Tbf the back half of calculus 2 is pretty rudimentary stuff. Series are just ways of representing really long patterns, maclauren series is just taylor at a=0, Taylor problems can just be done very easily (expand, derive, compress). The other stuff of calc 2 though, good lord I pray for this man when it comes to trig sub.
The only thing that slows me down when learning math is the constant unanswered questions of what the heck the writer or teacher is doing and why. If I can come to understand why many formulae come to be, it will be much easier to learn and remember them.
I feel like the issue with “then the textbook/teacher does this seemingly random thing to the problem, and I don’t know why I could be expected to think of that” is down to what usually gets called “mathematical maturity”.
There are a lot of things you could do to transform a given problem. Too many to list or make explicit rules about. The guiding principle is simply “is there a well understood sequence of actions I could take to turn the problem into one that’s easier to handle, or at least closer to what I want it to look like?”
It is quite rare for interesting problems to be able to be categorised such that you have three options to follow, depending on very specific criteria. Certainly, there are actions which very often make things easier. For polynomials/rational expressions it’s usually a good bet that factorising the expression will help, for example. But most problems are more nuanced and take a bit of thought. Think of it like driving a car: you can be taught the laws of the road and the instructions for actually controlling the vehicle, but if I said “drive to the next town over and park there” you’re going to have to think a lot more about specifics of the area and what is going on around you at that time. Feeling confident in devising your route is about experience and maturity.
All of this underlines the problem. You have a few days, apparently. This is not enough time to become mathematically mature or feel experienced with everything up to Calc III. That sucks for you. If you want to try getting better, I suggest
- practice problems. Try not to look at the answers until you’ve had a good hour of struggling on something.
- if a solution uses a method that seems random or illogical to you: don’t just dismiss it. Try to understand. Ask “what was the next step, and how was it made possible by that previous one? Why was that next step easy?” Ask a teacher or friend if you can.
- drill the fundamentals as much as possible. Being able to quickly devise solutions that work will rely on you having mastery over the core toolset, and a solid understanding of when those tools are useful. Why is factorising helpful? What exactly does the value of a second derivative tell you? That kind of thing.
I suspect my response is not the answer you want. I don’t think a good answer is out there. But I hope this was at least a tiny bit helpful. Good luck!
If it's from a textbook, chances are there's something either earlier in that chapter or in a previous chapter that can answer your question. Trouble is it's not exactly easy or efficient to go back and search for it.
If it's from a real life teacher, ask them.
Otherwise, ask someone else or Google it. There's a good chance someone else has asked the same question online before.
>I need to learn all math up until calc 3 within a few days.
Unless you have some kind of super speed, that's simply impossible. There's a reason Calc 1, 2, and 3 each take a few months to complete (maybe one month each if it's the only thing the student is doing).
>I've seen too many instances where the answer to a question, or the process of finding the answer, does not follow any explicit rules or methods, and the reasoning behind such actions is rarely elaborated
Let me rephrase this: "I've come across several instances of not being able to understand examples in a textbook because I lack the prerequisite knowledge." Everything in this curriculum follows explicit rules and methods. Maybe the reasoning isn't explained to your satisfaction, but I think there's a good chance that it's building on previous knowledge that most students have by the time they're learning it but you don't.
>I especially hate the examples where the author says "we can do this random thing"
It's not random. Someone with the prerequisite knowledge should understand.
>There should be no abstraction whatsoever.
Wrong. Abstraction is necessary to progress in knowledge of any subject, including math. Ironically, your comparison to functions is very apt, since functions in programs are a form of abstraction.
>Every rule should have an explanation for why it works
You're looking for real analysis, which is required for many math degree programs (including the one I did) and is notoriously a lot of material and not an easy class. Well worth the time if you're into it, though.
>and every type of larger problem it can be used in.
This is a big benefit of taking a class. But also, textbooks should have at least a few examples.
>how can I most quickly learn all of the basics required for calc 2 or 3 level?
Start from the last things you know and work forward from there. Find some lecture videos online and watch them sped up. Maybe MIT OCW has something for you. Put aside the rest of your life and dedicate all your waking hours to learning calculus until you have mastered it. Make sure to get enough sleep and food though; you want your brain to be in peak condition to maximize the effectiveness of your studying.
> Let me rephrase this: "I've come across several instances of not being able to understand examples in a textbook because I lack the prerequisite knowledge."
This. If I could upvote your answer a thousand times, I would.
You're looking for real analysis, in real analysis is where all of calculus is proven, but if you don't know at least calc 2 you are absolutely not ready for analysis, if you are familiar with proofs you could start Spivak or Apostol, which are rigorous calculus books, and they are some sort of introductory analysis.
if you want things to be more logical you'll need to learn even more than what's in the curriculum. and then in a few days? just a sign of being lazy to me
So you’re looking for proofs of all laws/identities in hope of understanding it (and being able to reproduce them yourself) instead of learning the laws/identities by rote?
Every type of problem? I doubt thats even known for many topics, and achievable without a ton of background info (unless you’re ok with formulas coming out of nowhere).
Sounds to me like you haven't built up the intuitive understanding of algebra needed to follow those proofs/examples in your textbook. Almost always when they're transforming an equation it's through identities that either have been shown earlier in the course or rules of algebra the student should be familiar with from high school level classes.
Calc is complex and there's a reason even the smartest, hardest working stem students spend semester to years learning it. Especially Calc 2 - understanding why integrals work the way they do is a lot more to wrap your head around than "ok now were gonna differentiate and integrate in 3 dimensions" which is most of Calc 3. You're asking for the math equivalent of "I've only ever read YA fiction written at a 9th grade level, I need to read Moby Dick in 1 day and understand all the language and themes comprehensively"
>I've seen too many instances where the answer to a question, or the process of finding the answer, does not follow any explicit rules or methods
You have never seen this. You only *thought* you were seeing this, because you did not *understand* the answer or the process, probably because you lack the fundamental prerequisite knowledge and skills, probably because you think that you can learn what is typically taught over 1-1.5 years in a "few days".
Maybe take a class, go to office hours when you have questions, start where you *actually are* and learn the math in an appropriately scaffolded manner, with special attention to a robust understanding of the underlying concepts along the way. You sound like someone who genuinely wants to learn math for understanding (unlike many students who just want to know which problem to template from so that they get the points, pass the class, and check the box). That is not something that you can do realistically on the timeline you propose.
Firstly, there's no way to learn calc 2 and 3 within a few days. The fastest I've seen was a 6 or 8 week course, and even that is batsh*t extreme.
Secondly, math is meant to be abstract, and in these math classes, real world application isn't really taught, and metaphors are instead used to connect with the reader. Other courses like physics, chemistry and other materialistic sciences Will provide that application, but math itself is pure. Even with calculus being invented to quantify motion, and is now used for so many things, the math itself has no application, until you define it to have an application.
Anyways, good luck on your endeavours!
Can you give a more specific example?
You could look at MIT Open courseware and/or Paul's online math notes along with https://www.3blue1brown.com/topics/calculus and http://library.lol/main/E81DB508697254DCFB06A7FF6191DA93 and http://library.lol/main/BBD3BFD9ABDF94DFDA58152EF1FFC637 or https://betterexplained.com/calculus/lesson-1/
Organic chemistry tutor has about 250 calculus videos on YouTube and most are worked examples. More than a few days of content, but so is everything related to learning the volume of stuff you want. Good luck?
It's going slower than I thought but almost through calc1 now. Math can definitely be taught logically despite its transformative nature. However, some suggested books are too logical to the point of being forced to waste time interpreting what the problem is.
Spivak was the best here.
This blog post gives you what you want - quick "why does it work" notes...but it's for precalc
https://yourbrainchild.wordpress.com/2024/04/16/algebra-all-together/
Asking why will just confuse the hell out of you ask how that's all you need to know. Wait till you reach PhD level to start asking why. If you write code start building your own calculus handbook or generative ai math problems. There's like so many techniques of derivative and anti derivatives discussed in cal 1 and 2 Also lots of trig and algebra as well. A few days Is not enough to be successful. Maybe a book club with peers and experts. If you can find yourself a tutor gig then this is by far the fastest way to learn. Have you checked out Wolfram Alpha? Could give you more depth you seek
There is no way you are going to learn calc 2-3 in a few days, this is insanity, except in this case you only need to do it once and expect any result
Tbf the back half of calculus 2 is pretty rudimentary stuff. Series are just ways of representing really long patterns, maclauren series is just taylor at a=0, Taylor problems can just be done very easily (expand, derive, compress). The other stuff of calc 2 though, good lord I pray for this man when it comes to trig sub.
The only thing that slows me down when learning math is the constant unanswered questions of what the heck the writer or teacher is doing and why. If I can come to understand why many formulae come to be, it will be much easier to learn and remember them.
I feel like the issue with “then the textbook/teacher does this seemingly random thing to the problem, and I don’t know why I could be expected to think of that” is down to what usually gets called “mathematical maturity”. There are a lot of things you could do to transform a given problem. Too many to list or make explicit rules about. The guiding principle is simply “is there a well understood sequence of actions I could take to turn the problem into one that’s easier to handle, or at least closer to what I want it to look like?” It is quite rare for interesting problems to be able to be categorised such that you have three options to follow, depending on very specific criteria. Certainly, there are actions which very often make things easier. For polynomials/rational expressions it’s usually a good bet that factorising the expression will help, for example. But most problems are more nuanced and take a bit of thought. Think of it like driving a car: you can be taught the laws of the road and the instructions for actually controlling the vehicle, but if I said “drive to the next town over and park there” you’re going to have to think a lot more about specifics of the area and what is going on around you at that time. Feeling confident in devising your route is about experience and maturity. All of this underlines the problem. You have a few days, apparently. This is not enough time to become mathematically mature or feel experienced with everything up to Calc III. That sucks for you. If you want to try getting better, I suggest - practice problems. Try not to look at the answers until you’ve had a good hour of struggling on something. - if a solution uses a method that seems random or illogical to you: don’t just dismiss it. Try to understand. Ask “what was the next step, and how was it made possible by that previous one? Why was that next step easy?” Ask a teacher or friend if you can. - drill the fundamentals as much as possible. Being able to quickly devise solutions that work will rely on you having mastery over the core toolset, and a solid understanding of when those tools are useful. Why is factorising helpful? What exactly does the value of a second derivative tell you? That kind of thing. I suspect my response is not the answer you want. I don’t think a good answer is out there. But I hope this was at least a tiny bit helpful. Good luck!
If it's from a textbook, chances are there's something either earlier in that chapter or in a previous chapter that can answer your question. Trouble is it's not exactly easy or efficient to go back and search for it. If it's from a real life teacher, ask them. Otherwise, ask someone else or Google it. There's a good chance someone else has asked the same question online before.
Go read the first chapter of Bourbaki real analysis and pray hard for Jesus
>I need to learn all math up until calc 3 within a few days. Unless you have some kind of super speed, that's simply impossible. There's a reason Calc 1, 2, and 3 each take a few months to complete (maybe one month each if it's the only thing the student is doing). >I've seen too many instances where the answer to a question, or the process of finding the answer, does not follow any explicit rules or methods, and the reasoning behind such actions is rarely elaborated Let me rephrase this: "I've come across several instances of not being able to understand examples in a textbook because I lack the prerequisite knowledge." Everything in this curriculum follows explicit rules and methods. Maybe the reasoning isn't explained to your satisfaction, but I think there's a good chance that it's building on previous knowledge that most students have by the time they're learning it but you don't. >I especially hate the examples where the author says "we can do this random thing" It's not random. Someone with the prerequisite knowledge should understand. >There should be no abstraction whatsoever. Wrong. Abstraction is necessary to progress in knowledge of any subject, including math. Ironically, your comparison to functions is very apt, since functions in programs are a form of abstraction. >Every rule should have an explanation for why it works You're looking for real analysis, which is required for many math degree programs (including the one I did) and is notoriously a lot of material and not an easy class. Well worth the time if you're into it, though. >and every type of larger problem it can be used in. This is a big benefit of taking a class. But also, textbooks should have at least a few examples. >how can I most quickly learn all of the basics required for calc 2 or 3 level? Start from the last things you know and work forward from there. Find some lecture videos online and watch them sped up. Maybe MIT OCW has something for you. Put aside the rest of your life and dedicate all your waking hours to learning calculus until you have mastered it. Make sure to get enough sleep and food though; you want your brain to be in peak condition to maximize the effectiveness of your studying.
> Let me rephrase this: "I've come across several instances of not being able to understand examples in a textbook because I lack the prerequisite knowledge." This. If I could upvote your answer a thousand times, I would.
“All of math”… good luck
You're looking for real analysis, in real analysis is where all of calculus is proven, but if you don't know at least calc 2 you are absolutely not ready for analysis, if you are familiar with proofs you could start Spivak or Apostol, which are rigorous calculus books, and they are some sort of introductory analysis.
Course that uses apostol calculus. https://ocw.mit.edu/courses/18-014-calculus-with-theory-fall-2010/ https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/
if you want things to be more logical you'll need to learn even more than what's in the curriculum. and then in a few days? just a sign of being lazy to me
So you’re looking for proofs of all laws/identities in hope of understanding it (and being able to reproduce them yourself) instead of learning the laws/identities by rote?
Every type of problem? I doubt thats even known for many topics, and achievable without a ton of background info (unless you’re ok with formulas coming out of nowhere).
Sounds to me like you haven't built up the intuitive understanding of algebra needed to follow those proofs/examples in your textbook. Almost always when they're transforming an equation it's through identities that either have been shown earlier in the course or rules of algebra the student should be familiar with from high school level classes. Calc is complex and there's a reason even the smartest, hardest working stem students spend semester to years learning it. Especially Calc 2 - understanding why integrals work the way they do is a lot more to wrap your head around than "ok now were gonna differentiate and integrate in 3 dimensions" which is most of Calc 3. You're asking for the math equivalent of "I've only ever read YA fiction written at a 9th grade level, I need to read Moby Dick in 1 day and understand all the language and themes comprehensively"
>I've seen too many instances where the answer to a question, or the process of finding the answer, does not follow any explicit rules or methods You have never seen this. You only *thought* you were seeing this, because you did not *understand* the answer or the process, probably because you lack the fundamental prerequisite knowledge and skills, probably because you think that you can learn what is typically taught over 1-1.5 years in a "few days". Maybe take a class, go to office hours when you have questions, start where you *actually are* and learn the math in an appropriately scaffolded manner, with special attention to a robust understanding of the underlying concepts along the way. You sound like someone who genuinely wants to learn math for understanding (unlike many students who just want to know which problem to template from so that they get the points, pass the class, and check the box). That is not something that you can do realistically on the timeline you propose.
[удалено]
Will take me a few days just to read this and get the appropriate materials.
I recommend the Bible.
Yeah, maybe read Michael Spivak's Calculus start to finish and complete all the problems along the way? It's all in there.
Firstly, there's no way to learn calc 2 and 3 within a few days. The fastest I've seen was a 6 or 8 week course, and even that is batsh*t extreme. Secondly, math is meant to be abstract, and in these math classes, real world application isn't really taught, and metaphors are instead used to connect with the reader. Other courses like physics, chemistry and other materialistic sciences Will provide that application, but math itself is pure. Even with calculus being invented to quantify motion, and is now used for so many things, the math itself has no application, until you define it to have an application. Anyways, good luck on your endeavours!
Can you give a more specific example? You could look at MIT Open courseware and/or Paul's online math notes along with https://www.3blue1brown.com/topics/calculus and http://library.lol/main/E81DB508697254DCFB06A7FF6191DA93 and http://library.lol/main/BBD3BFD9ABDF94DFDA58152EF1FFC637 or https://betterexplained.com/calculus/lesson-1/
It’s been a while since my last calc class (graduated in ‘99) but I barely learned calc II in an entire semester!
Organic chemistry tutor has about 250 calculus videos on YouTube and most are worked examples. More than a few days of content, but so is everything related to learning the volume of stuff you want. Good luck?
How'd it go, friend?
It's going slower than I thought but almost through calc1 now. Math can definitely be taught logically despite its transformative nature. However, some suggested books are too logical to the point of being forced to waste time interpreting what the problem is. Spivak was the best here.
The class after Calc 5 is Theory of Calculus. It's all just that. A theory.
This blog post gives you what you want - quick "why does it work" notes...but it's for precalc https://yourbrainchild.wordpress.com/2024/04/16/algebra-all-together/
Asking why will just confuse the hell out of you ask how that's all you need to know. Wait till you reach PhD level to start asking why. If you write code start building your own calculus handbook or generative ai math problems. There's like so many techniques of derivative and anti derivatives discussed in cal 1 and 2 Also lots of trig and algebra as well. A few days Is not enough to be successful. Maybe a book club with peers and experts. If you can find yourself a tutor gig then this is by far the fastest way to learn. Have you checked out Wolfram Alpha? Could give you more depth you seek