help-41

[ancient raft]

No, but the root changes with $(e^{i 2\pi/n})^k$, so how come they be the same?

[grizzled pagoda]

Good

[pallid canopy]

yes 1+2i is a double root

that's more than what you were asked

[ancient raft]

Alright, thanks for the help.

.close

[amber wasp]

Channel closed

Closed by @ancient raft

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.

[edgy cave]

Somehow both of these answers are still wrong

[edgy cave]

I plugged in my K and my initial conditions to solve for the first one

Which was the rate of change and it was still wrong

I tried -4.8 and -4.80

<@&286206848099549185>

[amber wasp]

@edgy cave Has your question been resolved?

Channel closed

Closed by @edgy cave

Use .reopen if this was a mistake.

[amber wasp]

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.

[frail holly]

help finding the range ?

[mint wraith]

parabolla part goes to $\infty$ right?

[grizzled pagoda]

MetuMortis

[amber wasp]

@frail holly Has your question been resolved?

[amber wasp]

@frail holly Has your question been resolved?

[amber wasp]

@frail holly Has your question been resolved?

[plain coral]

you here?

[frail holly]

Yep

i got the answer

but was wondering if you can help me with a new question ?

[plain coral]

sure

[frail holly]

Appreciate it

[plain coral]

what part about this are you stuck on

[frail holly]

everything

Dont even know where to begin

[plain coral]

what shape is the box?

[frail holly]

rectangle

[plain coral]

thats a 2D shape

this box is 3D

[frail holly]

rectangular

[plain coral]

which sides of the box are square?

[frail holly]

the front and back

[plain coral]

are you sure that its the front and back of the box that are square?

[frail holly]

like depnds where you look from

some can say the side

[plain coral]

what shape is the base?

[frail holly]

rectangle

[plain coral]

reread the first sentence of the question for me

[frail holly]

oh

square base

[plain coral]

now that youre rethinking the shape of the box,

which sides of the box are square?

[frail holly]

top and bottom

[plain coral]

yep, the shape of the box is rectangular with a square base
now does this box include a top side or leaves it open?

[frail holly]

open

on the top

[plain coral]

yep

now what does x represent?

[frail holly]

x represents one of the sides of the base

[plain coral]

the base is square

are all of the sides of the base the same?

[frail holly]

yea

[plain coral]

so whats the area of the base?

[frail holly]

x^2

[plain coral]

s?

[frail holly]

sorry

[plain coral]

whats the volume of the box?

[frail holly]

L times w times H

[plain coral]

whats l and w?

[frail holly]

I guess it would be x^2 times H

[plain coral]

now what value is the volume of the box?

[frail holly]

5 cubic metters

[plain coral]

can you solve for h in terms of x?

[frail holly]

yeah

[plain coral]

nice, now for the surface area

what formula would we usually use for the surface area of a rectangular box?

[frail holly]

2(wl+hl+hw)

[plain coral]

now since we're leaving the top out, what would that formula become?

[frail holly]

i guess no length or width ?

[plain coral]

...do you know where this formula comes from?

[frail holly]

No

[plain coral]

try thinking about it

[frail holly]

okok

like

the area of the surface

suronding the shape

if that makes sense ?

outside area

oh

area of the faces would be multiplying the length adn width

twice becuase theres 2 faces

and for the other one we would multiply 2 ( length times height ) twice

and then for the remainder one we would multiply 2 by the width and height

[plain coral]

what about the length and height

oh wait you already said that whoops

misread

now if we leave the top out, what does the formula become

[frail holly]

we would only multiply it once

since one of the faces is gone

[plain coral]

so what is the formula for the surface area

[frail holly]

lw + 2lh + 2wh

[plain coral]

now for our rectangular box, we know l = x and w = x

go place that into the formula

[frail holly]

so

[plain coral]

btw we're not done yet after this

theres still a few more steps

[frail holly]

x^2 + 2(x^2) + 2wh

[plain coral]

try doing that again

[frail holly]

x^2 + 2xh + 2xh

[plain coral]

now remember from earlier when we were talking about volume

we found volume = x^2 h and volume = 5

use this to solve for h in terms of x

[frail holly]

would

h = 5/x^2

[plain coral]

thats correct

[plain coral]

now with h = 5/x^2, place that into the surface area formula here

[frail holly]

x^2 + 2x ( 5/x^2) + 2(5/x^2) times X

[plain coral]

you dont have to worry about x being read as "times"

[frail holly]

okok

[plain coral]

youre in algebra, $\cdot$ is used for times instead

[grizzled pagoda]

mtt

[plain coral]

when you type it, you can use the * key to represent that dot

[frail holly]

appreciate it

[plain coral]

so x^2 + 2x(5/x^2) + 2x(5/x^2)

can you simplify this?

[frail holly]

yeah

x^2+ 20x/x^2

[plain coral]

thats not correct

try that again

the 20 is correct though

(so is the x^2 + )

[frail holly]

I mistyped

[plain coral]

now whats x / x^2

[frail holly]

1/x

[plain coral]

so x^2 + 20/x

this is your final answer

[frail holly]

x^2 + 20/x

yeah

sorry

didnt see you wrote it

Thanks alot mtt

[plain coral]

np

[frail holly]

really appreciate it

[amber wasp]

@frail holly Has your question been resolved?

Channel closed

Closed by @frail holly

Use .reopen if this was a mistake.

[amber wasp]

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.

[keen pawn]

How does this look?

[inner axle]

Yep

[keen pawn]

thanks

.close

[amber wasp]

Channel closed

Closed by @keen pawn

Use .reopen if this was a mistake.

[amber wasp]

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.

[regal swift]

not sure how to proceed

[crisp otter]

idk try usig chain rule in step 3

[regal swift]

and how would I do that

[pallid canopy]

This doesn't look exact

[regal swift]

im trying to make it exact by finding an integration factor but I dont think its working, I dont really have any other ideas

[amber wasp]

@regal swift Has your question been resolved?

[amber wasp]

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.

[azure ember]

Hi, i dont understand why x>0 for the asnwer? I could solve everything but i literally jut dont understand the x>0 thingys, are they necessary in my answer

[azure ember]

😮‍💨

[copper cradle]

Yeah it is. Observe that if x<0 then the original argument is going to be 7/6pi instead.

This sort of off-by-pi check happens all the time with polar form if we insist that r>0

So it’s worth being in the habit of looking for it as your last step

[azure ember]

OHH

[azure ember]

okay cool

thank you

[azure ember]

wait but if x is -5 or something

-5*(1/sqroot3)= around -2.5 or something

-2.5+1 = -1.5

and if x is negative ont he bottom

then it would be positive??

[copper cradle]

What would be positive?

x and y would both be negative, as you said

[amber wasp]

@azure ember Has your question been resolved?

[amber wasp]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

[amber wasp]

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.

[strange field]

!15m

[amber wasp]

Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.

[spare belfry]

Hint: sin^2 + cos^2 = 1

[amber wasp]

@viral solar Has your question been resolved?

Channel closed

Closed by @viral solar

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.

[dawn terrace]

I am not getting the right answer for a can someone help out

<@&286206848099549185>

[cunning birch]

!status

[amber wasp]

What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above

[cunning birch]

show your work if you didn't get it right

[cunning birch]

you did get it right

just... loosen up the bounds a bit

[dawn terrace]

Oh… right

Sorry I just expected it to fall out without any extra work

Mb

.close

[amber wasp]

Channel closed

Closed by @dawn terrace

Use .reopen if this was a mistake.

[amber wasp]

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.

[tough quail]

Hello! I don't know how to get the roots. please, help me! :>

Yes :>

Ty

I tried to search the answers from google to see if i was correct, but it says that i have to multiply the real and imaginary numbers, which are the i and 2

[grizzled pagoda]

faiyrose

[tough quail]

Yes, I'm not sure if it's correct.

I looked at different websites, there are also different answers.

Oh, okay. 👍

[native stag]

imaginary numbers are also commutative under multiplication so it's fine

commutativity is your friend. :D

[tough quail]

Oh okay :>

.close

[amber wasp]

Channel closed

Closed by @tough quail

Use .reopen if this was a mistake.

[tough quail]

Ty! I got the answer!

[amber wasp]

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.

[twilit heron]

I was scanning through my friends' notes and tried answering the same problem but her notes got cut off to -10±2sqrt30/2, so i got confused for a moment whether i have to continue the formula or leave the answer as is. I tried asking a friend what answer they got and their answer is farther from mines. Did i do it right or nah?

[twilit heron]

oh w8 nvm i think it was the 4(30)

.close

[amber wasp]

Channel closed

Closed by @twilit heron

Use .reopen if this was a mistake.

[native stag]

unrelated but you can do better than paint lmao

/lh

[left spire]

dont hate on paint

[twilit heron]

LMAO im on pc ok...

[amber wasp]

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 flower]

Hi

[exotic flower]

I need help on this, idk how to do it bc theres a square root

Question d

[tulip tapir]

,rccw

[exotic flower]

Solving slope

[tulip tapir]

you mean

differentiation

?

[exotic flower]

Ik that for others i just do power rule

[knotty reef]

You have to evaluate the derivative at x = 16?

[tulip tapir]

recall that sqrt(x) = x^0.5

[exotic flower]

Yes

[tulip tapir]

then you can just use the chain rule

[exotic flower]

Whats chain rule

[tulip tapir]

(f(g(x))' = f'(g(x) * g'(x)

[exotic flower]

I dont understand

So would i get rid of square root first?

[knotty reef]

First take the derivative with respect to x-7 and then take the derivative of x-7 wrt to x

[exotic flower]

Hmmm

[knotty reef]

I think what the author wants is that the slope remains same whether you take the derivative with respect to x or x-7

[exotic flower]

Yeah the slope

So what would i do then…

Can you vissually show me somehow

Bc for others all i did was lower the power down for all and move the exponent in front of the number

I cant use desmos

Or graphing anything

[tulip tapir]

okay look

the function is

[exotic flower]

I needa know how to do it

Like that

[tulip tapir]

correct?

[exotic flower]

Bc i have more questions way harder

[grizzled pagoda]

ren

[exotic flower]

Ryes

Yes

[tulip tapir]

we can say that f(x) = x^0.5

and g(x) = x-7

so y = f(g(x))

correct?

[exotic flower]

Ok..

[tulip tapir]

are you with me

[exotic flower]

Im thinking

[tulip tapir]

okay

[exotic flower]

Hmmm

How do i show my work then…?

Wait

[tulip tapir]

no no no

that's absolutely wrong

[exotic flower]

Ok

Im confused rn

I checked the answer book, its m=4 but idk how to get there

[tulip tapir]

ykw forget it

you're gonna need to learn the chain rule

[exotic flower]

Hmmm

What the chain rule

I need help

Anyone please

<@&286206848099549185>

[exotic flower]

Question d

Idk what chain rule is

[tulip tapir]

i just explained what the chain rule is...

[exotic flower]

Finding slope

[tulip tapir]

ykw

do you understand

[exotic flower]

Nope…

[tulip tapir]

that the slope of sqrt(x-7) at x = 16

is the same

as the slope of sqrt x

at x = 9

[exotic flower]

Sqrt is square root

Right?

[tulip tapir]

if you have to show ur working btw idt you can use this reasoning.

yes

[knotty reef]

Why ?

[exotic flower]

How is sqrt(x-7) at x=16 same as x=9…?

Ik if u sub 16 in

U get 9

[tulip tapir]

*sighs*

[exotic flower]

But that isnt the slope

[tulip tapir]

basic chain rule, check it out

[tulip tapir]

think about it

[knotty reef]

Mb ic 1/2√x and 1/2√x-7

[exotic flower]

Hmmmm

[tulip tapir]

mhm

[exotic flower]

Could u somehow

Like

[exotic flower]

Do this thing

For each step…?

[tulip tapir]

yes

we can

[exotic flower]

It makes it easier for me to understand

[tulip tapir]

but there're two things

[exotic flower]

Im listening

[tulip tapir]

either you can use the chain rule

[knotty reef]

I should stick to nt and combi calc isn't my cup of tea

[tulip tapir]

OR understand this

[exotic flower]

I want to learn chain rule

If its more benificial

For the fiture

Future

[tulip tapir]

ay fellow NT enthusiast
off track sorry

okay

this is the chain rule

[knotty reef]

Go see 3blue1browns video on it

[exotic flower]

Ok…

[grizzled pagoda]

ren

[tulip tapir]

@exotic flower

[exotic flower]

Is that a specific formula?

[tulip tapir]

yes

that's a formula you're gonna have to learn at some point

[exotic flower]

What does ‘ mean after the letters

[tulip tapir]

the letters are functions...

[exotic flower]

The comma

[tulip tapir]

and f'(x) means the derivative of f(x)

[exotic flower]

Oh

[tulip tapir]

it's called f-dash not f-comma btw

[exotic flower]

Alr

[tulip tapir]

so f'(x) is f-dash x aka the derivative of f

yeah

now what we need to do is apply this here

[exotic flower]

Ok

[tulip tapir]

f(x) = x^0.5

[exotic flower]

Im watching

[tulip tapir]

and g(x) = x-7

[exotic flower]

We have 2 functions?

[grizzled pagoda]

ren

[tulip tapir]

yes

[exotic flower]

Hmmm

[exotic flower]

Would solving this give us the slope

[exotic flower]

Ren…?

U still here?

[exotic flower]

I need to find the slope so do i plug x at 16 back in?

<@&286206848099549185>

Anyone here…?

[tulip tapir]

sorry im back

got distracted with other channels mb mb

[exotic flower]

Welcome back…

All g

[tulip tapir]

anyway

do you understand this?

[exotic flower]

Up to that point yes

[tulip tapir]

we differentiate f(x) and replace the x's in the resulting expression with g(x)

[exotic flower]

But then what…?

[tulip tapir]

then we multiply that with the derivative of g(x)

@exotic flower (chain rule)

[exotic flower]

To get this i understand

We just swapped x with g(x)

And gx was what was inside the square root

[tulip tapir]

yes

very good

[exotic flower]

And x being the square root itself

So if we have this equation

How will we use it for slope?

[tulip tapir]

recall

[exotic flower]

Do i use the power rule

[tulip tapir]

the derivative of f(g(x)) is f'(g(x)) * g'(x)

[exotic flower]

Where i shift power down by 1

And put 0.5 in front?

[tulip tapir]

so first let's figure out what g'(x) is

[exotic flower]

Alright

[tulip tapir]

g(x) = x - 7

then what's g'(x)

[exotic flower]

Ok…

Hmmmm

[tulip tapir]

g'(x) is the derivative of g(x) btw

[exotic flower]

I know its gonna be 1+x x at 0 but im sorta lost how

[tulip tapir]

g(x) = x - 7

[tulip tapir]

...?

what???

[exotic flower]

Idk

Im lost

😭

[tulip tapir]

okay okay it's fine

i'll explain again

[exotic flower]

How do we find derivatives…?

[tulip tapir]

we use the chain rule when we have a function composed of two basic functions.

[exotic flower]

Ok..

[tulip tapir]

(x-7)^0.5 is a function

but it's hard to differentiate

[exotic flower]

Yes

So we make it into 2

[tulip tapir]

so we write it as the composition of two functions.

correct

[exotic flower]

X^0.5 and x-7

[tulip tapir]

we write f(g(x)) = y = (x-7)^0.5

[tulip tapir]

mhm

x^0.5 is on the outside, x-7 is on the inside

[exotic flower]

Ok…

[tulip tapir]

now to find the derivative of a function of form f(g(x))

we multiply g'(x) with f'(g(x))

i.e. (f(g(x))' = f'(g(x)) * g'(x)

you with me so far?

[exotic flower]

Im thinking

So we do

(X-7) times (x-7)^0.5 ?

@tulip tapir

Please let me know if im correct

[tulip tapir]

no...

first of all

g(x) = x-7

we need to find g'(x)

what is the derivative of g(x) = x-7

[exotic flower]

Oh

[tulip tapir]

??

[exotic flower]

How do we find the derivative of g(x) tho

[exotic flower]

The chain rule provided me with x-7 and X^5 and this

But it dosent go forward

For the g(x)

Solving its derivative

[tulip tapir]

...

dude

x - 7

how can you not differentiate x - 7

[exotic flower]

Wait

G(x) = x-7….

G(x)’ cant also be x-7

Ik the formula is this

But what does d and n represent

@tulip tapir

[solemn escarp]

d/dx just represents the derivative of something, n means any number

[exotic flower]

What do i put in for n

If im trying to find d of x-7 ?

[solemn escarp]

x = x^1, so your n would be 1

[exotic flower]

Hmmmm

😭

I hate the chain rule

Or solving derivatives

Im too dumb to do it

Spent 1 full hour on 1 question…

So much ppl tried to help me but i still dont understand

No ones gonna help me

Ok…

[solemn escarp]

it should be much easier to understand if you watched a video, maybe khan academy

[exotic flower]

I need to solve this

The derivative

[solemn escarp]

for example the derivative of x^3 is 3x^2 because here, n = 3

[exotic flower]

Ok…

So x-7

Would be 1(x-7)^0….?

Bc it was originally (x-7)^1

Right?

[plain spindle]

🍭🍬🍫

[plain spindle]

Dm me all ur questions,I will try to help u

[plain spindle]

🥲

[exotic flower]

Alr

[amber wasp]

@exotic flower Has your question been resolved?

[amber wasp]

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.

[ruby dome]

Domain of $f(x,y)=\frac{(x^2+4y^2-4)^x}{\sqrt{x^2-4(x+y-1)}}$

[grizzled pagoda]

card

[glad pulsar]

hi

[ruby dome]

Hi

[glad pulsar]

show ur work

[grizzled pagoda]

leyla

Domain of $f(x,y)=\frac{(x^2+4y^2-4)^x}{\sqrt{x^2-4(x+y-1)}}$

[ruby dome]

$(x^2+4y^2-4)^x>0 \\ \sqrt{x^2-4(x+y-1)}>0$

[grizzled pagoda]

card

[amber wasp]

@ruby dome Has your question been resolved?

[amber wasp]

@ruby dome Has your question been resolved?

[amber wasp]

@ruby dome Has your question been resolved?

[amber wasp]

@ruby dome Has your question been resolved?

[amber wasp]

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.

[spice notch]

I'm starting out calculus 1 and didn't really know induction before the course started so I'm a bit behind in some sense
anyway I need to prove that n^(n+1) > (n+1)^n forall naturals n greater than 2 using induction

I showed that for the base case of 3 it holds (81 > 64)

assuming n^(n+1) > (n+1)^n for some n I need to prove now that it's true for n+1
(n+1)^(n+2) > (n+2)^(n+1)

I started from the assumption and wrote
n^(n+1)*(n+1)^2 > (n+1)^(n+2)

and so
(n+1)^2 * n^(n+1) > (n+1)^(n+2) > (n+2)^(n+1)
(n+1)^2 * n^(n+1) > (n+2)^(n+1)
then I opened the (n+1)^2 thing and got
n^(n+3) + 2n^(n+2) + n^(n+1) > (n+2)^(n+1)

and idk how to continue from here

[spice notch]

sorry for the lack of latex formatting idk how to do that well

[patent raptor]

,, n^{(n+1)} > (n+1)^n \quad \forall n \in \mathbb{N}_3

[spice notch]

yeah thanks
it's for all n>2 though

[grizzled pagoda]

bacc

[patent raptor]

Can you explain the step where you started from your assumption?

n^(n+1)*(n+1)^2 > (n+1)^(n+2)

the lower bound

[spice notch]

wdym

[patent raptor]

(n+1)^(n+2)

[spice notch]

wait a sec lemme remember

I multiplied by (n+1)^2 I think

[amber wasp]

@spice notch Has your question been resolved?

[patent raptor]

(n+1)^2 * n^(n+1) > (n+1)^(n+2) > (n+2)^(n+1)

Why is (n+1)^(n+2) > (n+2)^(n+1) isn't this the very thing you want to show?

[spice notch]

what I meant there is that if (n+1)^2 * n^(n+1) > (n+1)^(n+2)
is true then
(n+1)^2 * n^(n+1) >(n+2)^(n+1) is also true

because I thought it's the way

idk if it is

[patent raptor]

yea but that assumes (n+1)^(n+2) > (n+2)^(n+1) to be true which you haven't showed yet

I don't think it's that easy with induction

[patent raptor]

I can give you a nice hint

[spice notch]

sure

[patent raptor]

[ n^{(n+1)} > (n+1)^n \Leftrightarrow n > \left ( 1+ \frac{1}{n} \right )^n ]

[grizzled pagoda]

bacc

[spice notch]

ok
I'll think about it

thanks

[patent raptor]

Now you wanna prove
[ n+1 \left ( 1+ \frac{1}{n+1} \right )^{n+1} ]
You can start with
[ \left ( 1+ \frac{1}{n+1} \right )^{n+1} < \left ( 1+ \frac{1}{n} \right )^{n+1} ]

[grizzled pagoda]

bacc

[patent raptor]

Then use your assumption and done

[spice notch]

sure
I took a break though it's 1 am I'll come back to it tomorrow

.close

[patent raptor]

sure!

[amber wasp]

Channel closed

Closed by @spice notch

Use .reopen if this was a mistake.

[amber wasp]

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.

[woeful relic]

how do i start b

i know it should equal but i'm stupid and it doesn't

[quick ridge]

what’s the first step in induction

this is also just a geometric series btw

[woeful relic]

prove that p(1) is true

rhs = x+1

i am too dumb to find lhs = rhs

[quick ridge]

well they say x≠1

[woeful relic]

yeah

still not sure

because if you let n=1

[quick ridge]

for your base case

[woeful relic]

then rhs=(1-x^2)/1-x

then =1+x

but if n=1 then lhs doesn't = rhs

[quick ridge]

oh wait

don’t listen to me

yes

nevermind

you’re right

1+x

i thought you wrote something different my fault

so yes 1+x

what now

[woeful relic]

my question is how does rhs=lhs

[quick ridge]

so you assumed that when it is true for n=k

now you need to show it’s true for k+1

[woeful relic]

no

at the start don't you let n=1

and then you have to prove that lhs=rhs first?

[quick ridge]

oh you’re still on the base case wait maybe i was correct lol let n=0

sorry

that’s what i meant

because the 0th term is just 1

[woeful relic]

the way in the textbook says n=1

and you have to prove lhs=rhs

[quick ridge]

it isn’t always 1

for base case

or no

we’re dumb

lol it’s already 1+x

what are we saying

😭😭

god i’m delirious

i’m going to bed after this

you already had 1+x

[woeful relic]

1+x was rhs

[quick ridge]

the left side is 1+x

[woeful relic]

how

[quick ridge]

because n=1

it’s the sum of x^n

from i=0 to 1

thus x^0 + x^1

which is 1 + x

sorry i’m delirious

do you see it now?

[woeful relic]

yeah mostly

ty

and gn

[quick ridge]

oh you only needed base case

[woeful relic]

yeah

.close

[amber wasp]

Channel closed

Closed by @woeful relic

Use .reopen if this was a mistake.

[amber wasp]

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.

[strange field]

what

[strange field]

[pallid canopy]

That would be wrong of you then

[civic spindle]

you are correct

[civic spindle]

1. They have 2(X+3) + (X+3)^2

The image below depicts ^2 being removed by vaguely moving 2 into outer parenthesis and then adding it in the final equation.

I assume x+3 was multiplied by that equation to attain the result, but I dont understand how everything dissapeared afterwords resulting in (x+3) (x+5)

[tropic wadi]

they factored out the x+3 factor which was common in both the terms

[civic spindle]

would x+3 multiply by 2?

creating 2x+6

[tropic wadi]

it is true but they didn't distributed

[civic spindle]

wdym they didnt

define distribute in this context to so i know i understand properly

[tropic wadi]

the process to get this thing from 2(x+3) is called Distributing 2 over parenthesis

that's how we say it

[civic spindle]

alright

they didnt do that?

[tropic wadi]

no

[civic spindle]

confusing

over in math means division

and over which parenthesis?

both?

[tropic wadi]

kinda yeah

[civic spindle]

alright

[tropic wadi]

it is more English than Maths in that sentence

[civic spindle]

so explain how that is not what happened

2 is in the outermost parameter

how was it now multiplied by all of x+3

yet the innermost was?

[tropic wadi]

it was not multiplied in the image you sent

[civic spindle]

hmm... so there are ghost operations that werent written?

well

i guess not really

x+3 does multiply to 2 and x+3 yes?

[rancid raven]

Do you know normal factorisation in quadratic?

[civic spindle]

not really beyond completing the square

i never liked factoring

[tropic wadi]

generally they expect the reader to be Knowledgable enough in the context to understand some steps without mentioning it

[civic spindle]

infact its actually my MOST hated topic in math

its honestly the entire reason things like quadratics are painful

[tropic wadi]

Relative

[civic spindle]

whats that

[civic spindle]

oh wait

[civic spindle]

so this works because

2 times x+3 is 2x+6 and x+3 times x+3 would expand out

so combining them the 2x+6 would break into two

[rancid raven]

[civic spindle]

creating x+3 and x+5

?

is that it

[tropic wadi]

can you elaborate more

i didn't get what you mean

[civic spindle]

oh great so that automatically means im wrong then (pattern logic)

great...................

[tropic wadi]

I didn't say it

[civic spindle]

you didnt i mean that generally

thats just typical

[strange field]

$2(x+3) + (x+3)(x+3)$

[grizzled pagoda]

Astar777

[strange field]

do you see the common factor?

[civic spindle]

what im saying is

they took 2

and

because its a parameter operation

[strange field]

they didnt take 2

they took (x+3) common

[civic spindle]

x+3 must multiply by 2

making 2x+6

so then

because the inner parameter x+3 is also there

thats x+3 times x+3

so half of 2x+6

would be factored

divided into

x+3

then they would just eliminate

the like terms

[rancid raven]

Just understand in simpler terms, if you have

You can take b common and it will become 
b(a + c)

[tropic wadi]

there's no negative so how could you eliminate?

[rancid raven]

Correlate this to your main equation and you will understand what's happening, hopefully

[civic spindle]

i mean i would if i had any idea how to format it in the way you just did

but these questions are so abstract i dont even know how to make examples like what you just did

[tropic wadi]

okay see $2x+6$

[grizzled pagoda]

77²

[rancid raven]

You need to find relation

[tropic wadi]

what you can factor out from this?

[civic spindle]

1x+3

[strange field]

no, 2

1x+3 is what you would be left with

when you factor out 2

so 2(1x+3)

[civic spindle]

true

[civic spindle]

wdym

[strange field]

$2x+6$ -> $2.x + 2.3$

[grizzled pagoda]

Astar777

[tropic wadi]

hence if you have $ax+bx$,
what can you factor out from this then?

[strange field]

hence if you have $ax+bx \$
what can you factor out from this then?

[grizzled pagoda]

Astar777

[strange field]

its \\

[civic spindle]

i dont know

because everything is different

and there is no =

[rancid raven]

What's common term in ax and bx

??

[civic spindle]

x

[rancid raven]

Then that is the one you will factor out

[civic spindle]

a+b?

[rancid raven]

Yeah x will be factored out so (a+b) will remain
x(a+b)

[civic spindle]

hm

what has this achieved?

why would we want to add a and b?

[rancid raven]

Nothing it was for your understanding

[tropic wadi]

similarly notice $2(x+3)+(x+3)^{2}$

[rancid raven]

If you had ax+bx= 0
Then it could have made more sense

[rancid raven]

@civic spindle

[civic spindle]

alr

[grizzled pagoda]

77²

[civic spindle]

there might be two different missing concepts here but i can try

[tropic wadi]

wdym?

[civic spindle]

i think he means not expanding

absolutely everything out

[rancid raven]

Ohhh it's square mb

[civic spindle]

because then

THICK

[rancid raven]

Nah ignore my message

[tropic wadi]

@civic spindle can you apply the same process as we did in previous 2 examples?

[civic spindle]

(2x+6) + (x+3) (x+3)
dividing we get
(x+3) (x+3)```

[rancid raven]

[tropic wadi]

that's not what we did earlier

[civic spindle]

yeah but how is that wrong

this is one of the biggest problems with factoring for me

[tropic wadi]

No one said it is wrong

[rancid raven]

@tropic wadi I think let them first learn quadratic factorisation, else it will be difficult for them to understand

[civic spindle]

it is not?

is it not?

[tropic wadi]

not completely

[civic spindle]

what was wrong

[tropic wadi]

that Dividing part

[civic spindle]

how?

[tropic wadi]

what are you dividing with?

whivh term?

[civic spindle]

wouldnt 2x+6 split into (x+3) (x+3)

[tropic wadi]

no

[civic spindle]

nothing

oh oops

m

..

im out of time damn

[rancid raven]

x²+5x+6
Do you know how to find roots of this quadratic equation using factorisation method @civic spindle

[civic spindle]

i gtg

[tropic wadi]

https://tenor.com/view/gold-star-succession-gif-27712928

[civic spindle]

.close

[amber wasp]

Channel closed

Closed by @civic spindle

Use .reopen if this was a mistake.

[amber wasp]

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.

[rustic solstice]

Help me pls

[amber wasp]

@rustic solstice Has your question been resolved?

[strange field]

write $m^{th}$ and $n^{th}$ term in general form

[grizzled pagoda]

Astar777

[strange field]

then subtract them and find a (first term) and d (common difference)

then write the $(mn)^{th}$ term in general form and use the values of a and d

[grizzled pagoda]

Astar777

[amber wasp]

@rustic solstice Has your question been resolved?

Channel closed

Closed by @rustic solstice

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.

[gleaming magnet]

How can I do these on Desmos easily

[weak zinc]

Type them in as you see them, and choose to add a slider for a in the first image

(Assuming you only want to do them graphically, of course-)

[amber wasp]

@gleaming magnet Has your question been resolved?

[gleaming magnet]

Doesn’t matter what value the slider is right

[chilly jackal]

seems like a practice test and i believe u need to do those without desmos?

[weak zinc]

You want to choose your slider so that you find the value of a where you get only one point of intersection, just play around with it until you find it

[amber wasp]

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.

[rough trail]

How do I find the general solution?

$\sin(2x) - \cos(2x) = 0$

I've found $x = \frac{\pi}{8} + 2\pi n, x = \frac{5 \pi}{8} + 2\pi n \text{ where } n \in \mathbb{Z}$

But I want the general solution, and these are two general solutions. It needs to be one equation and I have no clue how to proceed.

I've tried treating my two solutions like simultaneous equations, but unsurprisingly that leads to a contradiction.

[grizzled pagoda]

eththorn

[worldly kite]

Whats the value of x = 23125 + 0.1y and y = 7875 +0.1x i just want the value of x .

[outer drift]

this channel already has a question wanting to be answered u gotta go to math help avaliable 🥶 #❓how-to-get-help

[amber wasp]

@rough trail Has your question been resolved?

[strange field]

how did u get the 5pi one

[rough trail]

I have memorised exact values and I know they are both equal to √2/2 at x = π/4 and 5π/4

[strange field]

$\sin(\theta) = \cos(\theta) ,, when ,\theta = \frac{\pi}{4} + n\pi$

[grizzled pagoda]

Astar777

[rough trail]

what about the third quadrant where they're both negative though??

that still evaluates to 0

I mean this also works:

$x = \frac{\pi}{8} + 3\pi n \text{ where } n \in \mathbb{Z}$

But my textbook had something completely different.

[grizzled pagoda]

eththorn

[full marsh]

sinx=cosx for this line

so 2x=pi/4+npi

x=...

[rough trail]

hence x = π/4 + πn? That's still quite different to my textbook answer though

[full marsh]

pi/8+pi n/2

[full marsh]

2x=pi/4 + n pi
not x

[rough trail]

well I have to find a rule for x not 2x, so I imagine dividing both sides by 2. But I had a feeling the πn doesn't need to be divided by 2

I think the solution was (4n + 1)π/8 iirc which I think is equal but not totally sure

[full marsh]

if the question would be sin(3x)=cos(3x)
you'd equate 3x=pi/4+ n pi

it depends on argument

[rough trail]

According to your diagram, can I say?:
$ where \sin(x) \notin {-1, 1} x = \sin^{-1}(x) + \pi n$

[strange field]

it is equal, yes

[rough trail]

because I think that might be a useful rule to remember for future questions

.close I'll just prove it myself

[amber wasp]

Channel closed

Closed by @rough trail

Use .reopen if this was a mistake.

[amber wasp]

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.

[keen pawn]

How does this proof look

oops

messed up

.close

[amber wasp]

Channel closed

Closed by @keen pawn

Use .reopen if this was a mistake.

[amber wasp]

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.

[daring plover]

Is this correct?

[grizzled pagoda]

faiyrose

[daring plover]

Why

Not the X

So its 15+49?

Okay

Can u help

Me with sum aswell?

Omg

Thanks

No way i did sumthing correct

.close

[amber wasp]

Channel closed

Closed by @daring plover

Use .reopen if this was a mistake.

[amber wasp]

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.

[calm pewter]

somone help idk wht to do

[royal mauve]

can u explain where you struggle

[calm pewter]

the average speed at the end

didnt they already give the speed?

[royal mauve]

average speed is different from speed bro

[calm pewter]

so how do i find average speed?

[worldly flame]

Since they went across one part at one speed and another part at another speed there’s a different number that’s the average

[calm pewter]

so 40-30

?

[worldly flame]

You need to find the total amount of time it took them to take their journey

[royal mauve]

you just need to take the total distance, and divide it by the average velocity

[calm pewter]

time is 1.5

and 1.125

[worldly flame]

No 1.5 to go from point a to point b not point a to point c

[calm pewter]

yh

so we add both of them?

[worldly flame]

So how long does it take to travel 20 miles while going 40mph

[calm pewter]

0.5

[worldly flame]

So you add 1.5 to .5 and now find the total distance

[calm pewter]

2

[worldly flame]

What’s the total distance now

[calm pewter]

65miles

[worldly flame]

So the average would be the total distance/the total time

[calm pewter]

65/2

[worldly flame]

Which is?

[calm pewter]

32.5

ye

[worldly flame]

Yes

[calm pewter]

thts the ans?

[worldly flame]

Yeah its the average speed for the journey

[calm pewter]

ok ty

[worldly flame]

Np

[calm pewter]

.close

[amber wasp]

Channel closed

Closed by @calm pewter

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.

[normal dove]

can anyone do this

[normal dove]

im not exactly sure

[limpid sun]

that doesn t seem true for a general f

[crisp stratus]

what is the task exactly

prove that? it doesn't look right

[tulip tapir]

it isn't true

at least idts

[lusty saffron]

f(x) = x is a counterexample

[tulip tapir]

from what it looks to be

[grizzled pagoda]

ren

[normal dove]

wdym?

[tulip tapir]

do you want to find a function satisfying this equation @normal dove ? or smth else?

[normal dove]

nah

just not sure of the integral result

[tulip tapir]

?

[normal dove]

cuz didnt get to differential equation correctly yet

[normal dove]

this

integrating both side wrt x

oh i get it

nvm

.close

[amber wasp]

Channel closed

Closed by @normal dove

Use .reopen if this was a mistake.

[amber wasp]

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.

[velvet junco]

I am stuck in this algebra question, i have the answer and the procedures done i just don’t understand one step

[velvet junco]

I didn’t understand the inverse x in the step 2

3=a^(1/x)

This part

[signal furnace]

Raise both sides to the power of 1/x

$(3^x)^{1/x}=a^{1/x}$

[grizzled pagoda]

Civil Service Pigeon

[signal furnace]

$3^{(x)(1/x)}=a^{1/x}$

[grizzled pagoda]

Civil Service Pigeon

[signal furnace]

$3=a^{1/x}$

[grizzled pagoda]

Civil Service Pigeon

[velvet junco]

Why?

Is that Conjugate?

[signal furnace]

Because you want to force stuff to have the same base

And 3*4=12 so your intuition is to multiply stuff involving that

[velvet junco]

Thanks

.close

[amber wasp]

Channel closed

Closed by @velvet junco

Use .reopen if this was a mistake.

[velvet junco]

.reopen

[amber wasp]

✅

[velvet junco]

why 3*4=12 for the eqn?

[velvet junco]

why that 3*4=12

[signal furnace]

These two multiply to make 12 = a^(1/z)

[velvet junco]

question is 3^x=4^y=12^z but why multiply to find 12=a^(1/z)

[signal furnace]

How else are you going to get a relationship

[velvet junco]

got it

thanks

.close

[amber wasp]

Channel closed

Closed by @velvet junco

Use .reopen if this was a mistake.

[amber wasp]

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.

[summer quarry]

Hii i could really use some help on this question :)

In a 250m^3 room, someone spills 20mL of an ammonia solution which has a density of 0.9 * 10^3/m^3 and a mass percentage of 7 of ammonia. This liquid evaporates completely,
a) Calculate the concentration of gas-form ammonia in the room. Give your answer in mg/m^3.

[summer quarry]

Like i do understand what they’re asking but i’m super confused on how to do that with the different units of measurement

[amber wasp]

@summer quarry Has your question been resolved?

[summer quarry]

Nvm I got it lol

.close

[amber wasp]

Channel closed

Closed by @summer quarry

Use .reopen if this was a mistake.

[amber wasp]

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.

[vast spade]

!15mins

[amber wasp]

Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.

[tulip tapir]

for a just integrate it

for b take the absolute value of the velocity

area b/w the curve and x axis

[tulip tapir]

i.e. make everything positive

and integrate it

[pseudo crescent]

fancy word for finding the area under the graph

[vast spade]

for your 1st question find the area of the graph (note :assingn positive area for above x axis and negative for below)

[tulip tapir]

do note that area below the x axis is negative

just break it up into triangles and quads here

[pseudo crescent]

he doesnt know what integrating means is the probelm

[pseudo crescent]

You need to explain it in simpler terms. Namely, area under the graph

[cursive apex]

This is the type of introductory physics questions where the graph is just a combination of simple geometric shapes so its fine to not understand integration yet

Its asking for the average velocity, the answer should be in m/s units

Do you know the formula for the average velocity?

Correct

[vast spade]

ok do you know how to find areas of squares,rectangles and triangles?

[cursive apex]

Not exactly

[vast spade]

physics uses math

[cursive apex]

You are making it harder by taking displacement, the average velocity is just final velocity subtracted from the inital one divided by the total duration

[pseudo crescent]

Are you sure?

If I drive at 10mps for 100 hours, and then at 100mps for 1 hour, is my average velocity (100-10)/101?

[thick apex]

if you had a simpler example where the velocity was a constant 2m/s over 5 seconds, how would you get displacement

[cursive apex]

Oh sorry i mixed things up, that was divided by 2 not the total time

[vast spade]

for the first one no

[pseudo crescent]

that works only if accel is constant

which it's not

[cursive apex]

Right

[pseudo crescent]

neither

[thick apex]

dot this’ll help with a

for every second you move 2m, and this takes place over 5 seconds

how many meters would you move

[pseudo crescent]

meters aren't even a unit of speed, so it's definitely not 900m. It also can't be 90m/s, since that's higher than any other speed on the graph

multiply what by 2?

[thick apex]

yes

another way to think of that is the area under the velocity graph

the height is 2m/s

the length is 5s

multiply them and you get 10m

the same concept applies to more complex velocity graphs

[pseudo crescent]

Yes, it is. But you dont yet know the distances. So focus on what ruby says, because he is doing a great job explaining ti

[thick apex]

you want to solve part a before part b

[pseudo crescent]

900m isn't correct for part a btw

[thick apex]

also, if the area is under the x-axis, it should be treated as negative

i.e. -2m/s over 5s gives a displacement of -10m not just 10m

the area under a velocity graph gives you the displacement

[pastel silo]

The sum of a velocity over time, in this graph, is displacement

[pseudo crescent]

the... infinite sum?...

[thick apex]

its velocity (m/s) times time (s), which gives displacement (m)

there are ways to calculate the area under a curve that involve calculus but for this example you have, you only need to split the area into shapes you can calculate, such as rectangles and triangles

yes

there isn’t a general formula for the area under a curve

[pseudo crescent]

Can I just ask you a question, are you able to determine the displacement in first second?