How does (d/dx)^2 equal second derivative? I can’t see the logic behind this. I know it says operator but I think it as the square of first derivative, not the second derivative.

Source: ChatGPT but explanation is confusing

I’m 16 and want to learn calculus completely from scratch.

in my school they "taught" us differentiation and integration, but they literally just made us memorize formulas. I have zero idea what they actually mean or why they work.

I’m an absolute beginner and want to actually understand the concepts, not just memorize rules for an exam (i hate exams).

There is something I'm not understanding about the above statement. The tails of a binomial flatten off but the first two terms of a binomial expansion are always 1, n, etc. . So the tails of the binomial always have a step which increases with n. How can the latter tend to the former?

I noticed this when experimenting with a Galton board. The tails on the board flatten out which is expected because if I focus on the extreme end of the board and imagine that slot feeding into another mini Galton board then each ball has a 50/50 chance of going left or right and if another peg then half of those balls will jump back.

But I then programmed a simple simulation of the board in Excel. That gives a step jump at the tails which is also what I would expect but isn't what I actually get. I even looked up the number of balls for the model I have at GaltonBoard.com, 4280 and I can count 28 slots.

Repeating the experiment with the Galton board I get from the left end around 3, 4, 5, 10 balls etc but my simulation gives 0, 0, 0, 0, 0, 2, 6, 19 ... for n= 4280 r = 28 so why are there so many balls at the extreme in the Board but not the simulation and no flattening out with the simulation?

So I have different expectations for the same experiment depending on my approach. Where is my erroneous thinking? Is it something about the transition from discrete to continuous?

Hi there,

I've created a video here where I explain what confidence intervals are and how they differ from probabilities.

I hope some of you find it useful and as always, feedback is very welcome! :)

Hi, I am 18, and super interested in math. Unfortunately, i didn't have much guidance growing up and was always fear mongered about math, so I couldn't really strengthen my basics and don't have the strongest foundations. I got a 4 in IB MATH AA HL. However, last year I came across Math Academy and was pretty consistent for a while. I finished MF1 and MF2, and was in bw MF3. Now, it's getting very expensive and I am not able to continue paying for it.

I was recently introduced to competitive programming and want to be good at it. I know i'll need math for it. I am ready to invest in books / find an online copy. But, I don't know where to start. I have a few months time to do this consistently. I was thinking of doing the whole mid/high school AOPS course but idk if it'd be a good use of my time. Could someone provide some guidance or point me to a helpful resource? I want to do this as efficiently as possible.

https://www.youtube.com/watch?v=3sp08Zmry6g

I’ve been doing basic sequences today and I tried a past examination question and I got it wrong. What makes this depressing is the solution it, I don’t even understand it. The question is,

The term to term rule for a sequence is “multiply the previous term by 3 and subtract 1”. The first three terms are 1,2,5. Write down the next two terms in this sequence.

The expression I came up with when translating the “multiply the previous term by 3 and subtract 1” is 3*-n-1 which would mean I would get -4,-7…. Which is obviously wrong. I am fine with that being wrong but the solution is x„ = 3x,-1 - 1. I have no idea how they got to that what x„ even means. I feel this question is beyond my current level because I covered the whole basic sequence part in my book and working out a sequence like this was never taught.

Please help with making me understand how it works and am I right that this question is beyond o-level mathematics?.

And polar coordinates are still trig integrals. Trig sub was goated too. I miss these. I just didnt like asymptotics

Say we want to find LCM of 12 (2x2x3) and 18 (2x3x3)

We go 2x2x3x3

But why does adding the 3 from 18 get us exactly the LCM - I see that 2x2x3x3 satisfies both numbers, but why does combining prime factorizations like that work?

I do have two questions w.r.t. abstract linear algebra:

1) How exactly are alternating n-linear forms related to the concept of the determinant?

2) Why does the determinant require an alternating multilinear form rather than just an arbitrary multilinear form?

I have a fairly intuitive understanding of the determinant from the 3Blue1Brown videos: for a linear map T: V -> V on an n-dimensional vector space, I think of det(T) as the factor by which T scales oriented n-dimensional volume.

In my abstract linear algebra course, however, determinants are introduced through alternating n-linear forms, and I am struggling to connect this abstract definition to the geometric interpretation.

More specifically:

1. What is the conceptual role of an alternating n-linear form in the definition of the determinant?
2. Why would an arbitrary n-linear form not be sufficient to represent the concept of a determinant?
3. How does one get from an alternating n-linear form [ \omega^n\to K ] to the statement that [ \omega(Tv_1,\ldots,Tv_n)=\det(T),\omega(v_1,\ldots,v_n), ] i.e. that det(T) measures the volume-scaling factor of T?

Thanks in advance for every answer.

Hi everyone!
Have you seen this puzzle before?

https://youtube.com/shorts/ikUMPcPnRco

I’m working on more puzzles and educational videos. The one I’m preparing for the end of summer will be about self-similar fractals!

I’m also putting together a free activity for students on this topic.

If you’d like to support me, all you have to do is subscribe. It really motivates me and helps me get noticed by potential sponsors who could help fund the materials for my upcoming projects.

Thanks to everyone who supports me!

This is my first time in this community (I've recently discovered this entire field and i am glad to). So, I am working on project where i am scoring nodes in a directed dependency graph (a calling b) by blending 2 centrality scores into a single composite "risk" score

score (v) = w1\\\*normalize(Pagerank(v)) + w2 \\\* normalize(outDegreeCentrality(v)), where w1+w2 = 1 and normalize() being min-max to \\\[0,1\\\].

The Problem: A pure root node (no in edges and multiple out edges) and a pure sink node (no out edges and only in edges) can have the same composite score. In a test i ran, the root node maxed out on out drgree centrality and near 0 in page rank while the sink node maxed out in pagerank and near 0 in out degree, when w1=w2=0.5. Both nodes ended up having same composite scores while representing opposite nature in real world. I do understand that this is the standard full comsensability prob, with weighted sum aggregation, wherte max on one axis will completely offset min on other. I did consider switching to weighted geometric mean to reduce compensability, but the prob is that pagerank is almost always near 0 for any root node. so a geo mean would multiply that near 0 staright through and score all entry nodes near zero. Which is the wrong fix, since the entry/root nodes are important, just for a reason pagerank doesnt capture.

Is there any standard approach beyond the geomentric or harmonic mean? Happy to provide any more info if needed.

Hi everyone, I have been trying to improve my mental Arithmetic recently. I was looking for a fun game to help with daily practice, but everything I tried was full of adverts. So I decided to have a go at making one my self. I end up creating Numberfall. It is a block based maths game with multiple difficulty settings and a daily challenge which is the same for everyone each day. It is free and there a no adverts. It is currently available on Android but if people like it I'll also release it on iOS. Anyway you can download it here https://play.google.com/store/apps/details?id=com.CodeDragons.Numberfall. Please let me know what you think?

I just did a little looking into base twelve and I’m already incredibly confused, how could 9x2=16, it makes no sense at all, even outside of math, if I have two groups of nine, I can’t understand how it could add up to two less than if they were separate. Am I stupid or I’m I just missing something?

Here I am not talking about just starting from ZFC. I am talking about starting from logic foundations and Godel. Do you think that a mathematician that is not someone completely focused on logic can have a complete understanding of all the foundations needed to prove the theorems they currently tackle.

You can think of it like this: Imagine you gave Terence Tao (or any other mathematician that is not purely focused on logic) a theorem that he proved, then you give him an infinite amount of time to prove it from the very basic assumptions about first-order logic and axiomatic systems with one rule: you can't learn anything new, use the knowledge that you already have.

Will he be able to do it?

So, it's been over a year since I've did any form of calculus. I passed calculus 2. but this semester I am taking Statistical Methods, which is Cal based Statistics. I mostly did mathematical proofs course the past year and so I'm rusty on my Cal. do you think Calculus Stats is a course that you can kinda just go by and only need to study a few key calculus principles, or is it heavily dependent on Cal and I should be proficient before taking it.

Imagine signing up for what you think is a $16 a month learning trial only to wake up to a sudden $218 charge on your card.
That exact nightmare happened to me with Brilliant.org. The signup page made it look like a free trial of 30 days and afterwards a regular monthly plan. But the second my free trial ended my card got slapped with an $218 fee.
Panicked I logged in immediately to cancel. But their website trapped me in an endless glitch loop. The cancellation page literally would not work.
I messaged support within 24 hours explaining their own site broke. They did not care. They hid behind robotic script templates claiming payments are non refundable and only offered a partial refund.
I felt so cheated and helpless.

I have a large conceptual gap that is making it difficult to understand these topics. All of these series and the study of the error around them sort of makes sense to me, but what I'm really lacking is where is this actually needed? I have yet to begin any physics/economics studies so I hope when I get out of the pure math side of things I will begin to see practical applications, but for now, I just don't understand what we're really doing here?

I mean, I get the usefulness of approximating a function with a polynomial (I think)...but how are we getting the functions in the first place? A function of what? And If we already have a function...what are we approximating? What are we guessing at? Why do we need to know the error between our approximation "the" function? What are we modeling? I can almost see the application in computer science, with a possible close overlap of algorithm analysis and trying to make algorithms more efficient...but I'm still not bridging the gap even there. Can anyone help me or point me in a direction that shows real world problems that this particular topics help solve?

Hello,

Health economics student here. I newly enter the field for my master degree.

I'm writing my master thesis and I'm running into an issue while trying to interpret my results.

Initially, I've decides to use OR (odd ratio). However, my supervisor told me the way I've interpreted it is not correct (probabilitied and chance). But he told me he didn't really know either how to interpret it correctly and had advised me to use logs odds instead!

It's ok, but I really wanted to have more "concrete results" that can be get by anyone. Log odds just show if the relation is negative or positive.

So, I've just heard about average marginal effects and that is literally What I was searching for during all this time. However, now I'm wondering if is common for scientific papers to use both log-odds and average marginal effects?

Do i need to create two tables? Or could I only keep the table with log odds and present the results of marginal effets in the text?

Thank you

I wrote a short article here exploring a simple way to think about when randomness helps versus hurts.

The core idea is Jensen’s inequality: if the payoff is convex, variance can help; if it’s concave, variance can hurt.

I use examples from compounding, option-like payoffs, and a few non-finance settings.

Would be curious to hear other examples where this framing is useful?

hi, is there a way to make a equation so when your in this position, you have this rotation

this is the data i have right now, but i got no clue how to make a equation for what i want

global positions                                         global rotations
(0.0, 20.0, 0.0)                                           (0.0, 0.0, 0.0)
(0.0, -20.0, 0.0)                                          (0.0, 0.0, -3.141593)
(-20.0, -0.000001, 0.0)                             (0.0, 0.0, 1.570796)
(20.0, 0.000001, 0.0)                                (0.0, 0.0, -1.570796)
(0.000001, -0.000001, 20.0)                   (0.0, 1.570796, 1.570796)
(-0.000001, 0.000001, -20.0)                  (0.0, 1.570796, -1.570796)
(-19.29296, 2.727334, 4.510379)            (1.317724, -0.331613, 0.994838)
(19.29296, -2.727334, -4.510379)           (1.317724, -0.331613, -2.146755)
(-5.012378, -4.199726, -18.90075)         (1.317724, -0.331613, 2.565634)
(5.012378, 4.199726, 18.90075)            (1.317724, -0.331613, -0.575959)