not sure since can't use remainder theorem

(x^2-1) = (x+1)(x-1)
means (x+1) and (x-1) are factors of f(x)

nvm i forgot that u can make it into 2 linear


whats the span of the set of the union of the x and y axes? (im guessing the x-y plane but i wanne be sure)

What about the span of the x, y and z axes? it should be R^3 right?

since a.(1,0,0)+b.(0,1,0)+c.(0,0,1) spans the entire R^3 space


How is this divergent? I separated the limit to the the numerator and denominator.  lim/lim

so if i ever encounter something like this i would say it diverges?

Just testing some things out and this one converges to 0

yeah, cause you have constant/(going to inf)

just to clarify if i were to draw a graph it would look like it's oscilatting and gets closer to zero

cause it alternates between negative and positive


Lol
I got it as cos(tan(x⁴)) sec²(x⁴) 4x³
Is that correct

Can u confirm please, im really confused cause I saw a different answer on youtube


Hi i'm new to this math topic. What is the method for solving these? :0

but actually you're probably supposed to multiply

I figured that, i was a bit confused at the time

if you have (a+b) * e, that's the same as ae + be
so now let e = (c + d)
then (a+b) * (c+d) = (a+b) \* e = ae + be = a(c+d) + b(c+d) = ac + ad + bc + bd


Is it possible to figure out if 3^9 is greater than or less than 4^7 without using a calculator?


can someone please explain how to solve this using partial decomposition

@kimchi feel free to ask questions about the video. I'm sure @a disappointing son has watched and understood it 10 times by now


I am thinking this because the bounds are the same

$\int af(x)+bg(x)\dd{x}=a\int f(x)\dd{x}+b\int g(x)\dd{x}$

cause the lower bound is a, you just said it was


It means, literally, x is not equal to r

Do you have some more context for this set? Or did you just make this one up?

okay my latex isn't working but whatever

It's saying the domain is all real numbers, except for 0

Yeah, domain is for the x values, range is for the y values

And yes, what you wrote means all real numbers except 1

Literally "the set of real numbers x such that x is not equal to 1"


okay so like, do you want *help* or do you want someone to do this for you

do you understand what "\_\_\_ property of equality" means

where the blank is one of the four arithmetic operations

no i am having a hard time understanding

has your teacher not explained such concepts as "adding the same thing to both sides", "multiplying the same thing to both sides" etc?


when it clearly isn't an equilateral triangle

That was evident from the right angle marking

ok I thought that X and Y are also going to be 30

reserve your own channel #❓how-to-get-help

> Arent they all =
no, this was already addressed by Mosh

Bruh I literally put it in simple terms for u to solve

consider the ratios of a 45°-45°-90° triangle
and/or set up some trig equations

or even properties of an isosceles right triangle (and apply pythagoras)


The only thing I can see is multiplying by T^2

if the numerator and denominator of the fraction both equal zero after doing that then you can take the derivative of the top and the bottom and try plugging t in again

if doing that makes it 0/0 again then take the derivative again

Is the derivative of sin(5t)^2 cost(5t)^2?

Sin(5t) is the inner, and x^2 is the outer tho

the first inner function being sin(5t), the second being 5t

How does chain rule work with more than one inner function

Wait how do I solve that without the derivative of f'(g(h(x))) when that is what we are solving for

Are the derivatives of trigs not real rules?

But how is H''(theta)=-theta sin(theta+ *2* cos (theta)


Can someone check my answers for b) and c) for #8?

Is the way i set up the shell diagram correct for 8?


You use the result from (1) and put it into (2)


A river flows at 3 km/h. At what rate should a boat sail in still water so that its actual speed upstream is 18 km/h?

When Jorrel was born his father was 25 years old. Now, his father is 3 years more than twice his age. How old is Jorrel now?

Let Jorrel be x

Is the equation "2x+3-x=25" correct for this problem?

when you say "Let Jorrel be x" do you mean "Let **Jorrel's present age** be x"?

if so, then yes, your equation is correct.


my work is above, although I think I'm not on the right track since ln of a negative is undefined and there is an answer in the textbook

what confuses me the most is the part at the beginning where (1+2x) is multiplied by dy/dx. I try to separate them so I can get y on the other side, but this results in ln when antideriving which isn't in the answer at all.

forgot the abs value bars when you integrated

also forgot the 1/2 that comes from integrating 1/(1+2x)

I thought since it's over 1 I would use ln


i.e you should have +1 instead of -1 after evaluating the bounds

the difference of the lns with the ts, yes

it'll be clearer if you consider the difference to quotient law

yes lim at inf of $ln|\frac{1+t}{t}|$ is $0$


Okay, so i have a task to do with basic maths and im paying 10€ if someone can solve it.
If you know Coding (Javascript) it will be perfect.
https://i.xpyth.dev/paclOR9l.png

Default rate:
a: 10
b: 60
c: 40 (ME)

If i put custom rate (Line 89)
a: ? (something like 8% ??)
b: ?? (something like 50% ??)
c: **60** (ME)

If i set my rate for c to **60**, how can i get the rate of a & b using their default rate and considering they will be shrinked because C is taking from their part.
If im not clear just tell me i will give mor details

**10€ On the line rn, willing to pay a lil bit more if someone can even bring the code with it**

Please read #❓how-to-get-help and #rules

I don't think im violating the rules considering thoses are not homework assignment, am i wrong ? @dldh06 :/

Oh, do you know where this type of request could be placed ?

I'm not sure sorry but definitely not this server


Hi, Stupid question.
What are the odds of picking one outcome out of three, if you have two attempts and the one you pick is removed from the pool? First time should be 1 of 3, or 33% second is 1 of 2, 50%. How is the total calculated?

Why is it lower than the 33.3% odds in the first attempt? Shouldn't it be higher? Sorry, I am just really confused.

Well I suck at probability so affirm my math with someone else

Because it's Chance B dependent Chance A

That's what got me confused. It seems odd to have lower odds when you get another attempt with even better odds.

@werr there are 2 cases, getting the number first try and getting it on the second

Chance of 2nd try is the chance of the desired number x still being there (2/3) times the chance of getting the number x out of the 2 remaining ones (1/2).
This is ⅔•½=⅓

Total chance is case 1 plus case 2, i.e ⅓ + ⅓=⅔

So the second event is dependent on the first one failing


are you able to draw something that looks like a rectangle

Should I just draw a line of 90 deg then a diagonal of 8cm with base 7cm?

you could start with what you described above


This lesson I learned how to find critical numbers

I mean like segment Connecting those two points

Ok so next would be to get the equation for the line

Which is less than 1 but how do I know if it's for all x? X isn't part of the derivative


can the radius of a circumscribed circle be equal to one of the side of triangle?

but visualize a right angled triangle ABC. C is the right angle

now surely somewhere along the way AC is equal to the radius


*just tell what you don't understand in this*

you agree that sec(x) = 1/cos(x) and tan(x) = sin(x)/cos(x) ?

the rest is just algebra and playing with trig formula

*basic algebra*
(a^2 - b^2) = (a-b)(a+b)

I didn't know you could do that including the sqrt

*again basic algebra*

a^m/a^n = a^{m-n}

On the left hand side or right hand side?

u should rewrite the left side as sec^2 a(sec^2 a) + tan^2 a(tan^2 a)

and then sub one of the sec^2 a with the identity, and one of the tan^2 a with the identity

so sec^2 a (1+ tan^2 a) + tan^2 a (sec^2 a -1)


h(x)=sqrt(x+4). h(x) is a composite function of f and g such that h(x)=f(g(x)). Give solutions for f and g.

what do you know about composite functions


I have to simplify terms but dont understand how the result came to be. What rules were applied and how?
What happened to the "-2" from the bottom "s^b-2" and the positive "r^3*a+b" in the bottom in the last step

okay i get now how the result came to be. By just subtracting them. But why? Is there a rule for when i subtract a bottom term with a  top term ?

by doing more problems that involves using it.

Not memorised more like know what they mean and from where they come from

i hear what you're saying. and i just disagree

bad idea for what? grades? i don't think so

i think doing more problems leads to memorizing

Memorising and understanding aren’t mutually exclusive

Feel free to convince salber to be able to derive each of the exponent rules in the list and understand it

i know most of these bust still fail to see which of those rules state what i can do in my problem.

If you dont care about understanding math you might as well just use wolfram alpha. I mean you will get good grades , right?

The logic is to be understood obv , but the the notation are simply just memorization work , and they are there purely for everyone to be able to understand

there are _some_ things you don't need to understand and better off memorizing. this is one of them.

ah great. i overlooked that. Thx for the help


f is function that take vector v and calculate the angle with the vector (1,0) counterclockwise. Is f continuous?

think about what the values are right before and eight after (1,0)

so the angle is 0 directly at (1,0), right, and as you go ccw, the angle gets bigger

by the same it's almost back to (1,0) it's at almost 2pi, then it jumps back down to 0 right at that point

so actually there is uncontinuous on the x axis?

I don't think how you said that is right

it's defined for all R^2 in the input, but the output isn't continuous

yeah I just wonder where it isn't continous

that's the only place where we have that jump


If the oil price is 100€ and the price goes up by 20% it now cost 120€. But if it goes back down to 100€ you have only a decrease by 17%. since the difference of 20€ is less percent of 120 then of 100


What do you need to find? Where are you stuck at?

Do you know what is the sum of all the angles in a triangle?


This is the limit definition you learned?

The limit definition is actually $\frac{f(x+h) - f(x)}{x+h-x} = \frac{f(x+h)-f(x)}{h}$

So it seems what you did above was right


The line segment in a triangle joining the midpoint of two sides of the triangle is said to be parallel to its third side and is also half of the length of the third side.”

and The converse of the midpoint theorem states that ” if a line is drawn through the midpoint of one side of a triangle, and parallel to the other side, it bisects the third side”

i was preparing for math olympiad but there was no solution to this problem

which basically means DM is DE by a half

you can say midpoint a special case of thales

use Thales theorem for DE // BC and EF // AB accordingly

from what lava said here we can say that DE = 1/2BC and EF = 1/2AB

applying converse of midpoint in DEF, we get MO=1/2EF

since D and F are midpoints of AB and B respectively, we can derive that BD=BF so DEFB is a square


what does it mean for this expression inside the limit to “not be differentiable at 0”

contrary |x| is defined at 0 but also not differentiable

To not be differentiable at 0 doesn't mean to not be defined at 0

An interval where the function is differentiable and an interval where the function is defined is very distinguishable

In fact, the function f(x)=|x| is differentiable at everywhere, except 0, so its interval of differentiability would be R*=R\\{0}

Well here, for any x different than 0, you have x/x = 1

Since the limit when x goes to a is about looking at a function infinitely close to the a value, but not exactly on it

u are only referring to the fraction, ignoring limit?

huh, why isn't it differentiable at 0 🤔
I'm not rlly with it today

I'm guessing there's a discontinuity there

Oh right, the limit doesn't exist because f(a) doesn't even exist

If $800 is deposited in an account that pays 9% annual interest, compounded semiannually, find the balance after 10 years.

I got 1929.37122 this I’m not sure if it’s right

f(0) gives nothing because ln(0) is not defined

so simply if a function’s not defined at a value, it’s derivative doesn’t exist?

it's not differentiable in 0 because there are multiple tangents to that point


why isn’t it
$R_n > \int_{n + 1}^{\infty} f(x) \dd x$ instead

it's an infinity thing, you can't say $R_n > \infty$

so if the thing on the right diverges to $\infty$

then you end up with $R_n = \infty$ also

there's also a second case, when $f(x) = 0$ for all $x$, in which case $R_n = 0$ and the RHS is also $0$, so it must be an equality

but the integral test wouldn’t apply :sully:

aren’t we assuming the integral test is applicable here

and that the series (hence the integral) converges in the first place

what is the exact statement of the integral test?

for +ve cont. dec f, series of f converges iff integral of f converges

i'm going to write that in words cause it's painful to read

in fact the true statement is somewhat more

Let $N\in\bN$ and $f$ be a monotonically decreasing function on some unbounded interval $[N,\infty)$. Then, $$\sum_{n=N}^\infty f(n) < \infty \iff \int_N^\infty f(x) dx < \infty.$$

continuity isn't strictly needed also, but in anywhere before advanced real analysis, you will assume continuity


What are the total number of 12 letter words that contain 3 'A's and 2'E's?

Dude we can make words from all the alphabet?

i need to make sure by word he means 'any combination of letters'

26c12 assumes that its all unique letters right?

@Not Trout in this case u r saying repitition is allowed for all alphabets

but you don't know how to approach this?

but im confused with how to calculate the question

so you're asking for the number of strings such that:
- contains 3 'A's
- contains 2 'E's
- is 12 in length

like... i dont know if i should do permutation for those 5 letters or do permutation for all or do multiplhying with all

now you would want to then change that to yk

try considering (conditions 1 and 3) and (conditions 2 and 3)

how many possible words you can form then do complimentary counting

We have 12 places i think the answer is 3c1 2c1 24c10/2!×3!

24^7 x [ways you can mess up 3As and 2Es]

U r taking power 7 cuz a and e aren't same?

the rest of the spaces can be anything but a or e

@Phoenix775 I'm getting that it  can be wrong

yeah i got that too and a friend of mine got that answer as well

for some reason i have a bad feeeling about this answer

cuz the last time i used this method of "eliminating" the requirements it didnt end up correct

but meh ig since the 3 of us got the same answer it must be correct


do you know the difference between coeffcient and variable?

basically, we add the exponents of all the variables of a term together

its degree is 5, because the variables are x and y, and adding their powers tgt we get 5

polynomials are just sums of multiple terms

deg of a polynomial is just the biggest of them

all terms are **arranged** in descending order of power

so you check which one is arranged in descending orders of power

and $x^3, xy$ have orders $3, 2$ respectively

i thought it would be the first option but it says thats wrong


A rectangular block is such that the sides of its base are of length x cm and 3x cm. The sum of the lengths of all its edges is 20 cm.

If it only gives you the base sides, what's the height?

I mean, if they don't give the height, you can solve it, but the answer will be in variables rather than a number.

@*___* 9 They tell you that the sides of the base (the length and width) are x and 3x.

The sum of the lengths of the edges is 20 cm.

So, there are four edges that are the height, four edges that are the length, and four edges that are the width of the block.

Yes. The front edge of the bottom (width) is x, the other edge of the bottom close to you (length) is 3x and the vertical edge (height) is h.

Yes, now you also know the sum of the edge lengths.

There are four that hold up the top side.

There are four that are 3x long, they go horizontally parallel to you.

There are four that are x long. They go from where you are away from you into the paper.


So, I've so far determined tan(alpha)=-7/3. Im not sure how I'd solve it from there, supposedly arctangent?

Okay, -66.80. I knew that partt, Now it's looking for an angle measure, how would that be determined?

No, Theyre looking for an angle measure to be determined from tan(alpha)=-7/3

Thats probably what it's looking for, I'm fairly lost in this part

you literally said the angle measure here

you have a coordinate system, i is your x axis and j is your y axis, then you plot the vector there, and determine the angle between your vector and the x axis

Yes. Hence why I was asking for a follow up. They answered, I'll do that

they just told you to do exactly what you already did

you found alpha. you just need to determine quadrant relevant to the question

If you read the problem, it asks for 360-66.90. It isnt just 66.80

Someone else was answering and having another person just made it convoluted. Thanks though.


idk why i find the slant height but my goal is to find Θ

can we like determine the Θ at the 3d cone? not the one that's been deform

theta is defined to be the angle of the sector of the unfolded cone, dunno why you're trying to get it from the 3d form

consider the which parts of the 3d cone correspond to the unfolded cone


hi guys does anyone know how to solve section d question 1

differentiate everything with respect to y?

really doesn’t matter. you both dx/dy and dy/dx have to multiply to give you -1 so it’s upto you really

because one is the reciprocal of the other and -1 is it’s own reciprocal

what if i substitute the sin(2x) into 2sin x cos x?

you cant cancel from outside the integral

if it cancels with du then you dont get du anymore

@moose if you solve the integral post your steps here

i got that because constant coefficients like -3/4 are easy to integrate since you can move them outside the integral

with this sub you end up with $\int 4/3du/u^8$

should i substitute it into v before i use the power rule?

its already substituted into a familiar form


I got this as my answer through trig sub

Post in a new channel - someone else is using it now

all good, I was too fast for it to update


Let $h: \mathbb{R}^4 \to \mathbb{R}$ be a function with the following properties: the domain of h is $\mathbb{R}^4$ and all partial derivatives of h from of 2 or lower exist and have the same domain as h. Let $H: \mathbb{R}^2 \to \mathbb{R}$ be a function defined by $H(x,y) = h(-3, e^{xy}, x^2 + y^2, y-x)$.

How can I prove that the partial derivative $\frac{\partial H}{\partial x}$ exists?

My idea was to prove $\lim_{h \to 0} \frac{H(x+h,y) - H(x,y)}{h}$ exists, but I don't know how I can prove this limit exists

write H as the composite of two partial differentiable function, then H has also a partial derivative ||one is the two functions is h||

So something like $\frac{ \partial H}{\partial x} = \frac{ \partial H}{\partial h} \frac{\partial h}{\partial x} + \frac{\partial H}{\partial ?} \frac{\partial ?}{\partial x}$?

if H(x,y)=h(g(x,y)) with $g(x,y)=(g_1(x,y),g_2(x,y),g_3(x,y),g_4(x,y))$, then  $\frac{ \partial H}{\partial x} = \frac{ \partial h}{\partial g_1} \frac{\partial g_1}{\partial x} + \frac{\partial h}{\partial g_2} \frac{\partial g_2}{\partial x}+\frac{ \partial h}{\partial g_3} \frac{\partial g_3}{\partial x} + \frac{\partial h}{\partial g_4} \frac{\partial g_4}{\partial x}$


I have another question if you dont mind

so the question was actually to do the minimal disjunctive NORMAL form, is that any different?

the "normal" refers to the fact that each term that's OR-ed is a conjunction

So for example A v (B(C v D)) is not DNF

Alright, no more questions im out for real this time 😆 see ya!


(its a volumetric revolution around y axis curve)

is the '1st quadrant' just top right quadrant or something?

The curve will be a border to the region in question

are the quadrants actually like 'officially' numbered?


left the original number of boys, girls and adults be 5x, 8x and 2x resp.

after k girls leave, the ratio is 10:9:4, so 50:80-k:20 :: 10:9:4


Can someone teach me find the equation to Thea table

are you trying to find the equation of a line? @itsmealex

Seems like it since I don’t think it can be like direct variation or smth

Yeah seems like a curve of sorts since the rise/run aren’t the same. Idk about those

i probably shouldnt say the equation but yeah its linear

Do you know how to find equation of a line?


Quite a general query. Is there a method to calculate the equation of a line with seemingly random fluctuations such as the one above without the use of integrals?

I'm not great with Mathematics and haven't been taught Calculus of any sort. I'd prefer not to share what the points represent, but my closest approximations have all been hyperbolas of which the points have fallen very close to.

The upper bound for the points is π (which is the Y-value of the highest point in that image) and the points decrease to 2√2 as x approaches infinity.

The x-value of the furthest-right point in that image is 4. I haven't plotted many more points as of yet.

I imagined creating a straight line between each integer for an approximation. Creating a straight line between every half would be twice as accurate, between every eighth would be 8x as accurate and so on. Therefore an infinite quantity of straight lines between each infinitesimally distant point would create an "exact" solution. This sounds like the sort of thing Calculus messes around with, hence why I presumed it would be the go-to.

Any advice would be very helpful, if necessary I'll show my closest approximation.

cant this be done some way using polynomials and vertex points?

im not sure myself but i think theres a way to do so

unloess you looking for the best aproximation of the lines that still hits all the points

I'm looking for an exact solution, you can't hit all the points with an approximation.

I have approximations that visually touch the points on Desmos to a degree, but that's frankly irrelevant.

No idea what that even was until now, I'll look into it thank you.

Interpolation unfortunately is limited to approximation also.

Interpolation hits every point. its unknowable what exactly happens between each point, you have to guess


how do i convert m squared to cm squared

i forgot but i couldnt find out the formula

1m^2 is 100x100 cm so  yes 10k, i always have to think about it because i never remember what the actual values are

@Kenes dont forget to close the channel if youre done


so I think I have the first multiplication correct but I'm kindof confused how to structure the product of the second set of multiplications

((1, u),m) or (1, u, m) or something else?

Technically it's the former, and what you did was correct

In practice, if you see someone insist on the distinction between the two outside of an introductory set theory class they deserve a slap to the face

Okay cool, thanks! The first feels a bit better but It had me concerned


Is there a common “way” to find the general form of a succession by having its values?

Or is it just about luck and some logic? 🤔

For example, by having: {-2; 6; -18; 54; -162; …}
How can I find out that one general form might be: An = (-3)^(n-1) * (-2)
?

Honestly, pattern recognition is just trying out things and hope they get you somewhere

There isn't any set method, otherwise many problems would be solved by now

In this case, perhaps you could have tried factoring everything by 2 since every term is divisible by it

You'd have gotten -1,3,-9,27,-81,... which are clearly powers of -3 in some capacity


I need help with the question at the top

@Van Can you find a right angled triangle inside the square that can help you find the side length of the square using pythagorean theorem?

Might take long because school wifi isn’t too good

ok just trying to figure out where your triangle is, your triangle should have lengths that are easy to count on the grid

what are the lenghts of the sides of your right triangle

so what are the side lengths? I'm assuming your teacher wants you to use the grid

But I have no idea what the side length is

you need to know that to know the third side

im assuming you can count each small square grid as 1 by 1

But what about the ones that are half in half out

it looks half in half out because of drawing

technically it is all the way to the vertex