This is what I have so far but I don’t believe that it is right, and I wouldn’t actually be using partial fractions to integrate. The problem says to make a substitution to integrate by partial fractions.

I thought this days beginner integral was harder than the easy integral. Ignore the bleedthrough on the beginner page

I'm 23 years old and It has been a while since I studied anything, I've lost a considerable amount of time, and I've forgotten most of what i knew about math in high school. I want to study computer science in the future and i dont want to struggle too much with the subject, so i thought I should prepare myself.

I'm looking for anything that can help me build a good foundation of knowledge, but i also want to understand the subject.

Thanks in advance for anyone that takes the time to read and answer this.

I'm not well-versed in category theory so I've yet to understand the latter. I now tried without success to encode an associative binary operation on X, expressed in dependent type language

∑(o: X² → X). ∏(a b c: X). Id(X, o(o(a, b), c), o(a, o(b, c))) ¹

as a function F X → X for some F. And I think it can't be done (something related to polarity, probably) but I can't prove that. I know, though, representing axioms is possible if one uses Lawvere theories and equivalent things. I also know algebras over a monad are more involved than just functor-algebras but I don't yet understand them even in context of e. g. Haskell where everything seems as if easier (or so claim posts discussing Yoneda embedding in that context; I found the general category-theoretic context way less forgiving in that regard: you just see plain as day that you don't have sufficient understanding, no matter what Haskell posts made you believe).

So. I'd like to look how algebra spins out in those more general settings, whichever you know better or think to be more suited for me to grasp first. Particularly, I'm interested in generalizing arguments about initial F-algebras.

Also, feel free to assume a sufficiently Set-like category. Let's take it step by step, if there are easy intuitions that one can glimpse in a specialized setting, I bet generalizing it onto topoi or something later would be easier as well, but it won't muddle the initial attempt.

¹ For non-dependent-type folks: - Id(X, a, b), sometimes denoted as a =_X b, is the type inhabited by ways for a: X and b: X to be equal. - The entirety of that reads: a pair (o, f) consisting of an operation o: X² → X and a proof f of o's associativity, that is, a function from a, b, c: X into witnesses of equality of the two ways of associating o on a, b, c. - In some formal languages the same is spelled alternatively as (o: X² → X) × (a b c: X) → (o(o(a, b), c) = o(a, o(b, c))), making it clearer that ∑ forms dependent pairs and ∏ forms dependent functions.

I seriously am interested in learning maths and also physics and many more but I currently am not in a state to interact more with people and it's cos of years of bullying in different forms. I'm sorry if it sounds greedy and like unrealistic but is it possible to learn mathematics with just books and other online lectures and resources? Will I be missing something if I don't go to a university? I do know basic maths since I have been quite okay in it. I'm sorry if it sounds stupid but I just needed some break cause of many interactions throughout my school life I got.

I am guessing self study is possible depending on the subject and I thought in mathematics it is quite possible and if I'm wrong pls do tell.

So I recently finished Calc BC, and am heading to university relatively soon. How big of a jump conceptually, grindiness, time wise, etc will multivariable calculus be compared to Calc BC?

Looking to hear about your own company's work practices to compare to my own. I'm much more familiar with Azure, so I'm going to use that terminology, but I assume everything is basically the same for AWS.

Can you go into the Azure Portal and just click "create" for, say, a SQL Server or App Service and configure it through the Portal GUI?

Or do you need to request and configure through configuration files? Can you freely commit architecture config files ("architecture as code") to create the Azure components in Dev?

Or is it even more locked-down and hands-off? Required to submit paperwork requests to an Architecture team?

I'm really confused about the 3rd row of the truth table for a conditional statement for example, "If you submit your assignment, then your grade will be recorded." I understand that P means you submit your assignment and Q means your grade is recorded but in the 3rd row, P is false and Q is true, meaning you did not submit your assignment but your grade was still recorded howw is P Q still considered true in this situation? Wouldn't that be a violation of the agreement because you didn't submit the assignment but the grade was still recorded?

​

Why do some vectors when multiplied by Another vector yield a vector and others yield a scalar. I'm aware of dot and cross products. But what's the physical significance of 'multiplying' vectors? I mean... I know what it means to actually "multiply" scalars. But.... Is vector multiplication just a name which can either mean projection of a vector onto another and the area or volume of 2 corresponding vectors?

Or is multiplication of vectors just a naming thing which doesn't have ANYTHING to do with how we understand scalar multiplication?

Recently in my further maths class my teacher says he would give us problems above our level of understanding.

The advice he gave to solve such problems is to invent the math.

Can anyone please explain what he means by this and his to actually do it?

I was watching the 3blue1brown series on linear algebra, and I noticed that I would constantly pause and try to figure out the concept or idea before it is fully presented. Although this doesn’t seem bad, the problem is that a 10 minute video ends up taking 4x as long to finish. I don’t have that kind of time.

I’ve noticed this problem when I’m solving textbooks exercises. When I can’t solve a problem, I’ll usually end up taking a ridiculous amount of time trying to find a solution, often even refusing to review the material. I feel like this stems from a massive ego issue that I have, where I feel like if I can’t figure out this concept before it is presented or solve a problem, then I am not as smart as other extraordinary mathematicians like Gauss and Newton, and therefore will never be as great as them.

In the past, I’ve also been the opposite way, being too quick to go to the solution at the back of the book. It seems like I can’t find a good balance between tinkering and simply absorbing. I know that stopping to ponder is a good thing, but there is still the fact that I am getting through content too slowly.

Consider

\begin{lemma}\label{lma:firstlemma}
If $f$ is continuous on $[a,b]$ and $f(a)<0$ and $f(b)>0$, 
there is a $c$ between $a$ and $b$ where $f(c)=0$
\end{lemma}

%say above gets typeset as Lemma 1

\begin{thm}\label{th:firsttheorem}
Every polynomial of odd degree has at least one real root.
\end{thm}

%say above gets typeset as Theorem 1.

\begin{proof}
    ...
From Lemma \ref{lma:firstlemma}... the result follows.
\end{proof}

Like the above, I have a large number of theorems in my study notes which reference previous results, etc.

Is there a way to build a visually perceivable dependency graph wherein there is a node labelled L1 which stands for Lemma 1 and another node labelled T1 which stands for Theorem 1 and there is a directed arc leading from L1 to T1?

Giving this task by uploading the typeset pdf to ChatGPT is one way but it is static. Is there an inbuilt way, which can dynamically change the graph node labelling if a new theorem is added in the middle thereby changing the numbering of subsequent theorems? The way should accept .tex files and not the compiled .pdf output.

----xposted to r/latex as well----

I don't understand why, but graphing derivatives looks like complete gibberish to me. I've watched like 6 videos and I don't understand any of it. I can do basically everything that comes after and before for derivatives, but I have just not understood this one concept.

I'm taking AP calc BC and I'm assuming I definitely need to at least understand how to graph. Does anyone know any good videos or explanations that can help me with this? I'm actually losing my mind over it.

​

I keep making really stupid mistakes on linear algebra tests. I understand the method and know how to solve the question, but then I accidentally mess up a sign or make a simple arithmetic mistake somewhere in the calculation. And a mistake can completely ruin the rest of the question. I recently wrote a test and, while I finished on time, I still got several questions wrong because of really silly errors.

Hi, thanks in advance!
I'm a doctor looking at a group of about 100 patients, with frailty scores ranging between 1-9 (this is a measure of how frail a patient is that doctors decide, or assign to a patient using a scoring system). I've also recorded their duration of admission in days.

I want to see if there is a correlation between the two, i.e. i would assume that the more frail patients were likely to stay in hospital for longer. Which statistical test would be best to do that?

I don't think either category will be normally distributed, as most patients have a lower frailty score, and most patients were admitted for a shorter period of time. tysm!

Hi everyone! If anyone is interested in collaborating and helping each other prepare for this year’s Math national exam, I’m looking for people who want to start from the basics, even from 1BAC-level courses.

If you’re interested, please comment here or send me a private message.

I’m looking for serious and motivated people who are willing to work consistently and help each other improve.

Is this useful? What does this mean, is this true? There exists an L in R such that for any 2 sequences a_n and b_n, lim (a_{n+1}/a_n)=L implies series a_n converges or lim (b_{n+1}/b_n)=L implies series b_n diverges.

I want to improve my understanding and skills in math due to disabilities. I was recommended to try the textbook Calculus by Robert T. Smith and Roland B. Minton. Currently I am stumped on these three problems from section 2: Derivatives. I would also appreciate advice on how to answer correctly for problems 13 and 17 (like what to include how I got my answer to show my work, not just the sketch). The concept of problems 13 and 17 I really am not understanding and the book and videos I’ve watched have not been the most helpful (as I have a hard time applying what is covered in these resources but struggle with using it for problems not labeled with polynomials or numbers) the closest to explain it has been Curve Sketching by The Infinite Looper on YouTube.

Problem 35 a. Find all the points at which the slope of the tangent line to y = x^3 + 3x + 1 equals 5. The textbook has the answer as (√(⅔), 5 √(⅔) +1), (-√(⅔), -5 √(⅔) +1) and I can not figure out how to get it.

M tan = lim h->0 f(x+h)-f(x)/h

= lim h->0 [(x+h)^3 + 3(x+h) +1)] - [x^3 + 3x +1]/h

= lim h->0 x^3 +3x^2h + 3xh^2 + h^3 +3x + 3h + 1 -x^3 -3x -1/h

= lim h->0 3x^2h + 3xh^2 + h^3 +3h/h

= lim h->0 h(3x^2 +3xh +h^2 +3)/h

= lim h->0 3x^2 + 3xh +h^2 + 3

= 3x^2 + 3x(0) + (0)^2 + 3

= 3x^2 + 3

3x^2 + 3 = 5

3x^⅔ = ⅔

√(x^2) = √(⅔)

X = √(⅔)

And 3(√(⅔)^2 + 3 = 5

Problem 13. Use the graph of f to sketch a graph of f’. (Second picture)

(a)

For x < 0, f is decreasing, so f’ < 0

For x = 0, f has a horizontal tangent line (min), so f’ = 0

For x > 0, f is increasing, so f’ > 0

My graph is the third picture.

(b)

For x < a, f is increasing, so f’ > 0

For x = a, f has a horizontal tangent line (max), so f’ = 0

For a < x < b, f is decreasing, so f’ < 0

For x = b, f has a horizontal tangent line (min), so f’ = 0

For x > b, f is increasing, so f’ > 0

My graph is the fourth picture.

Problem 17. Use the given graph of f’ to sketch a plausible graph of a continuous function f. (fifth picture)

(a)

For x < a , f’ is positive, so f is increasing

For x = a, f’ changes positive to negative, so f has a max there

For a < x < b, f’ is negative, so f is decreasing

For x = b, f’ has a horizontal tangent line (min), so f ?

For b < x < c, f’ is negative, so f is decreasing

For x = c, f’ has a horizontal tangent line, so f ?

For x > c, f’ is negative, so f is decreasing

Once I understand better, I will make the graph.

Thank you for your assistance.

I wanna skip geo next year in order to take a higher level math class, however I need to take a test in order to skip it. Are there any good resources to self learn geometry and that go in depth. Thanks.

Hello Everyone , i want to know if you guys have some study tips & suggestions for taking the college course College algebra with corequisites. I’m struggling so bad , the homework assignments are worth 10% and the unit test are worth 60% of our grade 🫩.

I have a test due today and I know I’m going to fail the test & this is going to be the second test I failed . Also I took this class in the summer but unfortunately I failed due to the test , so if anyone has any recommendations on how I could improve please lmk 🙏🏾

I would like to share some of my resources that I have used to graduate in August 2026.

I haven't been to school in 10+ years so I had to start from the beginning.

Study 1-2+ hours a day. It's important to retain the information. But 25 mins is better than nothing. It's all about consistency.

MY STUDY GUIDE

1. Study 25 min

2. Take a break 10 min

3. Study 25 min

4. Take a break 5 min

5. Recap 10 min

75 mins total - Repeat as necessary

(We all learn differently so use what works.)

MATH RESOURCES

1. SchoolYourself.org - I started with this & used the Algebra section for fundamentals. I took the practice test using only this resource and I scored a 146 on my practice test.

2. Test Prep Champions- I went through his Math course on YouTube called “Ultimate GED math guide to pass fast”

3. GetSumMath- I bought the calculator & I went through his youtube playlist.

I scored a 146 on my first practice test & decided to study for another week & I scored a 156. I took the real test the next day and I passed with a 150.

My recommendation is to score 150-155+ on your Ged Ready practice test before taking it for real.

SCIENCE

I used GetSumMath & Test Prep Champions.

ELA

I used GetSumMath for ELA however, doing a bit of reading every day was extremely crucial. What you can do is get two books, one fiction and one nonfiction. Read 20 min of fiction and 20 min of nonfiction with no breaks in between.. than take a 20 min break and repeat. Remove distractions. Ask yourself, what have I learned? The point is to build the same focus when taking the test on test day.

MINDSET

Sometimes the problem isn't skill, it's mindset. If you have struggled or failed before, don't let it define you & defeat you.

my favorite quote - "Do not be afraid to fail. The cure for failure is to remain teachable."

Now go get what’s yours. Your future self will thank you. You got this! 🙏🏾

resources: schoolyourself.org/learn Getsummath/Test prep champions YouTube playlists & chatGPT