

[broken otter]

depends on the context ig

[fossil pendant]

like if you have the points A , B

and you wanna be... goofy

and get BA from A+B for some reason

you can substract B twice

A+ B - B - B = A-B = BA

...ig? 😭

[broken otter]

lmao

yeah

[fossil pendant]

hey! expiermenting with math makes it a lot more intuitive

you'll def benefit if you do weird random stuff as long as you get the logic behind them

[broken otter]

logic is my problem in these things

[fossil pendant]

again, DEF try to not "remember how to solve this" and do "Hey I know this and that, let me think of how i can use the information ik to answer the question"

[broken otter]

yeah

[fossil pendant]

when you get into calculus, you'll realize that UNDERSTANDING what you're doing and what you're trying to achevie before proceeding to solve the question is what makes it "hard"

its not hard. Its just not taught properly, sadly.

[broken otter]

facts just gotta practice a lot to understand

[fossil pendant]

see me when you get itno trig sub for calculus 2!

its all about

"yeham, what can i do here 😭 "

anwaysy!

yes, any more questions?

[broken otter]

all g nah ima continue and see if im stuck anywhere

[fossil pendant]

ofcourse.

[broken otter]

ty man

[fossil pendant]

np opal!

[broken otter]

.close

[obtuse pebble]

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[obtuse pebble]

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[dusk venture]

hey, ive just got a few specific questions on my study guide for my upcoming exam. Can someone hop in vc to answer a few questions?

[dusk venture]

this is just an example

hold on ill send the whole thing

actually i might have to afk soon but just ping me

[obtuse pebble]

@dusk venture Has your question been resolved?

[obtuse pebble]

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[wind cloud]

.

[broken otter]

how do i do a)

the norm and orientation is 310 degree and 50 cm

[neon vector]

What happens if you scale a vector

(multiplying by a scalar)

[broken otter]

it becomes 0

[neon vector]

huh?

[broken otter]

like scalar product right

[neon vector]

No

Scalar product is a product of two vectors

[broken otter]

both coordinates multiplied

[neon vector]

Here you have a vector multiplied by a scalar

'scalar' is a fancy word for a number

[broken otter]

oh alr

[neon vector]

But okay, we have to consider orientation and norm

We have $\vec{u}$

[warm shale]

USS-Enterprise

[broken otter]

so 1/2(50)

[neon vector]

What happens to the direction when we scale the vector by $\frac{1}{2}$

[warm shale]

USS-Enterprise

[neon vector]

Try drawing it if you don't know

[broken otter]

it becomes 25

[neon vector]

Yes, the norm

The norm is simply multiplied by the ||absolute value|| of the scalar

Why the absolute value?

Say it was -2*u

Then the norm would be 100, right

[broken otter]

a vector cant be negatif

[neon vector]

We have to take the absolute value

Exactly

[broken otter]

yeah

[neon vector]

Otherwise we would get -100cm

Which is not really possible

Okay, but what about the direction

Draw a random vector u, and then 1/2*u

[broken otter]

i drew a vector and split it in half

[neon vector]

Yes

[broken otter]

the direction remains the same because its not -

[neon vector]

Correct!

When multiplying with a positive scalar, direction remains unchanged

1/2 * u has the same direction as u

But when multiplying with a negative, direction "flips"

[broken otter]

yeah

[neon vector]

-2 * u has opposite direction as u

[broken otter]

true

[neon vector]

So 360° - direction of u

[broken otter]

what if it was

-2 but the norm of u was -50

wait nvm

not possible

[neon vector]

Well you can't have a negative norm 😅

[broken otter]

yeah

the orientation remains the same?

right @neon vector

[neon vector]

If multiplied by a positive scalar, such as 1/2, yes

Think of it as just "extending" or "shrinking" the vector

You are keeping it in the same direction

When you multiply by a negative scalar, you do the same thing, shrinking and extending, but you also flip where it's pointing

[broken otter]

yeah

[broken otter]

opposite sides right

[neon vector]

Yes

[broken otter]

gotcha alr thanks

[neon vector]

Of course

[broken otter]

.close

[obtuse pebble]

Channel closed

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Use .reopen if this was a mistake.

[broken otter]

.reopen

[obtuse pebble]

✅ Original question: #help-10 message

[broken otter]

@neon vector sorry wait when its -3u

its -3(50) = |-150|

= 150

[neon vector]

Well

Saying -150 = 150 is not correct

Also -3 * (50) = |-150| is also not correct

[broken otter]

ah

[neon vector]

You have to bring the absolute value from the start

|-3| * 50

= 150

[broken otter]

gotcha

[neon vector]

Always the absolute value of the scalar

[broken otter]

alright bet

also the sides switched

[neon vector]

$\absolute$

[broken otter]

but how do i find the orientation

[warm shale]

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

[neon vector]

Well

Like in your example

Say 310°

[broken otter]

yup

[neon vector]

Then it's just 360° - 310°

Literally just the opposite side

[broken otter]

how do u know if its in the 4th quadrant

because when its in the 4th quadrant u do 360 - <

[neon vector]

Oh hang on

My mistake

You add 180°

[broken otter]

cause its in the 3rd quadrant right?

wait what

u add 180 to 310?

[neon vector]

490

-360

130

[broken otter]

alr but how can i know that i gotta do that

like how did u know that

[neon vector]

Give me a second

I'm drawing

[broken otter]

alr

[neon vector]

We have this right

The line to the bottom right

vector u is on it

310° angle

[broken otter]

ohh

@neon vector i see

[neon vector]

We just add 180°

to get to the other side of the line

And subtract 360° if we are over 360°

[broken otter]

i see

[neon vector]

Right so 310 + 180 = 490

[broken otter]

like this?

[neon vector]

And then 490 - 360 = 130

[neon vector]

Yes, exactly

[broken otter]

gotcha alright

[neon vector]

The angle between a vector and its opposite vector is 180°

[broken otter]

exact

490-360 = 130

[neon vector]

Yeah

[broken otter]

alright tysm again

[neon vector]

no problemo

[broken otter]

.close

[obtuse pebble]

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[obtuse pebble]

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[brave sky]

<@&286206848099549185>

[calm rose]

!15m

[obtuse pebble]

Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.

[exotic nest]

What do you need helpers for???

[brave sky]

.close

[calm rose]

Please send the question first, wait, if 15 minutes pass with no help then ping helpers

[obtuse pebble]

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[obtuse pebble]

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[dusk venture]

can someone check that ive done this correctly?

[dusk venture]

11 (a)

[pastel flint]

why is it written like integration by parts?

[dusk venture]

oh i thought thats how to do it

[pastel flint]

nah just use substitution

[dusk venture]

........uhm.......i....

thought thats qhat i did

[pastel flint]

ok i see what happened

2x+1 is the angle which is going in the sine function its not another function

[dusk venture]

uh

what that mean

like

do i not touch it??

[pastel flint]

sin(a) but a is 2x+1

[pastel flint]

you substitute it as something else you know integration by substitution ryt?

[dusk venture]

uh

no....well i thought i did

but like

u and v

and dv and du

[pastel flint]

thats integration by parts used when two functions are multiplied

[brave bramble]

You performed integration by parts, not u-sub. They are not similar methods. It happens! But you might want to find a source on u-sub.

[dusk venture]

sigh

ive been stuck on this for like a week

can you help me plsss

[pastel flint]

look substitute 2x+1 as u and differentiate both sides

[brave bramble]

I hope someone will! My personal suggestion: Youtube will be faster than us, since you've never seen this method be completed before.

[pastel flint]

yeah search integration by substitution it will be easier to understand that way cannot explain in text

[dusk venture]

hm

[dusk venture]

i have

i just dont get it

but ill look

thanks

[obtuse pebble]

@dusk venture Has your question been resolved?

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[obtuse pebble]

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[idle path]

Let f (x) = x2 + 19. Express f 0 (4) as a limit. I would just need to plug in 4 correct?

[spice trellis]

f0(4)?

[idle path]

oh

I can fix it

Let f (x) = x^2 + 19. Express f' (4) as a limit.

[spice trellis]

do u remember the first definition of derivative

[idle path]

not exactly

I would appreciate a refresher as I want to be able to apply it correctly

[spice trellis]

okay i give u a easy one f’(a)= lim x->a f(x)-f(a)/x-a

this is a simplified version of the definition

[idle path]

hmm'

thinking

a is 4 here

[spice trellis]

yeah

[idle path]

I'm confused

[ruby orchid]

You are finding the tangent slope of a point on the function, so you would pretend to take 2 points, find the slope between them and move the points closer to each other until the difference is infinitesimal

[idle path]

f’(4)= lim x->4 x^2+19-16+19/x-4?

oh wait

[spice trellis]

-16-19

[idle path]

Yeah

[spice trellis]

a^2-b^2=?

[idle path]

c^2

[spice trellis]

no

(a-b)(a+b)

[idle path]

-105

oh

is b 19 in this?

[spice trellis]

it is called difference of square

[spice trellis]

no

x^2+19-a^2-19/x-a

19 get cancelled

[idle path]

oh okay

[spice trellis]

x^2-a^2/x-a

now try to simplify

[idle path]

x^2-16/x-4

[spice trellis]

yes

x^2-16=??

use the difference of square formula

[idle path]

what is x in this instance?

or is it unkown

[spice trellis]

just think it is a number

[idle path]

16-b^3=?

wait

no

x^2-16=-16x^2

[spice trellis]

no

use the formula i gave

16=4^2

[idle path]

is that the answer?

[spice trellis]

no x^2-4^2=??

[idle path]

x^2-4^2=x^2-16

can I have the answer so I can work backwards and see where I'm misunderstanding?

[tardy epoch]

what's your question with this slide

[idle path]

so I'm just confused iif I should just follow this slide with what we are trying to solve above

[idle path]

do I repeat the formula shown in the slide with this?

[tardy epoch]

yes

[idle path]

ah gotcha

thank you

ill give it a go

so

it's x^2+19-16/x+19-4?

am I on the right track?

[tardy epoch]

No

Denominator should be x - 4. Not sure why you're adding 19

[idle path]

It's because 19 is still in the equation

Let f (x) = x^2 + 19. Express f' (4) as a limit.

[tardy epoch]

The equation is for f(x)

[idle path]

Oh

[tardy epoch]

There's no f(x) in the denominator

[idle path]

As in it represents everything

[tardy epoch]

see how it's just x-3

[idle path]

it's x^2-16/x-4?

[tardy epoch]

This is nonsense

[tardy epoch]

Just put parentheses and it's right

[idle path]

Well no f(x) is representing the entire function we're evaluating meaning yeah putting the +19 does nothing

I get what you mean now

it's (x^2-16)/(x-4)

Thank you

.close

[obtuse pebble]

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[timid silo]

Ahhhmmmm...

so part ii) are we meant to use the cosine rule? or

I tried using cosine rule it got me that z = z

which is great and all but

not quite what i was looking for

[obtuse pebble]

@timid silo Has your question been resolved?

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[heady wave]

Since 𝑨𝑫̅̅̅̅ is the median over 𝑩𝑪̅̅̅̅̅ of the triangle 𝑨𝑩𝑪,𝑬 is a point on 𝑨𝑫̅̅̅̅ such that 𝑨𝑬=𝟏/𝟑𝑨𝑫. The line 𝑪𝑬̅̅̅̅ intersects 𝑨𝑩̅̅̅̅ at 𝑭. If 𝑨𝑭=𝟏,𝟐 𝒄𝒎, Find the length of 𝑨𝑩̅̅̅̅. The drawing is mine

[heady wave]

help

[obtuse pebble]

What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above

[heady wave]

1

I don't know where to begin

[hardy widget]

Do yk mass points?

[heady wave]

Nop

[hardy widget]

uh probably shouldn’t use that then

Do you know Menelaus’ theorem?

[heady wave]

Either but yes harmonics

[heady wave]

Idk menelaus' but i know how to use harmonics

[hardy widget]

as in cross ratios?

[heady wave]

Kinda

Like harmonic points, harmonic beam, descartes relationship

newton relationship

harmonic quatrains

[hardy widget]

hm cause I was thinking of having you show ||AF/FB=(EA/ED)(DC/DB)|| but idk if imma be on the same page as you

So imma leave this to someone else

[heady wave]

No no

Please

Show me

[obtuse pebble]

@heady wave Has your question been resolved?

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[tranquil tide]

Proof Help Needed

[obtuse pebble]

@tranquil tide Has your question been resolved?

[oak bolt]

what if you show triangle ACF is congruent to triangle DBE?

[tranquil tide]

I get that you can porve angle FGB is congruent to CGE with vert. ang. congruence thm, and GBC to GCB with congruent supplement thm, but after that I dont know where to

how would you prove GB is congruent to GC?

[oak bolt]

if you drew a perpendicular from G to BC it should make congruent right triangles

so maybe it's a property of isosceles triangles

like, two congruent angles should be enough to make it isosceles

In geometry, the theorem that the angles opposite the equal sides of an isosceles triangle are themselves equal is known as the pons asinorum (/ˈpɒnz ˌæsɪˈnɔːrəm/ PONZ ass-ih-NOR-əm), Latin for "bridge of asses", or more descriptively as the isosceles triangle theorem. The theorem appears as Proposition 5 of Book 1 in Euclid's Elements.[1] Its converse is also true: if two angles of a triangle are equal, then the sides opposite them are also equal.

[tranquil tide]

.close

[obtuse pebble]

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[amber latch]

okay so what the hell is even happening here i was so sleepy during my math period

[tardy epoch]

https://www.youtube.com/watch?v=3G5WluJ7LFA

[amber latch]

hm

unfortunately i refuse to go on youtube anymore

[tardy epoch]

bleak

[amber latch]

sorry

[tardy epoch]

https://en.khanacademy.org/math/algebra/x2f8bb11595b61c86:exponential-growth-decay/x2f8bb11595b61c86:exponential-vs-linear-growth/v/exponential-growth-functions

can also read your textbook for graphing

[amber latch]

ok ok i'll see

[tardy epoch]

khan and org chem tutor are just good at summarizing

[amber latch]

ty

.close

[obtuse pebble]

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[obtuse pebble]

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[midnight jewel]

Can anyone help, I’m lost

[dense peak]

Which part are you lost actually?

[obtuse pebble]

@midnight jewel Has your question been resolved?

[obtuse pebble]

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[fading vine]

how can i solve the 6d question, that f(x)=x²-3x+2

[fading vine]

to ... then ...

like find a and b to let that limit equal to 2

oh no worry

[dense peak]

If I'm not misinterpreting, you're asking for a way to find values of a and b?

[fading vine]

yea yea

[dense peak]

Right, first you have to plug in f(x)

[fading vine]

yes

[dense peak]

Then since we have a+3b = 1

Manipulate it that one of them is a term of the other

In simple terms, make a in terms of b or make b in terms of a

[fading vine]

oh no, like check if that statement is true

[dense peak]

Right so, it isn't asking for values of a and b?

Wait, what's the actual required?

[fading vine]

like find a and b so that lim = 2

and when we find out a and b

[dense peak]

Right then, what's the purpose of a+3b=1?

[fading vine]

we check if a+3b=1

[dense peak]

If that also has to be true then we can assume that a+3b=1 can be used

Since we have to make sure the conditions of the limit is also meant, right?

[fading vine]

what if it is false

[dense peak]

But we have to find potential values first

And if we can't find it then we can say it's false

Right, what if we do it like this

We find the limit of the denominator?

Since a and b are in the denominator

Let's not make use of a+3b=1

Hmm actually we could just first find the initial limit of the function then go from there

[fading vine]

yea

[dense peak]

Let's find the function first, you should be able to get 0 on the numerator right?

0/(a+b)

So we need a value a+b to make it 2, right?

But no matter what we make it will equal to 0. Right?

But if we have a+b=0 then wouldn't it become an indeterminate form thus having the possibility to achieve the finite number 2 if simplified

So, what do we have so far?

What conclusion can you form on the value of b so far?

[obtuse pebble]

@fading vine Has your question been resolved?

[obtuse pebble]

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[random flume]

Hello, I need help with an exponential decay function problem, please!

The problem is: Cesium-137, with a half-life of 30 years, is one of the most dangerous by-products of nuclear fission. What is the annual decay rate for cesium-137? Write the answer as a percentage.

[iron edge]

Have you tried anything yet?

[random flume]

No, I'm having a hard time figuring out how to start the problem.

[iron edge]

Right, so for how many years, the amount of the compound(Cesium-137) will be halfed

[random flume]

30

[iron edge]

Can you write an equation demonstrating this idea?

Hint: exponential function

[random flume]

I'm not sure how to do that without an initial value

[iron edge]

Let’s say the initial value is k

[random flume]

I'm clueless

[iron edge]

I’ll give one more prompt before demonstrating it

Look, for every 30 years, the amount of the compound becomes half of its initial amount

[random flume]

k(0.5)^t/30

[iron edge]

yeah, was writing the equation

Doesn’t seem necessary now

[random flume]

that was a guess, I don't understand what t means in this case

[iron edge]

$\text{V} = \text{V}_I (0.5)^{\frac{t}{30}}$

[warm shale]

Approved Freak

[iron edge]

It means the years elapsed

[iron edge]

When t = 30, meaning that it has reached 30 yrs

The initial value gets halfed

Do you agree with this?

[random flume]

yes but I don't quite understand how t/30 expresses this

[iron edge]

30/30 =1

0.5^1 =0.5

Halfed

[random flume]

Ok I do get that

[iron edge]

When you plug in t = 60, you’ll find the same pattern

The value gets halfed and halfed again

Which is 1/4

[random flume]

right

[iron edge]

Alright, how about 7.5 yrs?

[random flume]

7.5/30?

[iron edge]

Don’t forget 0.5^

[random flume]

ah

[iron edge]

How about 1 year?

[random flume]

0.5^1/30

[iron edge]

That’s it

Checkmate

[random flume]

ah lol

[iron edge]

Anything else you wanna ask?

[random flume]

No, I think that covers it. Thanks so much!

actually

It says to write it as a percent. But my attempts are being marked incorrect

when I calculate (0.5)^1/30, I get ~0.9771599684

not sure what I'm missing

[iron edge]

Can I check the original question?

[iron edge]

,w calc (0.5)^(1/30)

[warm shale]

Results provided by WolframAlpha

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

Requested by skyandnight

[iron edge]

It says percent, what did you submit?

0.97716 is not the answer

[random flume]

I've tried 97, 98, and about 20 other variations

it's an exercise problem, so I have infinite attempts

<@&286206848099549185>

[past sand]

Hard to answer without know exactly what you typed

I could see it expecting 1-x, so like 2.28 or 2.3

[random flume]

I've tried exactly:
97
98
96
2
2.28
978
.978
7

just kinda doing random guesses at this point

2.3 is also incorrect

2.27 is incorrect

2.29 is incorrect

[past sand]

97.8?

97.72?

[random flume]

both incorrect

[past sand]

Sorry that's just wrong, it would be 97.7

How many characters can you input

[random flume]

97.7 is also incorrect

there is no character limit

[past sand]

(0.5)^(1/30) ?

[random flume]

but I believe my curriculum prefers 4 digits when rounding

[random flume]

so perhaps that's it

[past sand]

4 digits would be 97.72

[random flume]

I tried that as well, it was incorrect

[past sand]

or 2.284

[random flume]

sorry, I mean 4 digits after the decimal

[past sand]

You might as well try 0.97715996843424595493269814617765452366631293874787179003507567484943108652

[random flume]

oh wow, 2.284

that's it

[past sand]

[random flume]

the question is, why lol

[past sand]

Well it's either 4 digits or 4 digits after the decimal

Since it's 2.2840 anyway

[random flume]

ah that makes sense

[random flume]

I understand it all now. thank you so much!

.close

[obtuse pebble]

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[idle lynx]

Calculus again. I'm not sure how to find the derivatives or even if my original answers are right. I can't easily find the process in my book.

[elfin cosmos]

when the derivative is positive, what does that mean for the graph?

derivative is slope of tangent line right

[idle lynx]

I think so.

[elfin cosmos]

so do you understand how to tell whether f' is positive/negative from the graph?

you look at the slope

[idle lynx]

Ok. So how would I get the second derivative?

[elfin cosmos]

you look whether its a U shape or a Ո shape

concave up vs concave down

concave up -> f'' positive
concave down -> f'' negative

have you filled it out for the first derivative yet?

[idle lynx]

Not yet, but I think I understand what I have to do.

Is this right?

[obtuse pebble]

@idle lynx Has your question been resolved?

[tulip prawn]

i think i missed the context — what are you trying to check?

[idle lynx]

@tulip prawn

[tulip prawn]

sign of derivative just means increasing or decreasing

increasing means positive

decreasing, negative

[idle lynx]

I understand that, I'm not sure if it's correct or not now.

[tulip prawn]

its correct

[idle lynx]

@obtuse pebble

@tulip prawn

[tulip prawn]

what?

[idle lynx]

Oh. It loaded late. Apologies. Thank you.

.close

[obtuse pebble]

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[regal fable]

can someone show me the step by step proof? with explanation. I dont get it from my notes. i dont even think what i have on my book makes sense.

[wary badger]

where is the dt coming from?

f(t) = sin(t)/t dt

the dt is baked into the integral notation

what?

!help

[obtuse pebble]

To ask for mathematics help on this server, please open your own help channel or help thread. See #❓how-to-get-help for instructions.

[wary badger]

they call it Si(x) for a reason mate

[warm shale]

zedias

[wary badger]

jesus christ

[jaunty ocean]

[jaunty ocean]

\FT Why are you trying to find
[
int frac{sin x}x dd x
]

[warm shale]

クーリー

[wary badger]

"(f) find Si’(x). Hint : apply the FTC part 1"

[regal fable]

So I’m trying to find that integral but from 0 to x?

[wary badger]

no

you’re finding the derivative of the function defined in terms of the integral

[jaunty ocean]

You’re trying to find the derivative of Si(x)

[Si(x) = int_0^x frac{sin t}t dd t]

[warm shale]

クーリー

[regal fable]

Si'(x) = sint/t dt
?

i mean sin(x)/x

but then is the answer just sin(x)/x..?

[long dove]

Yep, by the FTC part 1 it's that simple

[regal fable]

oh it is

bruh

okmybad

[long dove]

If you want to check, solve the integral for Si(x) and seek for the derivative Si'(x) 😅

[regal fable]

thanks!!

.close

[obtuse pebble]

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[spark estuary]

hi could u help me with set

[obtuse pebble]

Please don't occupy multiple help channels.

[tardy epoch]

#help-20

!1c

[obtuse pebble]

Please stick to your channel.

[hardy widget]

.close

[obtuse pebble]

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[lusty plinth]

so i thought i knew how to do this question

[lusty plinth]

selecting 1 question from each section is (4C1)^3

and selecting remaining 2 from 9 is 9C2

the answer keys says A

i js wanna know if i did it the right way

[zenith raft]

well you gotta be careful with that

let's say you pick A1 B1 C1 in the (4C1)^3

[lusty plinth]

yes

[zenith raft]

then, idk, A2 B2 as the other 2

[lusty plinth]

yes

[zenith raft]

well what about when you pick pick A2 B2 C1 in the (4C1)^3, and then A1 B1 as the other 2

[lusty plinth]

one of the possivilities

[zenith raft]

then you'll be counting that collection of questions again

but you shouldn't be

[lusty plinth]

what yeah

oh yeah

whats the fix?

[zenith raft]

hm there are a few ways you could do it

[lusty plinth]

i have no idea how i could calculate the repeating cases

[zenith raft]

the possible amounts from each section are
1 2 2
1 1 3

so you can have e.g. 1 question from A, 2 from B, 2 from C

with a 1 2 2 "structure" there are 4c1 * 4c2 * 4c2 ways to select questions

there is also 2 1 2 and 2 2 1, so more like 3(4c1 * 4c2 * 4c2)

[lusty plinth]

oh so instead of doing them one by one

you just choose from 1 section by one expression

oh man thats like brute forcing

wait no

its the same numbers

ah still

[zenith raft]

,calc 3(4 * 6 * 6) + 3*(4 * 4 * 4)

[warm shale]

Result:

624

[lusty plinth]

if there was a way to count the repeating cases tho

[zenith raft]

that sounds more difficult

like my method gives rise to the expression 3(4 * 6 * 6) + 3*(4 * 4 * 4)

how much simpler could it get

[lusty plinth]

i dont get it

the first part is all 1 22 22 1 etc

[zenith raft]

same thing with 1 1 3

[lusty plinth]

oh second is that

k thanks

.close

[obtuse pebble]

Channel closed

Closed by @lusty plinth

Use .reopen if this was a mistake.

[lusty plinth]

.reopen

[obtuse pebble]

✅ Original question: #help-10 message

[lusty plinth]

hmm this is the question now

fixing 1 at one of the one place(out of n) and selecting 1 number out of two for (n-1) places gives me n[(2C1)^{n-1}] ways to ensure 1 is present , for 2 its also the same. So the total becomes 2n(2)^{n-1} which is just n(2)^n

if this is the right way then how do i solve n(2)^n = 510

[mortal blade]

There's an easier way to do this lol

How many numbers are there with n digits if you can only use 1 and 2 (no restrictions)

And how many of those numbers have no 1s or no 2s

[lusty plinth]

2^n

[zenith raft]

there are only 2 strings that don’t use at least one of each

[mortal blade]

Layla don't spoil smh

[lusty plinth]

oh js 2

[cerulean jolt]

slight interruption to say, smart move

[mortal blade]

Good

So how many are favourable

[lusty plinth]

oh so easy

n = 8

[mortal blade]

Yes

[lusty plinth]

512 - 2

[zenith raft]

sorry i was not really reading your messages, my message was just going to be my hint

[lusty plinth]

goddamit

[mortal blade]

It is often easier to work without restrictions and then exclude them later

[mortal blade]

No worries

[lusty plinth]

sometimes its easier to complete the restrictions first but clearly i should have tried two methods

[mortal blade]

Yup there isn't a general trick to which is better

[zenith raft]

“at least one” is a pretty good indicator

[lusty plinth]

i really need serious help with this one cuz i have no idea 💀

[zenith raft]

making rectangles and squares involves choosing some points on the board to be vertices

[lusty plinth]

clear version

[lusty plinth]

how do i ensure they are squares

[cerulean jolt]

perhaps it would be wiser to consider the number of lines involved in forming a rectangle

[zenith raft]

for example if you choose any two points that aren’t on a line, that corresponds to a rectangle

and there will be one other choice that corresponds to that rectangle as well

[lusty plinth]

wait i dont get it

[minor belfry]

this question is such a pain

[lusty plinth]

2 points have to be on the same line to be rectangle

[minor belfry]

so many to consider

[minor belfry]

not necessarily

[lusty plinth]

wait we have to count the diagonal rectangles too?

[minor belfry]

yes :P

[zenith raft]

let’s imagine a smaller board that look like this:
O O O O O
O O O O O
O O O O O
O O O O O

if i just choose 2 of them, let’s say

O O O O X
X O O O O
O O O O O
O O O O O

do you see how that corresponds to a rectangle?

[cerulean jolt]

no, I don't think so

[mortal blade]

Huh no

[minor belfry]

really?

[mortal blade]

It explicitly says the rectangles created by the horizontal and vertical lines

[cerulean jolt]

you are told that you are to form rectangles and squares with only the 9 horizontal and 9 vertical lines

[minor belfry]

ohh whoops

[lusty plinth]

yeah

[minor belfry]

i didnt see the question

[cerulean jolt]

that's why I brought up the idea of considering how many lines you need to form the rectangle

[minor belfry]

i thought it was just points

oopsiesss

[lusty plinth]

the other two points will be

X O O O X
X O O O X
O O O O O
O O O O O

[zenith raft]

yea

[lusty plinth]

right

[zenith raft]

and it’s worth noting that when you choose those other 2 points, you’ll get the same rectangle

[lusty plinth]

yes

so choose all pair points
minus pair of points lying on a line
divide by 2?

[zenith raft]

yep perfect

[lusty plinth]

i have to choose them 2 at a time right?

[zenith raft]

yes

all choices of two points that don’t lie on the same line, divided by 2

[lusty plinth]

k

[cerulean jolt]

wouldn't considering the lines directly be more straightforward

[zenith raft]

what do you mean by that

[lusty plinth]

id like to hear explaination about that too

[cerulean jolt]

you are given 9 horizontal and 9 vertical lines

you know a rectangle has to have two distinct horizontal and vertical lines each

[lusty plinth]

yes

just select them?

[cerulean jolt]

so do some combinations

and bam, no minusing needed

well some minusing needed to account for squares, but way easier to account for

[zenith raft]

it’s like the same thing i think

[cerulean jolt]

it should end up being the same, indeed, but you'd have to count the number of points collinear with the chosen point for the point-based method

but whichever method OP prefers

[lusty plinth]

how do i count squares?

[valid hearth]

Let's say your lines are divided into parts

[mortal blade]

Squares are rectangles

So you don't need to minus that ig

[zenith raft]

i have a dumb way if nobody suggests anything better

[mortal blade]

The wording of the question is ambiguous wrt that

[cerulean jolt]

that's fair as well

[mortal blade]

Same

[cerulean jolt]

ok, too crowded here, I'll back out

[lusty plinth]

k

[valid hearth]

You can choose same number of horizontal and vertical to form square

[lusty plinth]

so assuming 1 square to 1 unit square area, you are saying to choose lines based of their unit length

[zenith raft]

i’ll just share the dumb way in any case: ask yourself how many squares of area 1 there are, how many of area 4, and so on

you can count the number of points you can lay, say, the top left corner onto

[lusty plinth]

goddamn thats hard bro

[valid hearth]

Yes first 1by1 then 2by2

[valid hearth]

Ts easy trust me

At the end we get summation of n^2

[lusty plinth]

let me try the calculations

[valid hearth]

Yeah ts what I'm trying to suggest

[lusty plinth]

thats brute forcing and hurtful to do

tbh

[zenith raft]

doesn’t sound that bad to do in my brain

[zenith raft]

you can write a nice expression for how many squares of each size there are

[lusty plinth]

can i use some type of combination to count the squares

[zenith raft]

i’m trying to think of a better way but i don’t think the working way is that objectionable

[lusty plinth]

one part of the problem is intellectual and the other is pure mechanical work

very misleading

[zenith raft]

there shouldn’t be that much mechanical work

[lusty plinth]

let me try to generalize how many square can be made cuz im not counting them

[zenith raft]

for area 1, any of the 64 tiles will do. for area 4, you can choose 64 - 8 - 7 tiles as the top left tile of the square

for area 9, similar thing. just can’t use the bottom or rightmost 2 rows/columns

there is not really any brute force counting here

[lusty plinth]

49 for 4 units

[zenith raft]

yea

8*8, then 7*7, and so on

[lusty plinth]

hmm 👍

so im basically just going from easier to harder doubts , this ones the last i swear(for today)

[zenith raft]

you may wanna look into ‘double factorial’

that’s exactly what your question is about

[lusty plinth]

10000 factorialdivided by all factorials of even

[valid hearth]

can procced with this

[lusty plinth]

oh i didnt know that

[zenith raft]

but it can also just be reasoned through, you don’t need to look anything up

[lusty plinth]

howw

[zenith raft]

why don’t you start with this and factor some 2s out

[lusty plinth]

see i have an idea

exacylu

all power of twos

threes

5s

..

thats why i cant reason through

i know how to factor them out

but there are so many prime numbers

[zenith raft]

well not ALL

[lusty plinth]

there are so many

oh then you have divide them by lets say x prime factor and add the quotients up too

to get its power

[zenith raft]

we are working with 1000!/(1000 * 9998 * 9996 * … 2) right

[lusty plinth]

ten thousand

[zenith raft]

oops, pretend i wrote that correctly

factor one (1) two from each thing in the denominator

[lusty plinth]

k then

i can calculate them but what

its 10,000

[zenith raft]

it should look like one of the answers

[lusty plinth]

i dont understand how i can factor 2 from 5000 terms

then know whats inside it

[zenith raft]

10000 = 5000*2, etc

[lusty plinth]

2^5000(......)