help-27

[native briar]

for -x^8cosx

does it become

-8x^7-sinx

Or

-8^7cosx

[boreal karma]

so it might be easier if we pretend like the negative isnt there for now

and then just apply it at the end

[native briar]

Ok

[boreal karma]

heres another way to write it that might make it simpler for you

[native briar]

wait so I’m confused now

after we got rid of the 16

what do we do now

[boreal karma]

-(x^8 d/dx (cosx) + (d/dx x^8) cosx)

thats what you need to do to solve this problem

[native briar]

can you use the bot to type it

[boreal karma]

im new here and unfamiliar with the bot but ill write it out on paper

understandable?

[native briar]

._.

[boreal karma]

okay basically what im saying is

take the derivative of cosx and x^8

then multiply the derivative of cosx by x^8, this is one term

and multiply the derivative of x^8 by cosx, this is the other term

the two terms are added to eachother and then we apply the negative sign from earlier

also im sorry but i gotta leave for a class

maybe you can ping a new helper since im struggling to explain it anyways

[native briar]

hmm

<@&286206848099549185>

I'm still having issues understanding how to solve this problem

[silver fjord]

What

What do you have so far?

[native briar]

_> I can explain

so far I have

[silver fjord]

go ahead

[native briar]

we replaced the theta sign with x

and

i did

derivative of x^8

which is 8x^7

[tiny linden]

You have to use product rule

[native briar]

and derivative of cosx

-sinx

[silver fjord]

ok so far so good

now how do you use those facts to get the answer

[native briar]

no clue

[silver fjord]

I'm sure you do

you have $-x^8 \cos x$

[woven radish]

TooManyCooks

[silver fjord]

you don't have them by themselves

you have them multiplied together

[native briar]

yea

[silver fjord]

So what does that mean if you want the derivative

[native briar]

separate them?

[silver fjord]

you already did that

but there's a rule to use when you have two things multiplied together

you used it repeatedly earlier

allo?

[native briar]

yea?

sorry

Is it product rule ?

[silver fjord]

yes

that's exactly it

you separated them into x^8 and cos x

now i want you to use the product rule to give me an answer

you already told me the ingredients

you just need to put them together

if it helps, start by telling me what the product rule is

[native briar]

it’s fxg = f’•g+g’•f

[silver fjord]

(fg)', but yes that's it

So now I want you to use that to calculate the derivative

let's say f is x^8

and g is cos x

you don't have to give me the final answer. just tell me step-by-step what your solution is

[native briar]

ok

I got

(f•g)’=(8x^7•cosx)+(x^8•-sinx)

?

[silver fjord]

close! but good progress

you missed a minus sign on the second term

remember (cos x)' = -sin x

[native briar]

yea

oh yea

[silver fjord]

Got it?

[native briar]

yes

[silver fjord]

Ok

Now, there's an overall minus sign in your problem

Remember, it was $-x^8 \cos x$

[native briar]

yea

[woven radish]

TooManyCooks

[native briar]

yes

[silver fjord]

so you just bring that back by multiplying everything by minus sign

So with that, what's your final answer?

[native briar]

??

[native briar]

how?

[silver fjord]

you have a -1 multiplying your entire expression

that's the minus sign in front

I'm asking you to distribute it in your answer

$-\left( 8 x^7 \cos x - x^8 \sin x \right)$

[woven radish]

TooManyCooks

[native briar]

oh ok

[silver fjord]

that's what you have right?

[native briar]

hm

[silver fjord]

or no?

[native briar]

Yea

[silver fjord]

You sure?

[native briar]

Yea

[silver fjord]

Ok good

so now you just distribute that - sign

[silver fjord]

you have 2 minus signs on the second term

they should multiply each other

you're missing that

[native briar]

oh

[silver fjord]

first term is correct

[native briar]

ok

So like this ?

[silver fjord]

lol no

[native briar]

_>

[silver fjord]

you had the expression right

you just didn't multiply it properly

[native briar]

o

You mean this second one here

[silver fjord]

Ok now you need to distribute the minus sign

[native briar]

is it

-8x^7•cosx+x^8•sinx?

[silver fjord]

Yes!

So which answer was that?

Among the list

[native briar]

hm

is it

2nd one?

[silver fjord]

look carefully

look at all of them

[native briar]

hm

oh

last one

[silver fjord]

yes

[native briar]

yay

[silver fjord]

gj

[native briar]

tyty

how would I solve this

[silver fjord]

what do you think

you had a similar problem before actually

and i seem to recall you did that correctly

so what do you think you should do here

[tiny linden]

When you see that you have to find a derivative of a fraction what do you think you should do?

[native briar]

cry

[tiny linden]

Not after we teach you

[native briar]

but

[tiny linden]

You’ll be laughing with joy

[silver fjord]

https://tenor.com/view/out-disappear-bye-vanished-gif-4932063

[tiny linden]

Because that’s what this company stands for

[native briar]

i love ur enthusiasm

[tiny linden]

Our pleasure

[native briar]

ok

so

do I separate the fraction?

[tiny linden]

Have you heard of the quotient rule?

[native briar]

lord

ur right

[tiny linden]

https://youtu.be/DdV2UZV7AoA?si=lAzVr_BGMEbMPSj-

I’ve heard this helps people memorize it

[native briar]

Ok I got it

Like this ?

[tiny linden]

Exactly

[native briar]

nice

Is this right ?

<@&286206848099549185>

.close

[devout snow]

Channel closed

Closed by @native briar

Use .reopen if this was a mistake.

[tiny linden]

No

[native briar]

.reopen

[devout snow]

✅

[native briar]

No ?

[tiny linden]

You need to distribute the 3x^2 to (x-1) to multiply the two

[native briar]

How?

[tiny linden]

When you bring x-1 to the numerator it has to be in parenthesis

So it’s 3x^2 times x - 3x^2 times 1

For the upper left parenthesis

[ember pier]

f=√x+5 =x and f(a²)=aᵇ+c howwww

[native briar]

??

[ember pier]

do you know how

how to find b and c

[native briar]

I’m confused

[ember pier]

Bro this my math teacher killin my ass

[native briar]

what’s going on

[ember pier]

how to find b-c

[tiny linden]

See this picture

[ember pier]

i believe we workin on cartesian diagram or sum

but wth is thus

this

[native briar]

I’m lost

[tiny linden]

Do you see where you put down (3x^2 * x-1)

[native briar]

Yep

[tiny linden]

It’s supposed to be (3x^2 * (x-1))

Minor difference but it changes the result, you have to distribute 3x^2

[native briar]

??

[tiny linden]

How would you find 3x^2 * (x-1)?

[native briar]

no idea

is it

3x^2-3x^2?

<@&286206848099549185>

[junior bridge]

Ill help you give me a second

[native briar]

thx

[junior bridge]

The solution for this equation is zero

[native briar]

????

[tiny linden]

That’s close

[native briar]

This is the original question

[tiny linden]

It’s 3x^3-3x^2

So then the numerator would be 3x^3-3x^2-x^3

[native briar]

Yea

[native briar]

So all of this over (x-1)^2?

So is the answer

2x^3-3x^2/(x-1)^2

<@&286206848099549185>

[silver fjord]

Oh what's up

[native briar]

^^

Am I right

[main gull]

Yes

[native briar]

omggg hiii

and thanks

[main gull]

Well you need parentheses in the numerator when you write fractions in text form

[native briar]

oYea sorry

[silver fjord]

i member that

[native briar]

wrong one

Here we go

I tried to separate the fraction by writing it as

5x^2/5x+20x/5x

[main gull]

Yeah now you can simplify it

[native briar]

that’s where I’m stuck

ok can you simplify 5x^2/5x as 5x?

or does it simplify to 5x^2?

@main gull

[fleet eagle]

it simplifies as x

[native briar]

:0? Fr

?

[main gull]

Yes

x

[native briar]

How?

wait

[main gull]

Because $\frac{5x^2}{5x}$ the gcf of the numerator and denominator is 5x

[woven radish]

dldh06

[native briar]

is it bc 5x^2 simplifies to(5x*5x)

[main gull]

No

5x^2 = 5 * x * x

(5x * 5x) = (5x)^2

[native briar]

yea

so is it

x + 4x?

[main gull]

No

Because $\frac{20x}{5x} \neq 4x$

[woven radish]

dldh06

[main gull]

Notice how the x's are common and can be canceled out

[tiny linden]

Yeah

[main gull]

..... I guess better late than never

[tiny linden]

Lmao I wanted to make sure the question was answered

[native briar]

so then

what does 20x/5x simplify to

[main gull]

.

[native briar]

oh yea

So

yea idk

[main gull]

What are you left when the x's get canceled out

[native briar]

20/5?

[main gull]

Yes

Now simplify 20/5

[native briar]

4/1?

[main gull]

And that equals?

[native briar]

4?

[main gull]

Yes

[native briar]

it simplifies to x+4?

[main gull]

Yes

[native briar]

but it is that the derivative?

[main gull]

No

[native briar]

the derivative is 1 right ?

[main gull]

Yes

[native briar]

yay thx

This is my last question

<@&286206848099549185>

[fleet eagle]

Do you know how to derivate $\frac{1}{x^n}$?

[woven radish]

WilmiRosa

[native briar]

No

[fleet eagle]

ok mmm

do you know how to derivate $x^n$ ?

[woven radish]

WilmiRosa

do you know how to derivate $x^n$ ?

[native briar]

wait is it

1/x^n = x^-n

[fleet eagle]

Almooost!

[native briar]

yea sorry

negative right

[fleet eagle]

1/x^n = x^(-n)

[fleet eagle]

yeah!

[native briar]

yeeee

so is it

9/x^5 = 9x^-5?

[fleet eagle]

exactly

[native briar]

fr?

[fleet eagle]

100%

[native briar]

so then would 8/x be 8x?

[fleet eagle]

WAIT

NO

[native briar]

no?

[fleet eagle]

what about the negative? :v

[native briar]

:0

-8x?

[fleet eagle]

haha also, you write x/8 instead of 8/x jsjsj

[native briar]

_> wat

Oh yea my bad

[fleet eagle]

you got this! lets try again

[native briar]

ok so

I know that

9/x^5 can be written as 9x^-5

and so

can -8/x be written as -8x?

[fleet eagle]

no, you lift the x to the numerator, so you have to change the sign of the exponent ^^

-8x^{-1}

[native briar]

Ok

I see

[fleet eagle]

ok, so you can express 9/x^5 - 8/x as... ?

[native briar]

it can be expressed as 9x^-5-8x^-1

right? c:

[fleet eagle]

the sign!!!!!

[native briar]

ah sorry

I wrote it down but forgot to type it

so then do I differentiate it

And then switch it back to get the answer ?

I got y’= -45x^-6+8x^-2

[fleet eagle]

https://tenor.com/view/you-know-laughing-smile-gif-12881349

[native briar]

LOL

that’s the part I need assistance with 💀

[fleet eagle]

Thats peeerfect ^^

[native briar]

so

if I put it all back do it get

-45/x^6+8/x^2?

[fleet eagle]

and that's your answer ^^

[native briar]

https://tenor.com/view/ascension-sponge-bob-gif-15077046

SLEEP

[fleet eagle]

night!

[native briar]

ok thanks so much

goodnight !

[fleet eagle]

yaw

[native briar]

.close

[devout snow]

Channel closed

Closed by @native briar

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

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

[exotic marten]

a

[exotic marten]

https://i.imgur.com/r8cVcxS.png

I do not understand how (x^2 -8)^2 -8 goes to the next step

where does the -16x^2 come from?

[supple knot]

Do you know FOIL

[exotic marten]

yes

[supple knot]

(a-b)^2 = ?

[exotic marten]

i dont know that

i know like the (x+2)(x-2) foil

[supple knot]

(a-b)^2 = (a-b)(a-b)

Can you foil that right side?

[exotic marten]

hmm

i dont understand

ohh wait i see

you foil it okay

(x^2-8)(x^2-8)

.close

[devout snow]

Channel closed

Closed by @exotic marten

Use .reopen if this was a mistake.

[devout snow]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

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

[cerulean crater]

need help with this one

[cerulean crater]

If you reply plz tag me so I get a notification

[devout snow]

@cerulean crater Has your question been resolved?

[cerulean crater]

<@&286206848099549185>

Image of what I tried (a bit dark, sorry 😅 )

no one?

alright, it's getting late for me

.close

[devout snow]

Channel closed

Closed by @cerulean crater

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

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

[fringe bramble]

unsure where to start and where to go, i have aguess but it isnt feasible

[devout snow]

@fringe bramble Has your question been resolved?

[devout snow]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

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

[mint cedar]

help

[mint cedar]

am i doing this right

and what is the common ratio

<@&286206848099549185>

im sry im a bit stupid

i need help tho

so pls

[devout snow]

@mint cedar Has your question been resolved?

[devout snow]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

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

[tranquil lantern]

for 1.26, would a possible disadvantage be sampling bias because its only focusing on the eastern side of US?

[tranquil lantern]

or is this question looking for me to use the answers I obtained from 1.24 and 1.25 to come with a certain disadvantage?

[devout snow]

@tranquil lantern Has your question been resolved?

[devout snow]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

[devout snow]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

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

[neon folio]

If i have a graph 0.5x²-1 do i graph this graph first by moving it by -1 on the y axis and then stretch it, or do i first stretch it by 0.5 and then move it on the y axis

[neon folio]

Or are both ways correct?

[wooden veldt]

Why don't you try both and see what happens

[wheat pawn]

it is not the same.
check your order of operations and what they result in

[neon folio]

Wdym

[neon folio]

They re both the same

[wheat pawn]

you start with x^2

[neon folio]

Yes

[wheat pawn]

you move it 1 down
you got x^2-1

[neon folio]

Yes

[wheat pawn]

you stretch it by .5
you got 0.5(x^2-1)

[neon folio]

Yes ok i understand

.close

[devout snow]

Channel closed

Closed by @neon folio

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

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

[fringe bramble]

is this graph planar?

as far as i can see is no?

[devout snow]

@fringe bramble Has your question been resolved?

[fringe bramble]

<@&286206848099549185>

[silver coral]

I don't think this is planar

[fringe bramble]

okay i figured

[fringe bramble]

do you know to find subgraphs of k3,3 or k5?

[silver coral]

unfortunately no

[fringe bramble]

all goode

.close

[devout snow]

Channel closed

Closed by @fringe bramble

Use .reopen if this was a mistake.

[devout snow]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

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

[tepid berry]

is this question wrong worded?

[manic condor]

A bit confusing i think. I'm not sure whether each was cut into 16 equal pieces or total 16 pieces as in each into 8 equal pieces.

[tepid berry]

ya right and ans is saying this

[manic condor]

Ah. Lol. Badly worded then, yes.

[tepid berry]

ya i was confused what to do thanks for telling

.close]

.close

[devout snow]

Channel closed

Closed by @tepid berry

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

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

[bold cypress]

Can you guys help me with this?

[bold cypress]

I need an example of the answers

[restive river]

What do you mean exactly? Examples of solutions to this inequality? Examples of solutions to similar problems?

[gleaming scaffold]

I need help with a question

[restive river]

!help

[devout snow]

To ask for mathematics help on this server, please open your own help channel or help thread. See #❓how-to-get-help for instructions.

[gleaming scaffold]

!help

[devout snow]

To ask for mathematics help on this server, please open your own help channel or help thread. See #❓how-to-get-help for instructions.

[bold cypress]

Solutions to that inequality

[restive river]

x=0, x=-42, x=0.5, and many many more

If you need examples, just graph it

,w graph y=(3x+1)/x+4 and y=1

Oh

,w graph y=(3x+1)/(x+4) and y=1 from x=-3 to x=2

[woven radish]

Results provided by WolframAlpha

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

Requested by labyrinthsplash

[bold cypress]

It says that 3 of these given choices are the answers

[restive river]

,w graph graph y=(3x+1)/(x+4) and y=1 from x=-6 to x=-2

[woven radish]

Results provided by WolframAlpha

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

Requested by labyrinthsplash

[restive river]

dang

[restive river]

Then let's go through it arithmetically rather than graphically

@bold cypress Do you have any idea where to start? Have you perhaps already tried something

I'll be right back

[bold cypress]

Not rlly familiar with the solution

But i’ll try sumn

[restive river]

What did you try so far?

[bold cypress]

Stuck here so far

[restive river]

Hmmm okay, solving for the boundaries I see

That wasn't the method I had in mind but it's actually just as efficient so props for that

[bold cypress]

Are these correct?

[restive river]

Nothing tells you that all of these boundaries give valid solutions

In fact with your method right now every inequality would be true everywhere, I don't know if you even noticed that

These "boundaries" as I'm calling them just tell you where to check

Either all of the solutions in one of the intervals are valid or they're all wrong

@bold cypress For each of these intervals, pick a single value from it, and check it against your inequality

[bold cypress]

Idk man it just asks the intervals it disregards wether it is true or not

Im trippin balls

[restive river]

It doesn't care whether your interval is actually a solution to the inequality?

Why even bother then

[bold cypress]

So are these the intervals?

[restive river]

...I just answered that

The "boundaries" you found just tell you in which intervals you have to check

The points at which (3x+1)/(x+4)=1 tell you where it crosses the line y=1 and the points at which x+4=0 tell you all of the discontinuities

,w graph y=(3x+1)/(x+4)=1 and y=1 from x=-2 to x=2

[woven radish]

Wolfram Alpha doesn't understand your query!
Perhaps try rephrasing your question?
Click here to refine your query online

Requested by labyrinthsplash

[restive river]

??

[restive river]

Anyways, this

Where it crosses y=1, x goes from a valid solution to a non-valid solution

And where there is a discontinuity (where the denominator goes to 0), y jumps from -infinity to infinity

This is why those "boundaries" give you all the intervals to consider

Now you just need, for each interval, to pick one value from it

And whether that value is a solution tells you whether the whole interval is a valid solution

@bold cypress Does that make sense at all?

[bold cypress]

Hold on im tryna keep up

[restive river]

Yeah I was expecting that kind of response, I'll try to show it in a more intuitive way

whoops

[bold cypress]

It’s just that our teacher did not mention anything about graphing and boundaries. He just taught us how to find the solution set and intervals of a rational equation

[restive river]

He probably just taught you a different method

[bold cypress]

And wether it is true or not

[restive river]

By "true" and "valid" I just mean that it is a solution to the inequality

I'm going along with this whole "boundary" thing because it works quite well and because that's basically what you started doing

[restive river]

@bold cypress See how your "boundaries" divide the curve into 3 sections?

If you slowly change x, the only way for it to go from a solution to a non-solution or vice-versa is to go through the y=1 line or have a discontinuity

On the right, where you replaced your < with a =, you got a point where it crosses the y=1 line

[bold cypress]

Thank you for ur time man

[restive river]

This is why as x reaches 3/2 (3x+1)/(x+4) goes from being less than 1 to being greater than or equal to 1

And the point at which the denominator reaches 0 gives you a discontinuity as you can see on the graph

@bold cypress This is why, if you picture a point following along the curve's path, as it reaches x=-4, it will go from a non-solution to a solution, and why as it reaches x=3/2 it will go from a solution to a non-solution again

Points where the curve crosses y=1 and points of discontinuity are where non-solutions may become solutions or vice versa

And because there's none of that in the interval (-4,3/2), you know that if one of the x-values within this interval is a solution then all x-values within this interval are solutions

Let me say that again, since in the interval (-4,3/2) the function y=(3x+1)/(x+4) doesn't cross y=1 and doesn't have a discontinuous jump, there's no way that anywhere in there it goes from a solution to a non-solution or vice versa

So either they're all solutions or all non-solutions

@bold cypress You got all of that?

Ping me once you get back

[devout snow]

@bold cypress Has your question been resolved?

[devout snow]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

[devout snow]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

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

[pine rock]

!show

[devout snow]

Show your work, and if possible, explain where you are stuck.

[orchid wasp]

I have no idea what to do

[neon aspen]

as per the graph, what is f(-7)?

[willow helm]

Do you know what are constant functions?

[orchid wasp]

5?

[orchid wasp]

No

[neon aspen]

yes

f(-1)?

[orchid wasp]

5

What about the second function

It has 2 y

[neon aspen]

it's linear

in the form y = mx + b

m = slope

b = y intercept

[orchid wasp]

Y=-1x+3

?

@neon aspen

[neon aspen]

Check b

[orchid wasp]

3 or 6

[neon aspen]

No

What's f(0)

[orchid wasp]

5

[neon aspen]

Yes

[orchid wasp]

So the answer is 5 for first 2 questions?

[neon aspen]

Basically b=5 for the second bit

Cuz that's the y intercept

The first bit is constant at 5

The last bit is also constant

[orchid wasp]

So what’s the answer

@neon aspen so my first and third answer r correct

[neon aspen]

Gtg

[devout snow]

@orchid wasp Has your question been resolved?

Channel closed

Closed by @orchid wasp

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

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

[elfin grotto]

I haven't started doing this yet, I'm still pretty new to Calculus type math and are only going to the take the basic math course.

I was wondering if any of you have some tips, on some videos/problems to solve before tackling this problem? I don't understand how to solve this and would like to practice and understand first

[devout snow]

@elfin grotto Has your question been resolved?

[supple knot]

https://youtu.be/sdYdnpYn-1o?si=Rx_Q6DXLygBgfZWe

[elfin grotto]

Thank you so much 🙏🙏🤩

.solved

.close

[devout snow]

Channel closed

Closed by @elfin grotto

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

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

[restive river]

hi i kinda dont get it where i made mistake, help. at the first picture is where i do exercise and second of the right result

[violet wind]

You shouldn't be getting any y^4 terms

your -6y^4 term is wrong

[restive river]

oh

i see

is that all

[vagrant stump]

Last term also 2y⁴x

[restive river]

so

x^2*x^2 is not x^4?

for example

[vagrant stump]

Its 12y⁴x

[restive river]

oh

i see

thanks

.close

[devout snow]

Channel closed

Closed by @fiery hornet

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

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

[tranquil sand]

Hi everyone, I'm really struggling with an exercise in the middle of Armstrong, showing that if maps $f,g:S^{n-1}\to X$ are homotopic via $F$, then $X\cup_f S^{n-1}$ is homotopic to $X\cup_g S^{n-1}.$ \

The strategy online seems to be to see both are deformation retractions of the larger attachment space: $X\cup_F (D^n \times I)$.
(For example, a sketch here does it that way: https://math.stackexchange.com/questions/608908/ ) \

In the sketch one first constructs deformation retraction $d_f$ from $D^n \times I$ down to a space $D^n\times {0}$, along which $F$ looks just like $f$ (and similarly a deformation retraction $d_g$ to $D^n\times {1}$, where $F$ looks like $g$). \
Somehow this $d_f$ yields a deformation retraction on the attachment-spaces themselves: $(X \cup_F (D^n\times I))$ goes down to $(X\cup_f (D^n\times {0}))$. \

My question: how is this new deformation retraction on the attatchment spaces defined from $d_f$?

[woven radish]

enthdegree

[devout snow]

@tranquil sand Has your question been resolved?

[devout snow]

@tranquil sand Has your question been resolved?

[devout snow]

@tranquil sand Has your question been resolved?

[devout snow]

@tranquil sand Has your question been resolved?

[placid rover]

#point-set-topology

[devout snow]

@tranquil sand Has your question been resolved?

[devout snow]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

[devout snow]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

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

[mighty copper]

how to do part c

[mighty copper]

this is what I did:
Dx/dt = answer of part a
solved differential equaton, got x in terms of t. then i substituted part b to solve for x

but i ended up getting 0

[neat solstice]

we dont want x, we want v

F=ma=mv'
so we get
-ae^(bv)=mv'

[devout snow]

@mighty copper Has your question been resolved?

[mighty copper]

for c part too?

[neat solstice]

for c, we need x

[mighty copper]

yeaa i needed help with c

[neat solstice]

ah ok

[mighty copper]

i did dx/dt = the expression I got in a. solved the differential and substituted t from part b

[neat solstice]

do we have the same here?

[devout snow]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

[neat solstice]

also, assuming i did it correctly
at the end of the integration to get x(t), dont sub your u back in
instead try to rewrite the u(t) with the t you got from part b)
that makes it nicer

[devout snow]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

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

[placid jackal]

4x^4-4x^2+4x-1 =0

[river solar]

are you trying to solve?

[placid jackal]

yeah for x

[river solar]

have you tried to factorise?

[placid jackal]

yes i did

i also tried ruffini but it doesnt have any rational roots

[river solar]

have you been given any info?

[placid jackal]

that i have to do scomposition of first member and then use product cancellation law

[river solar]

so my first thought is, what does it look similar to?

[placid jackal]

a quadratic equation?

[river solar]

yes

i would agree

except its got something extra

a fourth term

[placid jackal]

yess

[river solar]

now, if it didnt have that fourth term

if one of the terms was gone

could you create a quadratic - type factorisation

[placid jackal]

ahh should i try but subtracting both sides by a negative member?

[river solar]

hm, maybe not

so

lets say you only took 3 of the values

is it possible to create a quadratic of some sort

something of form

(a+b) * (a-b)

[placid jackal]

yes

[river solar]

or (a+b) * (a+b)

[placid jackal]

i did try that

[river solar]

what did you try?

im not saying thats the solution

but what did you try?

[placid jackal]

i basically tried every possible thing like ruffini, binomial difference and playing with different members

[river solar]

okay 1 sec

ill try and work you through what my approach what

was*

so, take (a+b) (a+b) or (a+b)(a-b)

we have a constant, and also an x^4 value

these can only be obtained through two of the same type, right?\

[placid jackal]

yes

[river solar]

would we use (a+b) (a+b) or (a+b)(a-b), and are we using +b on both, or -b on one

[placid jackal]

i can proceed with (2x^2)^2+ 2^)

[river solar]

uhh

maybe rewrite, i think you made a mistake aha

[placid jackal]

(2x^2)^2+ 2^2)+4x-1

naaah

my bad

[river solar]

i would say, forget the rest of the equation

to get x^4

or 1

what will A and B be?

[placid jackal]

2x^2+1 2x^2-1

[river solar]

perfect start

and it will be +1, -1, since the final term is a negative

so if we expand that, what terms do we have?

[placid jackal]

4x^4-1?

[river solar]

great

so we are a little closer

now, what other terms could we add to the bracket

that doesnt create a larger power than x^4, or smaller than x^0 (or 1)

[placid jackal]

i'm thinking

[river solar]

we have (x^2, x^0)

[orchid sierra]

........Numan. 😑

[placid jackal]

yes??

[orchid sierra]

(Seinfeld reference. Carry on)

[river solar]

hhahahah

[placid jackal]

hahahaah

[river solar]

these are the powers of the terms we have in the brackets currently, what else could we play with

[placid jackal]

signs?? maybe

[river solar]

to try and create the remaining terms of the equation (x^2, and x^1)

maybe

(x^2, x^0, and we can add x^1)?

[placid jackal]

i've been working on this one for 3 hours so my brain is kind of fried rn

[river solar]

sso, just x

its okay ahahha

like

we could do something like

(2x^2 + ax + 1) * (2x^2 +bx - 1)

since ax * bx = c * x^2

[placid jackal]

okk

[river solar]

and ax or bx * 1 = cx?

do you get that?

now, we can either expand the general form of the brackets

[placid jackal]

nope

[river solar]

okay so

if you have (x+1) (x +2)

u can get x^2, x^1 and a constant

[placid jackal]

yes

[river solar]

the x^1 term is gotten by multiplying mixed powers right?

[placid jackal]

yess

[river solar]

when you have three powers

like (x^2 + 3x + 9) (x^2 + 5x - 18)

[placid jackal]

ahhh ok

[river solar]

u can get x^2 by doing x^2 and 18, or 5x and 3x

right?

[placid jackal]

now i get it

[river solar]

4x^4-4x^2+4x-1=0

thats why we can determine some guaranteed terms

you can only get the highest term by multiplying two things together, right?

[placid jackal]

yess

[river solar]

and the lowest by multiplying two, but the ones in between have multiple elements

now

i know that there must be a way to get 4x

and if we use x^2 and constants, can we get 4x?

are there any possible combinations?

at all?

i would suggest no, can you see why?

[placid jackal]

4x^2-1 = (2x-1 )(2x+1)

i think not

[river solar]

ur right

to get 4x

we must use some combination of x and constants

so

thats why

we have (2x^2 + ax + 1) (2x^2 +bx - 1)

[placid jackal]

ok

[river solar]

we already know we must have 2x^2 and 1 since they are used to obtain 4x^4 and -1

then ax and bx must somehow reach 4x and -4x^2

[river solar]

now, uve almost worked it out i can see already, but what is the highest term that ax * bx can achieve?

[placid jackal]

ax 4x^2 and bx 4x

[river solar]

what?

i was thinking if you multiply two x terms, you get x^2

[placid jackal]

honestly i didnt understand the qs

[river solar]

okay thats alright

ill try and explain

so we want ax*bx to hit the -4x^2 term

since (2x^2 +1) (2x^2 -1) = 4x^4-1

the only way to create -4x^2 is by multiplying some x terms

and i can guess

-2x and 2x

u actually found

[placid jackal]

yes

[river solar]

earlier u were very closer

[river solar]

maybe

[placid jackal]

yeah

how should i proceed after this?

ok

[river solar]

so actually

u can just say

since the only way to create -4x^2 is with 2x and -2x

you can just say

(2x^2 +2x -1) * (2x^2 -2x + 1)

is the solution

[placid jackal]

its like adding and subtracting nonesistent variables and breaking the existing ones

[river solar]

yes u can say that

[placid jackal]

i dont know which term is used in english to describe this method

[devout snow]

Channel closed

The owner is missing!

[devout snow]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

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

[marble quail]

What is the differenece between the formula for combinations and permutations?

[devout snow]

@marble quail Has your question been resolved?

[wooden veldt]

https://www.mathsisfun.com/combinatorics/combinations-permutations.html

[devout snow]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

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

[fair mantle]

Can someone explain what he did from step 2 to step 3? I'm feeling very stupid right now. I understand it up to step 2, but i don't understand what he did to get from step 2 to 3

[robust dust]

just factored out (n+1)/6 from both fractions

the left fraction is (n+1)/6 times n(2n+1), and the right one is (n+1)/6 times 6(n+1)

[fair mantle]

OH I see i understand now. Thank you

.close

[devout snow]

Channel closed

Closed by @fair mantle

Use .reopen if this was a mistake.

[devout snow]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

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

[round current]

Hi
Can anyone help me with this problem

An ellipse with its foci on the y-axis and passing through the focus of a parabola (4,0) , the ellipse pass through the Directrix of the Parabola (y²=8X) A segment with length 10 √3 Find the equation of the ellipse

The only thing i came up with was B = 4

[devout snow]

@round current Has your question been resolved?

[devout snow]

@round current Has your question been resolved?

[devout snow]

@round current Has your question been resolved?

[devout snow]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

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

[restive river]

This is my work. I dont know how to even start off with this question.

[devout snow]

@restive river Has your question been resolved?

[restive river]

<@&286206848099549185>

[white meadow]

can you scan the paper? it is not very readable

[restive river]

I dont know how to scan using discord. Is this better?

[white meadow]

not really, try using an app called "CamScanner"

[restive river]

This is it right?

[white meadow]

what topic is this?

[restive river]

Trig

Using the compound angle formulae Rcos(x+a) and Rsin(x+a)

Is the name of the sheet

[restive river]

close

.close

[devout snow]

Channel closed

Closed by @earnest finch

Use .reopen if this was a mistake.

[devout snow]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

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

[distant light]

whats the transformation for the 3 inside the root sqaure

[winter patrol]

note that sqrt(3x) = sqrt(3) * sqrt(x)

[distant light]

okay so what would the tranformation be?

[winter patrol]

,tex .transformation rules

[woven radish]

ℝam()n()v

[distant light]

ok tysm

.close

[devout snow]

Channel closed

Closed by @distant light

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

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

[idle tinsel]

can sigma notation like this be solve with integrals?

[idle tinsel]

i was taught sigma notation is like a "rough sum" and integral is a "smooth sum" (includes non-integers) but integrals are kinda easier to work with so curious if they can solve sigma notation problems

[wicked turtle]

you can certainly express it as an integral (of a step function, for example), but if you want to compute the actual value of the sum, i don't think using an integral is going to help you

[idle tinsel]

is there a faster way to calculate sums than manually adding it?

[wicked turtle]

some types of sums, yes

you can use facts like $\sum_{n=1}^{N} n = \frac{N(N+1)}{2}$ for example

[woven radish]

Bungo

[pine rock]

could apply distribution and association

[wicked turtle]

in this case you're only summing four numbers, so it may well be faster just to brute force it rather than expressing it in terms of simpler sums

[idle tinsel]

for this one I did:
(2*2 - 1) + (2 * 3 - 1) + (2 * 4 - 1) + (2 * 5 - 1)

if it went from 2 to 1,000,000, or some arbitary large number, how would I do a problem like that?

[wicked turtle]

something like this:
$$\sum_{j=2}^{1000000}(2j - 1) = 2\sum_{j=2}^{1000000}j - \sum_{j=2}^{1000000}1$$

[woven radish]

Bungo

[idle tinsel]

ohh

[wicked turtle]

on the right hand side, for the first sum you can use
$$\sum_{j=2}^{N} j = \sum_{j=1}^{N} j - 1$$
and use the formula I wrote above

[woven radish]

Bungo

[wicked turtle]

and the second sum is simply 999999

since you're adding up that many copies of 1

[idle tinsel]

i got 1,000,001,000,002

i mighta messed up somewhere'

oh its actually a pretty number

[wicked turtle]

hmm, let's see, 2(N)(N+1)/2 - 2 - 999999 = 999999999999

yep checks out

N = 1000000 in that formula

[idle tinsel]

ah yeah I made a couple mistakes, I think I added 1,000,000 and I changed N + 1 in the formula to N + 2 because I thought maybe + 1 was the lower limit of your sum

[wicked turtle]

the -2 in my calculation is the adjustment due to starting the sum at j=2 instead of j=1

[devout snow]

@idle tinsel Has your question been resolved?

Channel closed

Closed by @idle tinsel

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

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

[humble flare]

anyone could help me with this?

[graceful cosmos]

Can you put into simpler words, what makes a set "good"?

[still zephyr]

By "good" you mean well ordered set?

[humble flare]

no

its just like

a name of a set

it could also be "bad"

it doesnt matter it just has to fit the statement following

conditions

[humble flare]

i guess that every x value in the set has atleast one a value thats smaller than x

[graceful cosmos]

@humble flare well said!

[1,2) isn't good, because there's nothing smaller than 1 in the set.

[humble flare]

true

[graceful cosmos]

Any change we could make to it?

[humble flare]

maybe (1,0,-1,...)

?

[graceful cosmos]

Can't be a finite set

Note that [1,2) is a real interval

So 1.5 is in there

Oh wait whoops I am the one who misread

[humble flare]

so maybe make it A= Z

[graceful cosmos]

Yes okay I see what you're saying now. Yes that works

To just have a set of integers that trails off to negative infinity

[humble flare]

yep

how would u write that set

with this bracket right (

[graceful cosmos]

I was personally thinking of the real set (1,2)

[humble flare]

how would that work?

cuz there isnt any number smaller than 1

in the set

[graceful cosmos]

1 isn't in the set

[humble flare]

why

[graceful cosmos]

Nvm, let's just go with your suggestion.

So you should use {} and not () for a set

So you want {1,0,-1,-2...}

[humble flare]

okok thats our set

then i would go

[graceful cosmos]

Also Z is another example, that does work too

[humble flare]

yeah

i think A=Z is easiest

so i guess

start with WTS

every x part of A, there exists atleast one a whcih is part of A such that a < x

but after that idk

would proof by induction work for this?

[humble flare]

hello?

[graceful cosmos]

If you pick n, then n-1 is smaller

[humble flare]

what would be the base case?

[graceful cosmos]

Actually that's not even induction. Just a direct proof

[humble flare]

yea

so i could prove it without induction

is saying take a to be x-1

then a will always be smaller than x

[humble flare]

?

[graceful cosmos]

Yeah that works!

[humble flare]

okay great thanks

.close

[devout snow]

Channel closed

Closed by @humble flare

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

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

[pastel pasture]

integrate by parts

I suppose

[devout snow]

@urban hornet Has your question been resolved?

[urban hornet]

Doesn't work

<@&286206848099549185>

[fresh cosmos]

take secx + tanx = t

so (secx tanx + sec^2 x) dx = dt

[urban hornet]

How do i find secx in terms of t?

[fresh cosmos]

sec ^2 x - tan^2 x= 1

so

1/(secx - tanx) = sec x+ tan x

do you got it?

[urban hornet]

Yes

Thanks

[fresh cosmos]

so sec x can be wriiten as (1/2)(t + 1/t)

[urban hornet]

.close

[devout snow]

Channel closed

Closed by @urban hornet

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

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

[plain schooner]

Was watching my professors video

and was wondering how she was able to figure out what type of discontinuity based on the function

can someone please help me understand how come the answer is removable discontinuty just based off hte function

[echo vapor]

this too easy

so basically

[urban hornet]

If limit at the point exists then it is removable discontinuity

If limit dne then it is non removable

[echo vapor]

plug k into x2 and when k-1+ to the equation put it to the other side and see if the equation is still discontinous or not

[plain schooner]

ah okayyy

thank you guys

.close

[devout snow]

Channel closed

Closed by @plain schooner

Use .reopen if this was a mistake.

[devout snow]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

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

[tender grail]

need help

[tender grail]

7 is one right?

[pseudo basin]

yes f'(7) is indeed 0

f' is also zero at some other points tho.

[tender grail]

what makes it a 0?

[pseudo basin]

what's "it"

[tender grail]

what makes certain x points a 0

[silent mesa]

you see when the line flat lines

thats when the slope is 0

[tender grail]

wdym?

[silent mesa]

so

[tender grail]

is 0 and 2 a 0 too or?

[silent mesa]

you mean f'(0) and f'(2)?

[tender grail]

yes

[silent mesa]

yea those points also have derivative of 0

derivative is like the slope of a line at one point

[tender grail]

f'(3) and f'(5) would not be a zero correct?

[silent mesa]

f'3 and f'(5) dont have derivatives it kinda complicated but they are not 0

[tender grail]

so f'(0), f'(2), and f'(7) are zeros

is f'(9) a zero?

[silent mesa]

no

think of f'(number) as the slope of that point

at f'(9)

the graph line is going down

so its slope is negative and not 0

it is only 0 when the line is not moving up or down

[tender grail]

ok so f'(8) is also not a zero

[silent mesa]

yea

did you learn what limit is

like limit as x approaches 0

[tender grail]

yeah

[silent mesa]

you know how limit calculate for one point

[tender grail]

yeah i think so

[silent mesa]

well

you can just go to one point

and draw a line paralell to it

and approximate the slope of that line

its kinda what derivative is

i will send image

[tender grail]

I feel like I already have all the zeros though

[silent mesa]

yea i think those right

[tender grail]

but it keeps marking me wrong

Im just making sure im not missing anything

[silent mesa]

you see this is what derivative of 0 looks like

[tender grail]

yeah

[silent mesa]

you draw a flat line and its slope aligns

[tender grail]

like a tangent line

[silent mesa]

yea it is a tangent line

[tender grail]

how would you do it for f'(3) tho