What should I learn after learning Calculus?

[James M]

Hello.

I will be able to finish Calculus I, II, III, Linear Algebra, and Differential Equations, before I go to high school, yes, I am still in middle school.
So, what should I learn after I finish all of those?
Some people told me to learn probability/statistics, or some even more advanced math-topology, abstract algebra, discrete mathematics(since I am interested in Computer Science), etc..
But since I have 4 years(or possibly less) of high school science education ahead of me, should I self teach science also- biology, chemistry, and physics?
Do you think it's a good idea for me to self teach all of those before I move on to more advanced college level subjects(like Electrical engineering and Computer science)?

Thanks in advance.

 

[Suraj M]

Hi James, welcome to PF
I would suggest you to study what YOU want to study, not just high school subjects,

since I am interested in Computer Science

Then why don't you start from there, write codes,(if you haven't) go to a more advanced level, work towards, primarily Comp. sc.. if you like it so much.
But don't let it come in the way of your regular subjects and syllabus.
Hope this helps.

 

[Fredrik]

One option is to start with classical mechanics and then move on to special relativity and/or electromagnetism. These are pretty cool subjects that will also be covered (to some extent) in high school.

Don't forget to study the other great mystery of the universe:

 

[Suraj M]

Don't forget to study the other great mystery of the universe:

He's still in middle school. -_- hold it for a few years. Science needs him.

 

[Vanadium 50]

I will be able to finish Calculus I, II, III, Linear Algebra, and Differential Equations, before I go to high school, yes, I am still in middle school.
So, what should I learn after I finish all of those?

Is there any reason you need to know now, as opposed to when you've finished them?

 

[Cake]

AP Physics C has two courses which teach Physics at an intro college level. You'll probably want to be exposed to those before you jump into stuff that requires Diff. Eq. and Vector Calculus or any Engineering courses that require knowledge of those subjects. Also, take the opportunity to go to any early college programs your locale offers and enroll in some courses that interest you. I bet a bunch of places will be clamoring to get you in their classrooms if you know Differential Equations before 9th grade. Take advantage of that.

 

[ZapperZ]

Hello.

I will be able to finish Calculus I, II, III, Linear Algebra, and Differential Equations, before I go to high school, yes, I am still in middle school.
So, what should I learn after I finish all of those?

OK, I'll jump in.

When you say that you are "...able to finish...", what exactly does that mean? Surely, you didn't study this in school. Did you study this on your own? How? And how were you able to evaluate that you have actually understood and capable of, say, testing out of these subjects when you reach college?

So before I offer my suggestion on what you might want to do, I'd rather know exactly what you actually know, rather than what you think you know. After all, if this starting point isn't clearly established and verified, anything else coming out of it is moot and irrelevant.

Zz.

 

[jtbell]

So, what should I learn after I finish all of those?

It depends on what you're generally interested in: Pure math? Physics? Something else?

Also, I hope you're not doing this because of pressure from yourself or others to "get ahead" in your academic career. You're already way ahead of almost everybody else your age. There's no need at this point in your career to study anything in particular at the level that you've reached, so you should focus on things that you enjoy, and don't let them distract you from the other stuff that you're studying in school, and other aspects of life.

 

[QuantumCurt]

You're in middle school and you plan to complete calculus I, II, and III, linear algebra, and differential equations prior to starting high school? As some other posters have mentioned, how exactly are you going to complete this? What year are you in middle school? If you're in 5th grade and planning to self study these subjects before high school, then you're likely counting the chickens before the eggs hatch. How much of the material for these courses have you already worked through, and how has your understanding of the material been evaluated?

 

[James M]

OK, I'll jump in.

When you say that you are "...able to finish...", what exactly does that mean? Surely, you didn't study this in school. Did you study this on your own? How? And how were you able to evaluate that you have actually understood and capable of, say, testing out of these subjects when you reach college?

So before I offer my suggestion on what you might want to do, I'd rather know exactly what you actually know, rather than what you think you know. After all, if this starting point isn't clearly established and verified, anything else coming out of it is moot and irrelevant.

Zz.

I apologize for not establishing the starting point clearly.
According to my own predictions and my pace of learning, I will finish Calculus 1,2,3, and Diff. Eq. by the end of summer break.(All self-taught)
I am in 8th grade, and am currently on Pre-Calculus/Trigonometry right now, but I know that I can finish Calculus(1,2,3) at least before high school starts, which is August.

 

[James M]

You're in middle school and you plan to complete calculus I, II, and III, linear algebra, and differential equations prior to starting high school? As some other posters have mentioned, how exactly are you going to complete this? What year are you in middle school? If you're in 5th grade and planning to self study these subjects before high school, then you're likely counting the chickens before the eggs hatch. How much of the material for these courses have you already worked through, and how has your understanding of the material been evaluated?

I am in 8th grade, only a couple months away from graduating middle school.
And I am going to learn Calculus 1, 2, 3 by reading Thomas's textbook, Stewart's textbook, and Kline's Calculus: An Intuitive and Physical approach.
I have already mastered Algebra 1, Geometry, and Algebra 2 + trigonometry through the online courses which my school provided me. So, I am currently on Precal right now, but by next month, I will be done with precal and will start learning Calculus, and by the time the summer break is over(August), I will be done with Calculus 1,2,3, at least. Also, along with online courses, school has also given me tests once a month, to keep track of how I am doing, and I scored 100% on every single one of those, those tests were about geometry, algebra 2, and trigonometry. I am completely confident that I am ready for Calculus.

 

[Niflheim]

I am in 8th grade, only a couple months away from graduating middle school.
And I am going to learn Calculus 1, 2, 3 by reading Thomas's textbook, Stewart's textbook, and Kline's Calculus: An Intuitive and Physical approach.
I have already mastered Algebra 1, Geometry, and Algebra 2 + trigonometry through the online courses which my school provided me. So, I am currently on Precal right now, but by next month, I will be done with precal and will start learning Calculus, and by the time the summer break is over(August), I will be done with Calculus 1,2,3, at least. Also, along with online courses, school has also given me tests once a month, to keep track of how I am doing, and I scored 100% on every single one of those, those tests were about geometry, algebra 2, and trigonometry. I am completely confident that I am ready for Calculus.

Uh, no you won't. Feynman couldn't learn Calc 1-3 in 2 and a half months, and I doubt that you're significantly smarter than him.

 

[ZapperZ]

I apologize for not establishing the starting point clearly.
According to my own predictions and my pace of learning, I will finish Calculus 1,2,3, and Diff. Eq. by the end of summer break.(All self-taught)
I am in 8th grade, and am currently on Pre-Calculus/Trigonometry right now, but I know that I can finish Calculus(1,2,3) at least before high school starts, which is August.

So how are you evaluating your knowledge, which was the other part of my question?

Zz.

 

[James M]

Is there any reason you need to know now, as opposed to when you've finished them?

Definitely. I am interested in all kinds of science and engineering subjects, including, chemistry, physics, chemical engineering, electrical engineering, mechanical engineering, and computer science. My ultimate goal is to get into MIT and study these subjects, but to do that, I need to show MIT that I am capable of learning anything they throw at me, and the first step to showing that is to get the math first, since math is required for ALL of those subjects I am interested in. Also, I want to try to learn them before I go to college, so I know what will be hard or easy for me in those subjects.

 

[James M]

Uh, no you won't. Feynman couldn't learn Calc 1-3 in 2 and a half months, and I doubt that you're significantly smarter than him.

Well, I'll try my best, I guess.

 

[James M]

So how are you evaluating your knowledge, which was the other part of my question?

Zz.

My school has given me tests and quizzes once a month, to keep track of me, and I scored 100% on every single one of those(Geometry, Algebra 2, Trig).

 

[James M]

It depends on what you're generally interested in: Pure math? Physics? Something else?

Also, I hope you're not doing this because of pressure from yourself or others to "get ahead" in your academic career. You're already way ahead of almost everybody else your age. There's no need at this point in your career to study anything in particular at the level that you've reached, so you should focus on things that you enjoy, and don't let them distract you from the other stuff that you're studying in school, and other aspects of life.

Well, I want to develop 3D games, since game developing is the thing that I actually enjoy doing, but to create one, I need a very strong knowledge of geometry, trigonometry, Calculus(physics), and linear algebra, and more. So, nope, I am not doing this because of pressure to get ahead in my academic career, but for my own interest.

 

[QuantumCurt]

I sincerely doubt that you are going to manage to teach yourself Calculus 1,2,3, and differential equations over summer break. I can admire the motivation, but that is two years worth of college level math courses. The material is much more difficult and much more abstract than elementary algebra and introductory trigonometry. Rather than assuming that you WILL indeed self teach all of these subjects and asking what to work on afterward, it might be better to ask for advice on how to go about starting to learn all of this.

That being said...why? Is there some pressing reason that you need to get this done now?

 

[ZapperZ]

My school has given me tests and quizzes once a month, to keep track of me, and I scored 100% on every single one of those(Geometry, Algebra 2, Trig).

No, I mean how do you evaluate your knowledge of the stuff you learned on your own?

Zz.

 

[russ_watters]

That being said...why? Is there some pressing reason that you need to get this done now?

Right. How about starting with Calc I, see how that goes, then move on from there if it goes well. Just please don't flip through the book and think you've learned it. Do the problems on your own and do practice tests, as they would be given as real tests (without using the book).

Or V50 might be questioning the entire idea...being that far ahead, there may not be additional value to trying to learn more on your own. Unless you can get AP credit or take the classes at a community college for real college credit, it may not produce anything of value. You may be better off just getting a jump start on your next classes in school, to get good grades in them.

Or, better yet, do some non-academic things, like sports, music, scouts, etc. -- something to make you better rounded.

Or, even -- just have some fun over the summer. If you are that far ahead, you'll be ok taking the time off.

 

[Loststudent22]

Generally kids who have done this just start taking classes at local university depending on their interest and prep mainly for math contests You should be ready for a gold medal at IMO next year!

 

[IGU]

I am going to learn Calculus 1, 2, 3 by reading Thomas's textbook, Stewart's textbook, and Kline's Calculus: An Intuitive and Physical approach.

Is there some reason you feel these are good choices for you? What you choose as your source material makes a big difference when you are self-teaching. My son used Apostol when he was teaching himself calculus in 8th grade, which turned out to be an excellent choice for him.

I have already mastered Algebra 1, Geometry, and Algebra 2 + trigonometry through the online courses which my school provided me.

What online courses? It's a completely different thing if you learned them using Art of Problem Solving vs. something like Thinkwell.

So, I am currently on Precal right now, but by next month, I will be done with precal and will start learning Calculus, and by the time the summer break is over(August), I will be done with Calculus 1,2,3, at least.

It's a difficult thing to determine when one has achieved mastery of a subject. How do you do it?

Also, along with online courses, school has also given me tests once a month, to keep track of how I am doing, and I scored 100% on every single one of those, those tests were about geometry, algebra 2, and trigonometry. I am completely confident that I am ready for Calculus.

The tests you get at school are likely a poor check for mastery. They are usually at the level of a quick check for basic competence. If you are using a measuring stick on which you always measure 100%, it's not a good measuring stick for you. You might want to find something better.

 

[Entanglement]

I really do appreciate your ambitions and motivation, but calculus and physics are not that kind of science you just learn in a few weeks, to master them there must be a lot of interaction or a kind of "embracement" between you and the subject for a long while.
As previously mentioned, there's a difference between what you think you know and what you really know, there must be a decent evaluation to check on what you know.

Spivak is extremely difficult for beginners, I would only recommend that book if you were in college.