help-0

[oak chasm]

-0.1 < x - 1 < 0.1

Now, when we square that, what interval is (x - 1)² in?

[frosty cypress]

what do you mean by that?

[oak chasm]

We have what interval x - 1 is in.

[fringe socket]

is the matrix
1 2 1
0 1 2
0 0 1
in reduced row echelon form?

[oak chasm]

What interval is (x - 1)² in?

@fringe socket Sorry, this channel is busy.

[frosty cypress]

It would be in between?

[oak chasm]

In between what?

[frosty cypress]

-0.5<(x-1)^2<0.5?

[oak chasm]

How did you get 0.5?

[frosty cypress]

Oh from the other equation

[oak chasm]

That's what we're proving.

We can't usually use what we're proving to prove it.

[frosty cypress]

i see

[oak chasm]

So, using -0.1 < x - 1 < 0.1, get the interval for (x - 1)².

[frosty cypress]

so would we square both sides?

[oak chasm]

Well, try it.

@alpine sable Sorry, this channel is busy.

[alpine sable]

damn

[frosty cypress]

Hmm

I get 0.01<x-1<0.01

[oak chasm]

Right, but a number can't be both greater than and less than a number.

[frosty cypress]

true

[oak chasm]

So, 0.01 is the maximum.

What's the minimum?

[frosty cypress]

it would be -0.01, correct ?

[oak chasm]

Can you get a negative number when you square a real number?

[alpine sable]

What if x is a root

[frosty cypress]

Nope

[oak chasm]

What's the lowest squaring can get you?

[frosty cypress]

0

?

[alpine sable]

Real numbers?

[oak chasm]

OK, and do we have the thing we square being 0 in the interval for it?

-0.1 < x - 1 < 0.1

x - 1 is what we're squaring.

Is it allowed to be zero?

[frosty cypress]

yes i think so

[oak chasm]

Well, x - 1 is between -0.1 and 0.1.

Is 0 between -0.1 and 0.1?

[alpine sable]

yes

[frosty cypress]

yes

[oak chasm]

OK, so x - 1 can be 0.

So, (x - 1)² can be 0.

Does that make sense?

[frosty cypress]

Yes, i get that

[oak chasm]

OK, so 0 ≤ (x - 1)² < 0.01.

Does it make sense how I got that?

[frosty cypress]

could you explain it again?

[oak chasm]

OK, so we have -0.1 < x - 1 < 0.1.

So, x - 1 can be close to -0.1.

If it is, then (x - 1)² is close to (-0.1)².

So, (x - 1)² can get close to 0.01.

Does that make sense?

[frosty cypress]

Yes

[oak chasm]

And x - 1 > -0.1.

[keen temple]

pls

help

[oak chasm]

@keen temple Sorry, this channel is busy.

[keen temple]

someone

[keen temple]

what do i do then

[oak chasm]

@keen temple Find a channel that's not in use.

@frosty cypress x - 1 is closer to 0 than -0.1.

When x - 1 is close to -0.1.

Because -0.1 < x - 1

[stable warren]

Hello, I have a question plz. which number - belongs in -3/5 does it belong to 3 or 5

[frosty cypress]

yes i get that

[oak chasm]

@stable warren Sorry, this channel is busy.

OK, so that gives us (x - 1)² < 0.01.

What about the other endpoint?

x - 1 < 0.1

x - 1 is closer to zero when x - 1 is close to 0.1.

So, (x - 1)² will be less than (0.1)².

(x - 1)² < 0.01

[keen temple]

someone help me in #help-9

[oak chasm]

Which is the same as we got before.

[frosty cypress]

Yes

[oak chasm]

@merry coral Sorry, this channel is busy.

Then -0.1 < x - 1 < 0.1

So, x - 1 can be any number from -0.1 and 0.1.

So, x - 1 can be 0.

So, (x - 1)² can be 0²

So, (x - 1)² can be 0.

Does that make sense?

[frosty cypress]

Yes!

[oak chasm]

And 0 is the minimum value you can get when squaring a real number, so we can stop looking for a lower result.

So, 0 ≤ (x - 1)² < 0.01

Now, what we want to prove is that |(x - 1)²| < 0.5.

That means -0.5 < (x - 1)² < 0.5.

We have tighter bounds.

0 ≤ (x - 1)² < 0.01

We can do this:

[frosty cypress]

so the zero goes away?

[oak chasm]

-0.5 < 0 ≤ (x - 1)² < 0.01 < 0.5.

Do you see what I did there?

[frosty cypress]

yes i do

[oak chasm]

OK, so now we can remove some of them.

-0.5 < (x - 1)² < 0.5.

|(x - 1)²| < 0.5

Does that make sense?

[slow inlet]

How can i make this an equation to find t?
H= -5t^2+30t

[oak chasm]

@slow inlet Sorry, this channel is busy.

[slow inlet]

oh? what do i do then

[oak chasm]

Just find a channel where people say they're done or the last timestamp on something people said is at least 30 minutes ago.

[frosty cypress]

i see

[oak chasm]

So, that's almost what we want to prove.

Now we just expand.

And subtract 0.

And then we're done.

[frosty cypress]

expand the (x-1)^2?

[oak chasm]

Yes.

[frosty cypress]

alright

[oak chasm]

|(x - 1)²| < 0.5
|x² - 2x + 1| < 0.5
|(x² - 2x + 1) - 0| < 0.5

So, we started with the assumption and showed that it got us the conclusion.

[frosty cypress]

I see

thank you so much.

[oak chasm]

No problem.

[alpine sable]

anyone available

[oak chasm]

@frosty cypress Are you done with the channel?

[frosty cypress]

yes, thank you

[oak chasm]

@alpine sable OK, go ahead.

You're welcome.

[oak chasm]

OK, do you know which axis is the x axis and which is the y axis?

[alpine sable]

Yes I'm aware

[oak chasm]

OK, what is the x axis labelled with?

[alpine sable]

hours

[oak chasm]

Right, so that's time.

So, the first answer is time in hours.

What's the y axis labelled with?

[alpine sable]

miles

[oak chasm]

Right, so that's distance.

So, the second answer is distance in miles.

Now what's the slope of the line?

[alpine sable]

Not sure

[oak chasm]

OK, do you see where the grid lines cross?

The grid lines are the grey lines.

[alpine sable]

Yup

[oak chasm]

Find two places where the sloped line and two grid lines all cross at once.

[alpine sable]

20 to 2?

[oak chasm]

20 and 2 are where two grid lines meet.

But the graphed line doesn't go there.

Look at 40 and 1.

[alpine sable]

oh

[oak chasm]

Those are where two grid lines cross, right?

Also the graphed line crosses there.

[alpine sable]

Yeah

[oak chasm]

So, where's another place where that happens?

[alpine sable]

50 and 2?

[oak chasm]

No, the grid lines coming from the left are at 40 and 60.

So, there's no grid line for 50.

[alpine sable]

🤦‍♂️ I was looking at the wrong one

80 and 2?

someone help me with this assignment please

[oak chasm]

@alpine sable Sorry, this channel is busy.

[alpine sable]

what channel do I go in?

[oak chasm]

Good, 80 and 2 works.

[alpine sable]

Okay

[alpine sable]

Take x and y in one side and the constant value in the other side. Then substract them

Y will cut out

Hence, you'll find the value of x

[alpine sable]

this good?

[oak chasm]

#help-3 if you ask your question there before someone gets there.

@alpine sable Nope, we find the slope a bit of a different way.

We get the two points.

(1, 40) and (2, 80).

[alpine sable]

Ohh

[oak chasm]

We put the y values on top of a fraction. We put the x values on the bottom.

And we subtract them on top and bottom.

(40 - 80)/(1 - 2)

See how we have the first point's y minus the second point's y on top?

[alpine sable]

-40?

[oak chasm]

Right.

See how we have the first point's x minus the second point's x on the bottom?

[alpine sable]

-1?

[oak chasm]

So, we get -40/-1.

Now we simplify that.

So, the slope is about 40.

And the top was the y coordinates.

The y axis is miles.

The bottom was the x coordinates.

The x axis is hours.

So, it's 40 miles/hour.

[alpine sable]

🤯

[oak chasm]

That's about the speed they're going at.

[alpine sable]

Mind if we do another quick one?

[oak chasm]

OK.

[oak chasm]

OK, what's the x axis labelled as?

[alpine sable]

points

[oak chasm]

Nope.

[alpine sable]

games

[oak chasm]

Right, x axis is left and right axis.

[alpine sable]

so games in points?

[oak chasm]

So, the first answer is number of games.

What's the y axis labelled as?

[alpine sable]

points?

[oak chasm]

Right, now this one is tricky.

Points for what?

[alpine sable]

points for games?

[oak chasm]

Right, but do you mean points in one game or points in all games so far or what?

[alpine sable]

all?

[oak chasm]

Right.

The y axis is the number of points in all games so far added together.

Does that make sense?

[alpine sable]

Yes

[oak chasm]

OK, so something like that is the second answer.

Now we find the slope again.

Where do two grid lines and the graphed line meet?

[alpine sable]

15, 1?

[oak chasm]

Good. We need one more point.

[alpine sable]

20, 2?

[oak chasm]

No, 20 and 2 is where two grid lines meet, but the graphed line doesn't go there.

[alpine sable]

I don't spot anymore

[oak chasm]

OK, there are two more.

One is at 0 0.

See where the axes meet?

The graphed line also goes there.

[alpine sable]

Oh!

[oak chasm]

There's also 30 and 2.

The next line above 25 is 5 above 25.

Because the lines are all 5 above the last one.

[alpine sable]

So 15,1 0,0 and 30, 2?

[oak chasm]

Right.

So we need two of those.

Which two do you want to use?

[alpine sable]

15, 1 and 30, 2

[oak chasm]

OK, so we put the y values on top, the x values on bottom.

Then you subtract on top and bottom.

(15 - 30)/(1 - 2)

[alpine sable]

-15?

[oak chasm]

OK, what about the bottom?

[alpine sable]

and -1?

[oak chasm]

OK, what's -15/-1 simplified?

[alpine sable]

Have you learnt trigo? @alpine sable

Uhm 15?

[oak chasm]

Right.

So, the slope is 15.

[alpine sable]

Okay

[oak chasm]

The y axis is on top, and it's points.

The x axis is on bottom and it's games.

So, we have 15 points/game.

Now there's one thing you should know.

[alpine sable]

Yeah?

[oak chasm]

x comes before y in the alphabet.

So, when you write the points, you do (x, y).

Like (1, 15) and (2, 30).

Does that make sense?

[alpine sable]

Yup

[oak chasm]

So, the third answer is 15 points/game.

They get 15 points per game.

[alpine sable]

Hey, won't you mind if I mention a point here?

[oak chasm]

Sure.

Go ahead.

[alpine sable]

If he has learnt trigo, then we can find the slope quite easily.

We know by trigonometry that tan theta is the slope which is perpendicular / base, i.e. , 30/2= 15

[languid stone]

Help I got a test about numeric sequence and I am terrible can anyone help?
Numeric sequence is like
2 , 4 , 6 , 8 , ? Obviously this is 10 but it’s gonna be harder

[oak chasm]

@alpine sable Are you done with the channel?

[alpine sable]

Yes.

[oak chasm]

@languid stone There are a few different methods.

[alpine sable]

I can't see anything else without 10

[oak chasm]

You can find the amount of increase between the numbers.

Here they all increase by 2.

[alpine sable]

He/she's saying it's not 10

[oak chasm]

Then you have 2 2 2 as the list of increases.

[alpine sable]

Hmm...

[oak chasm]

Then you find the increases there.

[hoary shell]

That one was just an example?

[oak chasm]

0 0

Does that make sense?

[languid stone]

Yes but it’ll be harder

[oak chasm]

2 4 6 8
 2 2 2
  0 0

[glass lichen]

Yeah... do ask the question you need help with

[oak chasm]

See how each line is the increases of the previous line?

[glass lichen]

Not an XY problem

[oak chasm]

When you get the same thing each time, you can stop.

And that's the rule.

So, we got 2 2 2.

The numbers are all the same.

So, we can stop.

That's the rule.

[languid stone]

Like 8 16 20 40 ?

[oak chasm]

You add 2 each time.

OK, so let's look at that.

What is the increase each time?

[glass lichen]

That's only true for polynomial rules btw Chai

[oak chasm]

Right, but then you do successive factors and so on.

[languid stone]

I haven’t a clue it’s not like +4 every-time

[oak chasm]

OK, but get the list.

8 16 20  40
 8  4  20
  -4  16
   20

[hoary shell]

how come it’s not negative 4?

[oak chasm]

Oh.

[alpine sable]

Sorry

[glass lichen]

Yeah this is insufficient here Chai

[oak chasm]

OK, so that won't work.

Now there's something else you can do.

You can find what you multiply each time.

[languid stone]

I understand but that doesn’t give me the answer yk

[oak chasm]

@fast dagger Sorry, this channel is busy.

[fast dagger]

oh

[oak chasm]

@languid stone Right, not on this problem.

But it's useful in others, so you should learn it.

And sometimes, you can do combinations.

Like one level is what you multiply, the next is what you add.

And that will get you the pattern for a lot of sequences.

You find where one level is the same thing over and over.

And then you work back.

Like let's take this sequence:

[languid stone]

Explain…

[oak chasm]

1, 2, 6, 24, 120

What's the next number?

Well, let's do multiplications.

[hoary shell]

720

[oak chasm]

1 2 6 24 120
 2 3 4  5

Now let's do addition.

[thin pendant]

i hope i am not cutting into the convo, but i just want to check my answer for this question

[oak chasm]

1 2 6 24 120
 2 3 4  5
  1 1  1

@thin pendant Sorry, this channel is busy.

See how we got the same thing over and over?

Now we get the next number this way.

[hoary shell]

this is really cool I’ve never even seen a sequence that would use two rows of multiplication I don’t think

never learned this method

[languid stone]

I looked it up it’s 44 you multiply by 2 then add 4

[oak chasm]

1 2 6 24 120
 2 3 4  5
  1 1  1  1

We continue the pattern where it's the same.

That level is addition.

So, we add that to the last of the row above.

1 2 6 24 120
 2 3 4  5   6
  1 1  1  1

[languid stone]

6+4 ain’t 24?

[oak chasm]

No, the second row is what you multiply.

The third row is what you add in the second row.

Does that make sense?

[hoary shell]

And 24x5=120 120x6=720

Would this method work for Richis initial sequence?

[languid stone]

Yes

[hoary shell]

or a different method?

[oak chasm]

OK, now to get the next number.

1 2 6 24 120
 2 3 4  5
  1 1  1  1

We continue the constant pattern in the last row.

1 2 6 24 120
 2 3 4  5   6
  1 1  1  1

We use that to get the next number in the second row.

   1 2 6 24 120 720
×:  2 3 4  5   6
+:   1 1  1  1

We use that to get the next number in the sequence.

Does that make sense?

[languid stone]

Yes I only don’t understand why you have the 1’s

[oak chasm]

Well, the third row is what you add between each number in the second row.

   1 2 6 24 120 720
×:  2 3 4  5   6
+:   1 1  1  1

[alpine sable]

are you guys finding nth term of geometric sequence?

[oak chasm]

Well, sequences in general.

[languid stone]

So after 720 it’s 720 x 7? Ik it’s not needed but just to make sure I understand

[oak chasm]

@languid stone Yes.

Let's try it with the square numbers plus 1.

[languid stone]

This technique always works?

[oak chasm]

Well, it works for the common sequences.

[alpine sable]

why not use formula,
nth term = ar^(n-1)

[languid stone]

Have you an uncommon since I’ll get those on the test most likely

[oak chasm]

Well, you can have 8 5 4 9 1.

This method won't work on that.

Those are 0 to 10 in alphabetical order.

But on things like aptitude tests and college entrance tests and all that, when they do sequences, this method works on a lot of them.

Another it won't work with is 1 1 2 3 5 8 13

The Fibonacci sequence.

But you can still use it to figure it out.

[hoary shell]

yeah I have never been taught a method for figuring out sequences liek this usually the ones they test me on you can think of in your head

[oak chasm]

   1 1 2 3 5 8 13
+:  1 1 2 3 5 8

Notice that the addition line is the same as the sequence.

So, the next addition is 13.

   1 1 2 3 5 8 13
+:  1 1 2 3 5 8  13

   1 1 2 3 5 8 13  21
+:  0 1 1 2 3 5  8

So, we didn't get to a constant pattern.

But it still helped.

[hoary shell]

I wonder if there are ones similar to continued fractions where it goes forever

not just trivially like Fibonacci

[languid stone]

So the next is 21 after 13 just for my understanding (second row btw)

[alpine sable]

Is this channel occupied?

[glass lichen]

Yes, 13+8 is 21

[oak chasm]

Oh, sorry.

   1 1 2 3 5 8 13  21
+:  0 1 1 2 3 5  8

Made a bit of an error above.

But you can see that the sequence is repeated below.

[languid stone]

I'm looking at the second row 0+1 = 1 1+1 = 2 2+3 = 5 so 5+8 = 13 and 8+13 = 21

[oak chasm]

Right.

That's how the Fibonacci sequence works.

You start with 1 and 1.

Then you add the previous two numbers together.

[languid stone]

yes that one I understand

It's the ones as I mentioned earlier Like 8 16 20 40 ?

[glass lichen]

Yeah that's a bs one

[oak chasm]

That one I don't know the pattern of.

Maybe times 2 then add 4.

[languid stone]

yes I looked it up and that one's the answer.

[oak chasm]

8 16 20 40 44 88

[glass lichen]

Probably

[languid stone]

Especially where it isn't consistent like 16, 48, 12, 36, ?

[hoary shell]

So u couldn’t use the same method on those types of sequences where multiplication and addition are combined in the same step?

[oak chasm]

That looks like times 3, times 1/4, ...

[hoary shell]

Yeah

[oak chasm]

Well, you can use it to help.

[languid stone]

yes your eye sees that but I need one of them 1 1 2 3 5 8 13 21 +: 0 1 1 2 3 5 8 to see that

so how do I use your technique for that one?

[oak chasm]

16 48   12 36
  2  1/4  2

And then you can guess that it's just 2, 1/4, 2, 1/4, 2, ...

[languid stone]

16 x 2 is 32

[oak chasm]

Oh, sorry, 3, not 2.

16 48   12 36
  3  1/4  3

[languid stone]

Ahh thanks

[oak chasm]

You can also have a combined multiplication and addition row.

[languid stone]

I can't thank you enough I'll use your method and I'll just practice so I can do it faster and better, thank you so much 🙏

[oak chasm]

 ×2  +4  ×2```

And then you guess that it's just ×2, +4, ×2, +4, ...

You're welcome.

[languid stone]

F I might need your help again

[oak chasm]

Oh, OK.

[languid stone]

3968, 63, 8, 3, ?

I can't do it with big numbers beyond 100 for some reason

[oak chasm]

OK, that one's 3² - 1 is 8.

So, add 1 and take the square root.

[alpine sable]

20-7x = 6x-6
I know the first part
[-20-6] = -7x-6x
So what's before = gets an inverted sign while what's after remains with the same sign.
But the 2nd part doesn't work with the same logic as the first because it's
-20-6 = [-7x-6x]
why does -7x keep its sign while 6x gets a +?
Do "coeffiecients" work with an inverted logic? So what's before the = stays with the same signs while what's after gets an inverted sign?

[oak chasm]

sqrt(3968 + 1) = 63
sqrt(63 + 1) = 8
sqrt(8 + 1) = 3
sqrt(3 + 1) = 2

[languid stone]

AHH

thanks once again!

[oak chasm]

No problem.

For that one, you can do the multiplication row with decimals.

    62.9  7.9 2.6```

Then the second row is a lot like the first.

Multiplication in reverse, I mean.

Division, you could say.

[alpine sable]

bro, any term that is in addition/substraction state, when it changes side, it's sign gets changed

[oak chasm]

@alpine sable

20 - 7x = 6x - 6

What do you want to do first?

[alpine sable]

do stuff with 20 and -6

[oak chasm]

OK, which do you want to get rid of?

@alpine sable The idea is to put the variable you're isolating on one side and the rest of the stuff on the other side.

So, we need to get the non-x terms on one side.

We can get rid of the 20 or the -6 and then only one side will have the non-x terms.

[alpine sable]

yeah I get it

I found another method 😎

[oak chasm]

Oh, good!

[arctic blade]

Anyone any idea how to prove $\sum_{n=1}^{\infty}{\frac{1}{n\sin^2(n)}}$ diverges?

[ocean seal]

clr

[arctic blade]

Wolframalpha suggests a comparison test, but doesn't say what to compare it with

[sly mantle]

prove each term is >=1/n

[safe roost]

so would that become an induction proof?

[austere star]

Can someone help me?

[sly mantle]

not necessarily

[safe roost]

sheeit, that's the kind of question i should be able to answer right now and i can only partially get there

the range of sin^2(n) is (0,1]

and as n -> inf, 1/n goes to zero

[arctic blade]

@sly mantle oh god you'r eright, i forgot about the square of the sin

[safe roost]

clr, were you able to work it out? if so feel free to explain it to me 😄

[alpine sable]

$cos^{2}x=1-sin^{2}x$

[arctic blade]

@safe roost You can use the comparison test. As every term is >= 1/n my series diverges if and only if 1/n diverges

[ocean seal]

AB

[arctic blade]

and since 1/n is the harmonic series and can be easily proven to diverge, my series must diverge.

[hybrid plume]

@alpine sable how will that help to reduce the denominator? Sorry haven’t practiced these

[safe roost]

@arctic blade i haven't been taught the comparison test, guess i'll head to the google machine and learn some stuff 😉

[alpine sable]

$1-sin^{2}x=(1+sinx)(1-sinx)$

[ocean seal]

AB

[arctic blade]

@safe roost quite a useful test. seems to pop up whenever any other etst doesn't work, i'm always slow at spotting which series to use however

[alpine sable]

$sinx-1=-1(1-sinx)$

[ocean seal]

AB

[hybrid plume]

Life saver

[arctic blade]

@safe roost this is how i wrote it in my notes if it helps. i think it is correct haha https://i.clr.is/5aHRYwhcL

[hybrid plume]

So I can search up how to do them

@alpine sable you know what those are called?

[alpine sable]

i didn't encounter these type of questions yet

[hybrid plume]

Do you know how to do them?

Any idea

[alpine sable]

no idea

[hybrid plume]

<@&286206848099549185> anybody know how to do that or the name of the type of question?

@oak chasm Do you know how to do this? I’m ok with the integrals just need to understand which way to start

[alpine sable]

is this right

[hybrid plume]

Thank God these are due soon

[alpine sable]

I think you got to differentiate both sides

I've been trying to do these steps on other equations but it doesnt work out, the only difference is usually only the sign at the end result... what are constant terms and based on what do they change signs??

[oak chasm]

Differentiate the partial integration result and set it equal to what you're integrating:

f(x) sin(x) = (-f(x) cos(x))' + 4x³ cos(x)

@hybrid plume ^

f(x) sin(x) = f(x) sin(x) - f'(x) cos(x) + 4x³ cos(x)

f'(x) cos(x) = 4x³ cos(x)

f'(x) = 4x³

The derivative of the integral is what you're integrating.

The derivative of the original integral is what you're integrating.

The derivative of the partially-done integral is what you're integrating.

So they're equal.

Does that make sense?

[hybrid plume]

What’s the original integral here?

[ocean seal]

Chai T. Rex

[oak chasm]

The derivative of that is f(x) sin(x).

[hybrid plume]

Why isn’t a -cosx involved

?

[oak chasm]

Sorry?

[hybrid plume]

Wait nvm

[oak chasm]

Derivative is the inverse of integral.

So derivative undoes integral.

[hybrid plume]

Ohhhhhh yes

Integral and derivative cancel out

I understand that

[ocean seal]

Chai T. Rex

[oak chasm]

And because those two things you're taking the derivative of are equal, then the derivatives are equal.

[hybrid plume]

So to show workings I should put everything in brackets and a derivative sign to show I’m deriving everything?

(intf(x)sinxdx)’

Yes

[ocean seal]

Chai T. Rex

[oak chasm]

Something like that.

[hybrid plume]

So the derivative of the integral of 4x^3cosx is 4x^3cosx?

[oak chasm]

Exactly.

You need to do the derivative of all terms on the right.

You only did the integral term.

Or rather you didn't use the product rule.

[f(x) \sin(x) = (-f(x) \cos(x))' + 4x^3 \cos(x)]

[ocean seal]

Chai T. Rex

[oak chasm]

Use the product rule on (-f(x) cos(x))'.

[hybrid plume]

And in fg’+gf’ the derivative of (-f(x)) is (-f(x))’?

As in you dont simplify it further?

[alpine sable]

https://cdn.discordapp.com/attachments/269573202018041856/890315737997123594/unknown.png

any help, im stuck

[oak chasm]

@alpine sable Sorry, this channel is busy.

@hybrid plume -f'(x)

And we don't know the derivative of f yet, so we can't go further.

So we leave it as -f'(x).

Yes, just close the parenthesis before (-f'(x).

Should be cos(x) (-f'(x)) with a second closing parenthesis.

[hybrid plume]

Yes yes

[oak chasm]

Now simplify.

[alpine sable]

some1

[hybrid plume]

How do you simplify that??

I’m behind in my course I just need to finish these questions then I can catch up sorry for so many questions

[oak chasm]

Use algebra.

[alpine sable]

if u have thirty students and u have 3 groups, group A B and C and u have 12 in A 10 in B and 8 in C
to find the probability of two students being in the same group
would u just do something like
(12/30)(11/29) + (10/30)(9/29) + (8/30)(7/29)
or would u do that and then divide by like 30 Choose 2
or would it be like
[(12C2) + (10C2) + (8C2)]/(30C2)

[oak chasm]

For example, there's a simpler way to write -f(x) (-sin(x)).

There's a simpler way to write + cos(x) (-f'(x)).

@alpine sable Sorry, this channel is busy.

[hybrid plume]

I didn’t learn this what is the way?

[oak chasm]

What happens when you multiply a negative times a negative?

[hybrid plume]

Lol ohhh

[placid pollen]

yeah looks right to me

sorry i meant the bottom one

the [(12C2) + (10C2) + (8C2)]/(30C2)

[hybrid plume]

But the cosx(-f’(x)) how does that simplify?

[placid pollen]

139/435

[alpine sable]

thats what i was thinking too, tysm

[placid pollen]

that's exactly how i would've done it had i not seen ur method

[placid pollen]

np

[oak chasm]

@hybrid plume What happens when you multiply a positive times a negative?

[alpine sable]

you are actually adding/substracting same thing on both sides like,
$-2x+16=5x+9$
$-2x+16-9=5x+9-9$
quick way is that, addable things change signs when moved to other side

[ocean seal]

AB

[hybrid plume]

Oh I get that but are there any identities from the two subtracting?

[oak chasm]

• cos(x) (-f'(x))

That's a positive times a negative.

What's a positive times a negative?

[hybrid plume]

Negative

[oak chasm]

And what's adding a negative?

[hybrid plume]

Subtraction

[oak chasm]

• cos(x) f'(x)

See how that was simplified?

[hybrid plume]

Yes I would’ve don’t that but now I’m left with three terms on the right

[ocean seal]

Chai T. Rex

[hybrid plume]

Yeah I have that now

[oak chasm]

OK.

So, two of the terms are the same.

What can you do with that?

[hybrid plume]

Cancel them

Then move cos(f’(x)) to the left side?

[oak chasm]

Yes.

[ocean seal]

Chai T. Rex

[hybrid plume]

Then cancel the cosines

[oak chasm]

Right.

[hybrid plume]

Then get the integral

[ocean seal]

Chai T. Rex

[oak chasm]

What do you get when you integrate?

[hybrid plume]

x^4+C

[oak chasm]

Right.

And it says f(1) = 2.

So, get f(1) according to your integral.

And find C.

[hybrid plume]

C=1

[oak chasm]

Right.

So, f(x) = x⁴ + 1

[hybrid plume]

Man you are something else walking me through that thank you so much

[oak chasm]

You're welcome.

[alpine sable]

would this be right?

[wary stream]

Why are you using multiple channels? You're already in channel 9

[alpine sable]

ik

[oak chasm]

,w Log[6, 22]

[alpine sable]

but i need answers dont wanna bother the same people

[ocean seal]

Results provided by WolframAlpha

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

Requested by Chai T. Rex#7323

[quartz osprey]

does anyone know how ot do the second part

oh helo dldh

[oak chasm]

Looks good.

[alpine sable]

thanks

[wary stream]

One, we're not here to give answers, two, stick to one channel, don't use multiple channels

[vapid herald]

could I get some help with this problem

[hybrid plume]

Chai can you help with that word problem?

[oak chasm]

@hybrid plume Speed is how fast position changes. Acceleration is how fast speed changes.

[hybrid plume]

The integral of acceleration is velocity so int(a(t))dt=v(t)?

[oak chasm]

Yes, and the integral of v(t) is x(t).

So, v(t) = 88, x(t) = 900.

a(t) is constant.

v(t) = a(t) t + Cᵥ = 88
v(0) = a(0) 0 + Cᵥ = 0
Cᵥ = 0

v(t) = a(t) t = 88

Do you see how I got that?

[hybrid plume]

What’s Cv here?

[oak chasm]

The integration constant for the velocity function.

[hybrid plume]

So first you’re creating the equation for velocity

[oak chasm]

Yes.

Then I find the velocity at t = 0.

[hybrid plume]

A(t)t is like mx?

As in mx+b

[oak chasm]

Yes.

The integral of a(t) when a(t) is a constant is a(t) t.

Just like integrating all constants.

The integral of 3 dt is 3t.

[hybrid plume]

Have you integrated already?

[oak chasm]

The integral of a(t) dt is a(t) t when a(t) is a constant.

[hybrid plume]

a(t)t+Cv looks like the answer for an integral

[oak chasm]

It is.

v(t) = the integral of a(t) dt.

v(t) = a(t) t + Cᵥ, since a(t) is a constant.

[hybrid plume]

So in my workings should I write int(v(t))dt is equal to ^

[oak chasm]

Sorry?

[hybrid plume]

One sec I’ll write it

[oak chasm]

OK, is the integral of velocity the acceleration?

[hybrid plume]

Wait replace v(t)dt on the left with a(t)dt

[oak chasm]

OK.

But here's how it works in an overview.

You have position.

Velocity is the rate of change of position.

So, velocity is the derivative of the position.

Acceleration is the rate of change of velocity.

So, acceleration is the derivative of velocity.

[hybrid plume]

I understand

[ocean seal]

Chai T. Rex

[hybrid plume]

Yes I have that down

[oak chasm]

Then you find Cᵥ.

The way to do that is to get rid of the a(t)t term.

That can be done by making t = 0.

v(0) = a(0)0 + Cᵥ.

v(0) = Cᵥ.

And we know the starting velocity is at rest, or 0.

So, Cᵥ = 0.

We fill that back in.

v(t) = a(t) t.

Now we have a simplified expression for v(t).

[ocean seal]

Chai T. Rex

Chai T. Rex

[oak chasm]

Same thing, we get rid of the other term by setting t = 0.

[ocean seal]

Chai T. Rex

[oak chasm]

x(0) is 0, so Cₓ is 0.

So, you can simplify x(t).

[ocean seal]

Chai T. Rex

[oak chasm]

Now, we know v(t) = 88. We know x(t) = 900.

So,

88 = a(t) t
900 = ½ a(t) t²

[hybrid plume]

I understand that

Can I solve for the acceleration now using one of those?

Are you still there Chai? @oak chasm

[oak chasm]

a(t) t - 88 = 0
½ a(t) t² - 900 = 0

½ a(t) t² - 900 = a(t) t - 88
½ a(t) t² - a(t) t - 812 = 0

So we get both equations equal to zero.

Since they're equal to zero, they're equal to each other.

Then we get it ready for the quadratic formula.

Then, we use the quadratic formula.

We get t in terms of a(t).

Then we use a(t) t = 88.

We fill in t in terms of a(t).

And we have something in terms of a(t) = 88.

And then we solve for a(t).

[hybrid plume]

Just solve this?

[oak chasm]

Yes, get t in terms of a(t).

[hybrid plume]

Thank you 🙏

[oak chasm]

No problem.

[quartz osprey]

how do i find the equatrion of the tanget line?

y=mx+b i know

9rt2=rt2/2(9) +b

b=0 right?

[oak chasm]

@quartz osprey Use point-slope form.

[quartz osprey]

im ngl

i dont even remember that

[oak chasm]

y - the y coordinate = m(x - the x coordinate).

[quartz osprey]

(y2-y1)+m(x2-x1)

liekthat?

[oak chasm]

Right.

[quartz osprey]

is it = in the middle?

[oak chasm]

But y instead of y2.

[quartz osprey]

oh gotcha

[oak chasm]

Yes.

[quartz osprey]

(9rt2-y)=rt2/2(9-x)

like that?

[oak chasm]

Almost.

y - 9 sqrt(2) = sqrt(2)/2 (x - 9).

[quartz osprey]

ah i got it backwards

[oak chasm]

Now expand the right.

Then get y by itself.

[granite lily]

I am very stuck on this problem any help ?

[quartz osprey]

y = sqrt(2)/2 (x - 9) + 9 sqrt(2)

[oak chasm]

OK, that's fine. Now get it in slope intercept form.

[quartz osprey]

sec

how do i do that

[oak chasm]

Use the distributive property.

[quartz osprey]

ah

y=rt2/2x-9rt2/2 + 9rt2

[oak chasm]

Yes, now simplify the constant terms.

[quartz osprey]

😅

im ngl

[oak chasm]

You have something minus half of the something.

[quartz osprey]

i might have forgotton the rule on those

lemee write it down

[oak chasm]

Like if you have a pie and you eat half of it, how much do you have left?

[quartz osprey]

wait im combining the -9rt2/2 + 9rt2

[oak chasm]

Yes.

[quartz osprey]

oh its gonna be 9rt2/2 still

[oak chasm]

Right.

y = sqrt(2)/2 x + 9 sqrt(2)/2.

[quartz osprey]

ah i see

[oak chasm]

Now it's in slope-intercept form.

[quartz osprey]

is there a way doing it at point slope form?

[oak chasm]

Yes, we did that a while ago, but they don't want it in that form, so we had to convert it to the form they wanted.

[quartz osprey]

ohh

[oak chasm]

We got y - 9 sqrt(2) = sqrt(2)/2 (x - 9).

That was our point-slope form, which was easy to get, since we had a point and the slope.

[quartz osprey]

i see

tyty

[oak chasm]

You're welcome.

[errant wolf]

help pls

[oak chasm]

@errant wolf Use proof by contradiction.

[errant wolf]

so would i say that x is irrational and y is rational?

then xy is rational>

?

[alpine sable]

that xy is rational i believe

[oak chasm]

No, you don't change the sets they're in.

Oh, wait, never mind.

You have x rational ∧ y irrational → xy irrational.

[errant wolf]

oh

[alpine sable]

So it would not be proof by contradiction then?

[alpine sable]

Im just interested

[oak chasm]

¬(x rational ∧ y irrational → xy irrational)
¬(¬(x rational ∧ y irrational) ∨ xy irrational)
¬(x irrational ∨ y rational ∨ xy irrational)
x rational ∧ y irrational ∧ xy rational

So, x is rational and y is irrational and xy is rational.

You want to prove that leads to a contradiction.

If it does, then x rational ∧ y irrational → xy irrational is true.

@errant wolf Does that make sense?

Or, a simpler way.

[errant wolf]

how do i find the contradiction

[oak chasm]

→ is only false when you have T → F.

So, the left is true and the right is false.

[errant wolf]

yes

[buoyant leaf]

what are the properties of multiplication?

[oak chasm]

x rational ∧ y irrational ∧ xy rational.

[buoyant leaf]

one of them will be helpful here

[oak chasm]

So, now we represent x and xy as rationals.

x = a/b, xy = c/d.

[errant wolf]

ok

[oak chasm]

Solve for y.

What do you get?

[errant wolf]

do you just move the x to the other side?

[placid pollen]

ok, so by applying a similar argument to Zeno's Paradox of Achilles and the tortoise, first of all convert everything to the same units. The car is 0.11km behind the truck. By the time the car has caught up this 0.11km, the truck will have moved on another 75/88 of this distance, and by the time the car has caught up that distance the truck will have moved on 75/88 of that distance again. So the distance covered by the time they meet is equal to the sum of the infinite geometric series 0.11 + 0.11 x 75/88 + 0.11 x (75/88)² + .... = 0.11/(13/88) = 242/325km ≈ 0.745km. Since this is the total distance covered by the car, who is behind (since the first term included the 110m they had to catch up, to find the time taken, we divide the distance by the speed to get (242/345)/88 =11/1300h as an exact fraction ≈ 0.00846h which is 0.5 minutes or roughly 30 seconds.

[oak chasm]

@errant wolf y = xy/x

[errant wolf]

ok

[oak chasm]

So, what's y?

[buoyant leaf]

you... definitely don't need to use transfinite summation to solve this lmao, also it's being taken care of in #help-1

[placid pollen]

oh right i see, sorry i didn't realise

[errant wolf]

...

[jovial galleon]

Help?

[oak chasm]

@jovial galleon Sorry, this channel is busy.

[jovial galleon]

Oh ok

[errant wolf]

don't know man

[oak chasm]

OK, x = a/b, xy = c/d.

y = xy/x

y = (c/d)/(a/b)

y = (bc)/(ad).

Now b and c are integers, so bc is an integer.

a and d are integers, so ad is an integer.

So, we have an integer over an integer, which is a rational number.

So, y is a rational number.

[errant wolf]

oh ok

how did you make y = (c/d)/(a/b) turn into y = (bc)/(ad).

[oak chasm]

Dividing fractions is multiplying by the reciprocal of the bottom.

[errant wolf]

oh

[oak chasm]

(c/d)/(a/b) = (c/d)(b/a) = (bc)/(ad).

[errant wolf]

ok i get that

[oak chasm]

OK, so do you see how y is a rational number?

[errant wolf]

yes

[oak chasm]

But we had x rational ∧ y irrational ∧ xy rational.

So, by ∧ elimination, y irrational is true.

And by what we just discovered, y rational is true.

So, both y is irrational and y is rational.

Which is a contradiction.

[errant wolf]

yea

[oak chasm]

So, the original statement is true.

x rational ∧ y irrational → xy irrational

[errant wolf]

ok ty

[oak chasm]

No problem.

[errant wolf]

more questions are probably expected from me later

[alpine sable]

how would you check the left hand and right hand limit of an equation to know if the limit is +infinity, -infinity, or DNE

[oak chasm]

@alpine sable If the left and right limits agree, the limit is what they agree on. If they don't agree, the limit doesn't exist.

[alpine sable]

oh ok that makes sense

thanks

[oak chasm]

No problem.

[hybrid plume]

Isnt v(0)=88?

[oak chasm]

No, v(t) is 88.

v(0) is the velocity at t = 0.

The problem says it starts from rest.

So, at t = 0, v(0) = 0.

[hybrid plume]

The question didn’t say at rest

If it didn’t say at rest could v(0) be 88?

[buoyant leaf]

I would have used a slightly different method, although functionally equivalent

$a \in \mathbb{Q} \land b \in \mathbb{Q} \implies \frac{a}{b} \in \mathbb{Q}$

$ xy \in \mathbb{Q} \implies (\frac{xy}{x} \in \mathbb{Q} | x \neq 0) \implies (y \in \mathbb{Q} | x \neq 0)$

[alpine sable]

Can someone help me with box plot

[ocean seal]

Fawful

[hybrid plume]

Omg I’m so slow

Position of time at time=0 would position be 0?

Nvm the question answers that too

[oak chasm]

Yes, that's right.

You do that to make it easy.

0 to start, then 900 feet down the runway.

You could do -450 to 450 instead or something, but that just complicates it.

[hybrid plume]

Yes 👍

I don’t think I can use the quadratic formula yet

Since 2*a(t) is unknown

Same with 1*a(t)

[oak chasm]

You can.

The quadratic formula doesn't care whether you have variables you're not solving for in it.

After all, the quadratic formula itself has a, b, and c.

So, you can solve ax² + bx + c = 0 even if you leave them as variables.

[fallow kestrel]

please help me I got 3 hours left deadline

[oak chasm]

@fallow kestrel Sorry, this channel is busy.

[fallow kestrel]

I'm dead if i dont anß2er in 3 hours

[alpine sable]

maybe you could substitue the values

[hybrid plume]

@oak chasm Only just submitted it but you brought my grade on that from a 50% probably to maybe a 90%+. And I actually understand it all now. Thank you!

[oak chasm]

@alpine sable They moved to #help-2.

You're welcome. Glad it helped.

[fallow kestrel]

alright

[ocean seal]

?cris

@Chai T. Rex a)

\begin{gathered}f(x)=\sqrt{x+7}\\f(2)=\sqrt{2+7}\\f(2)=\sqrt{9}\\f(2)=3\end{gathered}f(x)=x+7​f(2)=2+7​f(2)=9​f(2)=3​

b)

\begin{gathered}f(x)=|x-4|\\f(12.5)=|12.5-4|\\f(12.5)=|8.5|\\f(12.5)=8.5\end{gathered}f(x)=∣x−4∣f(12.5)=∣12.5−4∣f(12.5)=∣8.5∣f(12.5)=8.5​

c)

\begin{gathered}f(x)=9-x^2\\f(-3)=9-(-3)^2\\f(-3)=9-9\\f(-3)=0\end{gathered}f(x)=9−x2f(−3)=9−(−3)2f(−3)=9−9f(−3)=0​

d)

\begin{gathered}f(x)=\sqrt{x+7}\\f(5)=\sqrt{5+7}\\f(5)=\sqrt{12}\\f(5)=\sqrt{4\cdot3}\\f(5)=\sqrt{4}\cdot\sqrt{3}\\f(5)=2\sqrt{3}\end{gathered}f(x)=x+7​f(5)=5+7​f(5)=12​f(5)=4⋅3​f(5)=4​⋅3​f(5)=23​​

e)

\begin{gathered}f(x)=9-x^2\\f(1.5)=9-1.5^2\\f(1.5)=9-2.25\\f(1.5)=6.75\end{gathered}f(x)=9−x2f(1.5)=9−1.52f(1.5)=9−2.25f(1.5)=6.75​

is this correct?
```Compilation error:```! Package amsmath Error: \begin{gathered} allowed only in math mode.

See the amsmath package documentation for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.57 \begin{gathered}f
                      (x)=\sqrt{x+7}\\f(2)=\sqrt{2+7}\\f(2)=\sqrt{9}\\f(2)=3...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.```

[oak chasm]

@fallow kestrel Yes, that's right.

[fallow kestrel]

ill submit it to my google class now thank you so much

[oak chasm]

No problem.

[oak chasm]

@errant wolf Expand (a - b)².

[errant wolf]

how do you know when to use contradiction or contrapositive

[oak chasm]

Well, you think about how you'd do each one.

Then you get an idea of how hard each one is.

Contrapositive looks easier here.

[errant wolf]

a^2-2ab+b^2

[oak chasm]

a² + b² = 2ab → a = b.

OK, so a² - 2ab + b²

How is that related to a² + b² = 2ab?

[errant wolf]

isnt it just added to the other side

[oak chasm]

Yes.

[errant wolf]

ok

[oak chasm]

So, subtract 2ab from both sides.

[alpine sable]

Is this room full?

[oak chasm]

@alpine sable Yes.

[errant wolf]

wouldn't that just go back -2ab

[oak chasm]

Yes.

But what would the equation be?

[errant wolf]

a² - 2ab + b²

[oak chasm]

That's not an equation.

[errant wolf]

oh

[oak chasm]

What equation do you get?

[errant wolf]

um

idk

[oak chasm]

You have a² + b² = 2ab.

What did you have to do to the left side to get a² - 2ab + b²?

[errant wolf]

-2ab?

[oak chasm]

So, do that to the right side.

[errant wolf]

but then doesn't it cancel out?

[oak chasm]

Sorry?

[errant wolf]

since 2ab was on the right side

[oak chasm]

Are we not allowed to have 0 on one side?

[errant wolf]

and then we -2ab

[oak chasm]

Right.

[errant wolf]

oh i guess so

a² - 2ab + b²=0

[oak chasm]

Right.

So, what's a² - 2ab + b² equal to from before?

[errant wolf]

2ab?

[oak chasm]

No, that's a² + b².

Not a² - 2ab + b².

[errant wolf]

before?

[oak chasm]

Yes.

[errant wolf]

idk man

was is not just = to 0?

[oak chasm]

Yes, that's one thing it was equal to.

But you did something else that got you a² - 2ab + b².

[errant wolf]

well i just -2ab from 2ab

[oak chasm]

Yes, that's where you got 0.

But you did something before you did any of the work to get it equal to zero.

[errant wolf]

what did i do

i expanded (a^2-b^2)?

[oak chasm]

Right.

Except it was (a - b)².

[errant wolf]

oh yea

[oak chasm]

That is equal to a² - 2ab + b².

So,

a² - 2ab + b² = 0
(a - b)² = 0

What's a - b?

[errant wolf]

2ab?

[oak chasm]

How did you get that?

[errant wolf]

i have no idea

[oak chasm]

OK.

So, let's finish this.

We have:

(a - b)² = 0

Square root both sides.

|a - b| = 0

So, the absolute value removes the sign from a - b, so we need to put it back.

a - b = ±0.

Now, no matter what sign a - b was, |a - b| = 0.

±0 is just 0.

a - b = 0

a = b

[errant wolf]

ok

[oak chasm]

You need to practice algebra.

If you're in a proofs class, you need it.

Khan Academy has a nice algebra course with video lessons and practice problems.

You can get a better understanding of how to solve algebra problems that way.

https://www.khanacademy.org/math/algebra

They also have an algebra 2 course.

So you can remember what you forgot from algebra and maybe even understand it better than when you learned it before.

[quartz osprey]

uhh

my work is kinda scuffed

i folowed along with the video and still got it wrong somehow

[alpine sable]

If anyone is good at this type of question I have a specific question about it. I tried to ask a few days ago but I think it got lost in the sauce

[zealous drum]

Hello

Does any know how to do 2 compliments notation?

[royal crane]

nope sorry

[zealous drum]

Oooof

[ivory zodiac]

can someone please explain how it becomes 1 on top in the 2nd line?

what cancels with what and how?

[alpine sable]

Just from looking at it in .5sec I'd say try to distribute it and you'll probably end up with 1

[plucky geyser]

I dont think you can pull it in a negative amount

[ivory zodiac]

i did and i got (4^2 - rootX^2) / (16x-x^2)(4+rootX)

so i put it in a equation simplifier and it got me that too

so top is 16-x

[alpine sable]

From when I took calc I'd say say that there is either an Identity that is = 1 from that, or you can simplify it more and you're probably using the calculator wrong

[ivory zodiac]

hm okay

ty

[alpine sable]

Calc is all about realizing you're actually dogshit at simple math, and re-checking your work 50 times

Is this open

[quartz osprey]

ok thanks

oh weait

youre right

LOL

the vicdeo did a - nubmer so i followed

[plucky geyser]

lol idek how to do that it just doesnt make sense that it would be pulled in at a negative rate

that would mean it was going backwards 🤔

[frosty loom]

How would I get the value of x?

[alpine nacelle]

x+45 = 4x+15

[plucky geyser]

ye

[frosty loom]

Thanks.

[plucky geyser]

get x on one side

solve for it

then plug it back in

[alpine sable]

I did it and got 16 - x on top as well so I'm assuming its def simplified more somehow possibly by the lower half of the eq

[royal crane]

can somone help

please

[plucky geyser]

solve for x?

[royal crane]

it just says "solve absolute value equations"

but im pretty sure its x

@plucky geyser

?

[plucky geyser]

idk thats above my level

[royal crane]

o lol

[plucky geyser]

im only on precal

💀

[royal crane]

idk what im on lol

this is math waaaaay above my grade level

im just learning it cause my parents make me

[plucky geyser]

lol

u know how to solve this?

[royal crane]

nope

we never learned abt i

do u know this?

@plucky geyser

[plucky geyser]

hmm

[royal crane]

or this

or this

[plucky geyser]

just use desmos

[royal crane]

i wanna know how to do it

[plucky geyser]

o idk

[royal crane]

any of them?

[plucky geyser]

well b could either be 1 or -1

actually a is one also 🤔

idfk

[alpine nacelle]

You posted a lot of exercises, I can help for some of them

[royal crane]

sure

@alpine nacelle which ones?

[alpine nacelle]

You can choose, just don't make me solve 5 problems in a row lol

[winter rock]

Wassap,

Is this channel being used rn?

[royal crane]

yes