help-0

[stiff hornet]

Set equations equal to each other

[rotund plank]

Sounds like a good idea

[stiff hornet]

Now I just solve it for x?

[rotund plank]

Looks good

[stiff hornet]

This right?

[rotund plank]

They want it in exact form

[stiff hornet]

Oh

So what do I do?

[rotund plank]

$y = \frac{1}{2}\cdot \frac{e^{\frac{7}{4}}}{2} + \frac{7}{4}$

Usually easier to work with fractions

[ocean seal]

dk.dkn

[rotund plank]

Try to simplify that

[rotund plank]

Yeah looks good

So the exact coordinates for the point Q are?

[rotund plank]

Yeah

You can see here everything you did

[stiff hornet]

Let’s go

[rotund plank]

Green is the normal, and the horizontal and vertical lines are your x and y values

You can see they meet where the normal crosses C_2

It’s probably good to write brief explanations for some stuff you’re doing too

[stiff hornet]

Wdym?

Like saying this is the normal etc

[rotund plank]

Like ‘find gradient of curve’, ‘gradient of normal is’, ‘set equations equal’

[stiff hornet]

Oh ok

Thanks for the help

[rotund plank]

Just a few words at each step, it makes your work clear even if you have bad handwriting

Yea np man

[stiff hornet]

I’m gonna work on the next question a bit, can I dm u if I need help?

[rotund plank]

Yeah sure

[stiff hornet]

Thanks

[rotund plank]

Np

[lone heart]

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[jaunty crane]

Can someone help me on this

On using the formula 1/2 r'2(0-sin0)

[last ether]

I'm assuming MO and NO are congruent since MN is a chord

So find the area of the sector, subtract the triangle

You can use law of cosines to find the angle MON

[jaunty crane]

use sin right

SOH CAH TOA

soh

50.06 degree

[last ether]

Well no

Because you don't have a right angle for sure

You gotta use law of cosines

[lone heart]

@jaunty crane Has your question been resolved?

[jaunty crane]

oh yearight

18.4 square = 12 square + 12 square - 2(12)(12) kos 0

i got 100.11 or 1.7475 rad

@last ether should i apply the next formula?

[last ether]

Ye

[jaunty crane]

54.9546cm

is it correct

[last ether]

Not sure. I don't have a calculator on me

But just find the area of the sector

You know that for every 360° ghe area will be 144π

so do $\frac{144π}{360} \cdot 100.11$

[ocean seal]

Umbraleviathan

[lone heart]

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[proud bobcat]

Im having trouble woth some proofing

[proud bobcat]

I have this and i need to show its odd when y is even

But i cant sene to put x in odd form

[marsh rapids]

You can't. If y=2, then x=1/3 is not an integer, hence not an odd number

[proud bobcat]

Yeah youre right, i thought this was true because for all integers i was still getting odds as an output

Let me show you the question

I made a chart and all were odds

[marsh rapids]

solving for x as a function of y is a good idea. What does it mean if some y values don't lead to an integer ?

[proud bobcat]

Then the function wouldnt be onto right

Well hm

[marsh rapids]

You'd also need to show that same value can't be reached from O. Then if it is reachable neither from E nor from O, it is not reachable at all

And if you always find it is reachable, than the function is actually onto

[proud bobcat]

How would i have realized the function wasnt onto if all values seemed to be outputting odd values

Oh im guessing itd have to be outputting every odd value

Could i have realized it wasnt onto from this not containing every odd integer,

?

[marsh rapids]

Are you sure it doesn't ?

f: Z -> O, not f: N -> O

[proud bobcat]

Thats true gmm

So why cant i do cases like i was a second ago

Because i was gonna do them bases on y being even and y odd

[marsh rapids]

You can. You just need to get to a result, which I don't think you did

[proud bobcat]

Put it in the form of odd eventually and show y is in the image and prove that in all cases its in odd

[marsh rapids]

Hint: what is f(O) ( = {f(x) \ x in O})?

[proud bobcat]

Its an odd isnt it

When you plug in an odd you get an odd in the piece wise

Unless im misreading that

[marsh rapids]

Yes, but what odd numbers ? Every odd number, or only some odd numbers ? And if only some, are the other ones in f(E) ?

[marsh rapids]

The function is defined as being from Z to O, so it tells you that the output is odd, so that shouldn't be surprising

[proud bobcat]

I think f(O) would cover all of them, theyre all in intervals of 2

And f(E) only covers some

[marsh rapids]

Correct

Now you need a formal proof for it

[proud bobcat]

can i not do that with cases?

[marsh rapids]

You don't need to

[proud bobcat]

Let me try to come up with something

Hmm i say let x = y-1/3

But then aftee that im confused

[marsh rapids]

To prove f is surjective (the real word for "onto") you pick a random element and show it's the image of at least one element

[proud bobcat]

Couldnt i do it with just the even function then since it covers all odds,

So something like

Hold on

I thought maybe if i used the even function itd cover it so like let x=y-4

But theres no way to make that odd

Hmm proofs give me so much trouble

I can do calculus just fine but this makes me confused

[marsh rapids]

That's almost all you need. Depending on the level of detail is a line or 2 at most

[proud bobcat]

Wait so would this be something like let x be that then because y is element of Z, then so is x sand so is the function

So then plug it in to the original and get y

I think

[marsh rapids]

That's the idea yes

If you leave out every detail you get: "let y in O. f(y-4)=y so f is surjective"

[proud bobcat]

Ill keep that in mind and try working it out to end up with that, thank you so much

[marsh rapids]

Considering your level (of study) you'd need to add quite a few details but I'll leave that to you

[lone heart]

@proud bobcat Has your question been resolved?

[proud bobcat]

I think i did it

[lone heart]

@proud bobcat Has your question been resolved?

[lone heart]

@proud bobcat Has your question been resolved?

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[alpine sable]

it's a question from code jam ,,
I'm new to competitive programming
I'm trying to solve this problem but I can't able to understand the question fully
I can't able to understand this part
A straight of length ℓ starting at x is the list of integers x,x+1,…,x+(ℓ−1). We want to choose some of the dice (possibly all) and pick one number from each to form a straight. What is the longest straight we can form in this way?

[alpine sable]

https://codingcompetitions.withgoogle.com/codejam/round/0000000000876ff1/0000000000a46471

[lone heart]

@alpine sable Has your question been resolved?

[alpine sable]

no

[prime badge]

basically ```
5 4 5 4 4 4

you need to find a subset

free order, free "connectedness"

where if you sort it, each number is not smaller than its index

your first number is anything, second is 2+, third is 3 or more, second is not smaller than 4 etc.

and you can pick from anywhere you want

so here the longest is 4 4 4 4 5 or 4 4 4 5 5

either one is 5 long

oh right they are never smaller than 4

[alpine sable]

why can't I able to understand question , is it the question tricky or am i lack of knowledge

[prime badge]

i don't know

[alpine sable]

okay thanks for the help

.close

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[open idol]

why can't i rewrite this inequality as (x-2)/(x+1) > 1

the inequality completely changes

[short cobalt]

how did you get that?

[simple turtle]

IMO, you should test the interval

[open idol]

i multiplied both sides by (x-2) then divided both sides by 4

[simple turtle]

x is not equal 1 and x is not equal 2

[open idol]

i could, but i have to solve for the interval

[slender horizon]

Is the problem not just that you have multiplied an inequality by a variable which could be positive or negative?

If you restricted it to the range in which what you multiplied is strictly positive or negative and then changed the sign accordingly it should be fine

[open idol]

so i have to consider the case where it could be negative too and create another case and flip the inequality sign?

[simple turtle]

Plug in x between negative infinity to 1 like 0, -1, and see if inequality is true or false.

[open idol]

i see

[slender horizon]

OR you can multiply both sides by the denominators squared as a variable squared is always positive and thus you don’t have to change the inequality

[simple turtle]

You are looking for domain where inequality is true

[slender horizon]

Although my preferred method most the time is to change the inequality sign to an equality sign and then find where the two curves intersect and then solve logically although that gets confusing when there are asymptotes

[open idol]

hmm i just did that but wouldn't i just end up having the x's cancle out and not being able to solve it

[simple turtle]

There is asymptote at x = 1 and x = 2 You want to check x values that makes the inequality true.

[slender horizon]

If the x’s cancel out and you get a true statement then that means it is true for any x

That doesn’t equal 2 or -1

Plot the two graphs on Desmos separately and look

[open idol]

ah, i see so i only have to test values to see the actual sections of where the inequality holds, ok

[open idol]

i'm don't quite see how that method works out

i would only see when they equal but not when they are greater or less than one anothher

[simple turtle]

Yes, there are 3 intervals you want to check. -infinity, less than 1, between 1 to 2, and greater than 2 to +infinity

[open idol]

i might've understood it wrong, but did you mean to say this when squaring both sides? it didn't look the same

[slender horizon]

This is assuming they do not intersect

You haven’t multiplied both sides

You’ve multiplied each side by something different

[open idol]

oh, your right

[open idol]

i think i'll just stick to this way

[slender horizon]

@open idol

Gotta trust the process sometimes

[open idol]

woah

that doesn't look as ugly as i thougt it would be

[open idol]

would this be your go-to method if you were to solve this question?

[slender horizon]

Probably the easiest way

Gotta expand a few brackets but if you keep it tidy that’s fine

there will sometimes be much easier methods

but this will always lead to the correct answer yk

[open idol]

yeah, it doesn't doesn't look that bad once things just start canceling out

[open idol]

yea, thats true

[slender horizon]

they wont necessarily cancel out though and thats when other methods are helpful

like if its a quartic you can be expected to solve it

[simple turtle]

There is no x such that the inequality is true No solution.

[slender horizon]

so its good being familiar with a few methods

[slender horizon]

what are you on

try x=0

4>-2

thus true for x=0

[open idol]

i agree, could you also explain the method where you switch the inequality sign to a equality sign instead

[simple turtle]

Be nice John. It could be the calculation error. If that is true, solution is (1,2) let me double check my calculation

[open idol]

doesn't that like mess up the process once get further down and your not sure whether your dealing with a less than or greater than when you mult things by negatives

[slender horizon]

the significant points are where there is an asymptote or they intersect

if you write these all down on a number line (in this case it is just the asymptotes as they dont intersect)

we have 3 areas

x<-1, -1<x<2 or x>2

and then simply plug in any number and test which regions the inequality is true for

for example we could put +5 and -5 and 0

and that would show it is only true for -1<x<2

[simple turtle]

you mean 1<x<2

[slender horizon]

no

unless i copied it wrong?

[simple turtle]

Oops, I wrote x-1 for denominator. yes, x can not be -1 and x can not be x =2.

[slender horizon]

yup

sorry for being rude also

its easy to make a slip up on inequality questions you need to be careful

[simple turtle]

It is ok John.

[open idol]

thx y'all for helping me out, i now know how to solve it

.close

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[ionic hazel]

hey, please can someone explain this? thanks

i don't understand where the -2 comes from

i suppose it oculd be an effect of integration

oh wait

i am dumb

[flint walrus]

it's just expanding the brackets. You have (a+b)^2 and the middle term is 2ab. so -2*e^x *e^(-x)

ah alright then lmao

[ionic hazel]

yeah yeah

i just realised that just then loool

thanks anyway <3

[flint walrus]

np good luck

[ionic hazel]

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[cerulean galleon]

can someone please give a general explanation of what series are? (currently learning it in calc 2)

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@cerulean galleon Has your question been resolved?

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[misty bobcat]

combinatorics question

[misty bobcat]

how many 7 digits number can we have only with strictly decreasing order ?

[rotund plank]

Separate the cases based on starting digit

Then where necessary do the same again for the second digit

It shouldn't be too bad

[lone heart]

@misty bobcat Has your question been resolved?

[alpine sable]

you dont need to do that

[misty bobcat]

i have tried but it seems to complicated for me, there is lot of possibilities

[rotund plank]

Not really

[alpine sable]

its just 9C7

isnt it?

[rotund plank]

Why

[misty bobcat]

only strictly decreasing...

[alpine sable]

select any 7 digits you can always order them to be strictly decreasing

oh wait

it decreasing not increasing

but you can work with same logic

i think

[rotund plank]

Yeah actually that's a pretty neat way

[alpine sable]

if it would be strictly increasing then it would be 9C7

[misty bobcat]

i think that does not work

[rotund plank]

I think it's 9C7 anyway?

[alpine sable]

but it can end in 0

so

9C7+9C6?

[rotund plank]

Why's that a problem?

[misty bobcat]

9C7 are only all combinations, there are all numbers included

[rotund plank]

Oh right nm

*Thought there were 9 digits involved not 10 🤦‍♂️

[alpine sable]

it should be 9C7+9C6

[rotund plank]

10C7?

[alpine sable]

no

or wait

[rotund plank]

I think you discarded it because it doesn't hold for increasing

But for decreasing it should be fine

[alpine sable]

yeah you are probably right

i will quickly write a program to test it wait

[rotund plank]

I wonder how far can that logic be extended though

Like alternating digits for example

Or a number with a fixed number of distinct digits defined by any rule

[alpine sable]

its 120

so 10C7

you were correct

[rotund plank]

I mean it makes sense

[alpine sable]

yeah

[rotund plank]

Our original set was 10 digits

[alpine sable]

just double checked with bruteforce

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[misty bobcat]

.repen

.reopen

[lone heart]

✅

[misty bobcat]

why ?

can you explain it please ?

[alpine sable]

select any 7 digits you can order them to be strictly decreasing

so you just need to find how many ways to select 7 digits and there are 10 digits

[misty bobcat]

hmm

but it may happen that after ordering you get multiple same numbers

[rotund plank]

No

Think about the choose function

And why we used it

Does the choose function allow permutation?

[misty bobcat]

if we choose 7,8,4,3,2,1,0 and 8,7,4,3,2,1,0

it would be the same

[rotund plank]

Build up the choose function

So we have a set of n objects

n! arrangements

[wary stream]

What dk is asking, what's the difference between permutations and combinations?

Because you have nPr vs nCr

[rotund plank]

Yep

with n! arrangements, we divide by (n-r)! because we want to choose only r objects

And we divide by r! because for any of those sets we can rearrange it in r! ways

That last step answers your question

[misty bobcat]

uhmm

thanks guys

[rotund plank]

Np

You should learn the choose function well

[misty bobcat]

yes, i will study it mooore

[rotund plank]

It's incredibly useful in combinatorics

[misty bobcat]

.close

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[solemn tendon]

I need help in Cartesian equation please

[solemn tendon]

it’s french but i can translate

[slender horizon]

If you translate specifically what it is you need help with I can help @solemn tendon

[solemn tendon]

solve the following equation systems (x; y) e R²

on the first pics

the exercise 2

[slender horizon]

expand the brackets?

if you do not know how to factorise confidently then work backwards

oh i was lookinmg at second pic exercise 2

S1 and S2 are both seperate sets of simultaneous equations, correct?

[solemn tendon]

yeah

its split

[slender horizon]

then for S1

2x-2y=-3
3x-4y=2

in this case multiply the top equation by 2

and you get 4x-4y=-6

and then subtract the second equation
3x-4y=2 from this to get

x=-8

[wary stream]

Don't do the problem for people or give out answers

[slender horizon]

oh mb

sorry

i was going to encourage him to do 2nd

I didnt know we couldnt use some as examples

[solemn tendon]

how ??

[wary stream]

You were pretty much doing the problem for the person

[slender horizon]

yh ik hes got like 8

i am sorry

wont do it again

@solemn tendon no pm

[solemn tendon]

sorry mb

[slender horizon]

its like if i had x+y=1

[solemn tendon]

ok

[slender horizon]

and x-y=-1

how would you solve

[solemn tendon]

y= -1-x ?

[wary stream]

FYI, the process that is happening is elimination

[rotund plank]

Usually better to prompt them through then tell them to go off on their own

[slender horizon]

yeah sorry

I shouldve created a similar example

[rotund plank]

It's not the end of the world

[slender horizon]

its kind of hard to explain exactly why elimination works but your method of substituting values will work just as well

[rotund plank]

You're the one helping them, if you really want to do use one as an example, you can

[solemn tendon]

you can use a another example pls ?

[slender horizon]

you can do that in the example you gave also

[solemn tendon]

ok

so i did it now

[slender horizon]

done?

[solemn tendon]

yeah

[slender horizon]

oh ok

[solemn tendon]

-3x+2y=1

2x-3y=5

so

its good ?

[slender horizon]

you were correct with 2y

[solemn tendon]

oh ok

so

2y=1+3x

y=1/2+3x/2

right ?

[slender horizon]

yes

[solemn tendon]

and after i didnt know

[slender horizon]

well you have an expression for y in terms of x

you should involve the other equation

[solemn tendon]

yeah

[slender horizon]

2x-3y=5

any ideas how?

think about what you can replace y with

[solemn tendon]

I don't know how to continue y=1/2+3x/2

[slender horizon]

you can substitute 2x-3(1/2+3x/2)=5

[solemn tendon]

by 2 ?

[slender horizon]

you know what y is equivalent to, so replace it

[solemn tendon]

AHH yeahhh

i can do this

put the first on the second

[slender horizon]

yes

[solemn tendon]

its that ?

[slender horizon]

you should try S1 now

that one doesnt work out as well as S1 does

you can prolly do it

[solemn tendon]

2x-2y=-3

3x-4y=2

so

2y=-3-2x

y=-3/2 -2x/2

[slender horizon]

try multiplying for 4y

instead of dividing for y

[solemn tendon]

so

4y=2-3x

[slender horizon]

nop

[solemn tendon]

why?

[slender horizon]

3x-4y=2

3x-2=4y

[solemn tendon]

why?

oh ok

[slender horizon]

thats just how the second equation rearranges

[solemn tendon]

i can put the 4y in the other side

[slender horizon]

yeah ur equation works only for -4y

[solemn tendon]

I can do it both ways

ididnt know that thanks

2x-2y=-3

2x=-3+2y

x=-3/2+y

[slender horizon]

yes

now use 3x-4y=2

[solemn tendon]

3(-3/2+y)-4y=2

?

[slender horizon]

yes

[solemn tendon]

omgggggggg

after this i'm noob

ahhhh

3x-3 = -9 -4

= -13/3

right ?

[slender horizon]

where has that come from

[solemn tendon]

nooo

[slender horizon]

use this

[solemn tendon]

yeah

[slender horizon]

solve for y not x

-9/2 +3y-4y=2

[solemn tendon]

ahh ok

[slender horizon]

-9/2-y=2

y=

[solemn tendon]

2 + 9x-2

y= 20

[slender horizon]

no

add y to both sides

subtract 2 from both sides

[solemn tendon]

ohhh YEAHH

[slender horizon]

-9/2-2=y

y=

[solemn tendon]

y=9

[slender horizon]

-9/2-2 does not equal 9

=-9/2-4/2=-13/2

you have to then use the initial equation

2x-2y=-3

and substitutey=-13/2

[solemn tendon]

2(-3/2 - 13/2) - 2(-13/2) = -3

[slender horizon]

no

simply keep x as x

[solemn tendon]

oh

i have

x= -3/2 + (-13/2)

but after ...

[slender horizon]

what does that simplify to?

[solemn tendon]

2 ?

[slender horizon]

no

[solemn tendon]

i dont know

[slender horizon]

-3/2-13/2

=-16/2

[solemn tendon]

-8

[slender horizon]

done

x=-8,y=-13/2

[solemn tendon]

x=-8 and y = -13/2

[slender horizon]

yes

[solemn tendon]

ily bro

[slender horizon]

same process for other one

im going to bed

[solemn tendon]

good night my hero

[slender horizon]

give it like 5m trying then someone else will probably help

[solemn tendon]

ok i go no w

-3x+ 2y=1

2x-3y=5

-3x=1 - 2y

X= -1/3 + 2/3y

2(-1/3+ 2/3y) -3y = 5

2x( ...)

so

-2/4 + 4/3y -3y = 5

euh

[lone heart]

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[sullen vortex]

can someone tell me how I can simplify this?

[lone heart]

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[short cobalt]

what part are u stuck on?

[vital ocean]

I'll send my work

[short cobalt]

are you trying to find x?

[vital ocean]

Yeah

I think I have a mistake somewhere

[short cobalt]

i see the issue

you must foil

[vital ocean]

Foil?

I think I see it aswell

I didn't divide both sides

I subtracted for 1 and divided for the other

[short cobalt]

in the second line

[vital ocean]

No nvm

[short cobalt]

you cannot distribute the the square

instead you must FOIL

are you familiar? or i can teach you

[vital ocean]

I might be familiar to the process but I don't understand the term you're using

I'm not American so I don't know the American names for stuff

[short cobalt]

its a method of multiplying two polynomials

no worries

[vital ocean]

Yes

[short cobalt]

basically it is the way to multiply and make sure everything gets multiplied together

[vital ocean]

we were told it's crab claws lmao

[short cobalt]

so with the first term (x+4)^2, how can u apply this method?

[vital ocean]

Because if you use curved lines you get crab claws

[short cobalt]

crab claws? Ive never heard that. If i may ask, where are you from?

[vital ocean]

Ireland

[short cobalt]

very cool! that is an interesting name/ way to teach math I've never heard of.

[short cobalt]

i see

[vital ocean]

So (x+4)(x+4) ?

[short cobalt]

lets apply the method to the first polynomial (x+4)

yes

[vital ocean]

Oh I see

Would you get x^2+16+8x?

[short cobalt]

yes

[vital ocean]

Ok I was missing the 8x thanks

[short cobalt]

np

just do that for each one

[vital ocean]

Yeah

[short cobalt]

and use the Pythagoras theorem

so instead of adding them all up

the two shorter sides should equal the hypotenuse

[vital ocean]

Ok I get ya

@short cobalt I'm getting 0=6x+x^2+25

Now I use Pythagorus theorem right?

[short cobalt]

Where is that from? Just so I know?

[vital ocean]

Shoot I added them all up

I wasn't meant to do that

[vital ocean]

I did foil on all of them and then added them

[short cobalt]

Which did u add?

[vital ocean]

All

Wait

I have this and now I use Pythagorus theorem

[short cobalt]

Hmmm

Now I just need to solve for x

Since u already have the sides equalling the hyp u should be good for that

I have to go but this is what I have

@vital ocean

[last ether]

Well it says that x is a natural number

[last ether]

What the fuck

<@&268886789983436800> change channel name to M4tty?

The bot is uh

Having its moments

[sly mantle]

ye its a bit borked rn

[last ether]

Should be $x^2 - 26x + 25 = 0$

[ocean seal]

Umbraleviathan

[last ether]

You should be able to solve from there

[short cobalt]

Ty for fixing my mistake. I was running out the door and probably made some calc error!

[last ether]

Door moment

You left room for error

Please laugh I require validation

[short cobalt]

Smh

Haha

Feel better?

[last ether]

Yes

[last ether]

@vital ocean and you should be good to go

[lone heart]

@last ether Has your question been resolved?

[last ether]

burn in hell

[lone heart]

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[alpine sable]

1. vertical translation 7 units down
2. horizontal translation 4 units left
3. vertical stretch by a factor of 3

this is what I have

[cursive badger]

that looks correct to me

although the order is wrong

since you translate it first, then stretch about the x axis, then it's different from stretching and then translating it (vertically)

and the horizontal translation is to the right

f(x-4) means that x needs to be 4 units greater (4 units to the right of normal) to do the same thing as just f(x)

or, well, nvm since it seems you're listing the translations to go from f to g

but it's asking for how you got f from g

[alpine sable]

thanks

.close

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[slim pebble]

how would we go about solving b? would we need to find a smaller function

prove that is divergent

to show that cos(1/n)/n also diverses?

[velvet pelican]

have a look at the limit comparision test

i think that should help

[slim pebble]

so you would use limit comparison right? not the ordinary comparison test

[placid zinc]

I'm trying to think of a way to do it with comparison and coming up short

[velvet pelican]

mhm im still trying to find a way for the ordinary comparision

[slim pebble]

there are other problems in my hw that specifically use limit comparison so i think im supposed to use ordinary for this but ive hit a wall

[velvet pelican]

wait

try comparing with ||cos(1) * 1/n||

[tight locust]

hint: the cosine is bounded above by 1

[placid zinc]

Since it alternates, it would seem to conditionally converge. But I can't imagine comparison should give this.

[slim pebble]

wait i thought this doesn't alternate

[tight locust]

it does

[velvet pelican]

^

[placid zinc]

Oh heh yeah it doesn't. Sorry I ignored the 1/n for a sec

[buoyant kayak]

how did you solve a?

[tight locust]

yes

[buoyant kayak]

seems you'd just do the same thing, no?

except with a slightly different b_n

[slim pebble]

i see how this works now, since cos(1)/x diverges the bigger original would as well

[slim pebble]

but how would u find this

[placid zinc]

Let x get large

[velvet pelican]

i realised cos(1/x) > cos(1)

for x > 1

[buoyant kayak]

uh

cos(1/x) <= 1 for all x

[velvet pelican]

mom thats why its cos(1/x) > cos(1)

[buoyant kayak]

yeah but why are you comparing with cos(1)

an unknown value

just compare... to 1

[slim pebble]

but 1 over x would be bigger

so it wouldnt prove whether original diverged or converged

[buoyant kayak]

ah true

hmm

[velvet pelican]

yea i needed a lower bound

[vague coral]

cos(1/n) <= 1 - 1/2n^2
when n goes to infinity

[slim pebble]

where did " 1 - 1/2n^2 " come from

[vague coral]

from daddy Taylor

[tight locust]

compare with (-1)^n/n

[tight locust]

you can add a constant so that it's always smaller

try 1/(2x) maybe

[slim pebble]

oh yeah nice

[vague coral]

its pretty much done by doing
$\left|\frac{\cos(\frac{1}{n})}{n^2} \right| \leq \frac{1}{n^2}$ right ?

[ocean seal]

Ｈｅｒｅｌｓ

[lone heart]

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[tribal geyser]

Can someone help me integrate 6)

[stiff hornet]

I can help

[tribal geyser]

yes pls!

[stiff hornet]

Multiply the coefficient by the power

Then subtract 1 from the power

Oh wait

[sudden hinge]

@tribal geyser Well what did you get when you differentiated x^2 e^x?

[stiff hornet]

This is for logs right?

Multiply the coefficient by the power in the constant and don’t change the coefficient

[tribal geyser]

2xe^x+x^2e^x

[sudden hinge]

Ah wait problem 6

did you learn substitution with integrals, yet?

If yes, try substituting $u = 2x^3$

[ocean seal]

nvx

[tight locust]

Don't

You need ibp here

[sudden hinge]

you don't

[tight locust]

Yeah you're right

I did not see

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[covert jungle]

Hi

[covert jungle]

Didn't they do this wrong?

how did the teacher get the pi on top

[wary stream]

No

[covert jungle]

pi is a constant

[wary stream]

Chain rule

[covert jungle]

so you can pull it out

[wary stream]

Not from ln like that

[covert jungle]

wait really, you can't do that?

[wary stream]

Can you pull out a 3 from sin(3x)?

[covert jungle]

so for trig and log functions we cant do that

but for stuff like derivative of pi times x^4

we can pull it out?

[wary stream]

Yep

[covert jungle]

alright sounds good

thank you

.close

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[alpine sable]

how?

[alpine sable]

when I did it I got 1±sqrt(1/2)

how do you get the 2 in the numerator?

[untold carbon]

sqrt(1/2)=1/sqrt(2)=sqrt(2)/[sqrt(2)xsqrt(2)]=sqrt(2)/2

[alpine sable]

i dont get it

[untold carbon]

you multiply the denominator and numerator by sqrt(2)

[alpine sable]

but why?

ah to get the 2 out

[untold carbon]

to rationalize the denominator

[alpine sable]

now what's that?

[untold carbon]

get rid of radicals in the denominator

[alpine sable]

ahh

makes sense

thank you

[untold carbon]

you´re welcome

[alpine sable]

.close

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[wary stream]

.$$\frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{\sqrt{2}}{\sqrt{4}} = \frac{\sqrt{2}}{2}$$

[ocean seal]

dldh06

[alpine sable]

thx for that

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[vestal aspen]

what is the meaning of that line over x in part4 of the question?

[lone heart]

Please don't occupy multiple help channels.

[wary stream]

Don't open multiple channels

[vestal aspen]

ok

.close

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[vestal aspen]

.reopen

[lone heart]

✅

[vestal aspen]

can I ask for different question in the same channel that I occupied before?

[wary stream]

Yeah you can ask other questions in your original channel

[vestal aspen]

ok, thanks

.close

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[pearl drift]

Is {b} an element of {a,b,c}??

[untold carbon]

no, but b is

[merry depot]

no, b is an element of {a,b,c}

{b} is a subset of {a,b,c}

[pearl drift]

oh okay. so it's a subset not an element

thank you for that

.close

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[tight locust]

What does this even mean

[merry depot]

b is in the set containing a, b, and c

[lone heart]

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[frosty sable]

It looks like they multiplied the denominator with z^4 and the numerator with z^2

[frosty sable]

Am I missing something?

[untold carbon]

seems that you are right to me

[frosty sable]

that should not be allowed

so the solution is incorrect?

<@&286206848099549185>

[untold carbon]

i think so, unless there is some other context of the problem that makes this legal

[lone heart]

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[silver current]

help??

[untold carbon]

what have you tried?

[silver current]

i dont know hoe to start

how

[untold carbon]

i think a general idea would be to do case work on the intervals were m can lie

to get rid of the absolute values

[silver current]

oh ok

.close

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[dim pine]

so i am unsure as to where my error is

actually nvm i am confused

can anyone walk me through their process on how to solve?

[wary stream]

Is this linear algebra?

[dim pine]

yes

[wary stream]

What did you try?

[dim pine]

so what i believe you are supposed to do

is but the system in terms of Ax=b

put*

[wary stream]

Yes that is right

[dim pine]

and then you put it in least square form

[wary stream]

Did you write that in Ax = b form yet?

[dim pine]

A=1 -5
1 0
1 5

b = 36
0
-24

correct?

[wary stream]

Yes

[dim pine]

what do i do after that

how do i solve for c

[wary stream]

$(A^T A)x = A^T b$

[ocean seal]

dldh06

[dim pine]

i did that i believe

i got

3 0 c0 = 12
0 50 multiplied by c1 -300

is this correct?

[wary stream]

Yes

Then you can solve that easily

[dim pine]

oh my goodness

i am so sorry

that i have taken your time for such an easy task

thank you for talking through it @wary stream

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[dim pine]

have i successfully closed?

[wary stream]

Yes

If you notice the bot's message

[dim pine]

ok

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@strange rampart Has your question been resolved?

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@strange rampart Has your question been resolved?

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[somber mango]

what are the 4 points for Utts & Heckard’s Guidance for Making Conclusions in Hypothesis Tests?

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[somber mango]

can someone explain point 1,2, and 4

[somber mango]

i really dont get it

<@&286206848099549185>

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[alpine sable]

How can I use wolframalpha to help me solve the problem?

I can integrate with wolframalpha but I cant sub in the initial conditions.

i want wolframalpha to give me the complete answer in one line

[lone heart]

@alpine sable Has your question been resolved?

[alpine sable]

i=dq/dt

.close

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[hollow pulsar]

Guys in example 2 can you explain to me where 1/2 came from?

[lament summit]

@hollow pulsar sqrt (u) means u^(1/2) and thus it's written so.

[hollow pulsar]

What is sqrt?

[lament summit]

@hollow pulsar square root.

[hollow pulsar]

How did it became 3/2?

Ahhh

The cube of u

Right?

[lament summit]

@hollow pulsar umm.. no..
Actually it's because u^(1/2) is in the form of u^n so using the rule of integration for u^n which is
(u^(n+1)) / (n+1)
Putting n= 1/2 u get (u^(3/2))/ (3/2).

[hollow pulsar]

Ahhhh ok

Wait my mom needs help

Excuse me

[lament summit]

@hollow pulsar ?

[hollow pulsar]

I'm still here just helpin someone

I get it now

(1/2)+1 = 3/2?

[alpine sable]

Yeah

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[alpine sable]

https://cdn.discordapp.com/attachments/874789238997147702/960083359420252170/unknown.png

[alpine sable]

how do you do this

[pine kettle]

@alpine sable use algebra

[alpine sable]

how

[pine kettle]

isolate x

[alpine sable]

how do i do that though

walk me through it

<@&286206848099549185>

Subtract 2 from both sides

yes

i can do that

<@&286206848099549185>

[red pecan]

I think like this

[alpine sable]

thanks

solved now

.close

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[tender wharf]

Hi! Im having trouble with a question concerning span{} in linear algebra. Im just re-learning linear algebra at the moment and so the current section I am on has not touched on subjects like linear independence/dependence etc. just plain old Span{} with linear combinations. So i'm struggling with the current prompt of proving that span{u} = span{cu} where c is a non-zero scalar. Not really sure about how to approach this. The textbook gives a hint to show that the span{u} is part of the set of vectors span{cu} and vice versa. I can see intuitively how they are equal but I just don't know how to put it in words.

[bitter vault]

To prove equality of sets, you show that each one is contained in the other.
To show span{$u$} is a subset of span{$cu$}, take an arbitrary vector $v$ in span{$u$}. Then $$v = \lambda u$$ for some scalar $\lambda$ in the field. Then, $$v = \dfrac{ \lambda }{c} (cu) \in \textrm{span}(cu).$$
This shows span{$u$} is a subset of span{$cu$}.

Try to show the reverse containment

[ocean seal]

1345631

[bitter vault]

Note that you use the fact that c is non-zero to rewrite lambda as lambda * (c/c)

[tender wharf]

Oh interesting! So in summary: take an arbitrary vector v that is known to be in the span{u} defined to be (as shown), solve for u? then back substitute to show that span{u} is just a subset of span{cu}?

I think I can vaguely understand what your saying I just cant put it into words😂

[bitter vault]

Not quite. To show A is a subset of B, you take an arbitrary element of A and show that it is in B.

[tender wharf]

Oh!! Yeah that makes sense now that you word it that way haha.

Thanks I think I understand it for the most part!

.close

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[versed lodge]

Anyone help with this please

Thabo is considering buying a car to the value of R233176. He expects to put a deposit down of R34000 and his repayments to be R3249. The bank has offered him an interest rate of 14% pa compounded monthly payable over 6 years. The problem is that Thabo is considering how much residual he has to pay on the vehicle. Being Thabo's bank consultant, Thabo wants to know how much the residual payment will be?

[lone heart]

@versed lodge Has your question been resolved?

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[alpine sable]

What is it mean when it says tangent to x=13

Its asking to use the information to write this in circle equation

But I don't understand what i means when it says tangent to

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[verbal compass]

"Identify the transformations that must be applied to y = x2 to obtain the graph y = -½(x - 2)² +3"

I got this homework questions, I tried googling what a Transformation is and watched videos but still don't understand, I'm not in the country so I can't ask my teacher for help physically and they don't have online Gmail's or anything. Can anyone explain what a transformation is and what I have to do here?

[lone heart]

@verbal compass Has your question been resolved?

[verbal compass]

Sadly no

[shell widget]

@verbal compass Okay so first of all, I recommend opening desmos and graphing y = x^2 and y = (x-2)^2

Tell me when ur done with this

[verbal compass]

Ok

[shell widget]

Replacing x by x - 2 in f(x) or y = x^2 here, the graph shifts to the right by 2 units

[verbal compass]

Done

[shell widget]

can you see that y = (x-2)^2 is just y = x^2 but shifted 2 units to the right?

[verbal compass]

Yes

[shell widget]

Okay good, this is something that you should know

Replacing x by x - a in f(x) (which becomes f(x-a)) means that we are shifting the graph of f(x) "a" units to the right

assuming a is positive here

[verbal compass]

Yes

[shell widget]

-1/2 (x-2)^2 + 3 is an equation for a downward parabola (this is something you should know since ax^2 + bx + c is the equation for a parabola and if a < 0 then the parabola is downwards)

[verbal compass]

It's downward because A is negative?

[shell widget]

so you can see here that a = -1/2 < 0 here so we need a downward parabola

Yes.

[verbal compass]

I see

And because it's 1/2 it's a wider parabola?

[shell widget]

uh not sure

hm

ill have to look into that

[verbal compass]

Wider than 1?

[shell widget]

for now we dont need that

[verbal compass]

Because it's half

Ok

[shell widget]

So we need to invert the graph of y = (x-2)^2

and this means multiplying it by a negative sign

[verbal compass]

Yeah

[shell widget]

inverting means all positive y values become negative, negative values become positive

that happens when u multiply by a negative sign

[verbal compass]

So I simply put a negative right after the = sign?

[shell widget]

so now we have y = -(x-2)^2

Yeah

[verbal compass]

Yes I see

It flipped

[shell widget]

In order to get that 1/2, we must divide every "y" value by 1/2.

[verbal compass]

So -½

[shell widget]

so the transformation will be: dividing every y value by 1/2, which indeed gives us a wider curve

you were right

[verbal compass]

Yay

[shell widget]

and then we add 3 to every y value

[verbal compass]

Yes I see it's wide

[shell widget]

and we get y = -1/2 (x-2)^2 + 3

[verbal compass]

And that will make it go up 3?

[shell widget]

Yes every y value goes up by 3

[verbal compass]

On the Y axis

[shell widget]

you should check each transformation one by one by graphing in desmos

Yes

[verbal compass]

I see

[shell widget]

First: y = x^2, then y = (x-2)^2, then y = -(x-2)^2, then y = -1/2 (x-2)^2, then y = -1/2 (x-2)^2 + 3

[verbal compass]

So moving it to the right 2 on the x axis, then flipping it (making it negative), making it wide, and moving it 3 up on the y axis

[shell widget]

Shifting it to the right by 2 units or horizontal shift to the right by 2 units, inverting it, widening it by a factor of 1/2, upward vertical shift by 3 units or shifting it upwards by 3 units

[verbal compass]

Yeah

That sounds good

Thank you!

[shell widget]

np

[verbal compass]

.close

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Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[pure tree]

guys i need help

[pure tree]

it doesn't matter if you don't understand the text part

can you check if my calculates are correct?

[lone heart]

@pure tree Has your question been resolved?

[pure tree]

<@&286206848099549185>

<@&286206848099549185>

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@pure tree Has your question been resolved?

[pure tree]

<@&286206848099549185>

[pure tree]

<@&286206848099549185>

<@&286206848099549185>

[lone heart]

@pure tree Has your question been resolved?

[versed lodge]

Not yet

[pure tree]

@versed lodge