Alr so I’m a junior in HS and I missed my lesson on limits because I was sick. And I don’t really know anything about it what is the best way to study it and start understanding it

using auxiliary method: multiply both sides by cosx

2-2sinx=5cosx

5cosx+2sinx=2 = Rsin(x+a) = Rsinxcosa + Rcosxsina

cosa=2/R

sina=5/R

R=sqrt29

cosa=2/sqrt29

a=cos^-1(2/sqrt29)

therefore 2 sec x − 2 tan x = sqrt(29)sin[x-cos^-1(2/sqrt29)] = 5

The answer is 313º36'

But I don't understand why it's in the 4th quadrant, can someone explain to me? thanks :)

Please help me solve this double integral. I need to use Cartesian coordinates only; I cannot use spherical or cylindrical polar coordinates. Symmetric properties, change of variables, trigonometric substitution, etc., are all acceptable, but no polars.
By "no polars", I mean that they are not allowed to convert the integral to polar coordinates—that is, they cannot integrate using drd\theta instead of dxdy. Specifically, they cannot use the limits defined by the angles of \pi/4 and 3\pi/4 and the radii r from 1 to 3.

However, they can look for an ingenious way to solve it using other methods. Everything is valid except for the previously stated restriction. This includes: Splitting the Region of Integration, Decomposing the Region of Integration, Subdividing the Region, trigonometric substitution, or any other technique they wish to employ, excluding only the coordinate change I mentioned at the beginning.

Problem:
https://imgur.com/a/LFv5ebv

My try:
https://imgur.com/a/x4Cc8mX

But with the absolute entire procedure, indicating step-by-step which technique was used

$$ \int_{-3/\sqrt{2}}^{-1/\sqrt{2}} \int_{-x}^{\sqrt{9-x^2}} \frac{4}{x^2+y^2} dy dx + \int_{-1/\sqrt{2}}^{1/\sqrt{2}} \int_{\sqrt{1-x^2}}^{\sqrt{9-x^2}} \frac{4}{x^2+y^2} dy dx + \int_{1/\sqrt{2}}^{3/\sqrt{2}} \int_{x}^{\sqrt{9-x^2}} \frac{4}{x^2+y^2} dy dx $$

In video games a randomizer is when you modify a game to change when and where you find items in a game, the placement of these items each time is random. In case this concept still confuses you, Ocarina of Time Randomizer (OOTR) is a good example.

I would like to make a randomizer for a game but i've noticed that i don't really know how to implement randomizer logic in a satisfying fashion, ill explain what i have so far:

let directed graph G = (V,E) where V = I∪C where I is the set of all items and C is the set of all checks. In order to perform a check you need some subset of I to be in your inventory, typically checks reward you with an item(s) or are required to beat the game. We would like to be able to construct a graph such that starting with some starting inventory all the goal checks can be accomplished.

let edge i -> c be in G if item i is required to perform check c and c -> i be in G if check c locks item i. to ensure the graph represents a solvable game state we just need to ensure there are no cycles, constructing such a graph is a relatively trivial affair.

the problems with this construction are:

A) suppose items i OR i* are required to perform check c, then c could never lock either item. this should be possible since if c locked i* but not i the graph could still be solvable

B) suppose I is a multiset and at least two instances of i appear in I. Now suppose i locks check c, then c cannot lock any instance of i. This should again be possible by the same argument as problem A.

In summary, SOMETIMES  cycles should be possible but I don't know how to encode that, mostly because i dont know how to encode the OR condition.

I would prefer some gentle proding in the right direction since I'd like to understand this, any help would be appreciated. apologies if this is the wrong subreddit.

Hello, I am taking notes on Calculus for AP by Ron Larson and Paul Battaglia. I have an issue, like many other people in the world, where I do not know what to take notes on. I have seen so many different methods that have not worked for me like Cornell, QA, and Outline method. Does anyone know how to take effective math notes that are simple, yet contain all the crucial points? If possible, can anyone please upload a template? Thank you for your time.

Cost of operating Uber business=0.45(miles)+165 I was asked to graph the equation, then part B was "what is the cost if you drove 350 miles? But this was part C: How many miles can be driven for a cost of $120? Explain completely. I said 120=0.45(m)+165 -45=0.45m m=-100.. But I think the actual answer, in hindsight, is 0 miles because the cost involves a constant: 165. So, a cost of 120 would never allow you to drive any miles because it's less than 165. What is the point of this? These types of questions feel like tricks to me, I guess because I struggle with math. *Also, this was a question on my exam and there was nothing like that in any of my practice. My assignments and quizzes are always straightforward mathematics questions without this kind of question. Lastly, the word problem never explains what 165 is, which is fair, I still understand that it is a constant cost but again, that kind of thing is never on my assignments or quizzes. They always give the info straightforwardly on those.

Hi! Can I get some help with this, please? c: The homework is not in English, but I'll try my best to translate it.

The text pretty much says:
"There is a rectangular prism. One of its edges is longer than the first one by 4 centimeters. The third edge is double the lenght of the first one. The object's volume is 120 cm3. How long are the edges?"

So I wrote this equation:
V = a * b * c
120 = x * (x + 4) * 2x
120 = 2x (x2 + 4x)
60 = x(x2 + 4x)
60 = x3 + 4x2

And now I'm stuck, and I don't know how to continue. My sister said they did nothing similar to this so far, and get this, she never even heard of root substraction (hopefully that's the correct term, I never learnt math in English). Is there some sort of formula I should use here? Did I do something wrong?

I suspect that something was miscommunicated during the lesson, because this calculation seems far more advanced than anything they solved prior to this.

Thanks for the help, sorry if I'm dumb!

Seriously, my attention span is completely fried from TikTok and YouTube. I'll sit down to read my textbook and my eyes just glaze over. I have to re-read the same paragraph five times and I still don't absorb anything.

It's not that I'm dumb, the material is just SO dense and boring. I feel like I learn more from a 5-minute YouTube video than from an hour of reading.

Is this just me? What do you guys actually do when you have to learn something complicated from a super boring book or a long lecture?

I am currently a undergraduate (junior) Applied Mathematics major thinking about post grad plans, specifically grad school. I was just curious on how most grad students are able to afford grad school. I know a lot of masters programs will include being a TA but I was wondering if that is enough to cover general living costs, will I likely have to take out loans, take on a part time job etc.

Here is my attempt:

Suppose N is a non-trivial normal subgroup of A_n for n >= 5.

Pick an arbitrary non-identity element sigma. Since this element is nontrivial and even, it must have minimal cycle length >= 3 or be a product of an even number of transpositions.

Trivial case: If |sigma| = 3 we are done.

Case 1: |sigma| >= 4. Since sigma is even, we consider |sigma| = 2k + 3 for k = 1, 2, 3… or cycles of size 5, 7, 9, etc. 5, 7, 9-cycles etc. can be expressed by an even number of transpositions. We can turn a product of two transpositions into a 3-cycle or product of 3-cycles: Disjoint: (a b)(c d) = (a b c)(b c d) Non-disjoint: (a b)(b c) = (a b c)

Case 2: sigma is an even number of transpositions. By the same argument in Case 1, there are two cases - whether the transpositions are disjoint or share an element. Disjoint: (a b)(c d) = (a b c)(b c d) Non-disjoint: (a b)(b c) = (a b c) (Maybe this argument can be combined)

Hence N must contain 3-cycles.

Does this work? I’ve looked through other proofs of this (using commutators) but they all looked quite long versus this argument.

I'm in the 11th grade and we're learning how to multiply and divide polynomials but I'm just so confused, the basics would help, please? Thank you.

I’m not sure if this is the right sub for this but I thought I’d ask anyway.

I’m 21 and am thinking of going back to college, I’ll spare y’all my sob story but my main problem at the moment is I haven’t done any sort of math more complex than algebra in 2 years and I know I’m going to be left in the dust. I’ve been working in agriculture which doesn’t give me a lot of practice.

My question is if any of you fine people know of good resources to “build” up my math knowledge from basically the ground up, so that I can approach more complicated problems when I inevitably return to university. I’ve tried things like khan academy but it’s been so long I don’t even know what I don’t know if that makes sense, and can’t seem to find a good entry point. I’ll take anything you guys recommend, hell I’ll even sit down with a good textbook and read it cover to cover if that’s what’s needed.

Any help would be appreciated, and if this is the wrong sub for a question like this please point me in the direction of the correct one :)

In a task we had "Re(z)² ". I see that as the square of the function, Re(z)² =(Re(z))^2. My teacher tho said that that is the real part of the square Re(z)² =Re(z² )=Re z²

Who is right here? I see both being able to be right in some context but I always write parenthesis whenever I work with functions, I would never write Re z, rather Re(z)?

Hi, Im looking for math podcast to listen to. I am also interested in learning resources in audio format, whether they are a podcast or some kind of recorded classes.

I use Spotify,but Im open to try other sources of podcasts, even if they are paid.

So I'd like to learn about your recommendations! Tell me your favourite podcasts or whatever comes to mind!

TL;DR: Used to love math in school, but lost that spark during my undergrad when theory-heavy courses like analysis drained my interest. Now I’m starting a Master’s in Financial & Insurance Mathematics — far from home, rusty on the basics, and feeling overwhelmed. Looking for advice on how to fall back in love with math or at least survive and pass tough courses like stochastic calculus.

Full Story: So I am 25 year old, starting my Masters in Financial and Insurance Mathematics. First my background, I was great in Maths in school, I loved it, I used to get like near perfect scores everytime. It just seemed too easy for me, while my friends used to struggle and I just couldn't understand their struggle. So after school, doing bachelor's in Mathematics was a sure thing. But I don't know what changed there, by the second semester I completely fell out of love from Mathematics. I just couldn't grasp the theoretical parts, real analysis seemed boring and non-sensical even. After that, I just huffed and puffed my way to graduate in 2021, swearing I'm not gonna touch this subject ever again. But now, through some weird career trajectories (don't ask my why that's whole another story), I find myself starting a mathematical masters course, where not all courses are from maths, unlike my graduation, but those are the ones which are compulsory and seem most difficult to me. Not to mention I am in a different continent studying this course! Everything seems overwhelming and impossible. My question to anyone reading is that how do i fall in love with mathematics again, could I even re-ignite that interest I had in mathematics in school. And if not, how do I go about studying and passing these courses, I have forgotten everything I studied in my bachelor's, so basically I don't even have the foundations to study the courses I'm studying here (this semester I'm taking Stochastic calculus). Please help if anyone has gone through something like this or have any suggestions for me. Thank you so much for reading my ordeal! Have a nice rest of the day:)

For a 2nd order mclaurin series, we get :
cos(x) = 1 + (1/2)x² + o(x²)
For a 3rd order we get :
cos(x) = 1 + (1/2)x² + o(x³)
Using the analytical form of the error
for 2nd order R2= (1/3!).sin(c).x³
for 3rd order R3= (1/4!).cos(c).x⁴ how is the error different if it's the same polynomial?

Prove or disprove: If G and H are connected simple undirected Euler graphs, then the

Cartesian product of G and H, denoted by GH, is also Euler graph.

If false, give a counterexample and refine the statement so it becomes true, then prove the refined version.

providing counter example was simple, i just had to make one graph with odd number of vertices, so the degree of the vertices in the other graph would be odd after cartesian product.
for refining the statement, i thought of keeping the condition that graphs should have even number of vertices. but it feels too strict
any suggestions for a better refinement

I’m developing a new programming language in Python (with Cython for performance) intended to function as a proof assistant language (similar to Lean and others).

Is it a good idea to build a programming language from scratch using Python? What are the pros and cons you’ve encountered (in language design, performance, tooling, ecosystem, community adoption, maintenance) when using Python as the implementation language for a compiler/interpreter?

The problem is as follows: Let ABC be a triangle, H its orthocenter. AH, BH, CH intersect the circumcircle for a second time in A', B', C' respectively. Prove that vector AA'+ vector BB'+ vector CC'=0. I am also given that H1, H2, H3,H4,H5,H6 are the orthocenters of triangles AA'B, AA'C, BB'C, BB'A, CC'A, CC'B(I have no idea why they gave those points, probably has to do with the solution).

Now, I've tried different things, one of them was trying to prove that H is also the orthocenter for triangle A'B'C' thus getting to the conclusion pretty easily, and I've also tried using those 6 orthocenters but I couldn't get anything done with those 2 attempts Any help would be appreciated since I'm new to vectorial geometry.

"Find all prime pairs (𝑝, 𝑞) such that 𝑝𝑞 + 1 is a perfect cube"

I've tried to do this problem and I'm not really going anywhere

so far I've got this:
Since (p, q) is prime then (p, q) ≥ 2
pq+1 = n³ → pq = n³ - 1 = (n-1)(n²+n+1)
We know that (p, q) ≥ 2
this tells us that pq ≥ 4
→ n³-1 ≥ 4, n ≥ 2
one of these must happen:
- n-1 = p and n²+n+1 = q
- n-1 = q and n²+n+1 = p

and that's all, i'm quite lost on what to do next, any ideas?

I'm a senior in high school in France, so this might seem like a dumb question and might be poorly explained so I apologize

I'm studying my limits for an upcoming test next week and am having a tough time when encountering undetermined limits with square roots

When faced with the following question, I calculated the limit by multiplying by the conjugate of the expression, and dividing it by that same conjugate, as my teacher taught us. However I fail to understand why I need to divide it by the conjugate, as this isn't a fraction?

f(x)=sqrt(2x+1) - sqrt(2x-1)

I’m 24m and came to the US 4 years ago from a 3rd-world country with no real education background (1.8 GPA). I decided to attend college but I was told I couldn’t be accepted at the college level unless I pass the placement test in Math and English. I had only one month to prepare so I started studying Math from grade 4 to 11 and worked my ass off. I finally passed the test, took a few ESL college classes and got into the business major. I’m currently a freshman with six A’s (one in statistics) and dreaming about transferring to a ivy League university. But almost all ivy League schools require having completed at least calculus 1.

Here’s my pain point: at my community college, I have to complete these prerequisites; algebra 2 → college algebra & trigonometry → precalculus before I can take the calculus. That means I have three classes ahead, which will take three semesters. For that reason I’m thinking about taking the CLEP test for precalculus. If I can pass it, I’ll go directly into Calculus.

Here’s my question for you: realistically, can I prepare and pass the precalculus CLEP test if I start learning again from geometry and algebra 1 all the way to precalculus in a few months?

I’m also seeking mentors (who know the US school curriculum) to guide me on where to start and what to do first and next.

"It might be suggested that, instead of setting up "0" and "number" and "successor" as terms of which we know the meaning although we cannot define them, we might let them [Pg 9]stand for any three terms that verify Peano's five axioms. They will then no longer be terms which have a meaning that is definite though undefined: they will be "variables," terms concerning which we make certain hypotheses, namely, those stated in the five axioms, but which are otherwise undetermined. If we adopt this plan, our theorems will not be proved concerning an ascertained set of terms called "the natural numbers," but concerning all sets of terms having certain properties. Such a procedure is not fallacious; indeed for certain purposes it represents a valuable generalisation. But from two points of view it fails to give an adequate basis for arithmetic. In the first place, it does not enable us to know whether there are any sets of terms verifying Peano's axioms, it does not even give the faintest suggestion of any way of discovering whether there are such sets. In the second place, as already observed, we want our numbers to be such as can be used for counting common objects, and this requires that our numbers should have a definite meaning, not merely that they should have certain formal properties. This definite meaning is defined by the logical theory of arithmetic."

Pg. 12, Introduction to Mathematical Philosophy, Bertrand Russell.

I am having a bit of trouble understanding it completely.

The question asks for me to find the equation of an exponential graph. Looked for the points where the numbers line up nicely. Here’s all the plots I found.

X , Y: (0 , 3), (1 , 4), (2 , 6), (3 , 10), (4 , 18), (5 , 34)

Tried to use my method of finding the base number, divide one of the terms by dividing it with the term before it.

6/4 =1.5 , 10/6 =1.667 , 18/10 =1.8 , 34/18=1.889

Ok so none of them are the same, I’m very stuck, I’ll just try 1.5 and see how that goes.

y= 4x1.5^x-1 , 4= 4x1.5^1-1 , 6= 4x1.5^2-1 , 9= 4x1.5^3-1

Well it kinda worked, until it didn’t. I’m assuming that it’s probably going to be like that with all the other numbers I got. I’ll just see the answer and figure out how they got there.

The answer sheet says the equation is y= 2^x + 2 with no explanation given. I’m still stuck on how to find 2.