Is Re-Learning Calculus Necessary for Future Mathematical Studies?

[Hirundo]

Hi, I'm a high school senior and I'm wondering if I should re-learn calculus. This is already my third year learning calculus in my high school, and I'm currently taking some easy version of multi-variable calculus, but doubt my high school calculus class is solid enough as the foundations of future mathematical studies. I'm also doing differential equations on MIT OCW, and I do not have any problem understanding the lectures and notes.
My biggest concern right now, is that I lack some basic calculus conceptions. I can compute and use all basic calculus equations, but I doubt there would be minor principles, or "mathematical thinking" involved that I missed. Since my goal is to do either research in either physics or math in future, I guess it would be important to build foundations. Also, my future college math classes would spend 90% of times on proofs, which I have utterly no experiences upon.
I've found the textbook my future college uses, if I decide to skip the class(99.9% chance), would you recommend me go over the textbook and do all the proofs out, or rather spend the time to do more advanced math? Another question, would minor details in basic calculus be important? Or would they be covered in future courses?

 

[symbolipoint]

YES, you should re-learn Calculus. Try to use a good college level Calculus textbook for Calculus 1 and Calculus 2. One learns this better each time one studies it/them.

 

[jedishrfu]

You can find a three-volume book on Calculus at Openstax:

https://openstax.org/subjects

 

[Stephen Tashi]

I've found the textbook my future college uses, if I decide to skip the class(99.9% chance), would you recommend me go over the textbook and do all the proofs out, or rather spend the time to do more advanced math?

If calculus instructors at your future college expect their students to understand the proofs in the textbook then you should go over those proofs. However, it's often the case (in the USA) that instructors do not expect students to understand the proofs in the text - or expect students only to understand the simpler proofs.

You should look at the exercises in the text that assign students to give proofs. These exercises will generally be simpler than the proofs expounded in the text.

If you are worried about proofs, I suggest you study elementary symbolic logic. Get familiar with the logic involving the quantifiers "for each" and "there exists".

 

[Mark44]

My biggest concern right now, is that I lack some basic calculus conceptions. I can compute and use all basic calculus equations, but I doubt there would be minor principles, or "mathematical thinking" involved that I missed.

Do you "doubt" that you've missed some minor principles, or do you think that you've missed some minor principles? From what you wrote, I think you meant the latter.
Can you give an example of what you think you might have missed? (It might be hard to list things you don't know about, but are there areas where you think you might be weak?)

I've found the textbook my future college uses, if I decide to skip the class(99.9% chance), would you recommend me go over the textbook and do all the proofs out, or rather spend the time to do more advanced math?

I would lean more to working the problems rather than doing the proofs. You'll have lots of opportunity to do proofs in analysis classes, many of which consist of doing nothing but proofs. It would be a good idea though to learn some of the basic proof techniques, however, e.g., direct proof, proof by contradiction, proof by contrapositive, induction proof.

Another question, would minor details in basic calculus be important? Or would they be covered in future courses?

Can you give an example of what minor details you mean?

Have your grades in these calculus classes been strong?

 

[Delta2]

I don't think its a good idea to skip proofs, maybe if you were going for engineering it would have been a good idea, but you say you going to major in math or physics. Proofs can reveal and emphasize a lot of the minor details you think you 've been missing.

 

[Scrumhalf]

You will relearn calculus "the right way" when you take your first Analysis class in college. It's probably the first class you will take anyway if you are on a math track.

 

[Hirundo]

Do you "doubt" that you've missed some minor principles, or do you think that you've missed some minor principles? From what you wrote, I think you meant the latter.
Can you give an example of what you think you might have missed? (It might be hard to list things you don't know about, but are there areas where you think you might be weak?)

I would lean more to working the problems rather than doing the proofs. You'll have lots of opportunity to do proofs in analysis classes, many of which consist of doing nothing but proofs. It would be a good idea though to learn some of the basic proof techniques, however, e.g., direct proof, proof by contradiction, proof by contrapositive, induction proof.

Can you give an example of what minor details you mean?

Have your grades in these calculus classes been strong?

The part I'm missing I guess is not some specific problem solving techniques, but rather definitions, or "how" they constructed calculus' picture. Since I have not yet read any textbook I don't know right now. I have very strong grades in my calculus class, but it's just too easy so I doubt I learned everything.

 

[IGU]

The part I'm missing I guess is not some specific problem solving techniques, but rather definitions, or "how" they constructed calculus' picture. Since I have not yet read any textbook I don't know right now. I have very strong grades in my calculus class, but it's just too easy so I doubt I learned everything.

Calculus only has a couple of basic ideas, so you're unlikely to be missing much. Are you thinking you want to learn calculus as a mathematician, or to use it as a tool like a physicist? If the former, you'll learn what you need when you learn analysis; the calculus is just a warm-up to get you thinking and to see it's useful. If the latter, don't worry about it as physicists don't care about proofs; calculus is a useful tool you can assume is right, so it's best to just push on and learn a bunch of linear algebra as it's another useful tool.

If you really want to learn calculus right, as a mathematician does, get a copy of Apostol and go through the book and work all the problems (you can skip most of the exercises if they're easy for you) and understand all the proofs. If you've had enough of calculus and want to go down the mathematical track, see if you can handle Terry Tao's analysis notes. Follow links here, or get his Analysis I book. This is serious stuff, but if you can handle it you can see if you like it.

 

[bestazy]

My suggestion.
Take some book like Spivak calculus, read the most important topics (limits, derivatives, integrals, series) and do some exercises of each one. You'll have no problem in see one entire topic in a day if you're mature enought for understand real analysis, even the concept of limit has his natural realm in topology, assuming you know it since you studied real analysis. With enought mathematical maturity, analysis isn't that challenging.
best regards
Bestazy