help-23

[keen coral]

I can whip a nice 3 step proof for the above statement

No

Look

2^(2n)=(2^2)^n=4^n

Now

2^(2n+1)=2^(2n) * 2^1=4^n * 2

Throw that 2 out of the summation

Now you have

$2*\sum_{n=1}^infy (\dfrac{4}{5})^n$

$ 2*\sum_(n=1)^oo (\dfrac{4}{5})^n$

https://cdn.discordapp.com/emojis/1145673226505179218.png?v=1&size=64&quality=lossless

[flat frigate]

Ericsson

[supple bramble]

okay thank you i got it

I was just confused on that exponet

[keen coral]

$2*\sum_(n=1)^oo (\dfrac{4}{5})^n$

[flat frigate]

Ericsson

[keen coral]

https://cdn.discordapp.com/emojis/1017709558065532948.png?v=1&size=64&quality=lossless

[supple bramble]

thank you very much 🙂

[keen coral]

How do you even write infinity

[supple bramble]

.close

[safe radish]

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[keen coral]

$2*\sum_{n=1}^{infty} (\dfrac{4}{5})^n$

[empty gyro]

$\infty$

[flat frigate]

SWR

[keen coral]

Ohh

Forgot the brackets

[empty gyro]

$\sum_{n=1}^{\infty} f(n)$

[flat frigate]

SWR

Ericsson

[keen coral]

And the backslash

[peak estuary]

you dont need brackets for the \infty

[keen coral]

I see i see

Thank you

[safe radish]

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[wooden oyster]

i have to find the eigenvalues, but this doesnt hae any solutions

[plucky elk]

No real solutions correct

[wooden oyster]

this has no solutons, right

[minor goblet]

it has complex solutions

[plucky elk]

Solutions to what question

[wooden oyster]

• Find all eigenvalues and associated eigenvectors for the matrix

the eigen vector is just where you put it in the vector

[plucky elk]

This doesn't say the eigenvalues have to be real

[wooden oyster]

what do you do it if isnt real

[plucky elk]

Nothing different

[wooden oyster]

the book says that complex solutions only make sense if you need complex scalars for your vectors, which we don't have

[plucky elk]

which we don't have
Wot

Did you not learn complex numbers?

[wooden oyster]

i have

so the solution is just 2i and -2i

[plucky elk]

Yes these are complex numbers

[plucky elk]

Which are the roots to
lambda^2 + 4 = 0

[wooden oyster]

yh yh

ok,

[safe radish]

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[umbral swan]

would like to check whether I approached a task correctly:

[umbral swan]

we're given a sequence and are supposed to find its limit as well as a starting index n0 for a given eps>0, such that for all n>n0 the absolute difference from the limit is less than eps

for limit I'd do:

and for finding the n0 I did:

so for eps > 1 one can choose n0 = 1, since the inequality is fulfilled that way. Let's assume eps <= 1

then I apply pq-formula:

yielding

the "oder" is german for "or"

can be chosen that way, fulfilling the inequality

every step was simple, just the procedure felt a bit long, am I missing something to solve it quicker?

[peak estuary]

without actually wanting to really look through this, usually its better to do lots of bounding steps along the way which make stuff easier

[umbral swan]

hm would you apply such a limitation right at the start?

[peak estuary]

as an example, note that 1-sqrt(n^2+7)/(n+7) is certainly smaller than just 1-1/(n+7)

[umbral swan]

yes, will try to do an optimized version

ah forgot to close

@peak estuary here in case

and thank thee 🦇

.close

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[vivid fable]

a

[vivid fable]

how do I figure this out?

[umbral swan]

how far are you

[vivid fable]

legit nowhere

I don't know how to do this

[umbral swan]

an elephant receives 1/2 = 0.5 barrels

a buffalo receives 2/3 the amount that an elephant receives

can you thereby calculate the amount a buffalo receives?

[vivid fable]

how do you know thats 0.5 barrels?

[umbral swan]

1/2 = 0.5 is the same

[gleaming coral]

1/2=0.5 its half

[vivid fable]

ohhh

[umbral swan]

1/4 = 0.25

3/2 = 1.5

[vivid fable]

.

uh

okay

[umbral swan]

regular fractions

[vivid fable]

how would I calculate it?

[vivid fable]

I never really learned fractions 💀

[umbral swan]

well if instead I gave person A 8 cookies

and I give person B half the amount I gave person A

how much did I give B?

[vivid fable]

umm

3

[gleaming coral]

How

Do

You got three

[umbral swan]

are you trolling, mistyping or confused

[vivid fable]

no i really dont know im sorry 😭

I guessed because I dont know the middle for 8

[gleaming coral]

You dont know what the half means?

[vivid fable]

no like

[umbral swan]

4 + 4 = 8

[vivid fable]

I dont know the half of eight

[umbral swan]

8 / 2 = 4

[vivid fable]

OH

ohhhh i get it now

[umbral swan]

and here you can see the suggested way to calculate

I first gave 8

and then I gave half: 8 * 1/2

= 4

if I instead only gave 1/4 of the original amount of cookies to person B

[vivid fable]

that would be uhh

[umbral swan]

then B would receive 8 * 1/4 = 2 cookies

[vivid fable]

oh

[umbral swan]

since 8/4 = 2 <----> 2*4 = 8

now your original question was

elephants receive 1/2 barrels

buffalos receive 2/3 the amount the elephants receive

[vivid fable]

so

um

1/2 is 0.5

so the buffalo gets 0.3?

[umbral swan]

intuitively on the right path, it may be better to stick with fractions in this case though

can you show how you tried to calculate 0.3?

[vivid fable]

...

I guessed, I don't know how to calculate it.

[umbral swan]

well before with the Person A/B example we multiplied by the fraction

[vivid fable]

it just made sense in my head

[umbral swan]

A gets 8, B gets half of what A gets ====> B gets 8 * 1/2, since 1/2 = a half

does that part remain unclear

[vivid fable]

no no i get that

[umbral swan]

kk, so now we just have different numbers:

elephants get 1/2, buffalos get 2/3 of what elephants get ====> ?

= 1/2 * 2/3

🐣

[vivid fable]

uhh

so

half

0.5

[safe radish]

Channel closed

The owner is missing!

[vivid fable]

WHAT

NOOO

um

[hardy lion]

Oh no

[umbral swan]

huh don't know that message either

[vivid fable]

so 0.5

0.25

[umbral swan]

careful, 1/2 * 2/3 = (1*2)/(2*3)

= 1/3

[vivid fable]

oh

[umbral swan]

if you're unfamiliar with fractions then this task may not be suitable

[vivid fable]

yeah I dont rlly know fractions

[safe radish]

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[loud shell]

simple question

log_10(f(x)) or f(log_10x)

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[hoary portal]

I don't have the vocabulary to descrive what I'm asking

[hoary portal]

I'm trying to get a function that describes a series of numbers I got from a process

It's like pythagorian triples but in all dimensions

how many pythagorean doubles, triples, quadruples, etc. result in x and how do I get a function for this?

For 1 there's 1 way, for 2 theres 2 ways, for 3 theres 4 ways, for 4 theres 8 ways but for 5 theres 19 then 6 is 43 then 7 is 98

I want to know why

also how to do it faster because brute forcing it took an hour

this is the work i did

<@&286206848099549185>

please someone have the answer

well I gtg ill be back in 2 hours i hope this gets answered

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[azure coyote]

help i dont understand these at all and i need the solutions

[torpid wasp]

https://discord.gg/s3jV7FzQ

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[azure coyote]

<@&286206848099549185>

[wise shell]

it looks like basically finding the volume of the small cone and subtracting it from the large to find the frustrum. the equation for the first is (1/3pi6^2(33))-(1/3pi2^2(33-22)

so it should be about 1198

[azure coyote]

tysm

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[peak estuary]

no wait

that would be to infty

well its just 1/0

or rather, 1/0^+

which is +infty

[white girder]

Was lhopital not used here because of 0+?

Instead of just 0

[peak estuary]

its not of the form 0/0 or infty/infty

thats when you use lhopital

[gritty hearth]

lhopital is for 0/0

[white girder]

Oh

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[desert pasture]

could somone help with this?

[desert pasture]

I'm having trouble proving b is true

[grand kraken]

Have you tried where f(t) = 0 or 1

[desert pasture]

no, will try that , Thanks

[desert pasture]

I did think of that, but I don't see how that helps

[bitter spear]

Wait how can f(t)=0 as its range is from (0,1) maybe you are talking about the limiting case

[stiff cradle]

Case b : g(0)=-int(0,pi/2) f(t) cos(t) dt , f(t)>0 , cos(t)>0 on ]0,pi/2[ , so g(0) <0

Pi/2-1<1 , so g(1)=1-A>0 , with A<1 (still because f(t)<1 and cos(t)<1.

g is continous , g(0)<0 and g(1)>0 so there is at least 1 point such that g(x)=0 ?

[desert pasture]

hmm, thanks!

.close

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[carmine panther]

how did they get -5

[quiet plume]

Just follow their explanation. At the last step, remember that a fraction a/b is equal to 0 only when a=0

[light shoal]

the fraction is zero if the numerator is zero

[carmine panther]

.close

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[last heath]

,, h^2\left(\frac{1}{\left(tan20:\right)^2}+\frac{1}{\left(tan:18\right)^2}\right)=100^2

[flat frigate]

Akira (Snowfall)

[last heath]

how do you solve this using a calculator?

what?

oh you're sending this to every channel

<@&268886789983436800>

[plucky elk]

divide by the term in parentheses

Then square root both sides

[last heath]

when dividing the parentheses I get 2.56822, is that right?

doing the calculation slowly cuz I have trust issue with my calculations

im stuck here

[plucky elk]

You're doing subtraction here?

[last heath]

yes

[plucky elk]

Use the calculator here

,calc tan(45 deg)

[flat frigate]

Result:

1

[plucky elk]

Read

[last heath]

it's tan 20

,calc tan(20 deg)

[flat frigate]

Result:

0.3639702342662

[last heath]

I did divide 1/tan20^2 to get 1.1917

and same thing for 1/tan18^2

[plucky elk]

Read the whole sentence

[last heath]

oh

divide by tan 20 + tan 18?

[plucky elk]

h^2 * (...) = 100

Divide by (...)

[last heath]

so 100 divide by (...)?

im doing it right or still wrong?

[plucky elk]

Algebra looks right. Didn't check your calculator values tho

[last heath]

I did but im getting 4.12562

hold on

ya I got it now

I did divide 100^2/17.02076, which is 24.2m

thanks for the help

.close

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[frank valley]

I'm working an exercise where I'm supposed to determine whether or not there exists numbers $a_k$ and a number $R>0$ such that $$\sum_{k=0}^\infty a_kx^k=\begin{cases} x^2 & \text{ when }0<x<R, \ -x^2 &\text{ when } -R<x<0.\end{cases}$$ My strategy so far has been to determine whether or not the function is smooth, since an analytic function is smooth. I'm unsure about the function above, since even though it could be smooth, it need not imply analyticity.

[flat frigate]

Philip

[drowsy karma]

If the radius of convergence of a power series with center at x = 0 is positive, then the power series is absolutely convergent for |x| < R
Think about the implications of this statement in the context of your idea

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[low vector]

!occupied

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

[cedar harbor]

use log

[dry jolt]

I just started logarithms I'm stuck at this

[fleet condor]

this is not your channel, open a new one and delete your messages here

[dry jolt]

aight man

[frank valley]

not sure what you mean, I was thinking of taking the derivative of the function and observing that it's not differentiable twice at x=0, hence not smooth --> not analytic

.close

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[frank valley]

According to WolframAlpha, cos(sqrt(-x)) is the same as cosh(sqrt(x)), however, when I graph them in Desmos, they do not look very similiar to me. Would anyone care to explain this more? Thanks.

[rich sinew]

A sqrt cannot be negative

So negative times negative is positive

[severe pond]

it’s the same just the restriction on the domain of sqrt-x

[rich sinew]

So all the negative x values are positive for sqrt -x

And Vice verses for sqrt x

Actually nvm

[peak estuary]

well desmos doesnt know complex numbers. which you kinda need for this conversion

it sees sqrt and a negative number and then doesnt plot it

[frank valley]

hmm, could you elaborate this a bit more?

[frank valley]

on which domain do the two functions that I plotted agree?

[peak estuary]

well if you allow sqrt(negative) then on all of R. otherwise, only {0}

[frank valley]

hmm, but how can we allow sqrt(negative) on the reals? that impossible...

[drowsy karma]

if x<0, then -x>0

why you are afraid here ?

and if -x>0, then x <0,

both funcitons of yours have only one common point x = 0

[frank valley]

This is from WolframAlpha.

[drowsy karma]

wolfram is not mathmetician, forgive me

neither desmos is

[frank valley]

true 🙂

[plucky elk]

Wolfram is using the complex definition of cosine, which you haven't learned yet

[frank valley]

ok, so is it actually true that cos(sqrt(-x))=cosh(sqrt(x))?

[severe pond]

for reals it’s only x=0

[drowsy karma]

for real x, only at x = 0

[frank valley]

ah ok

[drowsy karma]

lets leave complex numbers scenario in my opinion

[frank valley]

the reason for my question is, I'm given the function $$f(x)=\begin{cases} \cosh\sqrt{x} & \text{ when }0<x<R, \ \cos\sqrt{-x} &\text{ when } -R<x<0.\end{cases}$$ and I'm supposed to determine whether or not it has a power series expansion for some $R>0$.

[flat frigate]

Philip

[safe radish]

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[frank valley]

.close

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[desert pasture]

is there any pretty way to solve this? Of course one can solve it by equating it to an arbitrary integer x and then solve it, but is there any more elegant way

[marsh walrus]

its a factoring problem right

[desert pasture]

I was thinking I would have to multiply and divide by (7+3icos(\theta)) and then find theta

[marsh walrus]

well you can write equations

[desert pasture]

by equating the imaginary part to 0

[upper rivet]

yeah I cant really think of any other way

[marsh walrus]

theyre related y a single variable

1967 = k * 7

and the same for the imaginary part

but i think doing it by factoring is better

[desert pasture]

wait, why?

[marsh walrus]

because $\frac ab \in \mathbb N$

[flat frigate]

jan Niku

[marsh walrus]

so $a = nb$

[flat frigate]

jan Niku

[marsh walrus]

this splits into a real and an imaginary equation

[desert pasture]

Yeah, got that thanks, but is there no elegant way to approach this?

[marsh walrus]

im thinking through it too

,w pfactor 1967

[flat frigate]

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Requested by janniku

[marsh walrus]

yea so, thats the only value

well, the only candidate

[royal horizon]

I got cot θ = -2 or tan θ = -1/2 from the imaginary equation

[marsh walrus]

you have to check theres a way to make the imaginary parts equal

but otherwise n can only be 281, i think

[desert pasture]

I see. Thanks!

[marsh walrus]

dont suppose you have the answer

[upper rivet]

not really a neat value huh

[desert pasture]

Not from the official source no, this is from the 2023 JEE adv paper

[marsh walrus]

,w Solve[1686 Sin [x] == 281 * (-3) * Cos[x], x]

[desert pasture]

but there are sites out there with answers, yes

[flat frigate]

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Requested by janniku

[marsh walrus]

perfect, yea

so only 281

thats pretty elegant i think

[upper rivet]

damn

yeah 281 is right

[desert pasture]

huh, that was an elegant way , you didn't have to solve any messy equations

thanks

I'm going to screenshot this, have to think about this a bit more

thanks!

[marsh walrus]

all you really need is that it splits into two equations i think

[desert pasture]

yeah, pretty much, I'm still not sure how you got this though

what happened to the denominator

[marsh walrus]

its just how complex equations work

like a+ib = c+id

this means that a = b and c = d

because they cant contribute to each other, the imaginary and real parts

they have be equal separately

[desert pasture]

yeah, got it. Thanks!

.close

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[lavish belfry]

Please help me with the solution of the given limit function. explain how to solve it

[safe radish]

Please don't occupy multiple help channels.

[strong token]

If im not wrong its 1 xd

start with dividing by x

[lavish belfry]

xD nah, buddy, you have to solve through the second marvelous limit, only I don't understand how.

[strong token]

the answer is 1?

[lavish belfry]

nope, answer is e

[strong token]

ahh

u need to use the formula then

[desert pasture]

1^infty form ,, there's a formula for that

[strong token]

you know the formula?

[split ether]

You should get it to the form similar to (1 + 1/t)^t where t approaches infinity

[desert pasture]

yeah. Let $f(x)^{g(x)}$ tend to 1^infty at any point t

[split ether]

Automatically assuming it's e is not safe btw

[lavish belfry]

1 at infinity is an uncertainty, it doesn't work.

[flat frigate]

Why am. I here
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[desert pasture]

then the limit is $e^{g(x)(f(x)-1)$

where teh limit is evaluvated at x=t

[split ether]

[flat frigate]

Why am. I here
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[split ether]

I feel like there should be ln somewhere

[desert pasture]

I don't think so

this is the proof https://math.stackexchange.com/questions/1987215/1-to-the-power-of-infinity-formula

[desert pasture]

that's in the derivation of the formula

[lavish belfry]

did i hear you right?

[desert pasture]

yeah. now identify f(x) and g(x)

[desert pasture]

Hope this isn't an exam problem

or anythig like that

[lavish belfry]

oh, thank you bro, thank you all

[safe radish]

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[brazen helm]

find such a positive integer n for which n! ~ 10^9

[quasi bison]

what does ~ mean?

[brazen helm]

approx.

I thought of using sterling, but cant really solve the log equation

[quasi bison]

this feels imprecise

maybe you want to find n such that |n! - 10^9| is minimized?

[brazen helm]

sounds easier, the hint in the question says to use e

[quasi bison]

!xy

[safe radish]

Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.

[brazen helm]

yeah

ok great

[quasi bison]

.

[brazen helm]

wont help

[quasi bison]

why

[brazen helm]

its in a different language and all it says is to find a number n for which when you take n locomotives you can premute them at-least in 10^9 different ways

[quasi bison]

If the original problem is not in English, then post it anyway!

send it anyway even if you think it will not help.

[brazen helm]

there isnt a single number or letter

also im on a laptop so hard to take a photo

[peak estuary]

screenshot

windows snipping tool

[brazen helm]

please trust me when I say you have to find a positive integer n for whcih n! is closest to 10^9

[peak estuary]

well, closest in what sense

[brazen helm]

That when taking n objects you have atleast 10^9 premutations for the smallest possible n

[peak estuary]

well thats a different question

that doesnt ask closest, it asks at least

[brazen helm]

Well, but if you have a smaller choice you are supposed to take it

[peak estuary]

well 12! is the closest to 10^9 but its smaller

[brazen helm]

Yeah you are supposed to do this by hand, and I dont know how

[peak estuary]

10^9 is small

you can easily do this by hand

just calculating all the factorials until 13

[brazen helm]

Without a calculator?

.close

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[lean otter]

in this question, i got x^2 - 8x + 3 = y

but the answer was different from the mark scheme. however i dont think i did anything wrong.

[lean otter]

the method that i used

1. (x-2)(x-6) = y

when x = 0, y has to be 3
12 +z = 3
z = -9
(x-2)(x-6)-9 = y
x^2 - 8x + 3 = y

[torpid fable]

try plugging (2,0) or (6, 0) back into (x-2)(x-6)-9=y

[lean otter]

oh wait will the root change?

oh it does..

[torpid fable]

When you add a number at the end, it ruins the point of factored form (which is that everything should be multiplication)

[lean otter]

so how do i do this question?

i need to make y intercept = 3

[torpid fable]

The general factored form is a(x-b)(x-c)...=y

so instead of adding a value to account for the 3rd point, we multiply by a value (a)

[lean otter]

ohhhhhhhhhh so multiply by 1/4??

cuz y intercept has to be 3

and when we expand the bracket we get 12 for y intercept

[torpid fable]

yeah sounds right

[lean otter]

tyyy

merry christmas

[torpid fable]

thx <3 u too

[lean otter]

.close

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[lean otter]

c=3 if x = 0
and
a(x-2)(x-6)=0
let's plug in x=0
a(-2)(-6)=3
Therefore a=1/4
And sum of the zeros of a parabola is equal to -b/a
So. 2+6=-b/(1/4)
Therefore b=-3

a= 1/4
b= -3
c= 3

[safe radish]

Please don't occupy multiple help channels.

[lean otter]

ohhhhh thanks

thats very clear instruction

[lean otter]

Dud why did you close it

[lean otter]

oh cuz i finished it

Ok

.close

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[gritty pier]

What should my r be here?

[gritty pier]

see graph

(Im applying the shell method here to calculate volume, but that doesnt matter for my question)

It should be (2-x) , can someone explain why?

[safe radish]

@gritty pier Has your question been resolved?

[safe radish]

@gritty pier Has your question been resolved?

[lean otter]

Where's x

Where's r

On that graph

[forest gust]

2 = r + x

r = 2-x

[safe radish]

@gritty pier Has your question been resolved?

[safe radish]

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[gloomy cliff]

Hello, I need help to understand why this system of equations has (1,1,0) and (-1,0,1) as their solutions.

[granite idol]

well you can see they are valid by multiplying them by the matrix, right?

[quasi bison]

do you mean likee

"why is attention drawn to those specifically?"
or
"i think those vectors AREN'T solutions"

[gloomy cliff]

Im gonna be honest Im a bit confused in general, I am in the process of diagonalising my matrix and this system of equations is what I ended up with

but im not sure how to understand what im getting from it. except that as far as i remember 2 0 lines means theres infinite solutions iirc

[granite idol]

yes, there will be infinite solutions

you can parametrize for the general solution. e.g., let y = s, z = t

[gloomy cliff]

yeah u get

x-y+t=0

or rather x = y-t

[granite idol]

let y = s also (or y = r, or whatever)

[gloomy cliff]

Okay actually I understood what Im supposed to do know, thanks

[granite idol]

so you see why those 2 are particular solutions?

[gloomy cliff]

yeah, and I got my 2 base vectors i was looking for^^.

[granite idol]

cool

[safe radish]

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[gusty glacier]

Translation:

A curve in the plane given by the equation:

where K is a reell constant.

For what values on K is that equation a ellipse?

what i have:

[potent seal]

That curve should be an ellipse or ..?

[gusty glacier]

oh yea srry

[potent seal]

I just understood the last word lol

[gusty glacier]

xD

but what i have:

I got the determinant of:

2 - lambda 1
1 K-lambda

where lambda is a eigenvalue, then getting the determinant i have lambda^2 - (2+K)lambda + (2K - 1) = 0

let lambda = L, then we have a clear 2 degree equation L^2 - (2+K)L + (2K - 1) = 0 where we can use like Reduced quadratic equation, and somehow get what K is suppose to be?

i got:

L1,2 = (2+k/2) +- sqrt((2+k/2)^2 - (2k-1))

and now i have to deduce where L > 0, for it to be a ellips, but here is my question:

How do i deduce it?

[safe radish]

@gusty glacier Has your question been resolved?

[safe radish]

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[gloomy cliff]

whats a quick way to argue/prove there isnt a maximum to this, only a supremum?

[gloomy cliff]

I have to argue for all 4, min/max/inf/sup , only one left is maximum now

Only thing I got is that since my Supremum is 1, which is not included into the possible values of the sequence, therefore theres no Maximum

[safe radish]

@gloomy cliff Has your question been resolved?

[gloomy cliff]

So 1∉A ?

[obsidian oracle]

if 1 were in A, that would mean that there exists an n such that 1 = 1 - n/(7n^2) right?

also if maximum exists, then max(A) = sup(A)

and you've proved sup(A) = 1

[gloomy cliff]

I solve this to n=-1/7 and then I can say since n ∈ N this is not possible, correct?

[fickle pendant]

this is a monotonically increasing function, but for any n \in N, 1/7n is > 0 which means the set is bounded above by 1 but does not contain 1

[gloomy cliff]

Yeah ive proven monotonicity already

Just need to make sure i give a good enough reason why it doesnt contain 1

[safe radish]

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[safe radish]

@analog jewel Has your question been resolved?

[safe radish]

@analog jewel Has your question been resolved?

[safe radish]

@analog jewel Has your question been resolved?

[safe radish]

@analog jewel Has your question been resolved?

[warm patio]

Derivative F(x) first

[analog jewel]

yeah

but then what

f is not defined at 0

[warm patio]

Bro

Be careful

The tangent passes through (0,1) , but doesn't touch F (x) at this point

[analog jewel]

the derivative is -lnx - 1

which is m

the slope

then this

right

[warm patio]

Did you got me?

yes

[analog jewel]

y - f(x1) = (-lnx-1)(x - x1)

[warm patio]

K now plug with (0,1)

[analog jewel]

y - f(0) = (-ln0 - 1)(x-0)

[warm patio]

K continue

F(0) = ?

[analog jewel]

should be 0

right

there should be a plus there too

but still 0

but ln0 is -inf

[warm patio]

Bro come down

[analog jewel]

why do you call me bro 😭

[warm patio]

Here we're talking about Tangent function

[warm patio]

What's the problem here

Nvm

[warm patio]

F (0) = 1

The tangent passes through (0,1)
So X1 = 0 , F(X1) = F(0) = 1

[analog jewel]

y - 1 = (-ln0-1)x

[warm patio]

Yes

[analog jewel]

now what el 7ag

[warm patio]

But there's a little mistake

You should see the difference between
F( x ) in the question
And F(x) of the tangent line

They have different "X"

[analog jewel]

what do you mean

by that

are you saying f(x) is not -xlnx

[warm patio]

So it's going to be
y - 1 = (-ln(x)-1)x

[analog jewel]

o

[warm patio]

In this question
F(x) = -xlnx

We can't call the the tangent line function f(x) too

Because it'll make us confused

[analog jewel]

sure

so the equation of the tangent line is 1 + (-ln(x)-1)x ?

[warm patio]

Alright
f(x) = -xln(x)
f'(x) = -ln(x)-1
g(x) = [ y - y1 = F'(a)(x - x1) ]

a is any point on f(x)
x1 , y1 are points that the tangent line passes through

[warm patio]

Yes

[analog jewel]

and i have to check for x = 1 and x = 1/e

if i get 0

[warm patio]

So from the options in the question what is right about the function we got

[analog jewel]

right

if it passes through d

or like for x = 1/2 i have to get 1/2

[warm patio]

Amazing

[analog jewel]

and 1/e i get 1/e

right?

i dont mean to give spoilers but it says that A is correct

the issue how i get these solutions!

[warm patio]

Sorry 1 min

Here it's

[analog jewel]

i think they made a mistake then

when writting the thing

[warm patio]

When we plug in (x = 1/e) we don't get (1/e)

[analog jewel]

shouldve been (1, 0)

i think

[warm patio]

So D is wrong option

[analog jewel]

idk i think they got something mixed up somethings sus

[warm patio]

Actually I don't how when
x= .5 , y = .5

Must be something wrong in The slope m

m = (-ln(x)-1)

[analog jewel]

i think something's written wrong

anyways

do u have any ideas on this?

i got f(x) = f(-x)

[warm patio]

I've no idea

Sry i tried to help

[analog jewel]

ight

[safe radish]

@analog jewel Has your question been resolved?

[tardy mango]

Integrate both sides from 1 to -1

it should reduce to an equation in terms of the integral

[safe radish]

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[dreamy verge]

so this says that for ${v_1}^{v_2}$, v1 must be positive

assuming that both v1 and v2 are variable

i dont understand the logic behind this. why is this the case?

im learning limits btw

[flat frigate]

lily9369

[rustic goblet]

if you have (-1)^0.5 for example

that would mean sqrt(-1)

[bold ferry]

How about (-1)^2 tho

[rustic goblet]

that would work, yes

[bold ferry]

Is there any info about variable 2?

[terse lichen]

or (-2)^3

[rustic goblet]

but x^a doesn't exist for all real values of a if x < 0

[dreamy verge]

hmm, well so my original question was something else

[plucky elk]

! original

[safe radish]

Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.

[dreamy verge]

over here, a=2 is rejected, and a=0 is the correct answer

because base must be >0

[dreamy verge]

i understand the rest of the question, its just the a=2 being an invalid answer thats tripping me up

[safe radish]

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[gritty pier]

$x^{log(x)} = 1000x^2$

[flat frigate]

rainy

[gritty pier]

how would you solve this?

theres probably some log rule idk about

[clear blade]

try taking log of both sides

should give you a quadratic in log(x)

[gritty pier]

= $log(x)^2 = log(1000x^2)$

[flat frigate]

rainy

[gritty pier]

[clear blade]

can you rewrite the log of a product

[gritty pier]

= $log(x)^2 = 3+2log(x)$

[clear blade]

should be log(x)

[gritty pier]

Ah yes mb

[flat frigate]

rainy

[gritty pier]

well the final answer should be 1000 and 0.1

but i still dont know

how

[clear blade]

well now you have a quadratic

you have to solve:

$\log(x)^2 - 2 \log(x) - 3 = 0$

[flat frigate]

Tushar

[gritty pier]

does the quadratic formula work here?

[clear blade]

yep, just make a substitution

u = log(x)

[gritty pier]

Willdo

$u_1 = -1, u_2 = 3$

[flat frigate]

rainy

[gritty pier]

okay

wait

ok got it 😄

thx

.close

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[hallow knot]

How can i distinguish a local minimum/maximum and a global minimum/maximum ? ( Analitically )?

[hallow knot]

I did the first derivative of the function i got that the critical points are -1/2 and 1/2

After that i did the signs table i got that f(-1/2) and f(1/2) is a maximum , but how do i know if they are a global or local?

[lean otter]

[0.6\textwidth]Assuming $c$ is a critical point for the function $f$, then you can consider points in the neighbourhood of $c$. So like $c + \epsilon$ and $c-\epsilon$

[lean otter]

if $f'(c+\epsilon) = \t{something positive}$ and $f'(c-\epsilon) = \t{something negative}$ then there is a local minimum there

[hallow knot]

ok , but with that i can verify that c is a maximum or a minimum right ? in the first case i shouldn t have any point in the neighbourhood greater then c , and in the other case the opossite right ?

[lean otter]

Sorry i couldnt understand what you meant here?

[hallow knot]

You are saying something like this ?

Therefore f(c) is a maximum ? ( local )

[hallow knot]

Like if f(c) is a maximum for any value x in the neighbourhood f(x) must be less then f(c)

[lean otter]

yes exactly

the opposite applies for minima

[hallow knot]

But the samething applies here

[lean otter]

you have to be careful

[hallow knot]

So that means the other point lets call it f(d) is also a local maximum ? no right ?

[lean otter]

thats why i said in the NEIGHBOURHOOD of c

because if you notice like uh

there is another extermium here

[hallow knot]

yeah right in the neighbourhood of c , c is a local maximum cause the thing you said , but now lets consider other point d , which is the global maximum , that also happens rihgt ? So how can i know that d is a global maximum and c is a local maximum ?

[hallow knot]

Exacly

forget i already know how to do it xd

XD

.close

[safe radish]

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[quasi bison]

you mean subtract not substrate

and now you need to integrate their difference

(blue - red)

from -1 to 2

[safe radish]

@void onyx Has your question been resolved?

[queen parcel]

You can check the integral online

[queen parcel]

This

You can graph the equations normally (graphing doesn’t require a calculator).

You could also mathematically test points between where the curves intersect

[quasi bison]

you can subtract them in either order and then if the integral gives a negative number then multiply it by -1

this is assuming there's only two intersection points so the curves don't switch roles midway

which they don't so you are fine

[queen parcel]

You can probably find tutorials where you can practice graphing functions

[flat frigate]

Results provided by WolframAlpha

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

Requested by gitoo0451

[sage idol]

hi

[safe radish]

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[lime dust]

!status

[safe radish]

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

[zinc gull]

Well actually

I just want to check if 0.1 is correct

[zinc gull]

For the probability of losing the first game

And 58% as the final answer

[quasi bison]

show all of your work

we can't tell if your answer is correct or not otherwise

[safe radish]

@zinc gull Has your question been resolved?

[safe radish]

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[lean otter]

this is where I got' stuc

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[lean otter]

.close

[safe radish]

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[lean otter]

I have an another ruffini problem, now the disequation is x⁴+x²-3x-3<0 and I cant find the root

[lean otter]

, x⁴+x²-3x=3

nah I dont remember the command

,w x⁴+x²-3x=3

[flat frigate]

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Upgrade to WolframAlpha Pro!

Requested by hamsterlover6288

[lean otter]

,calc x⁴+x²-3x=3

[flat frigate]

The following error occured while calculating:
Error: Syntax error in part "⁴+x²-3x=3" (char 2)

[lean otter]

,calc x⁴+x²-3x=3

[flat frigate]

The following error occured while calculating:
Error: Syntax error in part "⁴+x²-3x=3" (char 2)

[lean otter]

,calc x⁴+x²-3x=3

[flat frigate]

The following error occured while calculating:
Error: Syntax error in part "⁴+x²-3x=3" (char 2)

[lean otter]

.close

[safe radish]

Channel closed

Closed by @rocky horizon

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[lean otter]

,w x⁴+x²-3x-3<0

[flat frigate]

Results provided by WolframAlpha

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

Requested by hamsterlover6288

[safe radish]

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[gray plaza]

May I please know how to solve this question?

From this figure I did calculation like this:- 0.1 / 100

[gray plaza]

But I know I am wrong. May I please know how to approach this question?

[quasi bison]

this is very low-resolution

how did you get from that figure to 0.1/100?

[gray plaza]

I subtracted 99.99 from 100.

[quasi bison]

... you misspoke just now definitely

[gray plaza]

Apologies.

[quasi bison]

i think you still misspoke

ok, and where did 99.990 come from?

[gray plaza]

The image I shared.

[quasi bison]

i do not see any percentage that says 99.990 specifically

[gray plaza]

I see.

Can you please help me to solve this question?

[quasi bison]

you want the probability of a value at or above +5σ

can you look at the image and tell me the percentage given for the interval between -5σ and +5σ?

[gray plaza]

99.99943%

[quasi bison]

ok great

so what's the probability of observing a value outside the range [-5σ, +5σ]?

[gray plaza]

will it be 0.0005699999999961847

Let me verify it in the book 🙂

[quasi bison]

....

why such precision

[gray plaza]

Python

[quasi bison]

if you are using python then why not find P(Z>5) using it directly...

[gray plaza]

I am not very much familiar with this.

[quasi bison]

what did you put into python to get this output?

[gray plaza]

>>> x = 100 - 99.99943
>>> y = 1
>>> x/y
0.0005699999999961847

[quasi bison]

oh so that's where those strange nines were from.

why did you even divide by 1? that was redundant...

this is all very strange stuff that you're doing.

[gray plaza]

I apologise!

May I please know what I have been doing wrong?

[quasi bison]

you didn't recognize that subtracting 100 - 99.99943 would have produced a number with only five decimal places

you should have seen all those nines and recognized the floating-point rounding error

also you're using tools inconsistently with their power

decide once and for all: do you want/need to use python, or do you not?

[gray plaza]

No

[quasi bison]

ok, then don't use python.

[gray plaza]

Okay 😀

[quasi bison]

this subtraction could be done by hand, and you would have gotten 0.00057

note this was in fact supposed to be 0.00057**%**. you should not forget about that.

this is the probability of observing a value outside the range [-5sigma, +5sigma]

you want only the upper half of that range. so because of symmetry, you divide this 0.00057% probability by 2.

[gray plaza]

Yes, that gives me 0.000285%

[quasi bison]

missed a zero.

[gray plaza]

Yes, sorry.

May I please know what will be the last step?

<@&286206848099549185>

[sage idol]

yess

@gray plaza

what do you want

[gray plaza]

I apologise that you will have to read the previous chat history.

[gray plaza]

.

[sage idol]

just tag the question

[gray plaza]

Here

@sage idol Apologies for tagging you.

Would you be able to help me?

[torpid fable]

Do you understand what this value represents?

[gray plaza]

I guess, yes!

[torpid fable]

What is it

[gray plaza]

It shows that 0.000285% is the area covering the normal distribution at or beyond 5 standard deviations.

[torpid fable]

Ok do you know what area under a distribution curve represents?

[gray plaza]

Integration

I am not sure what exactly you asked!

[torpid fable]

The area under a distribution curve on a certain interval is the probability of the value being in that interval

[torpid fable]

So you are done.

[gray plaza]

Ahh! Yes.

But the answer in the book is:-

[torpid fable]

That is basically what you got (once you account for the missing zero)

The image you originally sent rounded so it doesnt exactly match

[gray plaza]

I see.

Thank you for being patient with me and helping me out 🙂

[torpid fable]

np

[gray plaza]

.close

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[lean otter]

i dont know what to do

i was thinking because tan pi/2 is undefined to bring it back as a constant

but i dont think i can do that

so it will be 1/2 integral of .... tan(x)

[plucky elk]

How'd you get this

[lean otter]

i honestly do not now

[plucky elk]

????

[lean otter]

i was thinking of integration by parts

maybe

[lean otter]

no its just my brain speaking loud

i dont always say right things

is integration by parts the way to go?

[halcyon carbon]

It’s tempting to use weierstrass sub but i think a simple u = x/2 would do the trick

With a bit of trig identity.

[lean otter]

could i also use weierstrass sub?

or i am overcomplicating

?

[halcyon carbon]

Yeah

Well idk tbh haven’t really thought it through

[lean otter]

and could integration by parts could also be done on this?

[final halo]

I mean weierstrass is probably worthwhile just to get rid of the trig functions but what you're left with isn't pretty either

[lean otter]

$\int \frac {2u^3} {1+e^{2u}}} du$

[flat frigate]

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

[lean otter]

assuming my calculation are right through weierstrass i got left with this

which i dont know what to do now

u = ln(t) du = 1/t dt

yeah wolfram isnt happy about this either

then i guess i messed up somewhere

[drowsy karma]

that helps you, look:

$\int_{}^{}\frac{dx}{\sin\text{}x}\overset{\ast }{=}\\t=\tan\frac{x}{2}\Leftrightarrow x=2\arctan\text{}t\Leftrightarrow dx=\frac{2dt}{1+t^{2}}\\\sin\text{}x=\frac{2t}{1+t^{2}}\\\overset{\ast }{=}\int_{}^{}\frac{dt}{t}=\ln\left| t \right|+C=\ln\left| \tan\frac{x}{2} \right|+C$

[flat frigate]

Joanna Angel

[drowsy karma]

then your integral becomes quickly solvable

[safe radish]

@lean otter Has your question been resolved?

[lean otter]

solveddd ittttttt

thank youuu!!!

[drowsy karma]

yw)

[lean otter]

.close

[safe radish]

Channel closed

Closed by @fast wave

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[safe radish]

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[severe slate]

what is the difference between question a and question c lol? I tried checking google and they are the same? Or am I just missing something. Thanks!

[final halo]

well a is asking you what a cauchy sequence is, c is asking you to state a result involving cauchy sequences

[safe radish]

@severe slate Has your question been resolved?

[patent flame]

If I may ask is this a university past paper? feel like I’ve seen this exact question w A4 (3+3+2) before

[severe slate]

it is yes

its 2015 University of Bristol, Uk

[severe slate]

so for part a I stated the cauchy's criterion, and part c I state that if its a cauchy sequence then its convergent?

[final halo]

part a does not ask you to state cauchy's criterion

[severe slate]

but you need to state the cauchy's criterion to validate that it is a cauchy sequence tho?

[patent flame]

Oh cool yeah I’m in 2nd year there

[final halo]

its just asking you "what is a cauchy sequence"

i.e. what does it mean if i say a_n is a cauchy sequence

[severe slate]

Ohhh

I see

a_n is a cauchy sequence if mod (an - am) < epsilon

[severe slate]

oh shit rlly??

[final halo]

needs more

[severe slate]

oh yeah i cba but its like for all epsilon blah2

[final halo]

sure

[severe slate]

.solved

.close

[safe radish]

Channel closed

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[patent flame]

Yeah haha

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[lean otter]

How do I solve this

Does it involve log

[humble helm]

First of all

Simplify

[lean otter]

So division?

[humble helm]

Ye

[humble helm]

Yes, i think log is needed

[lean otter]

Ughhh what

Haven’t even learned it

[humble helm]

Okay

[lean otter]

Mats borrows SEK 400,000 to start his own business. He will not repay anything until after 15 years. Every year his debt grows at an annual rate of 8%.
a) How much does Mats owe after two years?
b) By how many percent has the debt increased in five years?
c) How many years will it take before the debt has doubled? Enter the answer in full years.

This is the whole thing

[humble helm]

It is enough for now

lol, cropped photo

log in the left

[humble helm]

You

[lean otter]

Yes

I’m on c

[humble helm]

I think there's a formula

For this

[lean otter]

Yeah it’s 400000*1,08^x

[lean otter]

Or idk

[humble helm]

Not sure

[lean otter]

Wtf

What’s principal

[humble helm]

Damn

<@&286206848099549185>

[sage idol]

yes

@humble helm

[humble helm]

I don't know too

[lean otter]

?😭😭

[sage idol]

prinicipal is the amount deposited

[lean otter]

But don’t we have to use log there too

If we want to find the number of years

[sage idol]

what is tghe question guysss