math-help

[sharp night]

https://www.youtube.com/watch?v=ASwOZYbfBjI

I think this video could help

And this

https://xaktly.com/GraphingPolynomials.html

Example 3

so you know you have a single root at -2, triple root at -1, and a double root at 1

[tall elk]

Hello there, any system engineering nerds around here?

[tropic bear]

can anyone tell mw how it changed from 0≤θ≤2π to −1≤sinθ≤1 ?

[tall elk]

perhaps this is helpful read it carefully

[slim juniper]

in questions for finding the integration by part for trig functions like arctanx, arcsinx, arccosx, do we always let u = arctanx and v = x?

[sharp ferry]

i think the option is d for 14th

[left knoll]

Yes, but why didn't we choose B?

[sharp ferry]

the angle is made with the greater force right

so since the force is greater the net force would be closer to the bigger force right?

[hushed ember]

90-70 = 20 > 10 I think

[ember kernel]

That's not the reason
As @sharp ferry said, theta must be < 45°

You can verify that graphically. Let's say we have perpendicular vectors u and v, with u being longer that v ===> |u| > |v|

If we measure theta (I'll call it t) relative to u, that means we have
|tan(t)| = |v| / |u|

Since I said |u| > |v| , that means |tan(t)| < 1, which means -1 < tan(t) < 1, which means -45° < t < 45°

Answer (d) is the only one that satisfies this inequality

[left knoll]

Thanks 🌸

[atomic dagger]

Hi! I'm a little confused at the calculation of this integral here. The primitive of sin = -cos, so I thought we should have -cos(2pi*ft) in the expression but instead the cosinus is positive.

in my textbook they say it's because the derivative of cosinus equals -sin but i'm not sure how that explains it

[weary arrow]

I haven't looked properly at it but I think that it's just that they switched the orders of the two expressions. Notice the order of the upper and lower limit on the integral and in the expression

so instead of writing -cos(a) - (-cos(b)) they've written cos(b) - cos(a)

[atomic dagger]

ohh you're right!! i completely missed that, thank you!

[weary arrow]

no problem 🙂

[sly sandal]

For 14, why is it not just 5!

[cinder yacht]

I'd guess u should still multiply by 2

because if u know that couple A is taking places 1 and 2, there are still 2 possibilities : person A1 taking place 1 and A2 taking place 2, or the opposiye

the question isn't perfectly clear though

[inland oracle]

Could someone help me to calculate the base of a vector? I have the matrix and I don't know what else I can do

LA

[sly sandal]

Ahhh, I didn't think of that

Thanks

[inland oracle]

b) I need to determine a base and dimension for W and for T

[cinder yacht]

for W, u can see that (-2,2,2)= - 2 * (1,-1,-1)

so u can "take out" the first vector, so
W = < (1,-1,-1),(1,1,1) >

those 2 vectors are not colinear, so they're a base of W, and that means W is of dimension 2

as to U a vector (x y z) is in U
<=> y-z=0 and x+y=0
<=> y=z and x=-y=-z
<=> there exists a c such that z=c; and x=-c and y=c
<=> there exists a c such that (x y z) = c * (-1 1 1)
so U = <(-1, 1, 1)> (dimension 1)

[atomic dagger]

i have another question about this integral T.T why do we have the 1/(2pi*fTc) in front of the cos? Someone told me it's because we have a composite function but it was very unclear to me T.T

[cinder yacht]

a primitive of t->sin(at) is t->-cos(at)/a

(considering a isn't 0)

to convince urself of that, try to derive t->-cos(at)/a

using the rule that the derivative of t->f(g(t)) is t->g'(t) * f'(g(t))

[in this case, the derivative of t->f(at) is t-> a * f'(at)]

in ur problem, a = 2 * pi * f

another way to see it is integration by substition ("changement de variable") : setting u = 2 * pi * f * s in that integral
AKA s = u / (2pi * f)
meaning ds = du / (2pi * f)

[atomic dagger]

thank you! i will try to calculate it again!

[next lark]

help plz

[sweet edge]

here you go 🙂

[cinder yacht]

or
13(x+4)^2 - 49 = 0
(x+4)^2 = 49/13
x+4 = +-sqrt(49/13)
x = - 4 +- sqrt(49/13)

[slim juniper]

For 3 a, x^4/8 is positive because of the ^2 right?

[cinder yacht]

yeh

[slim juniper]

but i don't understand, x is just a variable, we're not actually calculating a value, so how does it affect the sign of x?

[cinder yacht]

it doesn't affect the sign of x

(-x^2)^2 = ( [-1] * [x^2]) ^2
= [-1]^2 * [x^2]^2
= 1 * x^4
= x^4

the point is that (-1)^2=1

while (-1)^3 = -1

u need to understand there's a difference between the sign of X, and the sign next to X

for example if x=-3 then +x is negative, -x is positive
but the sign next to x in "+x"is plus (positive), the sign next to x in "-x" is minus (negative)

[slim juniper]

ohhhhhhh makes sense tysm

[hushed jacinth]

how

[left knoll]

how do i compute this limit pls i’m desperate

[sharp ferry]

hint 1: log(a/b)=loga-logb
hint 2: log(a^x)=xlog(a)
hint 3: 32=2^5 and 8=2^3
hint 4: (a^m)^n=a^(m*n)

[sharp ferry]

is this a 4 or a 1

[left knoll]

1!

[left knoll]

Nvm i think i did it, i got zero as result

[sharp ferry]

but how i think it diverges to -inf

[left knoll]

lmao sorry you’re right it should be -inf

but i still don’t know how to do it

[sharp ferry]

i did it like this idek if its correct or not
2^(n+1)-2^2(n-1) / 3^n= 2^n(2-2^(n-2)) / 3^n now since the 2 is very small compared to 2^(n-2) we can ignore it so we get 2^n(-2^(n-2)) / 3^n =- 2^(2n-2)/3^n =-4^(n-1)/3^n =-4^n/(3^n)4 now (4/3)^n tends to inf so the whole exp tends to -inf

[cinder yacht]

before computing I see that 4^(n-1) aka 4^n / 4 is the big boii in the fraction when n gets very big

so ik that he dictates where the limit is going, so ik that it's - infinity

but for actual proof, a common way to do it is to divide by a big boii in the numerator and the denominator : divide by 4^n in the numerator and divide by 4^n in the denominator

another way is to divide by 3^n in the denominator and in the numerator

[hushed jacinth]

tyy

[glacial dust]

can someone explain to me why is x-8 I thought area is length times width (this is very basic but forgot it can someone help?)

[glacial dust]

can someone help why that x-8

[icy spindle]

what are you trying to find?

[fossil token]

Tom works on 7% commission. Even after increasing his commission to 10%, his total income remained the same. Find the percentage reduction in his sales.

Can someone please help with this? Would be greatly appreciated

[icy spindle]

Let X be his total sales before change in commission, and Y his total sales after change. Then, C1 = income before change = X * 0.07, and C2 = income after change = Y * 0.1. Then X * 0.07 = Y * 0.1 => X*0.7 = Y, thus, change in income is by 30%.

[fossil token]

Thank you so much!!

[atomic dagger]

Hi, I was wondering, what is the general method to find the infimum or supremum of a given set ?

[icy spindle]

Whichever is the highest value is the supremum and the lowest is the infimum

[atomic dagger]

But sometimes I'm given sets like these:

I have the correction but it's in french and I don't understand it anyways...

[icy spindle]

As n increases the whole expression tends to 1

Well 1 and -1 depending on whether the number is odd or even

[inland sigil]

You mostly check in 3 ways

1. Put n=0
2. Put n=+inf and n=-inf
3. Differentiate the function and check its values at zeroes of the differentiated function

3. is so because zeroes of the differentiated functions are the local maxima and minimas of the actual function

[atomic dagger]

I think I understand you both @icy spindle @inland sigil thank you ^^

[glacial dust]

The question says that there are 2 numbers when multiplied = 36and when added = 13

my question is how did that a became 36 knowing that we know a times b = 36 not a = 36

and why is there a b on the other side that's supposed to be -b right cuz we transposed it

[icy spindle]

where are you looking? which line

[glacial dust]

the questions just ask what are the 2 number when multiplied is 36 and when added is 13

wdym what line

[icy spindle]

you said a became 36, where?

[glacial dust]

in the "substitute a in 1"

1 means the first equation

which is ab = 36

and 2 stands for the second equation

[icy spindle]

a = 13-b, so ab = (13-b)b = 36

im confused as to what the issue is

[glacial dust]

it says sub a into 1 equation so it means ab=36

[icy spindle]

yeah

[glacial dust]

not a=36

idk why a=13-b becam 36

[cinder yacht]

that works when we have smth that's differentiable

[icy spindle]

it says (13-b)b=36 not 13-b = 36

[cinder yacht]

here it's a sequence, n is in N, and u couldn't replace n with a x in R because (-1)^x wouldn't make sense

anyway here just try out u0, u1, u2, u3, u4
and see what it's like

maybe even draw them in a graph (X axis would be N, Y axis would be the sequence Un)

[glacial dust]

Doesn't make sense to me a=13-b then it became 36=b(13-b)

[inland sigil]

Yes, I'm aware, that was more of a general tip for anytime it's needed
In this case, the value of n is discrete

[cinder yacht]

and once u do these few terms u generally notice smth
in this case : that the absolute value of Un keeps increasing (we see that when n->infinity, it converges to 1), and that Un keeps switching between negative and positive

@atomic dagger

[icy spindle]

yes, because ab = (13-b) b, since a + b = 13 => a = 13 - b.

he is substituting the value of a into the other equation

[glacial dust]

yah i know this part

[icy spindle]

noone is saying a = 36

[glacial dust]

but the a turning 36 doesn't make sense

[icy spindle]

where do you see it? im not seeing anything like that

[glacial dust]

ill write it wait

[icy spindle]

Show me where it says that a = 36

[cinder yacht]

it never turned
look :
a = 13 - b
and
a * b = 36
so by replacing the value of a (which is 13-b) in this last equation : (13-b) * b = 36

[icy spindle]

been trying to explain that

[cinder yacht]

ik

[glacial dust]

ohhhhh

now i get we modified the 2 equation knowing that ab=36

making athe subject so we can sub it as(13-b)b=36

isn't that right?

[icy spindle]

exactly

[glacial dust]

okok thanks alot

[icy spindle]

actually we modified the first equation knowing a = 13 - b

which comes from the second one

[glacial dust]

yea thanks alot

[cinder ibex]

Hey, can anyone help me solve 13.6 pls? EDIT : Fixed it!

Ive done this but idk what to do next

[left knoll]

sorry i couldn’t answer before but tysm you made it a lot more clear

[left knoll]

ty alot!!

[rotund root]

i need help w factorizations

i sort of understand what i have to do but most examples show exercises that have the same common factor but everyone i get is like this and im confused asf

[chrome sierra]

isnt this done with the quadractic formula (think thats what its called in english) so that would be (y+x)*(y-x) (with the x bieng the solutions of that formula in the pic below)

Wait think i made a mistake

Keep my though on hold for a sec

[rotund root]

wait yeah that would be the result but what idk is the process of it but i think its around there

[chrome sierra]

in my calculations i think i made a - error somewhere, bc my results are correct but in the negative way XD

so trying to figure out why XD

[rotund root]

ok ok dw take ur time im doing other problems atm ^___^

[chrome sierra]

ah i think i found it

so from the formula above you get 2 solutions

lets call them x_1 and x_2

what we do is (y-x_1)*(y-x_2)

[rotund root]

a

but how do u get from point a to point b

[chrome sierra]

wym?

[rotund root]

OH

[chrome sierra]

oohh do you mean 'how do i use that formula'?

[rotund root]

OHHHH WAIT NEVERMIND

yesss omfg im dumb ok i get it tysmmm tysm

[chrome sierra]

Found it :)

Oh you got it haha, glad i could be of help!

[rotund root]

tyvm!!! 😄

[chrome sierra]

npnp, also just a quick question

you do know that somethimes you cant facorise right?

or at least i think XD

[rotund root]

yeah ik :(((( sometimes idk when though im scared of that part of the exam haha

[chrome sierra]

haha dw you got this

just asking so i know what you know ;)

now you will remember on your exam a stranger asked you that ;)

[rotund root]

ur a genius thnk u vinnie chemistry

[chrome sierra]

Good luck w the exam :)

[cinder yacht]

more specifically

it's 3 * (y-x1) * (y-x2)

u gotta keep the 3 that is next to y^2 in the formula

and find x1 and x2 the solutions of the quadratic equation

[cinder yacht]

3 * that

[chrome sierra]

May i ask why? bc filling in what I did i got the same line in geogebra

sorry forgot my thermonology 😅

[cinder yacht]

ehm

did u study equivalents?

like when x->+infinity

that could help me prove it pretty fast

[chrome sierra]

Nowp not that I can recall

can you do it in pure math form?

english isnt my first langue's so that might help me understand XD

[cinder yacht]

another way is to say that
3y^2+19y+20
= 3 * [y^2 + 19y/3 + 20/3]
and (y-x1)(y-x2) is the factored form of [y^2 + 19y/3 + 20/3]

[chrome sierra]

With all due respect but the quadratic formula includes that 3 in the a right?

So you kind of lost me

[cinder yacht]

ehmm basically when Y is very big (Y->infinity) , 3y^2 + 19y + 20 is approximately 3y^2
did u study smth like that for example?

[chrome sierra]

Not that i recal

[cinder yacht]

ok then I can't do that fast proof

hmm

[chrome sierra]

Sorry :(

wait maybe

ill explain where i am stuck/lost you

that might help

[cinder yacht]

do u agree that if x1 is solution of 3x^2 + 19x + 20 = 0, then x1 is solution of x^2 + 19x/3 + 20/3 = 0?
(because u can divide the equation by 3 left and right)

[chrome sierra]

so what i did was, I put that funtion in geogebra and it draws a line, then i did my stuff and put that in, both lines where the same

ohhh wait

[cinder yacht]

weird for me to say this but that's a bug xD

or u did it wrong

but im very sure of myself

[chrome sierra]

my x_1 and x_2 are the solutions of the quadratic formula

[cinder yacht]

yea ik

but u need to keep the number that's next to y^2

in the factored form

here, that's 3

so 3 * (y-x1) (y-x2)

because when u distribute that multiplication of (y-x1) (y-x2) you find that it gives u (y^2 + something * y + something)

and since u need the distributed form to have 3 next to y^2, it should be 3 * (y-x1)(y-x2)

anyway u can check more with geogebra (maybe u forgot to draw the curve of (y-x1) (y-x2), like maybe u just wrote it without telling geogebra to draw it)

[chrome sierra]

okay i got it :)

sorry was working it out on paper

and i do see i miss the facor 3

thanks for pointing that out ^_^

[cinder yacht]

yw

[chrome sierra]

but its wierd

bc geogebra does say we're wrong

so there is a bugg there XD

[cinder yacht]

I dont think it does

[cinder yacht]

yeah?

[chrome sierra]

this was wrong bc i used y

[cinder yacht]

I agree with this screenshot

what u gotta try is f(x) = 3x^2 + 19x + 20

u'll see its the same

as g(x)

[chrome sierra]

okay now i fully agree w you

thanks for explaining

actualy helped me out

Have a good evening ^_^

[cinder yacht]

you too :p

[atomic dagger]

why wouldn't (-1)^x work ? though I understand the rest of the answer, so thank you! Do you think in every case, I can find the supremum and infimum by looking at the limit in + infinite and - infinite ?

Also I was wondering, how can I calculate the derivate of x^x^x ? My teacher divided y' by y in this instance but I don't understand how or why.

[ember kernel]

You teacher just calculated the derivative of the second line with respect to x (y' = dy/dx) :
(log y)' = (1/y) * y' (via chain rule)

They don't divide by y, it comes from the derivative of log(y)

[tepid spade]

whats the value?

[cinder ibex]

What value?

[tepid spade]

uhmm both of these graphics

[tepid spade]

the value of x

[cinder ibex]

They are giving u the straight line(sorry dnt know how to say in english) equations so with any value of x u can get ur y or fx

[void gazelle]

in the images you shared you have 2 functions, x can be any value you want. for example for the graph on the left, if I choose x = 0, I would have y=4 because 0·(-2)+4 = 4

[tepid spade]

ty

[atomic dagger]

Okay thank you I understand !

[left knoll]

hi dudies im studying last year of middle school and i neeh help with square root numbers. how can i find answer of √ 2,5+√3-6

[icy spindle]

can you rewrite that with parentheses

[atomic dagger]

is 3-6 under the square root or just 3 ?

[left knoll]

under it

like that

[icy spindle]

2,5 = 25/10

and 3,6 is 36/10

so 5+6/sqrt(10)

[worthy wagon]

@left knoll

[left knoll]

thank u all

[worthy wagon]

welcome 😄

[vagrant sonnet]

Can someone explain this to me?

I'm kind've of lost...

[cinder ibex]

The instantaneous rate of change to x=2 i believe is (f(x)-f(2)) /x-2 but im not fulling understanding the question either so what i said might not be correct

[vagrant sonnet]

Oh ok.

[cinder ibex]

Cause the question is alto talking about the derivative right?

[vagrant sonnet]

I'm also lost here...

I'm not sure to be honest.

It's just asking for the instantaneous rate.

[cinder ibex]

I havent studied limits with x in that place yet so idk how to do that sorry

[cinder ibex]

If thats the case i believe it is what i said

Do u have solutions to check?

[vagrant sonnet]

I don't sadly.

[cinder ibex]

I was thinking and what i told u aint correct

I mean nevermind

Im getting confused

I had some good notes on my last year book tho

But im not home

Later today i will try to check it

Sorry im not very helpful

[cinder yacht]

-3

[cinder yacht]

it's f'(2) I think, so 4

[wheat oxide]

Have anyone known about linear programming minimization? with 4 constraints or more?

youtube videos explain if there are 2 constraints only 😦

[cinder ibex]

U right it would be what i said in case x belonged to [a,b] being a and b real numbers

[rotund root]

I PASSED MY MATH EXAM WOOOOOOOOOOOO I GOT MORE THAN EXPECTED OMFG FUCK YEAH

thank u 😭 coly and Vinnie if u read this i don't wanna bother u w a ping

[worthy wagon]

congrats

[left knoll]

Anyone doing university Integral calculus here?

[cinder yacht]

ahahaha good job!

[ember kernel]

Yuuup

What's your question ?

[slim juniper]

Why does it become e^kt ??

I don’t understand it at all

[tropic lava]

Anyone can help me finish this?

what should i do next

[median hornet]

Mention that since n^3 < 2^n (The hypothesis) => n^3+ [...] < 2^n + [...]
=> (n+1)^3 < 2^n+1 and therefore n^3< 2^n

[cinder yacht]

I think it's for n>=10 no?

cause from n=2 to n=9 it's wrong

that inequality

xD

[slim juniper]

can someone help me with this?

[cinder yacht]

it's a rule of

Differential equations

search in the theorems of ur lesson

[slim juniper]

Ok ty

For this, I don’t I understand why it’s 4 for the second part of the equation

Ignore the 2 circled

[left knoll]

Pls help ASAP

[storm thorn]

i think the 2 and 4 should be switched

[slim juniper]

It’s about the y-axis not x

[cinder yacht]

cause for a y between 1 and 2, the x goes from y^2 to 4

as u can see in the curve

[wheat oxide]

Morningg can I ask for help how did it becomes (2,1)

[fervent tide]

consider them as functions
y = 3 - x , and y = x / 2
these are equal when x = 2
and when x = 2, y = 1

[wheat oxide]

Thankk youuu

[glacial dust]

In the figure below, right triangle TUV has a hypotenuse of 6. If TU=5x−−√ and UV=x, find the perimeter of the triangle TUV.

5

Possible Answers:
10−25–√

15+35–√

10+25–√

15+25–√

3+35–√

Correct answer:
10+25–√

can someone help me with this

this is my first time seeing a side with a square root

[cyan night]

wait your square root notation makes no sense

whats 5x−−√ ??

[glacial dust]

side TU is sqaureroot 5x

inside a square root there is a 5x

[cyan night]

ah ok ok

[glacial dust]

https://www.varsitytutors.com/basic_geometry-help/how-to-find-the-perimeter-of-a-right-triangle

[cyan night]

use pythagorean theorem

[glacial dust]

u can check it here to be more clear

[cyan night]

well the perimeter is just adding the sides

ok

but use pythagorean theorem

[glacial dust]

but how can i square when theres a variable x nexto it

[cyan night]

sqrt(5x)^2 = 5x

[wheat oxide]

Can I ask how did the graphing calculator got the (3.429,1.714) and the (1,3.333)

Thank youu

[lilac kestrel]

hey guys how do i solve this problem?

grade 12 maths

could someone help me pleaseee? thank youuu

[ember kernel]

Quadratic formula to get a and b in terms of k

Then, the only quadratic function with roots a² and b² is (x-a²)(x-b²). Since you have already calculated the expressions of a and b in the first part, you just need to square them and sub them in

[lilac kestrel]

okok let me try it out

thanksss

[ember kernel]

Np

[lilac kestrel]

Can you also help me with this?

Where would the functions y=f([x]) and y= 1/f([x]) be

on this graph

[glacial dust]

guys question ii is wrong right?

if were looking for an angle then it would be inverse cos

not cos=6/10 then in that case that's a side

can someone elaborate this for me

[ember kernel]

Hi
Did anybody mention me then deleted ?

I got a notification apparently

[glacial dust]

I think i did 😅

[ember kernel]

The angle is just measured from the horizontal trait and clockwise

[glacial dust]

its angle so it should be cos invers?

[ember kernel]

Nothing wrong, it's just that this solution used a different convention for measuring theta

You mean arccos ?

Or 1/cos ?

[glacial dust]

cospower-1

because were finding theta

[ember kernel]

Cospower-1 could be either arccos or 1/cos depending on what you mean

I guess you mean arccos

Well

[ember kernel]

Yeah, and arccos(6/10) = 0.9273

[glacial dust]

The question asked for the angle

I meant this one

[ember kernel]

Yeah

[glacial dust]

its giving me 53.13

[ember kernel]

Degrees

Convert them to radians

You'll get 0.92

[glacial dust]

but why in radians tho its a trigonometric shape not circular one?

its radians because its reaching the other circle?

ohh I wasnt looking at the question lamaoo it literally said "IN RADIANCE" 😆

[fair harbor]

lines intersection point

[wheat oxide]

like can you help me how to solve it

[ember kernel]

You can describe ANGLES in degrees or radians, no context needed whatsover

"Oh but I can't use radians since we're not in the trigonometric circle. I have to use degrees :("
is exactly the same as
"Oh I'm measuring the size of a bottle, not a spoon, so I'm forced to use cm instead of mm :("

They're equivalent ways of measuring angles
Degrees are easier to visualise in graphical problems (eg. geometry)
Radians are far more efficient when used in analytical problems (eg trigonometry, physics, complex numbers, ...)
But you could choose either (although I would strongly recommend you to get comfortable with radians, as they become very very ve-ry useful in upper grades, and especially in engineering/scientific college courses)

[cinder yacht]

degrees are a dum dum way of describing angles

like "360°" for a full turn was just chosen because a year contains around 360 days

it's very arbitrary and doesn't have a good reason behind it

while deciding that a full turn is 2 * pi has a lot of meaning

a looot of good reasons to see it that way

so yea as Quentin said ° is nice cause it's easier to talk about degrees than radians, but in science radians are what we generally use

[sullen prawn]

no its wasnt

its derived from the egyptian and babylonian number base system

which was based in multiples of 60

so it just fit evenly into their number system

[queen magnet]

@here ANYONE CAN HELP

PLZ

IM DOING INVERSE COMPOSITIONS AND IM TOTALLY LOST

<@&717391911132069929> HELLO

DOES ANYONE KNOW

WHRE I CAN GET

SOME MATH HELP

[trail swift]

Hey, firstly, this is not a reason to ping the peeps. Secondly, please be patient when using this channel as I'm sure someone in here can help you

[queen magnet]

sorry i didnt know who to tag

my bad

[queen magnet]

i didnt mean to tag mods omg im sorry

[trail swift]

Don't worry, I won't warn you, but don'

t do this next time please

[queen magnet]

of course again im so sorry

[trail swift]

Also, you mean inverse function by composition?

[queen magnet]

yeah

[trail swift]

Do you have an example you don't get?

[queen magnet]

i have a test on it tomorrow and i was sick for review da

oh yeah totally i have a problem

lemme paste the instructions down and then the first one

For each, find g(x) = f^−1(x) and verify they are inverses by constructing and simplifying f(g(x)) and g(f(x))

and the first one is simply

f(x)=5x^2 -4

[trail swift]

are you given g(x) or expected to find the inverse

?

[queen magnet]

i think im just expected to find the inverse

[trail swift]

Do you know how to do that?

[queen magnet]

i think its saying that g(x) is equal to the inverse of f(x) but i dont really know where to go from there or if i found the inverse of f(x) correctly

[trail swift]

What did you get for g(x) ?

[queen magnet]

thats the thing i havent gotten that

cause i think i screwed it up

[trail swift]

To find a function inverse in the form f(x), sub y = f(x), then

use y = 5x^2 - 4
then replace every x with a y and replace every y with an x
then solve for y with new equation: x = 5y^2 - 4

[queen magnet]

how do i get rid of the y^2

[trail swift]

So y is something squared right, how do you undo that?

[queen magnet]

the root thingy

[trail swift]

correct

[queen magnet]

or divide by y

wait is it like

[trail swift]

square root, since y is to the power of 2

[queen magnet]

yeah

but where even is that other y

[trail swift]

https://gyazo.com/486b5639b7ca09879bd73c9da74025e4

[queen magnet]

wheres the plus minus come from

sorry im asking a million questions

[trail swift]

When you take a square root of something, the solutions can either be positive or negative

think of it like

2 square is 4

but

-2 squared is also 4

[queen magnet]

ahhh

ok

so

what now

[trail swift]

so, you want to show that the compositions f(g(x)) = g(f(x))

you know f(x), and g(x) now, so simply write f(g(x)) and g(f(x)) then simplify both, they should be the same

[queen magnet]

but how can i do that when there are two solutions for g(x)

do i narrow it down so only one is true?

[trail swift]

You may probably just assume the positive one

[queen magnet]

and then how do i get rid of the f and g outside of the parenthesis

[trail swift]

@queen magnet https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:composite/x9e81a4f98389efdf:verifying-inverse/v/verifying-function-inverses-by-composition

This is a good video that may be able to explain it in better detail

[queen magnet]

tysm

[trail swift]

[queen magnet]

if i have any more questions im totally asking you tho

[trail swift]

haha np, I usually sleep early but I'll still be here if you need something probably for the next hour or so

just ping me if you have another question

[queen magnet]

so what would the answer be im confused

like if f(g(x)) doesnt equal x is it not an inverse

[trail swift]

No, you're just simplifying

[queen magnet]

so i got x-16 over 5

is that like

correcrt

[trail swift]

for this one, they will both still simplify to x

[queen magnet]

h u h

so would i put it in like

5(gx)^2 -4

[trail swift]

https://gyazo.com/8a6f91d32fec7134ad458e11992b2e91

maybe that makes more sense

[queen magnet]

mathway totally screwed me over in trying to help me check

ok

[trail swift]

don't use mathway lol, use wolframalpha or symbolab

mathway is bad lol

[queen magnet]

so if they both equal x then what would be answer be

For each, find g(x) = f
−1
(x) and verify they are inverses by constructing and simplifying

f(g(x)) and g(f(x))

cause this is the question

[trail swift]

thats it

you verified it since they are equivalent

[queen magnet]

so my answer would be "they are inverses"

obviously next to my work

[trail swift]

no, your work is basically what i put in the picture, you would say something like "Therefore, the inverse has been verified by constructing and simplifying"

[queen magnet]

ok awesome

thank you

[trail swift]

[queen magnet]

ill try out the next few and report back😼

[queen magnet]

ok so

for the second one

i got x+2 for both

[trail swift]

as long as its for both f(x) and g(x), you are good

[queen magnet]

so like

wait

can you double check

f(x) = 2x
3 + 5

oh man that went in weird

f(x) = 2x^3 + 5

but if theyre both equal are they still technically inverse even if its not x alone

[trail swift]

@queen magnet https://gyazo.com/ab440ce9af2034219419e1ba2f7838fd

They are both x

[queen magnet]

I DONT KNOW HOW I MESSED THAT UP

god

[trail swift]

it will be okay!

never give upo

[queen magnet]

time for number 3

crying

[left knoll]

Please help me in this maths

[queen magnet]

hey whats the inverse of x=(y+5)/(8-y)

[queen magnet]

H

[tacit cargo]

hey guys can you explain me cancelation law by addition or a+c = b+c is a=b

[cyan night]

um subtract c from both sides?

[tacit cargo]

i mean i can do that

but what is the explination behind that

im gonna show you the text that im reading

[cyan night]

uhh ok

[limpid juniper]

hey yall im seeking some help please. basically we were to make an equation for a function with the same zeros as the graph pictured (function families) , and using the point (10,12) to find a k value for it. i did all the calculations but the numbers and the graphs im getting just really dont match up. if anyone could offer some assistance it would be greatly appreciated.

[ember kernel]

You forgot to multiply k by the 12 in the left side of the equation

[limpid juniper]

but i thought 12 is supposed to be used to figure out k? cause if its moved to the right (where k is) then wouldnt it be =0 for the equation then? so we still wouldnt know k.

[cinder yacht]

oh ok, my quick Google search lied to me

[ember kernel]

Only mind the blue part (black one is from my lecture)

[ember kernel]

^

Do you get your mistake now ?

[random flame]

Hi can someone help me with Q16? Any help is appreciated and do ping me, thank you!!

[limpid juniper]

thank you!

[sullen prawn]

👍

[sharp ferry]

is it 8 degrees ?

i did it this way
let the smallest angle be "a" then according to the question largest angle =6.5*a

now since we know the 12 angles are in an A.P we use the formula:
Sn=n(a+l)/2 here n=12 , a=a and l=6.5a
so we get sum of all the angles
Sn=12(a+6.5a)/2=67.5a
now we also know sum of all the angles should be 360 degrees as it is a circle and a complete angle is being divided so we now know
Sn=360 degrees =67.5a
therefore a=8 degrees

[fluid summit]

k! <= k^k

[cinder yacht]

what

I'd just say j<=k

so 1/sqrt(j)>=1/sqrt(k)
then u sum

[fluid summit]

works too

sum n=1 to k of 1/sqrt(n) >= k * (1/sqrt(k!))^(1/k) >= k/sqrt(k) = sqrt(k)

are the same idea

[cinder yacht]

that first inequality

how do u get it

[fluid summit]

average inequality

,tex $$x_1+x_2+\dots + x_k \geq k \cdot \sqrt[k]{x_1 \cdot x_2 \cdot \dots \cdot x_k}$$

for some reason the dots are using central dots

[cinder yacht]

oh

arithmetic avg >= geometric avg

[sullen elm]

who can help me with stereometry?

1.In the quadrilateral AVSD, the diagonals AC and VD are perpendicular and
are equal. Point M does not lie in the plane of the quadrangle, and MA
perpendicular to this plane. It is known that MA = MS = MD.
Find the corners of the quadrilateral.

plzzz

In a regular tetrahedron ABCD with an edge 𝑎, the point M
lies on the segment AC; MS = x.
a) Construct a section through point M and
parallel to straight lines DC and AB.
b) Find the perimeter of the section.
c) Find the cross-sectional area

help, plzzz

[sharp delta]

yes its 8

[vagrant sonnet]

can someone explain this to me?

Like how would I get the instantaneous rate of change when given a tangent formula.

[cinder yacht]

I guess what they want u to do is to compute [f(pi/4+0.001) - f(pi/4)]/0.001

for example?

[vagrant sonnet]

I'll give it a try.

[eager ibex]

Rate of change? Isnt that the slope of the func. at given point?

[glacial dust]

Guys what's the formula for finding the individual resistance in a parallel circuit google only gives me total resistance formula but I want is the individual resistance

[ember kernel]

Well you just need to isolate the resistance that interests you in the total resistance formula

[ember kernel]

That's basic algebra

(double check my answer pls I did it in a rush)

[blissful zinc]

Idek at this point

[knotty umbra]

could you send a picture again with d visible?

[left knoll]

Can someone explain this to me?

My brain is pretty sleep deprived and I can't seem to figure out how 2500/n = 25/9 becomes n= 2500x9/25

Please ping me when you explain!

[ember kernel]

Multiply both sides by 9n/25

[void gazelle]

isn't it just a simple cross multiplication?

[ember kernel]

That's basic algebra, so you might need to sleep very soon 💁‍♂️

[left knoll]

I just did 2500 x 9 = 25 x n -> 22500 = 25n -> 25n/25 = 22500/25 -> n = 900

But I don't understand how they did it in that example

[void gazelle]

@left knoll yeah, your answer is right, in both cases you get n=900, the calculations are the same as the ones you did, it's just that they took the 25 that was before the n and they placed it to the other side. Since they were multiplying they needed to divide after changing sides: 2500 x 9 = 25 x n -> 22500 = 25n -> n = 22500/25 -> n = 900

[random moat]

does anyone know how to help me with this?:

i understand part (a) but could use some assistance with the other parts

[lofty arch]

People are being served at a rate of 60/hour, which means that 60 people are leaving the line every hour

So the line will be getting longer so long as people coming in > people going out

[left knoll]

b-1300hrs
c-100
d-1600hrs

[random moat]

how did you get these answers?

[random moat]

thank you!

[left knoll]

comparing with the graph

[random moat]

thank you!

[lunar yarrow]

GUUUYS

It says: “what’s the intersection point of the following function”?

[cinder ibex]

intersecion with what? y=0?

[lunar yarrow]

idkk desn't specify

[cinder ibex]

well if u make the function equal to 0 it dsnt give any of those options

if u put x=0 it gives 4

so the question is kinda weird

this is ur graph

i dnt see how it is any of those options tbh

unless its d

but it dsnt make much sense either

but i might be wrong, if anyone knows please confirm

[lunar yarrow]

yuh its d

fanks bro

correct answer

[left knoll]

Does anyone understand how to find ARC (average rate of change)? This is calculus btw

[cinder ibex]

Great i think the question should say something like la intersección no es en los puntos arriba mencionados because it does intersect x and y axis tho, it isnt very explicit

[cinder ibex]

t.m.v = (f(x) - f(a))/(x-a)

That is the formula

[signal nimbus]

How can I make a + b turn into a - b algebraically?

[ember kernel]

Substract 2b from it

(a + b) - 2b = (a-b)

(The brackets are only there for clarity, I could've not used them)

[brisk pumice]

Hey guys I’m in pre-calc 2 in college and I have a test tomorrow and I’m totally fucked I really need help. If anyone is good at math please please PM me

[signal nimbus]

I can't change the value though, I need to manipulate it. It's trig identities. I actually have sin^2x + cos^2x and need sin^2x - cos^2x

[brisk pumice]

ill pay

i need a tutor

[sharp ferry]

but how will you go from sin^2x + cos^2x to sin^2x - cos^2x because like both are unequal so you would need some extra terms to compensate the inequality

sin^2x + cos^2x=1 whereas sin^2x - cos^2x=1- 2cos^2x=-sin2x

[sudden maple]

Can someone explain how I can do this?

[ember kernel]

Do you realise that you're asking us to prove something like 2 = 0 ? (Assuming cosx is not = 0)

a + b will NEVER be equal to a - b, unless b = 0

[tropic bear]

the answer is a but can anyone show me how ?

[sharp ferry]

you know the chain rule right ? so acc to the chain rule the derivative of f(g(x)) is the f'(g(x)) * g'(x)

so here first you differentiate the sin term to get cos(x^2) then u differentiate the x^2 inside to get 2x

so the answer becomes 2xcos(x^2)

[tropic bear]

oh yeah ok got that...thank you so much :)

[waxen nebula]

i really bad at math and i don't know how to get better at it
Do you have any tricks or tips to learn Math better?

help me
plz

[steel plover]

need some help with this question

[cinder yacht]

8000* (1-10/100)^7

[steel plover]

thanks 👍

[upper parrot]

I'm trying to prove euler's formula with the taylor series, and I have gotten all the way up until cos(x)+isin(x)=-1.

How do I continue proving from here?

[cinder yacht]

that would mean that x = -pi + 2 * k * pi where k is an integer

are u sure xD

[upper parrot]

fascinating

let me check again

found it

for some reason, you are supposed to plug in pi from e^i*pi as x into the trig equation

thx 4 the help

[cinder yacht]

huh

if im not wrong ur looking to prove that :
exp(ix) + exp(-ix) = 2 cos(x)

with Taylor series

so u write
exp(t) = 1+t+t^2/2+..
=> exp(ix) = 1 + ix - x^2/2 +...
and exp(-ix) = 1 - ix - x^2/2 +...
and u sum those 2 and u find the double of cos(x) 's Taylor series formula

[left knoll]

I did like this but it is a Long way for solving

[left knoll]

yo guys

see here i understand u have to integrate this

but why are the limits 2 and 3

integrals

i mean i get why its 3 but why 2

if anyones willing to help me it would be greatly appreciated

[ember kernel]

• The first century took place between year 0 and year 100
• The 19th century took place between year 1800 and year 1900

In the same fashion, the third second takes place before second 2 and second 3

So you integrate from 2 to 3

[left knoll]

smart guy

thank you my friend

i hope u do well in life

[ember kernel]

Well thanks, fortunately I do for the time being

[fluid summit]

cyclic quadrilateral

but i've never seen this notation: m < 1, that is the angle that points to the line 1?

[fervent tide]

the numbers seem to refer to the angles, while the letters refer to the points

this is the first time I see the "m" notation, too

so that would mean ∠1 = ∠OCV

[ember kernel]

Wow using the m<ABC angle notation
AND labelling angles with numbers, not parameters, was such an unconventional move from the teacher
I could never have solved that problem all by myself

And I just realised they used random letters for the vertices, with the center of the circle being neither O nor C (although they're both used as parameters)

That teacher really wanted to 1v1 maths conventions

[glacial dust]

guys can someone explain why the angle turned 21.33

from 21 20 (I don't even know what this kind of angle is)

what does 20' mean

plz ping me if u can help thankss

[cinder yacht]

I never saw this

but i guess from that image than 60' = 1° lol

so 20' = 0.33333.. °

[glacial dust]

same In my 3 years of high school trig never seen one of these lol

found another solution but still have no idea

@cinder yacht what does the comma on the top mean

so um guessing this is a conversion unit so for 60'(whatever that is)=1 deg.

mb there edited it

[left knoll]

that denotes smaller unit than degree

[glacial dust]

wdym

[left knoll]

Yes u r right

[glacial dust]

but what does the comma mean?

and then after I convert it I have to add it because it said 21.33 so that means 21+0.33=21.33

[ember kernel]

The ' is called a minute, and it's worth 1/60th of a degree
There are also seconds '' which are worth 1/60th of a minute

This convention of measuring angles is called DMS (degrees, minutes, seconds)

That means 21° 20' = 21° + (20/60)° = 21,33°

It follows the same pattern as our way of measuring time :
1h30min = 1h + (30/60)h = 1.5h
(Because 1min is 1/60th of an hour)

Personally I don't like it that much, but it exists

I hope this helps

[glacial dust]

It seems like you know everything lol

[blissful hill]

Does anyone know how to calculate the area of ​​this parallelogram? How do you calculate the hight?

[left knoll]

use Pythagoras for height
area = base * height * 0.5

[ember kernel]

I just happened to learn about it at school 🤷

[glacial dust]

Your actually right the practice advice you said the more practice I do the more I get comfortable at facing harder problems

[ember kernel]

That's only natural. I'm glad that works out for you

[blissful hill]

Thank you!

[atomic dagger]

how can I calculate the limit of x*e^-x as x-> infinity ?

I just don't understand why the limit of e^(-x) = 0 when x -> infinite but the limit is infinite with e^x

[left knoll]

np!

[sharp ferry]

e^(-x) is the same as 1/e^x now as you said the limit for e^x is inf so the lim 1/e^x is 1/inf=0

[atomic dagger]

thank you!!

[cinder yacht]

you should always have exponential's graph in mind, and think of it when thinking of limits

to see that exp(-99999999) is positive and very very small (very very close to 0), and exp (99999999) is very very big (going to "+infinity")

as for x * e^(-x), well the problem here is that when x->infinity, the first factor "x" goes to +infinity (like 9999) while the second factor "e^(-x)" goes to 0 (like 0.00000000001). And we don't really know what 0 * (+infinity) gives us..
but basically "exponential is stronger than polynomials like "x" " so it's the exponential that determines where the limit is going. In other words, the "0" from the exponential is much stronger than the "+infinity" from the "x". So x * e^(-x) -> 0

that's kinda the intuition behind it, never write this on a paper tho lol

[left knoll]

yo guys

last line of part a

is it correct?

wait nvm nvm

[atomic dagger]

got it!! thank you for your help as always!

[left knoll]

can someone give me a hand with part c?

please

im gonna die if i dont do it

thank you in advance

[left knoll]

dont bother now

[frank echo]

https://media.discordapp.net/attachments/326998503316520960/917312344240062484/unknown.png

could someone dm me working out

[tepid spade]

help pls

[limpid juniper]

is anyone able to help me with this question please? I keep on getting a linear inequality after simplifying but Im not too confident that its the right answer.

[glacial dust]

guys how can I find the angle of a sector if I only have a given arc length and radius

[lofty arch]

l = r*theta

[glacial dust]

theta is unkown

so its s/r=theta?

[limpid juniper]

thank you so much!

[vagrant sonnet]

how would i solve this?

i'm guessing at this point

[weary arrow]

Since there's no x your answer is corrext

[left knoll]

You’re welcome:))

[abstract mason]

Could someone help me with this problem?

[potent vapor]

This is the solution, I hope it's clear. Ask me about it if it's not

[left knoll]

Does anyone understand instantaneous rate of change (calculus)?
Ik I'm doing something wrong cause the 8 is supposed to be canceled out somehow.

I'm trying to factor out 2h^2 - 8 so that I could try to cancel only the H by its self

[ember kernel]

If you replace h by 0 in the last line, you get (0-8)/0 = 8/0 = infinity

I've just noticed the beginning of your work is wrong

⚠️ f(-2+h) ≠ f(-2) + f(h) ⚠️

Proof :

f(-2+h) = 2*(-2+h)² - 1 = 2 * (4 - 4h + h²) - 1 = 2h² - 8h + 7

f(-2) = 2*(-2)² - 1 = 7
f(h) = 2h² - 1

You can clearly see that (2h²-1) + 7 is not 2h² -8h + 7

The formula for the instantaneous rate of change at x=-2 is :

[ember kernel]

NOT :

[cinder yacht]

LOL

[inland oracle]

Someone explain to me how the example was solved):
I dont understand the procedure

[abstract mason]

I don't really understand how this proves the statement. both the left hand side and the right hand side of the inequality are larger than zero since x > 0, so arguments with +,- would probably first need some manipulation.

[potent vapor]

Lol you are right sorry

[cinder yacht]

another way is using Taylor series if u studied it

also it's wrong

(x+1)^(1/3) * (x+1)^3 isn't equal to (x+1)

a way to do what u wanted to do is to isolate (1+x)^(1/3), and lift both sides of the inequality to the power of 3

but it does give a really ugly polynomial

[potent vapor]

yep you are correct, I think Im too rusty for this lol

[cinder yacht]

that's why I can't think of a simple solution other than Taylor series

[potent vapor]

Thank you for pointing that out, I would've send this poor dude into a hell of a polynomial hahah

[abstract mason]

Lol

[atomic dagger]

Hi! I'm supposed to find values of a and b for which the system has a) no solution b) 1 solution c) an infinity of solutions
I'm assuming, using matrixes and the Gauss method. I already know the answers but I have no idea how we came to them.

here are the answers

[left knoll]

Does someone know what AD is?

It’s for homework

[tawdry sentinel]

What grade are you in i gotta revise these concepts too^

[median pendant]

hey i was told to apply root test for upper bound and the answer is e^-2 but I got 1

can someone help me please?

[lilac imp]

I can help you out

Just need to get home in a couple minutes

If u still need help

[left knoll]

im in 10th grade 1n

[left knoll]

*rn

[glacial dust]

Guys how do I prove the rad angle here

Can someone teach ms this 😓

Tried making a triangle tried to half it make a right triangle still didnt work out

Nvm I got it 😃😃

Btw will the angle of the biger sector will be the same for the smaller sector or the arc length of the smaller sector will be its angle

[cinder yacht]

i'm confused

if An ~ Bn, that doesn't mean Un^An ~ Un^Bn

AND

sqrt(n²+3n+1) is equivalent to sqrt(n²) which is n, not equivalent to n²

anyway whenever you have smth like Un^An, you need to write it : Un^An = exp(ln(Un^An)) = exp ( An * ln(Un) )
and then find an equivalent of An * ln(Un)

to find an equivalent of ln( (n-1) / (n+1) ), you write it as : ln ( (n+1 - 2) / (n+1) ) = ln( (n+1)/(n+1) - 2/n+1 ) = ln(1 - 2/(n+1) )
which is equivalent to -2/(n+1) because -2/(n+1) -> 0 when n->inf
and that is equivalent to -2/n

[glacial dust]

Deleted it might be reported for spamming

[median pendant]

the question is to determine if it converges or diverges
can you guide me to next step because i'm still stuck please?
you mean ln((n-1)/(n+1)) is equivalent to -2/n or -2/(n+1)? (sorry, i'm not study math in english)

[cinder yacht]

yes that's what i mean

and the proof is what i wrote

try to write the equalities i wrote on a paper

more clear than msgs

[left knoll]

Can someone help me with related rates?

[median pendant]

i'm still trying to understand why is it equivalent to -2/(n+1) (you explained because -2/(n+1) -> 0 when n->inf but i don't know is it a rule or sth)

[cinder yacht]

yeah when h->0, ln(1+h) is equivalent to h

@median pendant

[left knoll]

Here's the question:
The base x of a right triangle increases at a rate of 5cm/sec, while the height remains constant at h=10. How fast is the angle θ changing when x=10cm?

If anyone can help, please ping me!

[median pendant]

thank you so much ❤️ i get it now

[cinder yacht]

you're welcome!

[robust oak]

anyone know how to solve this question?

[left knoll]

If approximation is ok, 2.03756167

[robust oak]

so if i wrote 2.038 rad would be ok?

[left knoll]

Should be!

[robust oak]

great thanks

[atomic dagger]

hi thank you! I've re-made the exercice and I understand how we found that a = 3 and b = 8. I'm just unsure why does the correction say that when a = 3 and b = 8, then we have an infinite amount of solutions, but when a =/= 3 and b=8, we have no solution. I think I'm missing a property here

[robust oak]

can someone help me verify this?

[lyric pine]

Correct

[robust oak]

nice thanks

[royal reef]

hey

Can someone explain the trigonometric function of the tangent type?

[glacial dust]

it stated that x=10 cm so theta must be 2 since It emrges 5 cm persecond

[ember kernel]

You misread the question.
Theta is the angle, not the time elapsed

[glacial dust]

Oh lol I did

but whats is the “x=10” saying?

question says the other 10 is “h=10”?

[ember kernel]

Via trigonometry :
theta = arctg(10/x)

Arctg's derivative formula :
(Arctg(x))' = 1/(x²+1) * x'

Therefore :
(arctg(10/x))' = 1 / (100/x² + 1) * (10/x)'
= 1/(100/x² + 1) * 10 * (-1/x²)
= -10 / (100 + x²)

Therefore :
theta'(x) = -10 / (100 + x²)

And theta'(10) = -10 / (100 + 100) = -10/200 = -1/20

Correct me if I'm wrong pls, I did it out of head

[glacial dust]

if there is a sector inside a sector will both of their angles be the same?

or the angle of the bigger sector will be the arc length of tje smaller one

[ember kernel]

Picture pls, I don't know what you mean

[glacial dust]

Like this one

the angle of the smaller sector is pi by 4

so as the bigger?

[ember kernel]

Well yeah that's how angles work

You could stretch OR and OS to whatever length you want, it will never change the value of angle ROS

[left knoll]

Would we have to multiply dx/dt=5 to the answer -1/20 since we are finding the derivative in reference to time, but take the derivative of different variables?

[quiet raft]

so I know the solutions would be 2,-4 but how do I express this in interval notation?

[soft relic]

If anyone is good with integrals/derivatives can you explain these to me? I dont understand them😭

[hazy halo]

@quiet raft

[slim juniper]

what about it do you not understand?

[soft relic]

Everything? Idk i just dont get them 😕 I (think) i can solve the first one but idk what the professor means by symmetry? In her notes she would say if something was odd/even but idk where shes getting that from either. As for the derivatives idek where to begin😭

[ember kernel]

Symmetry :
The integral of an odd function is zero if the integration domain is symmetrical wrt x = 0 (the y-axis). That means, if you integrate from -p to p (p>0)

[reminder : f is an odd function <=> f(x) = - f(-x)]

This is very easy to understand graphically, let me plot you a graph

If you tried to integrate f from -1 to 1 using the general method (find the antiderivative, and substitute 1 and -1 into it), you could not find any solution

This function is waaaaaay too difficult to integrate analytically (and I'm not even sure it is possible)

BUT you can see that it is odd
!!! Life-saving fact !!!
Because, since you're integrating an odd function on a symmetrical domain around x=0, you know that the orange area (which is >0) and the green area (which is < 0 because f(x)<0 on that domain) will cancel each other out when summed (because they have the same "shape", but opposite signs)

And boom, without any calculation (besides those made to make sure the function is indeed odd), you were able to integrate a very difficult function from -1 to 1 => it's zero

"But Quentin, what if the boundaries aren't -1 and 1, where f(x) is conveniently = 0 ? What if we integrate from -1.75 to 1.75 ???"

It doesn't change anything :)
The orange area (f(x)) > 0) and green one (f(x)<0) still cancel each other out :)

Geometrically solving the integral :
Geometrically solving an integral means using the graph's shape to our advantage, to apply geometrical formulas

Take y = f(x) = sqrt(25-x²) that we would like to integrate from -5 to 5 for example.

• Either by plotting the graph y = f(x),
• Or by realising that f's expression comes from x²+y² = 25 (which is the equation of a circle of radius sqrt(25) = 5 and centered at (0,0))

You might realise that f(x) is actually the radius of the upper semi-circle centered at x=0 and of radius 5 !

We don't know how to find f's antiderivative (is it possible at least ?), but what we do know is the formula for the area of a semi-circle of radius 5.
Boom, geometrically solving saved us

[Here, the integral is positive because f(x)>0 on the integration domain, it would've been negative if we had had f(x)= -sqrt(25-x²)]

I hope this long explanation helped you understand integration using symmetry and geometry

I leave the rest up to you ✨

[left knoll]

Yo guys

I think the answer to this question is wrong

Can u guys solve it for yourself and tell me what u get

Bear in mind these powerpoints have quite a few errors in them

[lyric pine]

In reality, the solution is:
X is less or equal 2

But have to be different of -4

(-4,2]

[left knoll]

@lyric pine yo

[left knoll]

Can u solve that

[lyric pine]

Thats correct

[left knoll]

Just need someone to check it

I got all the values right except the last one

How is it 870

[lyric pine]

Remember:
Cossecx = 1/senx

[left knoll]

Whoops

[lyric pine]

I will try explain

[left knoll]

I just made a silly mistake

Naaaa bro dw I got it

[lyric pine]

Nice!

[left knoll]

I did 1080-30 to solve via synergy

Symmetry

But it’s 900

[lyric pine]

(Idk how to laugh in "English " Lmao)

[left knoll]

-30

That’s fine thank you my friend

[lyric pine]

:)

[left knoll]

How about this

Isn’t as clear but question is a proof from 1. To 2.

Where does the -cos x/sin x go

[lyric pine]

I will do this right now

[ember kernel]

The 1/sin is factored out of the numerator

like 1/sinx - cosx/sinx becomes (1-cosx)/sinx

[left knoll]

You know what

I understand it now

[left knoll]

I see it now

Thank you both

Big help

[quiet raft]

@hazy halo @lyric pine thank you guys

only thing is, what are the rules for writing them in interval notation?

like so I can do it on future assignments

just need a link or something if you dont want to type it all out

[lyric pine]

(y... is the same of y<x

[y... is the same of y minor equal x

...y) is the same of x<y

...y] is the same of y major equal x

[hazy halo]

[x, y] means in the interval, x and y are included
(X, Y) means in the interval, x and y are not included or rather everything between the extremities is included, except the extremities

[lyric pine]

yeah yeah

[lyric pine]

the U is the union of these two

[soft relic]

Yes it did !! Youve been so much help thank you SOSOSOSO MUCH!!!!

[hazy halo]

@lyric pine mate you gotta check again according to your interval the given constraints are not satisfying

[lyric pine]

Ohhhh thats true

sorry

[hazy halo]

Have a look at this

@lyric pine

[lyric pine]

yeah yeah

sorry mens

my bad

is just (-4,2]

because will be negative/negative

in -4 ahead

(bad english I know

)

LOL

[hazy halo]

[-4,2] buddy

[lyric pine]

no

[hazy halo]

It says less than or equal to 0

[lyric pine]

because will be any/0

u cant put 0 in the denominator

[hazy halo]

Oh lol ya

Damn

[lyric pine]

yeah yeah

lack of attention from both

[hazy halo]

Absolutely

[lyric pine]

@quiet raft correction

[quiet raft]

hai?

[robust oak]

can anyone help me with this question

[ember kernel]

I'm glad it helped :D

[cinder yacht]

do u wanna become a professor or an engineer?

u spend a lot of time helping here

you seem to like it a lot

[ember kernel]

I do ! I could become a tutor or something like that while in college, to earn some cash

[robust oak]

Can someone help me with this question

[ember kernel]

Have you seen where the axis of cot(x) is represented on the unit circle ?

(also called trigonometric circle)

[robust oak]

not sure what you mean by axis

like graph wise?

[ember kernel]

Yes

[robust oak]

like that?

[ember kernel]

Oh not like that

I'm talking about the trigonometric circle

[robust oak]

and then cot is just cosx / sinx

?

but specifically, transformation wise, how does tan(x + pi/2) change into -cot?

[ember kernel]

Ok do you know where the tan and cot axis lie on this circle ?

You see that sin's is the y-axis, and cos' is the x-axis

[robust oak]

ye

[ember kernel]

What's that "ye" for, the question or the statement ?

[robust oak]

statement lol

im still not sure about the question

@ember kernel srry are u there?

[ember kernel]

I am give me some time

[robust oak]

mb take ur time

[ember kernel]

I'm sketching what I mean by "cot axis"

[robust oak]

oh i c, thanks, would really help

[ember kernel]

That's what I'm talking about

(The dashed blue and red lines are obtained from the extension of the thick black line)

[robust oak]

whoa lol

so then how would I describe the transformation then?

[ember kernel]

Hold on

I'm checking I sketched it properly

Ok apparently Google has never learnt it that way, but it is indeed correct

[robust oak]

lol, icic, so then with this graph

how can I describe how tan(x + pi/2) = -cotx

[ember kernel]

Oh wow I've just realised that "graphically" meant you had to use the graph, it didn't mean "visually"

Alright

At least you've learnt a way of representing cot on the unit circl

Let's start over

tan(x + pi/2) = - cotx means that cotx = - tan(x + pi/2)

That means :
Take the graph of tan(x),
Shift it pi/2 units to the left (we get tan(x+pi/2))
Reverse it vertically (we get -tan(x + pi/2))
Boom, you can observe you end up with something that looks exactly like the graph of cot(x)

Therefore we conclude that cot(x) is indeed equal to -tan(x+ pi/2)

solved

[robust oak]

DAMN

wow popping off

greatly appreciated thank you

[ember kernel]

np

Sorry I misread the question in the first place

[robust oak]

its alright, but why is it negative tan(x + pi/2)

cuz of the flip?

cuz the original question says positive

[ember kernel]

Yeah but I just transformed tan(...) = -cot(x) into -tan(...) = cot(x)

[robust oak]

oh icic ok thank you

[ember kernel]

That's the same as proving the first equality, but at least we only need to transform one graph twice instead of two graphs once, which could be confusing

[robust oak]

lol makes sense

[robust oak]

can anyone help me write out the sin function for this?

[eager coyote]

i have a question

what does math mean

[slim juniper]

it's a game

[soft relic]

can anyone check my answers? if I got them all wrong don't be afraid to tell me 😅

[rain walrus]

can anybody send their work + answer for this? thank you

[soft relic]

hey @ember kernel if you have time can you check these for me?

[floral jungle]

Does someone need help with your math problems? I am using Gauthmath, it has LIVE tutors who will give answers and step-by-step explanations 24/7 for FREE. Here's my invitation code: KCGGJY.
https://s.mathme.xyz/lp/invite?code=KCGGJY

[quiet raft]

how would I find x for this problem?

[quiet raft]

I figured it out, nvm lol

[ember kernel]

1 and 2 look good to me
How did you proceed for 3 ?
4 seems wrong

integrate e^t from x² to x³ and you get
e^(x³) - e^(x²)

Differentiate this and, via chain rule, you get :
e^(x³) * 3x² - e^(x²) * 2x

(Correct me if I'm wrong pls)