IGNORING ALL COEFFICIENTS

How is all of this so connected?

Let's consider the homologous series of c, x, x^2, x^3, x^4... and so on: x^n where n belongs to a non negative integer only

Why is it that the graph for a particular expression not only explain the graph for the expression itself, but also the graph of literally *EVERY OTHER MEMBER IN THE SERIES*.

for example, the graph of x^3 (or simply x^3 itself, as graph is that thing itself) explain the graph of x^3 itself, but also x^4 (how f' graph explains a graph of f), and x^5 (how f'' graph explains for f)

How the graph for f' of x^3 is same as f'' of x^5 and f''''''' of x^10

How the graph for f' of x^8, f'' of x^9 and f of x^7 is all the same. A single graph is doing all of it

Yes, on paper, I know, and can prove this with my pen (I know the limit definition of derivative and what it is in terms of slope, in terms of rate of change in area/volume etc). But it still seems out of grasp on how this is possible.

[Only considering for the homologous series highlighted to keep it simple for now]

I made this in the shower about a year ago, and I don't really have a definitive answer for it.

In a random number generator that ranges from 0-Infinity, each number having equal odds, would it be possible to roll 1?

It is just as likely as the other numbers to be pulled, but that chance is infinitely small.
Any thoughts?

Hey everyone. I'm working on a browser extension that uses AI to help with math and solving problems, step-by-step explanations, that kind of thing.

Before I build too far, I want input from people who'd actually use it:

• What math do you struggle with most? (school homework, exam prep, uni-level?)
• What feature would make you keep it installed instead of deleting it after a week?
• Should I create an extension or make a website also?

 I want to build something people actually want.

in my case, i want to find how high i need n to get 2^n > 847 > 2^n-1, i can do it fine but i was wondering how i would do this with a higher number, trial and error would be really easy obv but is there something more principled?

Explanation:

This is for a game I'm playing with limited slots.

The two items do the following:

1. An infuser gives a 10% chance to give an item worth $50 on average (the default of getting the item is 0)
   
2. The magnifying glass increases the value of an acquired resource by 40%
   

I have 6 slots available, and I'm looking to see what combination of these two types of items gives the best estimated return. Anywhere from 6 infusers to 1 infuser & 5 magnifying glass is what I'm looking to see which gives the best return on average. 6 magnifying glasses does nothing because no item is gained without 1 infuser

Combination 1: 1 infuser & 5 magnifying glasses

-1 infuser with 5 magnifying glasses would be (.1)x((1.4)^5)x($50) = $26.8912
- item type infuser gives 10% of item, $50 is the outcome, and item type magnifying glass increases outcome by 40% for each 1
- Since there is only 1 infuser, there's only 2 outcomes with only 1 being more than 0. (There's obviously a 90% of getting nothing)

Combination 2: 2 infusers & 4 magnifying glasses

-2 infusers with 4 magnifying glasses (i think) would be
- outcome 1 (first infuser succeeds 2nd fails): (.1)x((1.4)^4)x($50) = 19.208
-outcome 2 (functionally identical) (.1)x((1.4)^4)x($50) = 19.208
- outcome 3 (both succeed) ((.1)^2)x((1.4)^4)x($50)x2 = 3.8416
- outcome 4 (both fail) 0
19.208x2+1.9208 = 42.2576

Problem:

I'm having issues with writing out and solving the other combinations, so I can't find out which combination gives the best average result.

Stats friends, show me how you organize your work when writing it out for assignments please! I've gone back to university at 32 years old and I haven't done math by hand since I was a kid. I feel like there must be a better way to organize this when doing it on a whiteboard/paper?

sSo, to be absolutely frank and upfront, I am not good at math and only have a GED. This may be a rock bottom basic question but I have to ask anyway.

Earlier this evening me and the wife were trying to find a movie and there was one that came out on 1947. I tried to do the math in my head for how many years ago that was and I like to make it even numbers to make it easier. Worth noting, it's the year my MIL was born.

Here's what I did:

1947 - 7 = 1940

1940 + 60 = 2000

2000 + 26 =2026

2026 + 7 = 2033

Which is obviously incorrect but I'm getting hung up on that 7 years; if I take it away do I not owe it back? My MIL is 79, so that can'tbe right. I figured I was doing something wrong (safe assumption) so I try it differently.

1947 + 3 = 1950

1950 + 50 = 2000

2000 + 26 - 3 = 2023

Respectfully, what in the frig am I doing wrong? I ran it through my calculator of the phone and I also looked up order of operations as while I know addition subtraction share priority so it's left to right but I just can't figure out where or why I'm going astray.

Again, this may be some ridiculously basic thing and I'm not to proud to admit I may be an idiot but I have to ask or it's going to drive me nuts. Please, if anybody could help me out would be greatly appreciated

Edit: I did the obvious thing and just did 2026 - 1947 and it's 79. But I'm still trying to figure out why my head math falls apart. Also, I should have read the room a bit better because you folks are clearly dealing with more advanced stuff than dipshit math I'm working with.

I was implementing an algorithm to calculate the Gauss-Legendre nodes/weights for arbitrary order and came across this solution:

https://rosettacode.org/wiki/Numerical_integration/Gauss-Legendre_Quadrature

which worked really well for me. Everything makes sense, except I can’t figure out why the initial guess to find the polynomial roots is x0 = cos( pi * (i-1/4) / ( n + 1/2 )). In practice Newton Raphson converges to 1e-16 tolerance in ~4 iterations on average, so it’s a really good guess.

Can anyone help me with a proof on this?

(I’m well out of school and this is not a homework assignment…)

I’m in linear algebra and I also just finished calc 3 last semester. I have seen Matrix product, Hadamard product, and dot product all using the same A•B look. What is the correct way to notate them?

Hello all

I've recently had to implement a pretty computationally intensive simulation and obviously only have so much computational power at my disposal.

The main reason for it being so computationally expensive is that several rather ugly integrals have to be evaluated, some of which end up being nested which seems to make things worse.

Now I recall from my numerical analysis courses that one can use interpolation to approximate a function. Interpolation also tends to be significantly less computationally expensive. As such I wanted to know whether one can replace the integration with interpolation under certain circumstances. I know these two concepts are very closely linked.

Take the following situation as an example (attached image):

Any help or clarification on this is greatly appreciated.

Thank you

For context, WGS-84 is the World Geodetic System 1984, used in all our modern GPS systems, and GCJ-02 is guó-cè-jú 2002, a proprietary coordinate system used in China. I know we have code to compute this nowadays, but if the only thing I had available was a scientific calculator, what must I do to approach this? I find the formulas too hard to understand and I only started Calculus II recently.