help-10

[lone harbor]

Hi my problem here is that im getting 2 different answers

maybe someone could point out the flaw in the ones that's wrong?

[obtuse pebble]

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[safe kite]

how would you know which root is the triple root out of x=2 and x=-1/2?

[safe kite]

normally i would sub it into p(x) and the first derivative but since theres also a and b i have to find idk how to determine the triple root

[wraith barn]

where did ur roots come from

[safe kite]

i found the second derivative of that equation

which was a quadratic

and then just factorised

[wraith barn]

im leaning towards x=-1/2 just not being it cause its a monic poly

but im sure theres a way in which you can prove it

[safe kite]

so what if its monic ?

do monic polynomials only have integer roots?

[wraith barn]

yea nah thats my bad

sub it into the 1st root, find a and then find b

ig

[wraith barn]

wait, whats the aim of the q

to find a and b?

[safe kite]

yeah

and hence find all of the roots

[wraith barn]

mmmmmmm

sum and product of roots maybe?

lemme try smth

huh

maybe they just both work lmao wait

are we sure only one of them works

cause i think i can give answers for both roots

[obtuse pebble]

@safe kite Has your question been resolved?

[safe kite]

yeah same

but i dont think x=-1/2 works

expanding it doesnt give the same polynomial

[wraith barn]

?

[safe kite]

u got the other root as 2 when the triple root is -1/2 right

[wraith barn]

nope

[safe kite]

oh nvm

[wraith barn]

got 9/2

[safe kite]

well i mightve messed up smwhere

but does expanding (x+1/2)^3(x-9/2) give you the same equation?

[wraith barn]

if you found b, you can do product of roots and the final root is b/(-1/2)^3

i mean, if the answer is 0 when i sub it in im guessing it works

ill expand it tho just for security

[tame python]

to save time, that should be (x**+**1/2)^3(x-9/2)

[wraith barn]

oh lmao didnt see that

[safe kite]

oh yeah my bad

[wraith barn]

yea it def works

only thing that can cancel it is if the q said (a and b are integers)

[safe kite]

ah okay i see

so theres two possible solutions

i was confused cos i thought only one of them could be a triple root lol

[tame python]

note that you can reflect one solution about x = 3/4 to get the other solution, where 3/4 is the midpoint of -1/2 and 2

[obtuse pebble]

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[silver finch]

not sure how to do this

[opaque dome]

Is it Z = (X - mu)/sigma ?

[silver finch]

i think so but its *2

and also idk what the x value is in this case

[alpine verge]

mm the weirdest part abt this problem is that it doesnt say anything about the distribution being normal or not

[safe haven]

it should be a t instead of z test no?

[alpine verge]

it gives you the mean and sd of the sampling distribution by the 1000 simulations

[safe haven]

also my gut just tells me that the test statistic is 0 because the mean diff is 0

[alpine verge]

but the sample size is only 9

[alpine verge]

i think its just a normalcdf problem ngl

[silver finch]

this is what it says when i got it wrong

[alpine verge]

0.9 is the z score i assume

[silver finch]

this is a diff problem w diff numbers

[alpine verge]

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

oh

[warm shale]

y0shi

[alpine verge]

yeah then for this, assuming the distribution is normal, you would just use this to find the z score

and then find the probability of it being greater than that z score and times it by 2

[silver finch]

so what is x in this problem

[alpine verge]

the 1.3 they gave

[silver finch]

so 1.3-0/1.69

then *2

[alpine verge]

mhm

well find the probability of the z score first

greater than that

and then times it by 2

[safe haven]

is that the population or sample sd though

[silver finch]

sample

[alpine verge]

well the sample size is 1000 so

[safe haven]

then we need to use t test because population sd is not known no?

[alpine verge]

shouldnt make much of a difference whether we use t or normal

[safe haven]

well yeah

[alpine verge]

theres not really a hypothesis here tho

its just asking for probability

[silver finch]

how do i do that

[alpine verge]

use a calculator

or a normal distribution table

[safe haven]

but then we cant just divide by 1.69

we need to divide 1.69 by 1000 first?

or sqrt(1000 sorry

[alpine verge]

im pretty sure the sd they gave there is uh

the sd of the sampling distribution

the actual samples we are taking are samples of size 9

and theyre running that 1000 times

to form something close to the real sampling distribution

[safe haven]

ah i see

fair enough

[silver finch]

ok im ngl im hella tired and i dont get this can u just give me the answer😭

[alpine verge]

weird problem

ill put it in desmos for you

[silver finch]

bless

[alpine verge]

i got 0.2209

so times that by 2 and that should be it

or if you want more digits its 0.220878

[silver finch]

so 22%

[alpine verge]

yeah but it says to round the nearest hundredth as a percentage

so like 22.09%

but we need to multiply that by 2

since it couldve been just as extreme if it was -1.3 or less

[silver finch]

and for this i woudl say it is difficult right

[alpine verge]

yeah i would say so

weird using a sampling distribution to show a correlation

but its whatever

[silver finch]

alr lets see

nope

44.18%

oh well

[alpine verge]

is the number wrong or

yeah i told you to 2x it

[silver finch]

oh u did

im stupid

hahaha

[alpine verge]

all good

[silver finch]

thanks thats on me

[alpine verge]

yw!

[obtuse pebble]

@silver finch Has your question been resolved?

[obtuse pebble]

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[lethal trench]

i dont understand why you would use 7 x 6 x 5 x 4 x 3 instead of (7 x 6 x 5 x 4 x 3)/(5 x 4 x 3 x 2 x 1) to calculate the problem

[lethal trench]

the answer is D which is 7!

[whole saddle]

its 2520

[lethal trench]

wait it isnt 7! it's 7 x 6 x 5 x 4 x 3

[lethal trench]

yes thats the answer in the answer key

[whole saddle]

yeah just multiply those

[lethal trench]

but i dont understand why we woudln't divide by 5!

[kind hawk]

why should we divide by 5!

[lethal trench]

because i dont understand why order would matter in this situation

wait

yeah nevermind

.close

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[lethal trench]

.reopen

[obtuse pebble]

✅

[lethal trench]

nvm heres another problem

idk how to approach this one

[kind hawk]

do you know how to convert such a number to a fraction

[lethal trench]

yes

it would be xy/99

but idk where to go from there

[timid silo]

well what possible ways can you express 99

what are its factors

[lethal trench]

11, 3, 9, 33, 99

oh i see

so the answer would be 3?

[kind hawk]

how many numbers did you list

also, why did you not list 1

[lethal trench]

i forgot

yeah 1

1, 3, 9, 33, 99 are its factors

and it says the values are neither nine or both zero

wait

1 can't be a denominator i think

uhh

wait

oh i see

you can only have 1, 3, 9, and 33 be equal to xy

which means the possible denominators would be 99, 33, 11, and 3 i believe

but that's 4

[kind hawk]

why should I not be able to write x=5 and y=7

[lethal trench]

what do you mean

[kind hawk]

well you said xy can only be 1,3,9 or 33. why should it not be able to be 57

[lethal trench]

oh no it can be equal to 57

i think what i meant is that those are the only numbers that can factor out the 99 in the denom

and be able to reduce the fraction

[kind hawk]

well what about xy=06

and from the way the question is worded, I am actually not sure if they allow x or y to be 9

the both is only in front of the zero

[lethal trench]

i believe neither can be nine but one can be zero

the answer given to the problem is 5

[kind hawk]

ok lets recap what we have

before reducing the fraction it will be xy/99

so after reducing, it could only ever be a/b where b is in {1,3,9,11,33,99}

you said b couldnt be 1. why

[lethal trench]

because i think would mean that the fraction wouldn't be a fraction anymore

everything divided by 1 is itself

[kind hawk]

its still a fraction but what you mean is that it would be an integer itself, yes

[lethal trench]

yes sorry integer

[kind hawk]

so it could be 0/1 or 1/1

[lethal trench]

but then the decimal wouldn't repeat if that was the case

[kind hawk]

well, 0=0.00000000...

[lethal trench]

i think

[kind hawk]

and 1=0.999999....

[lethal trench]

oh

[lethal trench]

this woudln't work tho since it's given both xy can't be 0

[kind hawk]

yes exactly

[lethal trench]

oh i see

[kind hawk]

these are exactly the forbidden numbers

[lethal trench]

so 1 would work?

[kind hawk]

0.999999.... is also forbidden

[lethal trench]

right

1 wouldn't work in that case?

[kind hawk]

so thats good. its always a good sign when you find why some restriction was in the problem

yes

so our denominator b cant be 1

[lethal trench]

okay i see

so now we are left with 3, 9, 11, 33, and 99

[kind hawk]

yes

and now the question is, are there values xy such that xy/99 would reduce to a fraction with those as denominators

[lethal trench]

if the numerator was 33, denom will be 3

[kind hawk]

yes

[lethal trench]

if 11, denom is 9'

if 9, denom is 11

if 3, denom is 33

and if anything without those factors, it would be 99

so that gives 5

i see

i thought 9 wouldn't work for some reason

i forgot 9 could mean x = 3 and y = 3 instead of one of them being 9 and the other 1

[kind hawk]

uhm xy doesnt mean multiply

for 9/99 you would have x=0,y=9

you could also choose 18/99 so that you dont have to choose either of them as 9 itself

[lethal trench]

uhhh

oh xy just has to be either the number itself or a number with one of those factors

[kind hawk]

yes

[lethal trench]

okay i understand now

heres another question

im not sure how to approach this one either i tried doing the cube rule(?) of (a-b)(a^2 + ab + b^2) but dont know where to go from there

[kind hawk]

hmm yeah not sure either

[lethal trench]

finding the trick to these is weird

also this one

it's not too difficult to find at one occurrence of this but finding how many ways is tripping me up

[kind hawk]

if the numbers are a, a+1, ..., a+(n-1), then you can find the closed form for that and set it equal to 55. then it will depend on 55-some number being divisible by n or something. and then a bit of casework I would guess

[lethal trench]

what do you mean by closed form and being divisible by n?

[kind hawk]

you can find a formula for a+(a+1)+(a+2)+...+(a+n-1)

if you set that equal to 55 you can subtract some stuff and arrive at n*something = 55-some number

and then its a question of whether the right side is divisible by n

I have to go now

[lethal trench]

okay thank you

[obtuse pebble]

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[zealous bobcat]

hyia! can you please help me find the largest absolute value of integer n so that p is prime

[zealous bobcat]

huge thanks! ❤️

[opaque dome]

Have you try something ?

[mint nebula]

i have squared rhs

i think that's how it's gonna solve

[zealous bobcat]

let me try that, thank you

[mint nebula]

just add 4 but make sure to substract it afterwards

[zealous bobcat]

so we've got p = (n+2) - 3^2

[mint nebula]

Very well

[obtuse pebble]

@zealous bobcat Has your question been resolved?

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@zealous bobcat Has your question been resolved?

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[honest forum]

Two masses are arranged with one, of $50$ kg, on a table with a friction coefficient of $\mu = 0.20$, and the other of $30$ kg, attached by a cord and hanging, over a frictionless pulley, off the edge of a table. Find the acceleration of the system.

[warm shale]

Dork9399

[honest forum]

In the solution, it says $\mu N = \mu m g = 0.20(50 \mathrm{ kg})(9.8 \mathrm{m}/\mathrm{s}^2) = 98 N$

[warm shale]

Dork9399

[honest forum]

When calculating the forces acting on the 50 kg mass

However, I don't understand why we multiply friction with gravitational acceleration when friction is opposing the horizontal movement of the mass

.close

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[ebon sable]

,w integrate -1/(x+x^2*arctanx + x^3 + (x^4)arctanx)

[warm shale]

Results provided by WolframAlpha

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

Requested by mustafashahzad

[ebon sable]

how

.close

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[cunning burrow]

!status

[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

[reef pike]

sorry

my teacher went through this problem in class

but i don’t know how he arrived at this answer

why didnt we take the anti derivative here?

[cunning burrow]

Do you a problem with a particular step

[mystic lintel]

It's one of fundamental theorems of calculus

[reef pike]

yeah the final step i guess

well on other questions like this one

we took the anti derivative of t

and then did fb-fa

[cunning burrow]

differentiation and integral reverse of each other

You can perform calculations in different ways by making use of certain theorems

[reef pike]

to be clear my question is like why didn’t he take the antiderivative of square root of 9-x^2 before doing the subtraction thingy?

[cunning burrow]

You can just apply the fundamental theorem of calculus. Also as I mention before the derivative and the integral are the reverse of each other.

[reef pike]

i understand that bit

[cunning burrow]

This means you only need to consider the integrand not calculate the antiderivative

[reef pike]

oh

wait

what

but the formula in my notes says that you have to take the antiderivative then apply the subtraction thing

what do you mean you only need to consider the integrand?? how should i consider it

[cunning burrow]

Yes but the formula you are referring to does not consider the case derivative "and" integrand

There are different fundamental theorem of calculus

If you are interested in the technical details you can refer to the proof though it is not necessary.

[reef pike]

i think part 1 is what i’m learning right now

i took this calculus class just to avoid proofs

i don’t think it’s like required here

um but why am i using different formulas for these questions?

[cunning burrow]

You can complete both questions using second or the first fundamental theorem of calculus. It does not matter

[reef pike]

i didn’t realize we were using part 2 for our questions. do you know which one my teacher used here

[cunning burrow]

I am having trouble understanding the notation your teacher is using. It does not seem conventional

Here is how I like to think about it

[reef pike]

yeah the brackets through me off

threw

[cunning burrow]

Let f(y) = sqrt(9-y^2)

integrating that we get F(y)

We don't need to actually calculate F(y)

Applying the second fundamental theorem of calculus we get

d/dx (F(x) - F(3))

We had the d/dx outside at the beginning

[reef pike]

oh

[cunning burrow]

But by definition derivative of the anti-derivative is just the original function

so d/dx F(x) = f(x)

[reef pike]

that makes sense

[cunning burrow]

F(3) is a constant and the derivative of a constant is zero.

So that how we end up with:

sqrt(9-x^2)

[reef pike]

so i would be able to tell that i can skip the step of integrating because the d/fix is on the outside?

d/dx*

[cunning burrow]

correct

[reef pike]

got it got it

[cunning burrow]

the derivative will cancel out the integration

[reef pike]

ohhh

thank you for help!

it really cleared things up 👍

.close

[obtuse pebble]

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[reef pike]

.reopen

[obtuse pebble]

✅

[reef pike]

wait a second

then is this the final answer?

[reef pike]

would this be wrong?

[cunning burrow]

When you write integral you should always indicate what you are integrating with respect to e.g. dt, dx etc

[reef pike]

oh my bad

[cunning burrow]

There some weird technical issues with having an integrand and bounds of integral being of same variable.

[reef pike]

right

uhm but it’s just what my teacher told me to solve

[cunning burrow]

This does not seem to be the same question from earlier

[reef pike]

that’s the answer to the question

i think if i did it properly

wait but i followed all the steps you told me to

oh sorry it’s question 2

separate from the square root 9 question

if thats what you mean

[cunning burrow]

So this was the original question except there was a d/dx in front of of the integral

[reef pike]

yes!

[cunning burrow]

I think in this case I would apply the second fundamental theorem of calculus and not the first. As the form is slightly different from 1st fundamental theorem of calculus

Doing this we get:

t^2/2 evaluated at sqrt(x) and x

So we get x/2 - x^2/2

We then take the derivative

[reef pike]

oh

we need to take the derivative?

[cunning burrow]

And we get 1/2 - x

[reef pike]

wait we just took the anti derivative

[slate monolith]

i think it means you take the derivative of the integral after you solve the integral

[cunning burrow]

In the original question you posted there was a derivative sign in front of the integral

[reef pike]

oh

oh

but you said it cancels out

😭

man

[cunning burrow]

Yes but we can't apply the 1st fundamental theorem of calculus because the form is slightly different

[slate monolith]

i think its when its an indefinite integral

and they have to be with both respect to the same thing

e.g. both dx

[reef pike]

sorry what has to be with respect to the same thing?

[cunning burrow]

there is 1st fundamental theorem of calculus involving boundaries. I guess that is another way to thik about it has well

Something of the form d/dx integral f(x) dx = f(x)

I think that is what gooji means

[cunning burrow]

https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus

[reef pike]

oh i must have gotten it twisted. so the FTC1 is what you just stated and FTC2 is taking the antiderivative then the subtraction thingy?

[cunning burrow]

correct

[reef pike]

got it

and the derivative and integral only cancel out when it’s an indefinite integral and when the dx at the end of a function is the same as the d/dx at the front of the integral symbol?

😭

sorry

[slate monolith]

PRETTY sure

might need to fact check

but 98.753% sure

[reef pike]

that’s high accuracy

i trust

[cunning burrow]

You can have bounds but only certain bounds are work for the first fundamental theorem of calculus. You can refer to the wikipedia article

[reef pike]

that makes sense

[cunning burrow]

Also refer earlier to the explanation I gave for the first question

sqrt(9-x^2)

[reef pike]

by certain bounds you mean it can’t have 2 variables as bounds?

[cunning burrow]

You can have two variable in a bound it is just calculation will be more complicated e.g. the second question we did.

[reef pike]

oh okay

wait

for the first question you said that the integral and the derivative cancel out

but dt and d/dx out in front are different

[cunning burrow]

Yes the fundamental theorem of calculus the main thing is you a bound that is just x and not some weird function like x^2 etc

[cunning burrow]

The meaning of the symbols will depend on the precise context. Here dt and d/dx are not related. By itself dt means nothing

[reef pike]

okay so i should take in the context when it comes to like word questions

but for these solving ones i can just imply it cancels out

[cunning burrow]

Yep if the integral in the right form

[reef pike]

got it

sorry one last question

for question 2

i use FTC2 where i find the integral and then subtract Fb-Fa

then i gotta go back in and take the derivatives of those two?

do i not subtract until i find the derivative?

[cunning burrow]

I can't understand your question

[reef pike]

oh so i mean

[cunning burrow]

Yes the process for the working out is correct

[reef pike]

that’s not the final answer, we have to take the derivatives of these

right?

[cunning burrow]

Because in the original question you had a d/dx. That is why I took the derivative at the end

[reef pike]

yeah i was just gonna ask what happens to the cancelling out of the integral symbol and the derivative symbol here

[cunning burrow]

If add a derivative sign then x/2 -x^2/2 then you going to need to do work

[reef pike]

that makes sense

[cunning burrow]

You can do something similar to question 1 but you will need apply chain rule

[reef pike]

oh what

[cunning burrow]

So we would end up with d/dx (F(sqrt(x)) - F(x))

[reef pike]

but they cancel out why do we do it again?

sorry referring to question 1

[feral tide]

can someone help me in priv

[cunning burrow]

I am not referring to question 1, I am referring to question 2

[reef pike]

i was replying to this message here

[cunning burrow]

Yep that is for Q2

[reef pike]

oh okay

[cunning burrow]

You can do something similar but the working out will be different due to additional requirements of chain rule

To give an analogy

Let say we have f(x) = x^2

Note that f(x^3) = x^6

note that d/dx F(x^3) does not equal d/dx F(x)

[reef pike]

okay

[cunning burrow]

I have use F(x) to indicate we are considering the antiderivative instead of f(x)

[reef pike]

that makes sense

i think i get it

[cunning burrow]

I think take some time to practice the content and you will be good. Always ask yourself what are the key theorems

[reef pike]

yeah, i’ll go try my own but thank you for helping me answer my questions i really appreciate it

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[karmic hedge]

This is the second half of a question

The first half was to get the area between f_4(x) and f_3(x)

which = cos(2pi) - cos(pi) = 2

Im not really sure how to prove this tho

If I just leave it as (2sin^2(x)-(p+1)•sin(x)+3) - (2sin^2(x)-p•sin(x)+3) and integrate that between 2 pi and pi I get 2 again...

Which is what I think they want me to do

Idk if that proves it tho

[opaque dome]

Taking some values of p dont prove for every p, i think maybe be for all p use periodicity

[karmic hedge]

Wym sry

I get that subbing in values for p doesnt prove it

I need to prove for all values of p

[opaque dome]

The periodicity

?

[karmic hedge]

Im not really sure what you mean by that sry

[opaque dome]

Like cosx is 2pi periodic, since cos(x+2pi)=cosx

Like this

Showing that f is pi periodic could lead to something

Wait

[unique cove]

@karmic hedge Your reasoning is true, and it works. By taking a random p and showing that it works, it generalizes that it works for all p 👍

[karmic hedge]

Oh?

I dont see how what I did proves it tho

Like I got 2, im guessing that means I will get 2 for every p and p+1, but how do I prove that

[unique cove]

Well you definitley need to be clearer, such as saying: "Let p be a natural number." Then you calculate like you did for ANY p, which if works, therfore works for ALL p.

[karmic hedge]

Ah ok. What line would you write at the end of the proof, after I pick a random p and integrate to get 2?

[unique cove]

Also you should show that f_p+1 >= f_p for all p, so that calculating the integral makes sense

[unique cove]

Your proof should be constructed as:

1. I take ANY p
2. I check if the relation works for just this p.
3. Therfore if it works for ANY p, then it works for ALL p

ie. if you use the variable "p", you're pretty much testing FOR ALL p

[karmic hedge]

I get you. Thank you so much you explained it really well!!

Thats pretty much all my questions haha

❤️

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[dull sequoia]

how do i go about doing this problem?

[forest sinew]

think about the order

lets pretend theyre polynomials

can you identify the order of each graph

[dull sequoia]

B, C, A?

[forest sinew]

no, order, like, highest power of x in the function

[dull sequoia]

oh

[forest sinew]

a quadratic ax^2 + bx + c is order 2 for example

[dull sequoia]

so from most zeros to least zeros?

C, B, A?

[forest sinew]

lets try a different way

say i said that one of these graphs was of a quadratic

which one is it

quadratics describe parabolas right

[dull sequoia]

yeah

A

x^2

[forest sinew]

alright

how about, which one is a cubic

what does a typical cubic look like

[dull sequoia]

C

[forest sinew]

yea

okay, what order is B

can you tell just by guessing

[dull sequoia]

4th

[forest sinew]

yea

so, can you organize these?

i mean dont worry about the graphs, for a second

if you know one is something like x^4

and another is something like x^3

and the other is something like x^2

which is f, f', and f''?

what is $\dv x \qty( x^4 )$?

[warm shale]

jan Niku

[dull sequoia]

4x^3

[forest sinew]

yea

so, idk, lets say f(x) is a quartic (x^4)

then f' would be cubic, like you say

then whats f''?

[dull sequoia]

3x^2

[forest sinew]

12x^2 but yea

its a x^2

[dull sequoia]

Oh yeah

[forest sinew]

so, make a guess which graph is which

you identified which one was quadratic, cubic, and quartic

so itd stand to reason that f is ...

[dull sequoia]

B

[forest sinew]

yea

and derivative of a quartic is a cubic

[dull sequoia]

then C, cubic

[forest sinew]

[dull sequoia]

then A, quadratic

so f = B, f' = C, f'' = A

[forest sinew]

yea, i think so

there are other ways to do these problems

but this i think is the easiest

[dull sequoia]

yeah

[obtuse pebble]

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[lone bobcat]

so

it says in the second image

"using all other information we know the function crosses the x-axis once"

at a point left of the y axis

what is the thought process of finding this conclusion

we determined that (2,30) is a maximum

so left of it is increasing

and 4,26 is minimum right of it goes up

and those points are above y=0 (x axis)

so we know the function goes down to the left, and also y can take any value to -infinity

is that it?

[fervent tusk]

sorry just want to clarify one thing, (2,30) isn't a maximum, its a local maximum in its neighbourhood of points

[lone bobcat]

my book says maximum, but yes it is

it hasnt yet made the distinction

[lone bobcat]

to refresh, in this particular function the maximum would be infinity, or if say in the set x = [2,4], it would be 2,30 ?

[fervent tusk]

well i mean infinity would technically be the supremum

but your second assertion is correct

i think what your book is saying is that as x gets sufficiently big, the rest of your terms kinda stop mattering

only the x^3 term mattes

[lone bobcat]

yes

it dominates the effect

[fervent tusk]

meaning that the behaviour is determined purely by the x^3

just to reiterate, as x gets sufficienty big enough, the function simply acts like x^3

and as we know from x^3, as x -> Inf, y -> Inf

and as x -> -Inf, y -> -Inf

did that answer your question?

[lone bobcat]

is it basically this?

we know to the left of 2,30 it will reach -Inf

and thus cross the x-axis

but how do they know it would cross for negative values of x, without inputting numbers

[fervent tusk]

oh okay i think i know what your query is now

so you are essentially asking why it only cuts through the x-axis once right for all x < 2

[lone bobcat]

i want to know the thought process behind how they found it crosses the x-axis for x<2 yes but also how they know it crosses for x<0

[lone bobcat]

it says here

crosses x axis to the left of the y axis

do they just put numbers in the function, or is there another way

it says "clearly from the other information" so i assume we dont have to

input numbers

ah i know

its because the y-intercept

is (0,10)

so for y=0, x<0

okay now i get it

brain shart

[fervent tusk]

cool

[lone bobcat]

ty for your time James

[fervent tusk]

nw

[lone bobcat]

.solved

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[heady turtle]

a function with a removable discontinuity does not have an antiderivative

Is this true?

[versed stratus]

at that point, I think

[heady turtle]

What about at the interval?

[versed stratus]

I'm not too sure tbh

[forest sinew]

what does that mean at the interval

I think theyre just citing the requirements of FTC

[obtuse pebble]

@heady turtle Has your question been resolved?

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[royal pendant]

been stuck here for a while

[full maple]

try using binomial probability

[royal pendant]

thats what i tried

[timid silo]

show your work

[royal pendant]

ok just a sec

[drifting wraith]

that's insane

[royal pendant]

i came up with a binomial probability formula and its :

[full maple]

let's try it from the start?

[full maple]

that's not quite right

[royal pendant]

thought so

[full maple]

firstly, what is the sample size?

[royal pendant]

100?

[full maple]

yes, and what is the probability of success?

[royal pendant]

0.5

[full maple]

let X be the random variable

you need to find P(X > 60)

or P(60 < X <=100)

[royal pendant]

alright

[full maple]

on the right side of the question, it says CAS

you have a CAS calculator? because you're going to need it

[royal pendant]

yes i have one

[full maple]

do you know how to perform a binomial cumulative distribution calculation on it?

[royal pendant]

no i'm not very good with the cas

[full maple]

click menu

[royal pendant]

ok

[full maple]

then statistics

[royal pendant]

ok

[full maple]

distributions

[royal pendant]

yep

[full maple]

go down to binomial cdf

we are using cdf, which is a cumulative distribution function

[royal pendant]

yep

do i then just sub in the numbers we had

[full maple]

yes, lower and upper as well as n and p values

[royal pendant]

n=100 p=0.5 lower bound is 61 and upper bound is 100?

[full maple]

yes, correct

[royal pendant]

thank you so much

i got the test tmrw

[full maple]

best of luck

[royal pendant]

thx (:

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[opaque topaz]

How do I express y in terms of x for this equation?

[latent walrus]

any idea?

[opaque topaz]

this is the final answer but I don't know how get from there to here

[latent walrus]

well yeah, i asked if you had any ideas

[opaque topaz]

not really, I know that ln is log e, but from here I don't know how to proceed

<@&286206848099549185>

[latent walrus]

alright so if i have ln(y)=x, it must be true that y=e^x, right?

[opaque topaz]

yup

[latent walrus]

thats all you need to do here

then simplify

[opaque topaz]

can I use other variables for x because I already have another x in the equation

[latent walrus]

x in my equation is just the entire right side of your equation

[opaque topaz]

ohhhh

thanks a lot 🩷

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[ebon sand]

Is the deravative here 2lny?

[full maple]

show your working

[ebon sand]

oh I was just thinking mentally

I meant is the derativive of 2^y

2lny

cuz k would be 1 here?

and y = k

i dunno

[full maple]

not quite

it would be ln(2)*2^y

[ebon sand]

oh that would be yeah

i was gna say

ok cool thanks

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[vale tundra]

Hi all, I have this matrix M defined in the first screenshot, and the second screenshot is an example. Now, can someone prove the third screenshot, given that r is the smallest integer such that a^r is congruent to 1 (mod N)

[vale tundra]

I'll break it down because it's kind of long.

This is the definition of M_a

Here is an example

And prove that this is true, given that r is the smallest value such that $a^r \equiv 1 \pmod{N}$

[warm shale]

Decoway

[mighty geyser]

hold on, it seems that it's already all laid out already

are you trying to understand what this is saying

at what step are you confused?

[vale tundra]

I'm trying to understand the third screenshot

[vale tundra]

yes

[mighty geyser]

okay, where along the 3rd screenshot are you stuck

there are 4 equal signs there, what do you understand from each of them

[vale tundra]

${M_a}^r = M_{a^r}$ is getting me confused

[warm shale]

Decoway

[vale tundra]

I guess it's probably something to do with the number theory

[mighty geyser]

okay, this can be proven by induction

clearly true when $r=0$, because $M_a^0=\mathbb{I}=M_{a^0}$

[warm shale]

Element118

[mighty geyser]

then, suppose true for some $r$, we want to show it is true for $r+1$

$M_a^{r+1}=M_aM_a^r=M_{a\times a^r}=M_{a^{r+1}}$

[warm shale]

Element118

[mighty geyser]

so the issue is showing $M_aM_b=M_{ab}$

[warm shale]

Element118

[mighty geyser]

Proceed by $M_aM_b|x\rangle=M_a|bx\rangle=|abx\rangle=M_{ab}|x\rangle$

[warm shale]

Element118

[vale tundra]

okay I understand 👍

thanks

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[honest forum]

proving that it is not divisible by 3 or 11 are very straightforward

different configurations will block divisibility by 2 or 5

i dont know how other primes will work

[wooden cipher]

you sure its not divisible by 3?

oh ok yeah it isnt

[honest forum]

i checked bru

[wooden cipher]

you forgot to count a number

[honest forum]

wdym

[wooden cipher]

i meant yeah* forgot a number

my mind auto typed

my bad

for a question like this it just becomes "find the trick that makes this not prime" because otherwise they wouldnt ask

and im struggling to find it

[honest forum]

yea ik

[honest forum]

me too lol

btw this is a usamts problem

from last year nov

[wooden cipher]

ah

[honest forum]

incase anyone wants to read sol and give hints

[wooden cipher]

brb

101

[honest forum]

is that my hint?

[wooden cipher]

yes

i could have done my math wrong but im pretty sure

the idea stems from the idea is that it would be easy if it was mod 9999 or some factor of it

[honest forum]

so we can pair off the integers

im just tyrna check divisibility

so 2024 and 2117

2025 and 2116

[wooden cipher]

?

[honest forum]

20242117 is divisible

by 101

right?

,calc 20242117 mod 101

[warm shale]

Result:

0

[honest forum]

and so is 202411112117

,calc 202411112117 mod 101

[warm shale]

Result:

0

[honest forum]

so we can use that

to check divisibility

[wooden cipher]

use mod 9999

[honest forum]

wdym

how so

[wooden cipher]

20240000 mod 9999 is 2024

[honest forum]

yes ofc

[wooden cipher]

you can reduce the concatenation to a sum

[honest forum]

how so?

when I check divisibility by 101 by taking mod 101

we can pair off 2024 and 2117

2025 and 2118

that way everything from 2024 to 2117 will be paired

[wooden cipher]

when checking divisibility by 3, sum of digits is mod 9

youre doing something similar

[honest forum]

yes, im using the fact that 2024abcd2017 mod 101 is abcd

where abcd is any integer

[honest forum]

after that you can easily show that 101 won't divide

[wooden cipher]

sure that works

i was thinking abcdefghijkl = abcd+efgh+ijkl (mod 9999)

then you make it an arithmetic series

[honest forum]

yes but that will just rule out the factors of 9999

so just 3,11, and 101

now what?

[wooden cipher]

that shows its divisible by 9999 always

[honest forum]

huh?

how is it divisible

[wooden cipher]

oh

sorry

it reduces to 101(2024+2124)/2

which is divisible by 101

so the original number is diviisble by 101

divisible by 101 always

[honest forum]

then why does pairing numbers and summing them show otherwise

oh wiat

ohh

yea

that makes sense

[wooden cipher]

i think it should work

[honest forum]

yes it does

[wooden cipher]

50 pairs and 1 divisible by 101

[honest forum]

thanks!

thats what they have written as the solution as well

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[timid silo]

I am confused with this question. How can the answer be 0?

[hidden compass]

Because j is 6

And 30 / 6 = 5 with remainder 0

[timid silo]

How did you solve j

[limber portal]

21 mod 1 = 0
21 mod 2 = 1
21 mod 3 = 0
21 mod 4 = 1
21 mod 5 = 1
21 mod 6 = 3
21 mod 7 = 0
21 mod 8 = 5

So j = 6

30/6=5 -> 30 mod 6 = 0

[obtuse pebble]

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[void arrow]

.open

whats 2x190x26x1092x98 squared?

[void arrow]

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[pale brook]

i really need help with finding the positive and negative set notation for this graph, i already graphed it and i need help finding it in set notation. thanks!

this is the graph btw

[obtuse pebble]

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[honest chasm]

hi can i have some help with quadratic expressions

[honest chasm]

i have a test tmrw and i just want to prepare

pls someone i need help 😭

[scarlet atlas]

Help what

[honest chasm]

with quadratics

expressions

like factoring

[scarlet atlas]

I mean what’s the question

[honest chasm]

are you good at quadratics tho

i just need help with decomposition, breaking up the middle term and stuff like that

[scarlet atlas]

Yes

So do you know this?
X=-b+-sqrt(b^2-4ac)/2a

[subtle sinew]

You can use khan academy, they have practice questions and videos

Like this https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratics-multiplying-factoring/x2f8bb11595b61c86:factor-quadratics-intro/e/factoring_polynomials_1

[obtuse pebble]

@honest chasm Has your question been resolved?

[honest chasm]

what

i dont know how to write that on paper

where did the s come from

its grade 10 quadratics

[subtle sinew]

That's the quadratic formula, if you haven't seen it, don't worry about it

[honest chasm]

no i know the quadratic formula but thats used to solves quadratic EQUATIONS

i need quadratic EXPRESSIONS

[subtle sinew]

.

[honest chasm]

sorry

thank you

[obtuse pebble]

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[full saddle]

id:customize

[full saddle]

.close

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[timid silo]

help

[timid silo]

want to prove =>

i have a proof where we start with some sequence xk in C that converges to some a in C since C is closed. Then consider yk = f^-1 xk

then as k-> infinity

yk = f^-1 a

thus every arbitrary sequence in f^-1 c converges to a point in C thus it is closed

should i have started with some sequence xk in f^-1 C that converges and shown that it converges to x and then x must be in C instead

[shut lagoon]

Which definition of continuity are you working with? Preimages of open sets are open?

[timid silo]

sequential

[wild swallow]

what is your argument here

[timid silo]

since yk = f^-1 xk

[wild swallow]

various problems with your argument

[timid silo]

do you mind explaining

[wild swallow]

1. f is not necessarily invertible, so y_k = f^-1(x_k) does not even make sense. the best you can do is possibly choose a point in the preimage
2. just because f is continuous does not mean f^-1 is continuous (if it even exists), so you are not free to pull the limit inside f^-1

[timid silo]

yea i was too optimistic

should have used the topological characterization of continuity

thanks

.close

[obtuse pebble]

Channel closed

Closed by @lime harbor

Use .reopen if this was a mistake.

[wild swallow]

you chose your sequence in the wrong space

[wild swallow]

this will work better

[obtuse pebble]

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[native grail]

I cant understand what they are trying to ask for

[candid yarrow]

can you find the equation of the plane containing those three points?

[native grail]

PQR?

do i just add them up

[candid yarrow]

No

[candid yarrow]

Yes, that plane

[native grail]

so i need to find the area equation of PQR

[candid yarrow]

No, the equation of the plane

like ax + by + cz + d = 0

[native grail]

does PQ x PR work

i didnt go for lecture so i dont really understand the 3D vectors thing

[candid yarrow]

Maybe you can see if this helps with that https://www.khanacademy.org/math/multivariable-calculus/thinking-about-multivariable-function/x786f2022:vectors-and-matrices/a/multivariable-calculus-prerequisites

[candid yarrow]

by PQ, do you mean Q - P

[native grail]

PO+OQ

[candid yarrow]

well

[candid yarrow]

do you understand 2D vectors then

[native grail]

yea i understand 2D

[candid yarrow]

so how would you calculate PQ in 2D, if P and Q were two points in 2D

[native grail]

PO+OQ

i was taught that way

O is the origin

[candid yarrow]

not PO - OQ?

[native grail]

wait

oh

[candid yarrow]

or QO - OP

[native grail]

why cannot be PO +OQ

Ah

omg my brain is fried

[candid yarrow]

because if P = Q = (1,1), then PQ would be (2,2) =/= (0,0), but PQ represents the difference between P and Q, which is (0,0) because they are the same

If PQ = QO - OP, then we get that PQ = (0,0), which is correct

[native grail]

yea that makes sense

So if I want to find PQR

i need to use PQ -QR

[candid yarrow]

PQR is a plane, and you are trying to find its equation

[native grail]

.-.

[candid yarrow]

the equation of PQR is different from PQR

[candid yarrow]

not quite

[candid yarrow]

but you do need to find PQ and QR first

[candid yarrow]

and you can do that with this formula

[native grail]

yes

[candid yarrow]

so, can you find PQ and QR in the question?

[native grail]

PQ is (-5,0,-7)

QR is (-3,-2,-6)

omg

its wrong

[candid yarrow]

yep

why?

[native grail]

I dont know

let me try again

is it not PO + OQ

i used PO-OQ

[candid yarrow]

are you sure you computed that correctly

[native grail]

then i got this

let me see

PO is -2,1,-5

no?

[candid yarrow]

wait it should be OQ - OP

so you just subtract the coordinates

you can try drawing a similar situation in 2D to convince yourself

[native grail]

if i flip it, its +PO+OQ

[native grail]

this makes more sense to me

[candid yarrow]

I guess so

[native grail]

ye

[candid yarrow]

what do you get if you use that?

[native grail]

(1,2,-3)

[candid yarrow]

Yes

and for QR?

[native grail]

QR is (-3,0,2)

[candid yarrow]

Yes

Next, to find one of the normal vectors of the plane, you can take the cross product of PQ and QR

[native grail]

why cross product

[candid yarrow]

because the cross product of two vectors is a vector that is perpendicular to both of them

[candid yarrow]

you could try reading some of these to learn more about 3D vectors

[native grail]

ye im reading them

4i+7j-6k

[candid yarrow]

is this the cross product of the PQ and QR?

[native grail]

yep

[candid yarrow]

then you should just have to divide it by its length to get one of the answers

that will make it a unit vector

[native grail]

unit vector is

what does the unit vector represent

[candid yarrow]

it means the vector has length 1

[native grail]

wat

[candid yarrow]

A unit vector is a vector that has length 1

[native grail]

so a unit vector is 'a' unit of the vector

one unit of a given vector

?

[candid yarrow]

No, a unit vector is a special kind of vector

[native grail]

oh

i saw this one question about finding the equation perpendicular to the plane

and

they want to find the shortest distance to (0,0,0)

how to do

[native grail]

if I were to use this question

[candid yarrow]

you mean if you were using the points from this question?

[native grail]

ye

if i have found the perpendicular