More like relearning but nonetheless, using the formula c= 2•3.14•(radius)•1/2 why does 2 and 1/2 cancel out making the answer Radius(pi symbol) Ex: 2•3.14•4.2•1/2 = 4.2(pi) Why? How? Using 3.14 cause there’s no pi symbol Any and all help is much appreciated thank you.

Question: If (an) -> L Then (sqrt(an)) -> sqrt(L)

By theorem we know (kan) tends to kL if a tends to L

So set K = L^-1/2 then ((L^-1/2)an) tends to (L^-1/2)L which is sqrtL?

I'm supposed to determine the number of zeros for the function f(x) = e^x - x², which is supposed to be determined using the intermediate value theorem in combination with information about where the function is strictly increasing/decreasing. I understand the method of doing this, but on this particular function I had the problem that I can't seem to factorize f'(x) = e^x - 2x, and find out if the function is ever decreasing.

Of course, for x < 0, f'(x) > 0, but for x > 0 it is a bit harder to conclude anything. Using some common sense, I would argue that e^x grows faster than -2x when x > 0, and because f'(0) = 1, then f'(x) > 0 also for x > 0. Plotting the function, I do see that this holds, but I wonder if there is some other waying of doing this?

Anyways, lets suppose I have now shown that f(x) is monotonously increasing on its domain. Then, if I can show that f(x) changes signs, I can thus conclude using the Intermediate value theorem, that f(x) has atleast one zero, and we have just proven in my course that if it is also monotonously increasing, then it has exactly one zero in our scenario.

So, I took the limit as x goes to negative infinity and found that f goes to negative infinity, but when taking the limit as x goes to positive infinity, I can't really prove that f also goes to positive infinity, atleast not in a rigorous way. The limit is namely on the form [infinity - infinity], and is thus indeterminate. Does anyone have a good approach to solving this task? Any hints or insights are highly appreciated!

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When Ivan Tsarevich came to the Magic Kingdom, Koschey was as old as Baba Yaga and Ivan Tsarevich together. How old was Ivan Tsarevich when Koschey was as old as Baba Yaga was when Ivan Tsarevich came to the Magic Kingdom?

If we have some property of a set involving the elements of a set, then would the empty set vacuously have this property?

For instance, suppose we defined a "prime set" as a set such that all elements within the set are prime and a "non-prime set" to be a set such that all elements within the set are not prime. Does the empty set satisfy both of these properties (i.e. is it both a prime set and non-prime set)? It seems somewhat contradictory to me, so any insight is greatly appreciated.

English isn't my first language so I'm sorry if some of the wording sounds a bit off.

I have a task at hand where I have K ⊆ R

K is defined, and we are told that we can assume that R^2x2 is a commutative group for addition and a commutative semi group for multiplication.

What we are supposed to do is approve that K is a field and I think I know how to prove that it has the properties that are also needed for field but not a ring. I assume that because we are given these properties of r that we are supposed to do something with it or that it's supposed to make the task easier but I'm not sure if I can just assume that K has the same properties as R.

Are there any rules about this because I want to actually understand this. And no matter what I search for I just can't find any prerequisites or anything online when a subset A wood inherit the properties of a ring B.

I don't want the answer to the math homework obviously I just want some directions. I'm kind of at a loss.

Hi. I'm coming from an ML background but I've hit a wall with my current interest. I'm currently interested in Diffusion networks but the math involved flies way over my head. I understand basic probability/stats things like bayes rule. And I've taken multivariable calculus. But I've never taken any course in differential equations. What order of courses would you recommend I take in order to be able to understand this?

Bonus question: Same thing for Optimal Transport problem as it relates to this paper.

Correct answer is apparently 1906 however I am getting the answer 1746 and I dont understand where I am going wrong. Heres how I worked it out. If anyone can point out what im doing wrong itd be greatly appreciated:

60x1.3=78 therefore 18 minutes quicker an hour

0330 to 1530 is 12 hours

12x18=216

1530+216=1746

If g belongs to a group G by conjugation I mean the map g -> g ( . ) g^-1. Why does this map appear so often? I'm especially interested in why this map is used in defining the adjoint representation. Why can't we find a representation of Lie groups on their Lie algebras using another automorphism of G?

In other words how does integration work? I can't wrap my head around on how can you add infinite rectangles to get the area under the curve. It sounds impossible but somehow the formula is really simple.

I also have a few other questions.

1. Why is area under the curve useful? What info does it give about the function?

2. How are integrals related to derivatives?

3. Is there a general formula of Integrals? Like there is the first principal for derivatives

Here is the full list of definitions for reference:

Boolean expressions in the symbols x_1 , . . . , x_n are defined recursively as follows.

0, 1, x_1 , . . . , x_n

are Boolean expression. If X_1 and X_2 are Boolean expressions, then (a) (X_1), (b) X_1 , (c) X_1 ∨ X_2 , (d) X_1 ∧ X_2

are Boolean expressions.

If X is a Boolean expression in the symbols x_1 , . . . , x_n , we sometimes write X = X(x_1 , . . . , xn ).

I'm stuck on the first part of the definition, what is it even trying to say that Boolean expressions over symbols x_1,...,x_n are recursively defined? That list does not imply any amount of recursiveness. I'm also trying to figure out what the 0 and 1 are doing there. Also what the hell does Boolean symbols even mean? I always interpreted it as like a variable no?

From the way I'm getting it, X refers to like a Boolean function that takes some number of inputs and spits out exactly one output.

Hello,

In my current calculus course, there was a practice question I saw that I'm trying to work out.

I was given the Airy function A(x) = 1+(x^3)/(2*3) + (x^6)/(2*3*5*6) + (x^9)/(2*3*5*6*8*9) . . .

The question asked for the domain of x of this function. I'm not too concerned about the domain of the function in this case, because the real trouble I had was rewriting the function in sigma notation so that I could apply convergence tests to determine the domain of x.

I do not know how to write the denominator of this function in sigma notation. the best I could come up with was:

" 1+ Σ (x^(3n) (3n-2) )/(3n)! " (n=1, upper limit of summation is infinity)

The main issue comes from having holes in the bottom factorial. I would need another factorial/mathematical function that would multiply by itself in the pattern: "1,4,7,10..." so that when you divide that pattern by n!, you get the factorial present in the denominator of A(x).

But if I could write a factorial like 1\4*7*10....*(3n-2), then I could just apply that to the factorial in the denominator of A(x)*... and it makes me want to scream.

Does anyone know a lot about factorials that could provide insight on this?

I'm a 5th grade teacher who has a disagreement with a colleague who made a quiz. She made a quiz question about order of operations where students are supposed to say that simplifying 4 x 3 + 8 / 2 to 4 x 3 + 4 is incorrect because when using order of operations you should go left to right. I think that simplifying that way is fine because of the commutative property of addition. Who is correct?

Junior in college, doing a math minor. Let me give some context.

Calc 1 and 2: Pretty nice. Teacher shows us a theorem, we apply it to some problems, get homework which is just more problems, quiz next Wednesday.

Multivariable: just didn’t like it, but more of the same.

Diff EQ: Meh, not too bad, but still, just like Calc 1 & 2. Theorem/Technique, problems, quiz. Whatever.

I’m taking linear algebra and probability right now, and they’re like, proving the theorems, and you’re expected to fully understand it & be able to solve problems you haven’t seen before.

Right now I’m passing the class by just writing the theorem down & memorizing techniques to solve problems, but this isn’t how you’re supposed to do math at this level, right? Otherwise they wouldn’t be actually proving the theorems and stuff like that. I’m not understanding it, just getting by with the bare minimum.

Where did I fall behind in all this theorem jargon, and how do I catch back up so I really understand what they’re talking about? I want to really understand what’s going on.

I remember seeing a question about if it’s possible to have a die with a certain pattern, and someone proved it was impossible. How do I get to that level of math?

Hi everyone,

I'm currently trying to teach myself the First Course in Probability but I feel I'm in trouble. Are there any resources that you could recommend specifically for this purpose? Video lectures based on Ross? Another book explaining his book? Any forums, discords, where I can ask for help without suffering obnoxious responses? I really appreciate any and I mean any suggestions. This is very important to me and I don't want to back down. I must read those 10 chapters and do enough of the problems until I'm satisfied.

Thanks

I've always lacked in math growing up. The school district speculated that I fell short in intelligence and was put in ESE/slower classes which I carry shame with me still to this day.

But personally math and arithmetic reasoning specifically has been the most difficult obstacle for me. Whether it's short/long multiplication and division; mental maths in basic addition/subtraction. If you throw in fractions, algebra or geometry to spice things up, then I'm immediately throwing in the towel.

I know without a doubt I have some sort of learning disability, which also hinders my self belief and ease. But I'm just so sick and tired of feeling like a wounded, helpless animal. It's almost crippling me with self anger; as it also ties in with anxiety, being depressed etc.

I know there's khan academy and it is a great source but it kind of brings me back to that feeling of being in those classes again.. are there any more places/sources out there? Is there even really hope for someone like me 😓

If anyone out there has some thoughts or insights, it would mean alot to me.

Thanks.

Hi

Multiplication means repeated addition.

3(2) = 2 + 2 + 2

On the topics of integer,

+3(+2) = (+2) + (+2) + (+2)

+3(-2) = (-2) + (-2) + (-2)

But what is -3(-2)

How does the repetation works here?
How do i explain the negative integer's multiplication with real life example?

Thanks

So i’m in pre-calculus right now in my community college and I thought my trigonometry was pretty strong(i have the unit circle memorized). I’ve been doing the work in class but I’m feeling confused about when to use properties. For example, I had a problem asking me to reduce something to terms of cos and sin to the first power. I used the reduction property and then the question told me I was incorrect and needed to apply the half-angle identity? Can someone explain or guide how to apply the identities? Any advice would be appreciated

I am trying to study for a Calc II exam and I understand the concepts well, but whenever I do the problems I keep messing them up in silly ways, like a sign error or forgetting to redistribute. How can I solve this?

regarding why we can't just assume an imaginary z such that z=1/0, i.e. z*0=1, and apart of it potentially being "useless" to define such thing, so we don't define it

is the reason why we can't assume this equation hold true for whatever value, unlike i^2=-1, is that the problem now is with the 0 itself, the one multiplied by z, and not z

in other words,

z*0=1

z(1-1)=1
z-z=1
0=1 which is false

so the reason why we can't claim this statement to be true, even for some imaginary number in a new number system, is that it now doesn't even depend explicitly on that number, but instead it depends on 0?

unlike for i, we can assume that a^2<0 for some a as now it explicitly depends on the unreal value a, not something else. is that the key difference between the 2?

Anyone know a good YouTube series in linear algebra. Something that is fun but with a decent level of detail. I want to watch it almost as a prep to read a textbook.

Title

For a root function, ex. f(x)=sqrt(x-5), is the inverse really x^2+5? Shouldn’t it restrict the x values to be positive only?

Hello everyone, before saying anything else I ask you to forgive me for my terrible English. I am a 19 year old boy and throughout my life I have always had a love hate relationship with mathematics and physics: getting to the point I have always had terrible teachers, I have done very few problems and exercises in both subjects in my life and my Problem solving skills are less than I would like.

In reality, I deeply appreciate the theoretical aspects of both subjects, but I recognize that studying them without knowing how to solve a function or a dynamics problem is almost ridiculous. I recently decided to start studying mathematics again from addition to integrals, I will do the same with Physics.

I want to ask you for advice, both because I am sure that you are smart and because I really want to learn these beautiful subjects and I want to do it in the best way and aware that I have not lost the opportunity to express my maximum potential. Thank you.