What's Next After AP Calculus BC?

[Char. Limit]

I am a high school senior seriously interested in mathematics (and physics, and chemistry, but that's beside the point). I am wondering what would be the best path for me to learn involving mathematics now that the AP Calculus test is over (I don't want to have to wait until September to continue...). So, now that I've learned the material in AP Calculus BC, where do I go from here?

FWIW, AP Calculus BC taught me differentiation, including implicit, integration, the methods of which included intuition, parts, substitution, and partial fractions, infinite sequences and series including Taylor and Maclaurin series, and finally an introduction to vectors, parametrics, and polars. I'm also using Khan Academy to teach myself the basics of Differential Equations (It's not really that much harder than calculus was, really).

Any thoughts?

 

[Cyosis]

Looking at the amount of calculus you've had in high school I'd say go for linear algebra.

 

[xxsteelxx]

Or you could look into Multivariable Calculus.

 

[jeffasinger]

I took multivariable calculus and a computations based linear algebra class my first semester, and the highest math I had in high school was a similar BC Calculus course. I was a little less prepared than people who had taken Calc II in college, but I did fine.

 

[Char. Limit]

Looking at the amount of calculus you've had in high school I'd say go for linear algebra.

Well, I began to look into this area (thank God for KhanAcademy), and I noticed immediately how much it relied on the matrices I had forgotten. So I'm brushing up on those now.

Or you could look into Multivariable Calculus.

Actually, this was the first thing I went to after Calculus, since it was in the same playlist. I think I got lost around the time he introduced curl. Those matrices again... Always the matrices...

I took multivariable calculus and a computations based linear algebra class my first semester, and the highest math I had in high school was a similar BC Calculus course. I was a little less prepared than people who had taken Calc II in college, but I did fine.

Sounds good to know. I'm hoping to both not lose all of my skill at math over the summer, and be more ready than the other students next year.

What is different about a computation-based linear algebra course? Do they spend less time on proofs than normal?

 

[l'Hôpital]

Honestly, learn some Linear Algebra. A proof version of it. Then take Multivariable Calculus, preferably a rigorous treatment(with Differential Forms) and it'll make perfect sense. Alternatively, pick up Hubbard and Hubbard's Vector Calc book. It teaches you a little linear algebra, multivariable, and other random crap. It's a great book because it teaches advanced mathematics at a not-so-high level.

 

[thrill3rnit3]

Spivak

 

[Fragment]

Spivak

I have to agree with this, Spivak is what everyone should try to work through atleast once in their life.

 

[zooxanthellae]

I recently picked up Spivak, 4th ed. and am working through it. Just wondering: should I be doing every single problem? Because there are...a lot. Especially in the first section, it gets kind of tiresome :x

 

[thrill3rnit3]

Are you getting them all right?

 

[FieldDuck]

I'm actually reading through Spivak right now too and I think it'll be a great way to solidify what you learned in your more computational based AP Calculus class. I feel that if you can work through Spivak on your own and obtain a reasonable grasp of the material, when you do start mathematics in college, you will probably have a greater amount of insight than your peers who have not gone pass computational mathematics yet.

However, if you don't wish to do that, Linear Algebra that revolves around proofs, discrete math, or going onwards to multivariate calculus would probably be beneficial to you. If I were you, I would start to learn how to write proofs now so when you have to do it for a grade it won't be as stressful.

What is different about a computation-based linear algebra course? Do they spend less time on proofs than normal?

In my experience, a computational based linear Algebra class do cover some proofs, do ask some questions on it, but mostly ask you to use what the theorems state to help answer computational problem. While a more proof based linear algebra book will have very little exercises that ask you to take this Matrix and apply this theorem and then get a numerical answer, but rather will ask you, given this theorem, can you prove that if such and such is true then this and that is true. If I had my linear algebra book on me I would give you a more solid example, but it's lost somewhere, but I hope you can get the gist of what I am saying.

 

[zooxanthellae]

Are you getting them all right?

I'm not sure, since it's the first section and it's kind of hard to think solely in terms of the ~10 theorems he gives.

 

[FieldDuck]

I'm not sure, since it's the first section and it's kind of hard to think solely in terms of the ~10 theorems he gives.

If you ever feel inclined, since I'm working through the same book, you can message me if you're not sure what you're doing is correct. I can't promise that I'm correct most or even half the time, but sometimes it helps to talk things out with another person to see if your line of thinking is correct.

 

[Char. Limit]

I've started on Spivak's Calculus (assuming that's the book you meant) and I'm on the first chapter. And you're right, it's very difficult to restrict myself to his ten theorems. It's also difficult to prove things you know to be true, while restricted to those ten theorems. I keep wanting to say "if x^2=y^2, then of course x=y or x=-y! Everyone knows that!"