help-36

[tranquil pine]

Factor this trinomial:

[tranquil pine]

?

[rancid mural]

so $(5x-2)(x-6)$

[soft zealot]

e_waste

[tranquil pine]

Idk 😭

[rancid mural]

I use cross method

[tranquil pine]

My mom told me to multiply 5 and 12

And AC is 60

[tranquil pine]

And the 2 numbers need to add up to -32

🗿

[rancid mural]

you see $-2 \times -6$ is 12 and $1 \times 5$ is 5

[soft zealot]

e_waste

[rancid mural]

and $5 \times -6 + 1 \times -2 = -32$

[tranquil pine]

Yes, I see that

[soft zealot]

e_waste

[rancid mural]

so this can be $(5x-2)(x-6)$

[soft zealot]

e_waste

[tranquil pine]

Let me ask my friend

[rancid mural]

this is the cross method

[tranquil pine]

The chapter is called

Factoring ax2 + bx + c where a doesnt = 1

[rancid mural]

why 60 and -61?

[tranquil pine]

Okay

[rancid mural]

huh?

Don't get it?

ok but the result is the same

[tranquil pine]

Okay

Lemme try to do it

…

(2x - 10x) (-3x +15)

Wait

I forgot to simplify

(X-5) (2x - 3)

[tranquil pine]

This

Yea

Makes sense

. . .

I think it’s correct cause I checked it

Wait

[rancid mural]

This is the right answer

[tranquil pine]

I’m so f3cking dumb

I was looking at another problem

(x - 6) (5x - 2)

[tranquil pine]

Same answer but do I have to flip it?

So it doesn’t matter?

Okay thanks

.close

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[rancid mural]

no cuz it just the same

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[wise glacier]

Hi can anyone please help me with this question, I dont even know how to get the ball rolling

<@&286206848099549185>

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[tranquil pine]

In the second step, x^2 is taken equal to Ix^2I. But shouldn't it be wrong because, x^2=IxI^2?

[digital steeple]

if you think its wrong then you should be able to name a counterexample.

[twilit vapor]

It's true that x^2 = |x^2| but its also true that x^2 = |x|^2. A number squared is always positive, so the absolute value here doesnt change anything

[tranquil pine]

I think it's wrong because what if we substitute x=√-1

[twilit vapor]

Youre likely only looking at real solutions

[tranquil pine]

Ohhhh

[digital steeple]

you didnt mention what x could be.

so your question is ambigous.

[tranquil pine]

So you mean x^2=Ix^2I for real solutions?

[twilit vapor]

yes

for real x

[tranquil pine]

Ahh

Thanks

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[elfin quartz]

How do I do part b and d? I tried part d by doing (9!x3!)/11! To get 3/55 but the correct answer is 6/55. For part b, I don’t know how to do it

[final saddle]

@elfin quartz Has your question been resolved?

[obtuse flame]

for part d, consider (T, H, E) as a single unit

now arrange the 9 units to form (THE, M, A, T, M, A, I, C, S)

$$\frac{9! \times 3!}{2! \times 2! \times 1! \times 1! \times 1! \times 1!} = 544320$$

[soft zealot]

misha

[obtuse flame]

dividing this number by the total permutations 4989600 that you obtained correctly in part a results in 6/55

[obtuse flame]

as for part b, the word must start with M and end with S. this leaves us with 9 remaining letters ATHEMATIC

keep in mind that A and T are repeated twice

$$\frac{9!}{2! \times 2! \times 1! \times 1! \times 1! \times 1! \times 1!} = \frac{362880}{4} = 90720$$

[soft zealot]

misha

[obtuse flame]

also, just in case, your answer for part c is correct

[elfin quartz]

thank you

[elfin quartz]

but why is it 1! in the denominator? is it the rest of the letters

[obtuse flame]

A appears twice: 2!
T appears twice: 2!
H, E, M, I, C each appear once: 1! each

[elfin quartz]

Ooh okay thanks

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[hexed sedge]

do i use 4m/s as replacement for gravity in this case

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@hexed sedge Has your question been resolved?

[hexed sedge]

where the help at

@help

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@hexed sedge Has your question been resolved?

[cerulean igloo]

Maximum deceleration corresponds to maximum kinetic frictional force

Which is equal to uN

U is the coefficient of friction and N is the normal force

Normal force can be obtained by considering the equilibrium in the vertical direction

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[stray spruce]

is this application of vectors? i swear i literally just saw this on my textbook

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[lunar grove]

how do i solve this

it doesnt say positive or negative

[warm python]

expand both sides to start

[final tangle]

shouldn't expand for efficiency

[warm python]

or split it into two cases

[lunar grove]

i expanded and cant solve

[final tangle]

rearrange to get both terms on one side,
then factor out (x-3)

[lunar grove]

how do i rearrange

[final tangle]

subtract same thing from both sides

[lunar grove]

rearrange to (x-3)(x-3)>(x+5)(3x-2)?

[final tangle]

no

[lunar grove]

then what

[final tangle]

subtract same thing from both sides

[tawny halo]

a<b a-b<0

[lunar grove]

take x-3 out?

[flint kettle]

cooked

[final tangle]

wdym by take x-3 out

[tawny halo]

(x-3)(a-b)<0

[lunar grove]

huh

[lunar grove]

which x-3

theres 2

[final tangle]

subtract same thing from both sides to get either
stuff > 0
or
0 > stuff

[tawny halo]

Both of them

[final tangle]

do subtraction in this step ONLY, nothing else
don't expand

[tawny halo]

ad-cd<0 d(a-c)<0

[lunar grove]

like this?

[final tangle]

yes

[lunar grove]

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[azure fox]

Hullo

[azure fox]

Merrp

Uhhh

Does something that had been derivated have maximum value if it equals to zero?

Please ping

[marble agate]

if the derivarive of a function has a zero it means there is a local maximum or minimum

but you cant tell if its max or min solely from that

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@azure fox Has your question been resolved?

[azure fox]

BUT why my book tell me like it must be maximum?

And

Does a derivated thing ate equals to its original?

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[round oar]

whats wrong?

[round oar]

nvm

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[feral siren]

can someone help me with the last question please

[feral siren]

c

im confused how to answer it

what does it mean by series of expansions

[young bridge]

series expansion is taylor series/maclaurin series

[warm python]

series expansion

or taylor expansion

[feral siren]

ohhhhhhhhh

ok that makes sense thanks

[young bridge]

looks like a level further maths

maybe im wrong

[feral siren]

nah you right its makes sense

i just gotta find the 3 derivative which will be annoying

[young bridge]

hmm

i was about to suggest another method but it might take longer

i think differentiating is definitely faster

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[fiery prism]

Can someone help me solve 5b on this past paper

[undone nymph]

Triangle abc is similar to triangle ade

AC/ AE = BC /DE = 3/7

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[tranquil pine]

i've got a question

[tranquil pine]

say i have to construct a 3x3 determinant from {-1, 0}. what would be the maximum and min values for the determinant

now my teacher used a 'trick'

that you take the smallest element(-1 in this case) as the diagonal value and the largest value (0 in this case) for the rest of the positions, and that solving this determinant gives you max value

but that didnt work in this case, i'm getting max value as -1, and he marked 2

[sudden matrix]

well the max value has to be bigger than or equal to 0

[tranquil pine]

so i'm wondering

1. where this 'trick' comes from
2. what is the proper approach for this

[sudden matrix]

since you can just take all 0's

[tranquil pine]

yes but he gave this technique as an approach so im confused as to why it failed and more so that why had it been working so well with the problem up until this point

[sudden matrix]

i see hmm

his trick shouldnt work at all for odd matrices since $(-1)^n$ is negative for all odd n

[soft zealot]

AlphaNull

[tranquil pine]

im sorry i dont follow

not its not necessarily {-1,0}

he said this for general case scenario

[sudden matrix]

but then you'd expect it to work for the specific case

which it doesn't

so his method fails

this question is really hard i believe

you need to do an exhaustive search

[tranquil pine]

oh

[tranquil pine]

how do i do that

<@&286206848099549185>

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[fathom walrus]

@void crown

[fathom walrus]

Let's start over

[void crown]

alr

[fathom walrus]

so we want to solve for 'n'

sum of finite geometric series is

first term * (1 - ratio^n)/(1-ratio)

n being number of terms

so our first term is

15 * 6^i

ratio is 6

and number of terms is n-i

[void crown]

B = (15 * 6^i) * (1 - 6^n)/(1-6) right?

[fathom walrus]

$B = (15 \cdot 6^i) \cdot \frac{(1 - 6^{n-i})}{1-6}$

[void crown]

oooh this makes it so much easier

[fathom walrus]

alr so do u understand this

[void crown]

yeah this makes so much more sense now

[fathom walrus]

ok now we're just solving for n

$\frac B{15 \cdot 6^i} = \frac {6^{n-i} - 1}{6-1}$

[fathom walrus]

$\frac B{3 \cdot 6^i} = 6^{n-i} - 1$

[void crown]

where did the fraction go?

[fathom walrus]

the 6-1 became a 5, then i multiplied both sides by 5

then i simplified the left fraction

[void crown]

hm can you show me how it looked like before simplification?

ive lost track

[fathom walrus]

$\frac {5B}{15 \cdot 6^i} = 6^{n-i} - 1$

[void crown]

ooooh okay i get it

alr ty

continue

[fathom walrus]

ok so then we go to here

by simplifying the left fraction

then we add 1 to both sides

$1+ \frac {5B}{15 \cdot 6^i} = 6^{n-i}$

[fathom walrus]

then we get

$1+ \frac {5B}{15 \cdot 6^i} = \frac{6^n}{6^i}$

[void crown]

isnt this the non-simplified fraction?

[fathom walrus]

yea its fine

[void crown]

alr

[fathom walrus]

$1+ \frac {B}{3\cdot 6^i} = \frac{6^n}{6^i}$

[fathom walrus]

got it?

[void crown]

yeah

[fathom walrus]

so now multiply both sides by 6^i

$6^i+ \frac {6^i \cdot B}{3\cdot 6^i} = 6^n$

[void crown]

i knew this wasnt right lol

[fathom walrus]

now we simplify a bit

[void crown]

wait

oops replied to the wrong message

lol

but i knew it was wrong at first lol

meaning im still on board lol

[fathom walrus]

$6^i+ \frac {B}{3} = 6^n$

[fathom walrus]

now see if u can do the rest

[void crown]

nope this is as far as i can go lol

we havent gotten to logorithms in school yet

the best i can do is quadratics lol

or whatever theyre called

[fathom walrus]

yea now take log of both sides and we get

$\log_6 {(6^i+ \frac {B}{3})} = n$

and there we have it

[fathom walrus]

and add the floor to make it a integer-valued

[void crown]

oh its as simple as that

never knew

this actually helped a lot

[fathom walrus]

👍

[void crown]

ty

[fathom walrus]

np

gl on the game dev

[void crown]

gonna be able to not only adjust the values in my formula now lol

but also probably solve things easier later on lol (if i dont forget)

because this is way above what we are learning rn in 10th grade

[fathom walrus]

u shd be reaching it soon honestly

if you havent alrdy

but anways i gtg

pce and gl

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[void crown]

alr cya

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[summer birch]

I need to define f(0,0) so that the function below is continuous at (0,0) using the sandwich theorem for multivariable functions

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@summer birch Has your question been resolved?

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@summer birch Has your question been resolved?

[solid cobalt]

What have you tried?

What is your candidate for f(0,0)... How you tried to sandwich it?

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[fierce osprey]

I am so close to done

[celest crane]

Swapping the interval changes the sign.

[fierce osprey]

ah, so negative 13/45?

[vital surge]

int from 3 to 6 of f(x) is 45/13, not 13/45

[fierce osprey]

-45/13?

[solid cobalt]

Yes.

[fierce osprey]

thanks

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[tranquil geode]

can some body explain it in simpler terms

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[versed crater]

Looks like some rearranging of the gamma function

And maybe an IBP to the right side

[tranquil geode]

man pls explain in paper or something every thing is bouncing off

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[pale snow]

yo

[pale snow]

i need help i dont undertsnad the end behaviour and behaviour at NPV

idk why or how to write it

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[mint abyss]

I need to integrate that with the partiall integration mehtod (i hope its called like that in english)

[mint abyss]

And i am just lost on where to start which one is f(x) and which one is g(x)

[flint kettle]

what did you try to take as u

[flint kettle]

Which one are you differentiating and which one are you integrating

Just so we're on the same page regarding notation

[mint abyss]

i read somewherer that the ln is always the u

or v? i dont know we use different variables

[flint kettle]

you can take ln^2 x to be u and 1 to be dv

[mint abyss]

oo oke

oke i will try that and if it doesnt work i will come back

thank you

[flint kettle]

dont forget the chain rule

[mint abyss]

yes thank you now i understood it

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[faint monolith]

how to intergrate this without substitution

[magic sparrow]

sin(2x)/cos(2x) = ?

[faint monolith]

marking scheme rearranged to this and after intergration it is 2sec 2x

[magic sparrow]

yup

[faint monolith]

tan2x

[magic sparrow]

yup

So

what don't you understand

[faint monolith]

how does it become 2 sec 2x

[magic sparrow]

$$\frac{\sin(2x)}{\cos^{2}(2x)}=\frac{\sin(2x)}{\cos(2x)\cos(2x)}=\tan(2x)\cdot \frac{1}{\cos(2x)}=\tan(2x)\sec(2x)$$

[soft zealot]

Austin

[faint monolith]

yea i understood the arranging part

but how to intergrate that without substitution

[magic sparrow]

what is the derivative of sec(x)

[faint monolith]

💀

eh wait

[magic sparrow]

I asked for the derivative

[faint monolith]

derivative

its -cosx^-2 (-sinx)

[magic sparrow]

no

[signal shell]

🔠⁉️

[faint monolith]

sinx/ cos^2 x?

[magic sparrow]

or equivalently...

[faint monolith]

tanx secx

[magic sparrow]

yes

so you're trying to integrate something

which is the form of the derivative of something you already know

integral of 4 sec(2x) tan(2x)

but you just told me

d/dx sec(x) = sec(x)tan(x)

so d/dx sec(2x) = ?

[faint monolith]

sec2x tan2x

[magic sparrow]

⛓️

don't forget

[faint monolith]

chain?

[magic sparrow]

as in the chain rule

[signal shell]

I like to call it the Chen Lu

My math sir black pen red pen calls it that

[magic sparrow]

he's also my math sir, what a guy

[faint monolith]

im so confused af wait

[magic sparrow]

d/dx sec(2x)

what is it

you did it wrong earlier because you forgot to use the chain rule

[signal shell]

🔠

[faint monolith]

ok so differentiation of sec x= sinx/cos²x ,

[magic sparrow]

there's no need to keep expressing it like that

[faint monolith]

and integration of sinx/cos²x should be sec x

but the question is 2x so i juz make it sec 2x?

[magic sparrow]

Not exactly

for example

if it asked for

$\int \cos(2x) dx$

[soft zealot]

Austin

[magic sparrow]

and you just said

sin(2x)

you'd be wrong

because d/dx sin(2x) = 2cos(2x)

[faint monolith]

but at first how do you know its related to sec x after intergration

[magic sparrow]

However the idea is the same. You see in this example cos(x) in the integrand, and you're like, hey, I know that's the derivative of sine, so I know what the antiderivative should be, aslong as I sort out the multiplicative constants due to the chain rule

In your problem, your integrand is 4sec(2x)tan(2x)

and if you know your trig derivatives

that should look awfully similar to

d/dx sec(x) = sec(x)tan(x)

[faint monolith]

oh shit

[magic sparrow]

you just have to sort out the constants, due to the chain rule

[faint monolith]

i didnt memorise this

[magic sparrow]

you can always rederive it using the quotient rule, as sec(x)=1/cos(x). However, it's an easy one to remember

[faint monolith]

as in i cant directly tell the pattern whether should i change to tan2x

[magic sparrow]

well of course you should

it's simpler that way

[faint monolith]

also why cant i use this

[magic sparrow]

well the first question I have is

why would you want to use that, when the answer without it is much easier

but the reason you cannot use it

is the denominator is cos^(2)(x)

[faint monolith]

ik the answer is 2sec2x after checking marking scheme

[magic sparrow]

so to use that formula

the numerator must be d/dx cos^(2)(x)

but that is not sin(x)

so it doesn't apply

[faint monolith]

ok wait

i write down and take a pic

[magic sparrow]

see how the numerator is the derivative of the denominator

it isn't in your problem

so the rule doesn't apply

[faint monolith]

,,rw

,rotate

[magic sparrow]

Are you listening to what I'm saying

I don't need a picture, no offense, your formula doesn't apply here

Read what I said above about the numerator being the derivative of the denominator

in your problem

that isn't true

so you can't use the formula

it isn't just saying the integral of any random fraction of functions is = ...

it's saying if the numerator, is the derivative of the denominator, then the integral is = ...

[faint monolith]

but i multiple a 1/cos2x outside to make it the same tho

[magic sparrow]

you cannot do that

[faint monolith]

only constants?

[magic sparrow]

yes

Again, I have no idea why you would want to even try it this way

if the question was integrate cos(x)

you'd just say sin(x)+C

[faint monolith]

cuz i didnt memorise d/dx secx so i have no idea how to approach lmao-

[magic sparrow]

Okay but now I've told you many times

and you know the approach

that's what you should use

and if not, use substitution

[faint monolith]

yep

ok thanks alot g

[magic sparrow]

No problem

[faint monolith]

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[sand heron]

@old kettle, the answer for the simplification question is 8x².

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[feral siren]

where did the p come fron?

[feral siren]

the sin p and p cos

[fast heart]

P is the force pushing the mass up the slope

[feral siren]

ok ty

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[wary blade]

what is a differential coefficient

is it dy/dx ?

[marble agate]

i think it is

just go with quotient rule

[spring haven]

jesus christ what the hell is that font

does that say $\frac{\sqrt{x}}{\cos(x)}$

[soft zealot]

kheerii

[wind glen]

ye

[steel bolt]

Yes

I did quotient rule but my teacher said it was wrong

(the answer was wrong)

[wary blade]

can you show your work

[steel bolt]

@steel bolt

[wary blade]

derivative of sqrt(x) is not x^-1/2

remember power rule

[steel bolt]

Updated version

@wary bladecan u solve it becz im tired after 40 minutes trying 😪

[wary blade]

@steel bolt is this the right answer?

[steel bolt]

How to simplify it like that?

Is it like that?

@wary blade

<@&286206848099549185>

I want to simplify it like this

|||
|||
V

simplify first equation to the second down it

[jade mason]

What part are you stuck on?

[soft zealot]

Niklaus

[final saddle]

@steel bolt Has your question been resolved?

[steel bolt]

Just I want this to be like this

[steel bolt]

.

[jade mason]

Yea this is the second term

[steel bolt]

.

[soft zealot]

Niklaus

Niklaus

Niklaus

[jade mason]

You cant ignore that 1+ in front of 2xtanx

Or combine it without first getting a common denominator

[steel bolt]

ok then how did we get that 1?

[jade mason]

Its in your original Q... looks like this

[soft zealot]

Niklaus

[jade mason]

This can also be written

[soft zealot]

Niklaus

[steel bolt]

How you said cos/cos

It is for the left side

why did you add it to the right side here

[jade mason]

Because cos/cos = 1

[steel bolt]

no..

I want to know why,

you put in the nomenator cos

[jade mason]

Because this....

[soft zealot]

Niklaus

[jade mason]

Which is the same as

[soft zealot]

Niklaus

[steel bolt]

So this is incorrect?

🫥

[jade mason]

No that technically correct

Why did you split it back out?

[steel bolt]

to remove the cos and add tan

Here why u add cosin in nominator

[jade mason]

Right but it reduces back down to your original question

[steel bolt]

uh

Did you take common factor

??

like cos(x) (1 + 2xsinx)

@jade mason

[jade mason]

From the top, down to the red arrow, what are you trying to show. It is correct but where are you trying to get to after that top equation?

[steel bolt]

The cosin from left how it got removed and replaced in the place that the red arrow indicates?

From that to 1/2square root

[jade mason]

You multiple accross

1 times anything it itself

So for the numerator you are left with cos x + 2xsinx

And then the denominator is where you get cos x squared

2sqrt(x) cos x squared

[steel bolt]

oh now i got it

thanks teacher

[jade mason]

Right. Multiplying fractions versus adding fractions.

[steel bolt]

If I did like this how will i remove the cosin on the right side denonirator

[jade mason]

From you paper?

Itll turn back into tan

And youll still be left with one cos

You would get

[soft zealot]

Niklaus

[jade mason]

And those 2's would cancel

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[eager obsidian]

hi

[eager obsidian]

index notation and inequalities ig

$ 52 $

@spring depot

<@&286206848099549185>

oops

[wild rover]

$52$

[soft zealot]

amw

[pliant shore]

Yep, so do you know what the last digit could be if you have to round down?

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[feral siren]

i thought the co-efficient of friction was 0.5 not 0.15

[feral siren]

why is it different inthe answers?

is it a mistake are am i missing something?

[faint monolith]

Is it ur coefficient

?

Why is it 0.15 Instead of 0.5

I think the answer might be wrong

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[wide fog]

I saw this primary school/grade 7 algebra question from china (bilibili) and was wondering how they derived the answer that
x = 10^(1/5) for the question x^x^5 = 100.
i tried lambert W function and i was only be able to get
(e^W0(5ln(100)))^(1/5) assuming x >= -1 and i know there might be more solutions

[thorn shell]

do you have a screenshot

[wide fog]

screenshot of the question or my solution?

the chinese title for this video says Solving Equations Cleverly for Junior High School Competition Problems
according to google translate

[thorn shell]

yeah yeah I got it

[wide fog]

these are the solutions from wolfram alpha which is pretty similar to my answer

[thorn shell]

feel like there has to something far simple you're missing

(x^x)^5 = 100

x^x = 100^1/5

x = sqrt 100^1/5
x = 10^1/5

@wide fog this is how they got it

[wide fog]

that is what i thought before but the problem is that x^x^5 != (x^x)^5
think 2^2^5 != 4^5

x^x^5 is x^(x^5)

[thorn shell]

can you not factor that differently

[wide fog]

i dont think so

but we can take ln() on both side

[wide fog]

which is why we cant do what you did for step 1 i think of (x^x)^5 = 100

[thorn shell]

maybe take the log base 10

[wide fog]

care to show your thoughts/steps?

since it is a grade 7-9 question, i think we can ignore the complex branch of the lambert w function

[thorn shell]

logbase10 would get us something like

nah it doesnt work

take the ln

would get us something like x^5 ln x = ln 100

[wide fog]

i did ln and substitution and got the result in terms of the principal branch of lambert w function

[thorn shell]

sometimes i am pretty certain the chinese are lying

[wide fog]

why? are you interested in the original video link?

https://www.bilibili.com/video/BV1Gt421w7ra/?spm_id_from=333.880.my_history.page.click&vd_source=0848ba4644fc9798011c6237f8f36ecc

[thorn shell]

i mean

not to be that guy, but i am not stupid

you dont seem stupid

yet we cant solve 7th grade algebra?

[wide fog]

i know, id probably fail math if i were in china (im currently doing my master degree but not in maths)

did you check out their solutions?

i think they used multiple coincidence that about the number 100

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[fallen rune]

(College Math) I already moved the 9 to the other side with the x then i ^2'd oth sides to take the square root out what should i do next

[tranquil pine]

you would have a quadratic equation, if im right

not in college but wouldnt that give you 2x+26= x²+81?

then would be -x² +2x-55

and you could use quadratic formula to find the roots, aka solution set

sorry not 26

thats really it

really hard to comprehend if u start like that though

so i understand ur confusion

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[sinful flume]

k

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[sinful flume]

.reopen

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[old hornet]

Given a connected graph G with n vertices and positive edge weights, how would one find a connected subgraph of G with n-1 vertices that has minimum total cost?

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[gentle trout]

i got 2ab+2b

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[young verge]

How do i do this

[shadow aspen]

Do you know what a critical point is?

[young verge]

when the derivative = 0 or undef

I have hte derivative down

its just the algebraic manipulation im having trouble wiht

[shadow aspen]

Alright what did u get for the derivative

[young verge]

xe^(kx) ( 2 + xk)

that is after factoring

[shadow aspen]

Yeah that is good

Use the fact that if a*b=0 then a=0 or b=0

[young verge]

so if x = 0

the whole thing is 0

[shadow aspen]

Well x is given as 2/3

So you are finding a value of k such that xe^(kx)(2+xk)=0

What happens when (2+xk)=0?

[young verge]

would k be -3

or wdym by that

[shadow aspen]

Yeah

[young verge]

alr makes sense

thanks

[shadow aspen]

np

[young verge]

you got any tips for the ap calc exma

[shadow aspen]

Nah idk whats on it

But prob just look for practice exams for it and do those problems till your confident

[young verge]

do u have any practice exams

[shadow aspen]

Looks like college board website has some past exam questions

[young verge]

arent they all frqs tho

[shadow aspen]

Wym by frqs

[young verge]

free response questions only

i tried looking for mcqs

but i cant find any

besides the 2012

[shadow aspen]

U taking ab or bc?

[young verge]

bc

[shadow aspen]

Hmm yeah I can’t really find any but I don’t think freq and mcq should be too different. I think studying the freq should be just as good

Ive never taken an AP test though so idk for sure the best way to study for it

[young verge]

alright sounds good

thanks

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[lost drift]

Hi, Got this from Gilbert Strang Differential Equations and Linear Algebra. I am unsure of how he got this solution, in particular where did the highlighted term come from?

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[maiden lark]

I don't understand why: if two experiments are identical (same outcomes, same R.V.) and independent (i.e. the outcomes of one do not affect the outcomes of the other), then the probabilities of the outcomes of experiment one are the same as the probabilities of the outcomes of experiment two.

My father tells me it's evident but I couldn't find a proof

Thanks in advance.

[maiden lark]

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[tranquil pine]

One of my friend shared me this question from calculus... i tried solving it and tried eliminating a complex number i. can you tell me what to do after that?

[late yarrow]

.

[tranquil pine]

this is how i went

divided sqrt(x squared -1) numerator and denominator
i^87 -> i^86 * i
but i^2=-1 therefor i squared raised to power 43 = -1
therefor -1*i=-i and minus minus cancel

sqrt(-1)/(sqrt(1-x^2)
thats how i removed i
after that idk how to do i am thinking

[fiery bluff]

I'm seriously considering a x=sin u substitution

alternatively I'll try differentiating that numerator to see what I get

[tranquil pine]

i thought similar but i am not very good in calculus (advanced level) so i thought it might go complex

[fiery bluff]

looks like we might be able to recognise this as something of the form $f'(x)/f(x)$, but I'm not sure

[soft zealot]

Element118

[tranquil pine]

on differentiating num i got $e^{sin(x)}(1+x cos x)$

[soft zealot]

de.p_ele.dele.deleted_usr._1234

[fiery bluff]

hmm, maybe that's not very promising...

kinda just want to pull out the complex numbers and see what happens

still looks bad

[tranquil pine]

i tried pulling out (i mean eliminatng i) refer my approach (3 steps)

[tranquil pine]

the red one is final integral

[tranquil pine]

@fiery bluff just in case you missed

[fiery bluff]

maybe you can try inputting to wolfram alpha, if there's an easy solution it should find

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[tranquil pine]

am closing it, thanks

.close

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[old bramble]

Could anyone help me understand how I got from the first step to the second one?
I understand the chain rule is being used but somehow I don't know how they got there.

[solid cobalt]

What is the derivative of kx ?

In this case k=ln(a)

It is not the chain rule.

[old bramble]

with respect to x?

[solid cobalt]

Then, what is the definition of ln(a)?

[solid cobalt]

yes

[old bramble]

$\dv{x} kx = 1 \cdot \dv{k}{x}$

[solid cobalt]

What?

[old bramble]

no

[solid cobalt]

k is constant

[old bramble]

well I would just use the product rule probably

[solid cobalt]

What is the derivative of 2x with respect to x?

[tranquil pine]

is this derivative of $a^x$?

[old bramble]

2

[solid cobalt]

So what is the derivative of ln(a)x?

[old bramble]

ln(a)

[soft zealot]

de.p_ele.dele.deleted_usr._1234

[solid cobalt]

Ok so you have one factor of the second step

Now think why e^(xln(a)) equals a^x

If you don't get it, let me know

[old bramble]

well, its about what power do I need to raise e to get a to the power of x
which is a^x

[solid cobalt]

e^(xln(a)) = e^(ln(a^x))=a^x

OK

so a^xln(a)

[old bramble]

I suppose this is the main takeaway

[solid cobalt]

I don't know what a takaway is

sorry

[old bramble]

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[blazing pewter]

Correct ?

[blazing pewter]

Ad - cb = non zero

Xy = 0

:<

<@&286206848099549185>

Someone!!!!!!

😦

😭

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[lunar grove]

Just wanna check if my question 3 is correct or not

[lunar grove]

i got 5 and my friend says its 3

anyone?

[nova iron]

what does g^2 (-2) mean?

composition?

also isn't that like pi?

[lunar grove]

replace the x with -2

[nova iron]

no idea why you're doing this

[lunar grove]

my friends solution which is right

[nova iron]

but... the question has $g(x) = 1 - \frac {\pi}{2}$ not $g(x) = 1 - \frac x2$

[lunar grove]

yeahhh

we do it like this

[soft zealot]

nyxie9151

[lunar grove]

if ur using ur method what answer do u get tho

[nova iron]

????

there is no method here

this is wrong because you can't just do that

it's 1- pi/2 not 1- x/2

your friend is also wrong cuz he just changed the question lmao

where does it say g(x) = that?

[lunar grove]

the question

was right

its correct

[lunar grove]

its not pie

[nova iron]

the question is g(x) = 1- pi/2

[lunar grove]

its x

[nova iron]

:huh

[lunar grove]

its x

[nova iron]

it wrote pi

[lunar grove]

no

its x

[nova iron]

take a clearer picture

[lunar grove]

ok

bet

[nova iron]

that's for sure a pi 😭 i refuse to believe that's an x

[lunar grove]

alr

[nova iron]

take a clearer picture yeah and post it

only that part is okay

[lunar grove]

[nova iron]

bruh

i'm blind

anyway

how did you just move the 1 into the numerator

[lunar grove]

oh shit

i moved wrongly after i saw my friends answ

[lunar grove]

.

this was before

[nova iron]

,w g(x) = 1-x/2, find g(g(x))

[soft zealot]

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Requested by nyxie9151

nyxie9151

[lunar grove]

i got it

its 3

i did some mistakes

[final saddle]

@lunar grove Has your question been resolved?

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[cobalt mango]

how do i solve this differential equation? this is example given

[final saddle]

@cobalt mango Has your question been resolved?

[pliant shore]

Go watch a video if you don't know how they came to m^2 - 1 = 0

They literally tell you to sub in y = Ae^(mx)

If you do that for your quesiton, you get m^2 + 1 = 0 ahhhhh so it has imaginary roots and that's probably where you're stuck

Yes so recall you can take any linear combination

So $e^{ix} = \cos x + i \sin x$ and $e^{-ix} = \cos(-x) + i \sin(-x) = \cos x - i \sin x$

[soft zealot]

south

[pliant shore]

Add these two and you get 2 cos x, which is a multiple of cos x

Subtract and you get 2i sin x, which is a multiple of sin x

So a general solution can also be expressed in the form of these 2 linearly independent solutions

$A \cos x + B \sin x$

[soft zealot]

south

[pliant shore]

https://www.youtube.com/watch?v=u-eQaef1EWw

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@cobalt mango Has your question been resolved?

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[flint wind]

Help me explain where h comes from and what all this means graphically

[final tangle]

several resources have nice diagrams for this

iirc wikipedia has an animation

[lucid marsh]

^

If you want a quick explanation, this is basically calculating the average rate of change over smaller and smaller intervals

[flint wind]

Thank you

.close

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[final saddle]

Please don't occupy multiple help channels.

[hybrid heath]

please stay in one channel

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[worthy furnace]

I lost the other one sorry ><

[hybrid heath]

#help-28

[worthy furnace]

Thank uu

[tulip coyote]

.reopen

[final saddle]

✅

[tulip coyote]

Oh you literally made another one the same second

.close

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[ancient monolith]

can somone help me with cartesians

[ancient monolith]

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[static plinth]

which part are you stuck on, a or b?

[narrow shuttle]

@neon yacht

[static plinth]

so basically we need to use normalcdf

do you want to do it with a calculator or a normal distribution table?

[narrow shuttle]

you can use ,w normalcdf btw

,w normalcdf(0, 0.5, 0, 1

[soft zealot]

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Requested by praneel.g

[static plinth]

so that’s the empirical rule

not exactly using the normal distribution table

[narrow shuttle]

68-95-99.7

[static plinth]

empirical rule is just used to approximate things

mm what class is this for

i’m guessing it’s algebra 2

i remember statistics is at the very end of that

they don’t really go in depth with statistics i suppose

i see

so you have only learned the empirical rule?

and using your calculator yeah?

[narrow shuttle]

graphing calculator?

[static plinth]

alright so look for something called normalcdf

maybe go into the stat section first

yep that’s it

so the mean and the standard deviation is just what we’re given up there

and the lower and upper bound is the area/probability of interest

so it says that we want to find the probability of getting something less than 250

we essentially need to find the area/probability under the curve from negative infinity to 250

we can’t really put in negative infinity into our calculator

so just spam a bunch of 9’s and it’ll do the trick

does it tell you what each parameter there corresponds with?

it might be different with each calculator

alright so let’s try it then

the lower would be something like -999999

upper would be 250

and then the mean and standard deviation

mm i got around 0.933 when i did it on desmos

alright good

so that’ll be the answer to the first one

yep that’s it

so do you know what a z score is?

$z=\frac{\bar{x}-\mu}{\sigma}$

[soft zealot]

y0shi

[static plinth]

x bar is the number that we have

mu is the mean

and sigma is the standard deviation

essentially it’s a measure of how many standard deviations away a number is from the mean

well so

we don’t know what the mean is

it says that if we have a value of 250

the probability of anything less than that is 80%

we essentially need to find what z score aligns with that 80%

ik you might not be familiar with this but i’ll show you what a normal distribution table is

we kinda need that for this

cuz i’m not sure if your calculator can calculate this without the mean

so the z score is the column and the rows

so let’s say we have a row of 0.2 and a column of 0.01, that’ll correspond with 0.21

basically we want to look for the z score that gives us 80%

it says we want the probability to be 0.8

z is the z score

[static plinth]

this

yep

yep

there’s probably not going to be a number that is exactly 0.8

but find the ones closest to it

yeah and if you go one lower, it’s 0.7995

yeah so what z score corresponds with that?

0.8 yes but also need the hundredths place

that is given at the very top

yep so 0.84 is our z score